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Difference between revisions of "PEH:Fluid Flow Through Permeable Media"

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{{Infobox Book
 
{{Infobox Book
|series       = Petroleum Engineering Handbook
+
|series = Petroleum Engineering Handbook
|editor-in-chief   = Larry W. Lake
+
|editor-in-chief = Larry W. Lake
|volume       = Volume V – Reservoir Engineering and Petrophysics
+
|volume = Volume V – Reservoir Engineering and Petrophysics
|vol editor = Edward D. Holstein, Editor
+
|vol editor = Edward D. Holstein, Editor
|date         = 2007
+
|date = 2007
|publisher   = Society of Petroleum Engineers
+
|publisher = Society of Petroleum Engineers
|image       = [[File:Vol5REPCover.png|center|120px]]
+
|image = [[File:Vol5REPCover.png|center|120px]]
|imagestyle   =  
+
|imagestyle =  
 
|chapter = Chapter 8 – Fluid Flow Through Permeable Media
 
|chapter = Chapter 8 – Fluid Flow Through Permeable Media
 
|ch author = John Lee, Texas A&M U.
 
|ch author = John Lee, Texas A&M U.
|ch author info =  
+
|ch author info =  
|page numbers   = 719-894
+
|page numbers = 719-894
|ISBN   = 978-1-55563-120-8
+
|ISBN = 978-1-55563-120-8
}}
+
}}<br/>This chapter discusses fluid flow in petroleum reservoirs. Basic concepts, which include flow equations for unsteady-state, pseudosteady-state, and steady-state flow of fluids, are discussed first. Various flow geometries are treated, including radial, linear, and spherical flow. The pseudosteady-state equations provide the basis for a brief discussion of oil well productivity, and the unsteady-state equations provide the basis for a lengthy discussion of pressure-transient test analysis. For pressure-transient test analysis, semilog techniques, type curves, damage and stimulation, modifications for gases and multiphase flow, the diagnostic plot, bounded reservoirs, average pressure in the drainage area, hydraulically fractured wells, and naturally fractured reservoirs are included. The chapter also discusses transient and stabilized flow in horizontal wells and gas-well deliverability tests. It concludes with considerations of coning in vertical and horizontal wells.<br/>__TOC__<br/>
<br>
+
<div class="toccolours mw-collapsible mw-collapsed">
This chapter discusses fluid flow in petroleum reservoirs. Basic concepts, which include flow equations for unsteady-state, pseudosteady-state, and steady-state flow of fluids, are discussed first. Various flow geometries are treated, including radial, linear, and spherical flow. The pseudosteady-state equations provide the basis for a brief discussion of oil well productivity, and the unsteady-state equations provide the basis for a lengthy discussion of pressure-transient test analysis. For pressure-transient test analysis, semilog techniques, type curves, damage and stimulation, modifications for gases and multiphase flow, the diagnostic plot, bounded reservoirs, average pressure in the drainage area, hydraulically fractured wells, and naturally fractured reservoirs are included. The chapter also discusses transient and stabilized flow in horizontal wells and gas-well deliverability tests. It concludes with considerations of coning in vertical and horizontal wells.  
 
<br>
 
__TOC__
 
<br>
 
<br>
 
<div class="toccolours mw-collapsible mw-collapsed" >
 
 
== Basic Concepts ==
 
== Basic Concepts ==
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
=== The Ideal Reservoir Model ===
 
=== The Ideal Reservoir Model ===
  
Many important applications of fluid flow in permeable media involve 1D, radial flow. These applications are based on a model that includes many simplifying assumptions about the well and reservoir. These assumptions are introduced as needed to combine the law of conservation of mass, Darcy’s law, and equations of state to achieve our objectives.  
+
Many important applications of fluid flow in permeable media involve 1D, radial flow. These applications are based on a model that includes many simplifying assumptions about the well and reservoir. These assumptions are introduced as needed to combine the law of conservation of mass, Darcy’s law, and equations of state to achieve our objectives.<br/><br/>Consider radial flow toward a well in a circular reservoir. Combining the law of conservation of mass and Darcy’s law for the isothermal flow of fluids of small and constant compressibility yields the radial diffusivity equation, <ref name="r1">_</ref><br/><br/>[[File:Vol5 page 0719 eq 001.png|RTENOTITLE]]....................(8.1)<br/><br/>In the derivation of this equation, it is assumed that compressibility of the total system, ''c''<sub>''t''</sub>, is small and independent of pressure; permeability, ''k'' , is constant and isotropic; viscosity, ''μ'', is independent of pressure; porosity, ''ϕ'', is constant; and that certain terms in the basic differential equation (involving pressure gradients squared) are negligible. The grouping 0.0002637''k''/''ϕμc''<sub>''t''</sub> is called the hydraulic diffusivity and is given the symbol ''η''.
<br>
 
<br>
 
Consider radial flow toward a well in a circular reservoir. Combining the law of conservation of mass and Darcy’s law for the isothermal flow of fluids of small and constant compressibility yields the radial diffusivity equation, <ref name="r1" />
 
<br>
 
<br>
 
[[File:Vol5 page 0719 eq 001.png]]....................(8.1)
 
<br>
 
<br>
 
In the derivation of this equation, it is assumed that compressibility of the total system, ''c''<sub>''t''</sub>, is small and independent of pressure; permeability, ''k'' , is constant and isotropic; viscosity, ''μ'', is independent of pressure; porosity, ''ϕ'', is constant; and that certain terms in the basic differential equation (involving pressure gradients squared) are negligible. The grouping 0.0002637''k''/''ϕμc''<sub>''t''</sub> is called the hydraulic diffusivity and is given the symbol ''η''.  
 
  
 
=== Line-Source Solution to the Diffusivity Equation ===
 
=== Line-Source Solution to the Diffusivity Equation ===
  
Assume that a well produces at constant reservoir rate, ''qB''; the well has zero radius; the reservoir is at uniform pressure, ''p''<sub>''i''</sub>, before production begins; and the well drains an infinite area (i.e., that ''p'' → ''p''<sub>''i''</sub> as ''r'' → ∞). Under these conditions, the solution to '''Eq. 8.1''' is<ref name="r1" />
+
Assume that a well produces at constant reservoir rate, ''qB''; the well has zero radius; the reservoir is at uniform pressure, ''p''<sub>''i''</sub>, before production begins; and the well drains an infinite area (i.e., that ''p'' → ''p''<sub>''i''</sub> as ''r'' → ∞). Under these conditions, the solution to '''Eq. 8.1''' is<ref name="r1">_</ref><br/><br/>[[File:Vol5 page 0720 eq 001.png|RTENOTITLE]]....................(8.2)<br/><br/>where ''p'' is the pressure at distance ''r'' from the well at time ''t'', and<br/><br/>[[File:Vol5 page 0720 eq 002.png|RTENOTITLE]]....................(8.3)<br/><br/>the''Ei ''function or exponential integral.
<br>
 
<br>
 
[[File:Vol5 page 0720 eq 001.png]]....................(8.2)
 
<br>
 
<br>
 
where ''p'' is the pressure at distance ''r'' from the well at time ''t'', and
 
<br>
 
<br>
 
[[File:Vol5 page 0720 eq 002.png]]....................(8.3)
 
<br>
 
<br>
 
the'' Ei ''function or exponential integral.  
 
  
The ''Ei''-function solution is an accurate approximation to more exact solutions to the diffusivity equation (solutions with finite wellbore radius and finite drainage radius) for 3.79 × 10<sup>5</sup> ''ϕμc''<sub>''t''</sub>''r''<sub>''w''</sub><sup>2</sup>/''k'' < ''t'' < 948 ''ϕμc''<sub>''t''</sub>''r''<sub>''e''</sub><sup>2</sup>/''k''. For smaller times, the assumption of zero well size (line source or sink) limits the accuracy of the equation; for larger times, the reservoir’s boundaries affect the pressure distribution in the reservoir, so that the reservoir is no longer infinite acting.  
+
The ''Ei''-function solution is an accurate approximation to more exact solutions to the diffusivity equation (solutions with finite wellbore radius and finite drainage radius) for 3.79 × 10<sup>5</sup> ''ϕμc''<sub>''t''</sub>''r''<sub>''w''</sub><sup>2</sup>/''k'' < ''t'' < 948 ''ϕμc''<sub>''t''</sub>''r''<sub>''e''</sub><sup>2</sup>/''k''. For smaller times, the assumption of zero well size (line source or sink) limits the accuracy of the equation; for larger times, the reservoir’s boundaries affect the pressure distribution in the reservoir, so that the reservoir is no longer infinite acting.<br/><br/>For the argument, ''x'', of the''Ei ''function less than 0.01, the''Ei ''function can be approximated with negligible error by<br/><br/>[[File:Vol5 page 0720 eq 003.png|RTENOTITLE]]....................(8.4)<br/><br/>For ''x'' > 10, the''Ei ''function is zero for practical applications in flow through porous media. For 0.01 < ''x'' < 10,''Ei ''functions are determined from tables or subroutines available in appropriate software. <ref name="r4">_</ref>
<br>
 
<br>
 
For the argument, ''x'', of the'' Ei ''function less than 0.01, the'' Ei ''function can be approximated with negligible error by
 
<br>
 
<br>
 
[[File:Vol5 page 0720 eq 003.png]]....................(8.4)
 
<br>
 
<br>
 
For ''x'' > 10, the'' Ei ''function is zero for practical applications in flow through porous media. For 0.01 < ''x'' < 10,'' Ei ''functions are determined from tables or subroutines available in appropriate software. <ref name="r4" />  
 
  
 
=== Altered Zone and Skin Factor ===
 
=== Altered Zone and Skin Factor ===
  
In practice, most wells have reduced permeability (damage) near the wellbore resulting from drilling or completion operations. Many other wells are stimulated by acidization or hydraulic fracturing. '''Eq. 8.2''' fails to model such wells properly. Its derivation includes the explicit assumption of uniform permeability throughout the drainage area of the well up to the wellbore. Hawkins<ref name="r2" /> pointed out that if the damaged or stimulated zone is considered equivalent to an altered zone of uniform permeability. ''k''<sub>''s''</sub>, and outer radius, ''r''<sub>''s''</sub>, the additional pressure drop, Δ''p''<sub>''s''</sub>, across this zone can be modeled by the steady-state radial flow equation
+
In practice, most wells have reduced permeability (damage) near the wellbore resulting from drilling or completion operations. Many other wells are stimulated by acidization or hydraulic fracturing. '''Eq. 8.2''' fails to model such wells properly. Its derivation includes the explicit assumption of uniform permeability throughout the drainage area of the well up to the wellbore. Hawkins<ref name="r2">_</ref> pointed out that if the damaged or stimulated zone is considered equivalent to an altered zone of uniform permeability. ''k''<sub>''s''</sub>, and outer radius, ''r''<sub>''s''</sub>, the additional pressure drop, Δ''p''<sub>''s''</sub>, across this zone can be modeled by the steady-state radial flow equation<br/><br/>[[File:Vol5 page 0720 eq 004.png|RTENOTITLE]]....................(8.5)<br/><br/>'''Eq. 8.5''' states that the pressure drop in the altered zone is inversely proportional to ''k''<sub>''s''</sub> rather than to ''k'' and that a correction to the pressure drop in this region must be made. Combining '''Eqs. 8.2''' and '''8.5''', we find that the total pressure drop at the wellbore is<br/><br/>[[File:Vol5 page 0721 eq 001.png|RTENOTITLE]]....................(8.6)<br/><br/>For ''r'' = ''r''<sub>''w''</sub>, the argument of the''Ei ''function is sufficiently small after a short time that we can use the logarithmic approximation; thus, the drawdown is<br/><br/>[[File:Vol5 page 0721 eq 002.png|RTENOTITLE]]....................(8.7)<br/><br/>We can conveniently define a dimensionless skin factor, ''s'', in terms of the properties of the equivalent altered zone:<br/><br/>[[File:Vol5 page 0721 eq 003.png|RTENOTITLE]]....................(8.8)<br/><br/>Thus, the drawdown is<br/><br/>[[File:Vol5 page 0721 eq 003.png|RTENOTITLE]]....................(8.9)<br/><br/>'''Eq. 8.9''' provides some insight into the physical significance of the algebraic sign of the skin factor. If a well is damaged (''k''<sub>''s''</sub> < ''k''), ''s'' will be positive, and the greater the contrast between ''k''<sub>''s''</sub> and ''k'' and the deeper into the formation the damage extends, the larger the numerical value of ''s'', which has no upper limit. Some newly drilled wells will not flow before stimulation; for these wells, ''k''<sub>''s''</sub> = 0 and ''s'' → ∞. If a well is stimulated (''k''<sub>''s''</sub> > ''k''), ''s'' will be negative, and the deeper the stimulation, the greater the numerical value of ''s''. Rarely does a stimulated well have a skin less than –7, and such skin factors arise only for wells with deeply penetrating, highly conductive hydraulic fractures. If a well is neither damaged nor stimulated (''k'' = ''k''<sub>''s''</sub>), ''s'' = 0.<br/><br/>The altered zone near a well affects only the pressure near that well; that is, the pressure in the unaltered formation away from the well is not affected by the existence of the altered zone. Thus, use '''Eq. 8.9''' to calculate pressures at the sandface of a well with an altered zone, and '''Eq. 8.2''' to calculate pressures beyond the altered zone in the formation surrounding the well. See '''Sec. 8.4''' for more information on damage and stimulation.
<br>
 
<br>
 
[[File:Vol5 page 0720 eq 004.png]]....................(8.5)
 
<br>
 
<br>
 
'''Eq. 8.5''' states that the pressure drop in the altered zone is inversely proportional to ''k''<sub>''s''</sub> rather than to ''k'' and that a correction to the pressure drop in this region must be made. Combining '''Eqs. 8.2''' and '''8.5''', we find that the total pressure drop at the wellbore is
 
<br>
 
<br>
 
[[File:Vol5 page 0721 eq 001.png]]....................(8.6)
 
<br>
 
<br>
 
For ''r'' = ''r''<sub>''w''</sub>, the argument of the'' Ei ''function is sufficiently small after a short time that we can use the logarithmic approximation; thus, the drawdown is
 
<br>
 
<br>
 
[[File:Vol5 page 0721 eq 002.png]]....................(8.7)
 
<br>
 
<br>
 
We can conveniently define a dimensionless skin factor, ''s'', in terms of the properties of the equivalent altered zone:
 
<br>
 
<br>
 
[[File:Vol5 page 0721 eq 003.png]]....................(8.8)
 
<br>
 
<br>
 
Thus, the drawdown is
 
<br>
 
<br>
 
[[File:Vol5 page 0721 eq 003.png]]....................(8.9)
 
<br>
 
<br>
 
'''Eq. 8.9''' provides some insight into the physical significance of the algebraic sign of the skin factor. If a well is damaged (''k''<sub>''s''</sub> < ''k''), ''s'' will be positive, and the greater the contrast between ''k''<sub>''s''</sub> and ''k'' and the deeper into the formation the damage extends, the larger the numerical value of ''s'', which has no upper limit. Some newly drilled wells will not flow before stimulation; for these wells, ''k''<sub>''s''</sub> = 0 and ''s'' → ∞. If a well is stimulated (''k''<sub>''s''</sub> > ''k''), ''s'' will be negative, and the deeper the stimulation, the greater the numerical value of ''s''. Rarely does a stimulated well have a skin less than –7, and such skin factors arise only for wells with deeply penetrating, highly conductive hydraulic fractures. If a well is neither damaged nor stimulated (''k'' = ''k''<sub>''s''</sub>), ''s'' = 0.  
 
<br>
 
<br>
 
The altered zone near a well affects only the pressure near that well; that is, the pressure in the unaltered formation away from the well is not affected by the existence of the altered zone. Thus, use '''Eq. 8.9''' to calculate pressures at the sandface of a well with an altered zone, and '''Eq. 8.2''' to calculate pressures beyond the altered zone in the formation surrounding the well. See '''Sec. 8.4''' for more information on damage and stimulation.  
 
  
 
=== Inertial-Turbulent Flow and Rate-Dependent Skin ===
 
=== Inertial-Turbulent Flow and Rate-Dependent Skin ===
  
The diffusivity equation, '''Eq. 8.1''', assumes that Darcy’s law represents the relationship between flow velocity and pressure gradients in the reservoir, an assumption that is adequate for low-velocity or laminar flow. However, at higher flow velocities, deviations from Darcy’s law are observed as a result of inertial effects or even turbulent flow effects. In 1D radial flow, these inertial/turbulent effects (often called non-Darcy flow effects) are confined to the region near the wellbore in which flow velocities are largest. This results in an additional pressure drop similar to that caused by skin, but the additional pressure drop is proportional to flow rate. The apparent skin, ''s''′, for a well with non-Darcy flow near the wellbore is given by<ref name="r3" />
+
The diffusivity equation, '''Eq. 8.1''', assumes that Darcy’s law represents the relationship between flow velocity and pressure gradients in the reservoir, an assumption that is adequate for low-velocity or laminar flow. However, at higher flow velocities, deviations from Darcy’s law are observed as a result of inertial effects or even turbulent flow effects. In 1D radial flow, these inertial/turbulent effects (often called non-Darcy flow effects) are confined to the region near the wellbore in which flow velocities are largest. This results in an additional pressure drop similar to that caused by skin, but the additional pressure drop is proportional to flow rate. The apparent skin, ''s''′, for a well with non-Darcy flow near the wellbore is given by<ref name="r3">_</ref><br/><br/>[[File:Vol5 page 0722 eq 001.png|RTENOTITLE]]....................(8.10)<br/><br/>where ''D'' is the non-Darcy flow factor for the system. ''D'' can be regarded as constant, although, in theory, it depends slightly on near-well pressure. In practice, non-Darcy flow is ordinarily important only for gas wells, which have high-flow velocities near the wellbore, but it can be important for oil wells with high-velocity flow in some situations.
<br>
 
<br>
 
[[File:Vol5 page 0722 eq 001.png]]....................(8.10)
 
<br>
 
<br>
 
where ''D'' is the non-Darcy flow factor for the system. ''D'' can be regarded as constant, although, in theory, it depends slightly on near-well pressure. In practice, non-Darcy flow is ordinarily important only for gas wells, which have high-flow velocities near the wellbore, but it can be important for oil wells with high-velocity flow in some situations.  
 
  
 
=== Radius of Investigation and Stabilization Time ===
 
=== Radius of Investigation and Stabilization Time ===
  
Radius of investigation is the distance a pressure transient has moved into a formation following a rate change in a well. This distance is related to formation rock and fluid properties and time elapsed since a rate change in the well. Consider this concept by visualizing the pressure distributions at increasing times as '''Fig. 8.1''' shows for a well producing at constant rate from a reservoir initially at uniform pressure. (These pressure distributions were calculated using the ''Ei''-function solution to the diffusivity equation.)  
+
Radius of investigation is the distance a pressure transient has moved into a formation following a rate change in a well. This distance is related to formation rock and fluid properties and time elapsed since a rate change in the well. Consider this concept by visualizing the pressure distributions at increasing times as '''Fig. 8.1''' shows for a well producing at constant rate from a reservoir initially at uniform pressure. (These pressure distributions were calculated using the ''Ei''-function solution to the diffusivity equation.)<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0722 Image 0001.png|'''Fig. 8.1 – Pressure distributions at increasing times.'''
 
File:vol5 Page 0722 Image 0001.png|'''Fig. 8.1 – Pressure distributions at increasing times.'''
 
</gallery>
 
</gallery>
<br/>
 
  
 
Important observations about this figure include the following:
 
Important observations about this figure include the following:
<br>
 
* The pressure in the wellbore, at ''r'' = ''r''<sub>''w''</sub>, decreases steadily as flow time increases; likewise, pressures at other fixed values of r also decrease as flow time increases.
 
* The pressure drawdown (or pressure transient) caused by producing the well moves further into the reservoir as flow time increases. For the range of flow times shown, there is always a point beyond which the drawdown in pressure from the original value is negligible. This time-dependent point of "negligible drawdown" can be considered to be a radius of investigation.
 
<br>
 
  
Analysis shows that the time, ''t'', at which a pressure disturbance reaches a distance, ''r''<sub>''i''</sub>, which is called the radius of investigation, is given by the equation<ref name="r4" />
+
*The pressure in the wellbore, at ''r'' = ''r''<sub>''w''</sub>, decreases steadily as flow time increases; likewise, pressures at other fixed values of r also decrease as flow time increases.
<br>
+
*The pressure drawdown (or pressure transient) caused by producing the well moves further into the reservoir as flow time increases. For the range of flow times shown, there is always a point beyond which the drawdown in pressure from the original value is negligible. This time-dependent point of "negligible drawdown" can be considered to be a radius of investigation.
<br>
+
 
[[File:Vol5 page 0722 eq 002.png]]....................(8.11)
+
 
<br>
+
 
<br>
+
Analysis shows that the time, ''t'', at which a pressure disturbance reaches a distance, ''r''<sub>''i''</sub>, which is called the radius of investigation, is given by the equation<ref name="r4">_</ref><br/><br/>[[File:Vol5 page 0722 eq 002.png|RTENOTITLE]]....................(8.11)<br/><br/>Investigators differ on the numerical constant in '''Eq. 8.11''', but this difference is of little practical importance if the radius of investigation is used as a semiquantitative indicator of the distance into the reservoir to which formation properties have influenced the response of a well in a pressure-transient test.<br/><br/>The radius of investigation has several applications in pressure-transient test analysis and design. A qualitative use is to help explain the shape of a pressure buildup or drawdown curve. For example, a buildup test plot may have a complex shape at early times when the radius of investigation is in the altered zone near the wellbore, where the permeability is different from formation permeability. Or a buildup test plot may change shape at long times when the radius of investigation reaches the general vicinity of a reservoir boundary.<br/><br/>The radius-of-investigation concept provides a guide for well-test design. For example, you may want to sample reservoir properties at least 1,000 ft from a test well. The radius of investigation concept allows you to estimate the time required to achieve the desired depth of investigation.<br/><br/>'''Eq. 8.11''' also provides a means to estimate the time required to achieve "stabilized" flow; that is, the time required for a pressure transient to reach the boundaries of a tested reservoir. For example, if a well is centered in a cylindrical drainage area of radius ''r''<sub>''e''</sub>, then the time required for stabilization, ''t''<sub>''s''</sub>, is<br/><br/>[[File:Vol5 page 0723 eq 001.png|RTENOTITLE]]....................(8.12)<br/><br/>For other drainage shapes, the time to stabilization can be quite different, as discussed later.
Investigators differ on the numerical constant in '''Eq. 8.11''', but this difference is of little practical importance if the radius of investigation is used as a semiquantitative indicator of the distance into the reservoir to which formation properties have influenced the response of a well in a pressure-transient test.  
 
<br>
 
<br>
 
The radius of investigation has several applications in pressure-transient test analysis and design. A qualitative use is to help explain the shape of a pressure buildup or drawdown curve. For example, a buildup test plot may have a complex shape at early times when the radius of investigation is in the altered zone near the wellbore, where the permeability is different from formation permeability. Or a buildup test plot may change shape at long times when the radius of investigation reaches the general vicinity of a reservoir boundary.  
 
<br>
 
<br>
 
The radius-of-investigation concept provides a guide for well-test design. For example, you may want to sample reservoir properties at least 1,000 ft from a test well. The radius of investigation concept allows you to estimate the time required to achieve the desired depth of investigation.  
 
<br>
 
<br>
 
'''Eq. 8.11''' also provides a means to estimate the time required to achieve "stabilized" flow; that is, the time required for a pressure transient to reach the boundaries of a tested reservoir. For example, if a well is centered in a cylindrical drainage area of radius ''r''<sub>''e''</sub>, then the time required for stabilization, ''t''<sub>''s''</sub>, is
 
<br>
 
<br>
 
[[File:Vol5 page 0723 eq 001.png]]....................(8.12)
 
<br>
 
<br>
 
For other drainage shapes, the time to stabilization can be quite different, as discussed later.  
 
  
 
=== Pseudosteady-State Flow ===
 
=== Pseudosteady-State Flow ===
  
The ''Ei''-function solution to the radial diffusivity equation is valid only while a reservoir is infinite-acting; that is, until boundaries begin to affect the pressure drawdown at the well. For the constant rate flow of a well centered in its drainage area of radius, ''r''<sub>''e''</sub>, and modeled by the ''Ei''-function solution, these effects begin at ''t'' = 948 ''ϕμc''<sub>''t''</sub>''r''<sub>''e''</sub><sup>2</sup>/''k''. Before these boundary effects, the regime is called unsteady-state flow. After boundary effects are felt fully, the solution to the radial diffusivity equation for a well centered in a cylindrical drainage area and producing at constant rate is<ref name="r4" />
+
The ''Ei''-function solution to the radial diffusivity equation is valid only while a reservoir is infinite-acting; that is, until boundaries begin to affect the pressure drawdown at the well. For the constant rate flow of a well centered in its drainage area of radius, ''r''<sub>''e''</sub>, and modeled by the ''Ei''-function solution, these effects begin at ''t'' = 948 ''ϕμc''<sub>''t''</sub>''r''<sub>''e''</sub><sup>2</sup>/''k''. Before these boundary effects, the regime is called unsteady-state flow. After boundary effects are felt fully, the solution to the radial diffusivity equation for a well centered in a cylindrical drainage area and producing at constant rate is<ref name="r4">_</ref><br/><br/>[[File:Vol5 page 0723 eq 002.png|RTENOTITLE]]....................(8.13)<br/><br/>This equation for calculating pressure in the wellbore becomes valid for ''t'' > 948 ''ϕμc''<sub>''t''</sub>''r''<sub>''e''</sub><sup>2</sup>/''k'' at the same time at which the ''Ei''-function solution becomes invalid.<br/><br/>Another form of '''Eq. 8.13''' is useful for some applications. It involves replacing original reservoir pressure, ''p''<sub>''i''</sub>, with average pressure, [[File:Vol5 page 0723 inline 001.png|RTENOTITLE]], within the drainage volume of the well. The volumetric average pressure within the drainage volume of the well can be found from material balance. The pressure decrease [[File:Vol5 page 0723 inline 002.png|RTENOTITLE]] resulting from removal of ''qB'' RB/D of fluid for ''t'' hours (a total volume removed of 5.615''qBt''/24 ft<sup>3</sup>) is<br/><br/>[[File:Vol5 page 0723 eq 003.png|RTENOTITLE]]....................(8.14)<br/><br/>Substituting in '''Eq. 8.13''', the time-dependent terms cancel, and the result is<br/><br/>[[File:Vol5 page 0724 eq 001.png|RTENOTITLE]]....................(8.15)<br/><br/>'''Eqs. 8.13''' and '''8.15''' are more useful in practice if they include skin factors to account for damage or stimulation. In '''Eq. 8.15''',<br/><br/>[[File:Vol5 page 0724 eq 002.png|RTENOTITLE]]....................(8.16)<br/><br/>[[File:Vol5 page 0724 eq 003.png|RTENOTITLE]]....................(8.17)<br/><br/>and [[File:Vol5 page 0724 eq 004.png|RTENOTITLE]]....................(8.18)<br/>
<br>
+
 
<br>
 
[[File:Vol5 page 0723 eq 002.png]]....................(8.13)
 
<br>
 
<br>
 
This equation for calculating pressure in the wellbore becomes valid for ''t'' > 948 ''ϕμc''<sub>''t''</sub>''r''<sub>''e''</sub><sup>2</sup>/''k'' at the same time at which the ''Ei''-function solution becomes invalid.  
 
<br>
 
<br>
 
Another form of '''Eq. 8.13''' is useful for some applications. It involves replacing original reservoir pressure, ''p''<sub>''i''</sub>, with average pressure, [[File:Vol5 page 0723 inline 001.png]], within the drainage volume of the well. The volumetric average pressure within the drainage volume of the well can be found from material balance. The pressure decrease [[File:Vol5 page 0723 inline 002.png]] resulting from removal of ''qB'' RB/D of fluid for ''t'' hours (a total volume removed of 5.615''qBt''/24 ft<sup>3</sup>) is
 
<br>
 
<br>
 
[[File:Vol5 page 0723 eq 003.png]]....................(8.14)
 
<br>
 
<br>
 
Substituting in '''Eq. 8.13''', the time-dependent terms cancel, and the result is
 
<br>
 
<br>
 
[[File:Vol5 page 0724 eq 001.png]]....................(8.15)
 
<br>
 
<br>
 
'''Eqs. 8.13''' and '''8.15''' are more useful in practice if they include skin factors to account for damage or stimulation. In '''Eq. 8.15''',
 
<br>
 
<br>
 
[[File:Vol5 page 0724 eq 002.png]]....................(8.16)
 
<br>
 
<br>
 
[[File:Vol5 page 0724 eq 003.png]]....................(8.17)
 
<br>
 
<br>
 
and [[File:Vol5 page 0724 eq 004.png]]....................(8.18)
 
<br>
 
<br>
 
 
=== Productivity Index ===
 
=== Productivity Index ===
  
The productivity index, ''J'', of an oil well is the ratio of the stabilized rate, ''q'', to the pressure drawdown, [[File:Vol5 page 0724 inline 001.png]], required to sustain that rate. For flow from a well centered in a circular drainage area, '''Eq. 8.17''' allows us to relate productivity index to formation and fluid properties:
+
The productivity index, ''J'', of an oil well is the ratio of the stabilized rate, ''q'', to the pressure drawdown, [[File:Vol5 page 0724 inline 001.png|RTENOTITLE]], required to sustain that rate. For flow from a well centered in a circular drainage area, '''Eq. 8.17''' allows us to relate productivity index to formation and fluid properties:<br/><br/>[[File:Vol5 page 0724 eq 005.png|RTENOTITLE]]....................(8.19)<br/><br/>Thus, if a well is tested at several different stabilized rates and the stabilized flowing bottomhole pressure (BHP), ''p''<sub>''wf''</sub>, is measured at each rate (that is, if pseudosteady-state is attained at each rate), '''Eq. 8.19''' implies that a plot of test data should produce a straight line with slope ''J'' and intercepts ''q'' = 0 when ''p''<sub>''wf''</sub> = [[File:Vol5 page 0724 inline 002.png|RTENOTITLE]] and [[File:Vol5 page 0724 inline 003.png|RTENOTITLE]] when ''p''<sub>''wf''</sub> = 0. (See '''Fig. 8.2'''.) In practice, actual field data will fall below the theoretical straight line for pressures below the bubblepoint pressure of the oil because of increasing gas saturations and oil viscosities that increase the resistance to flow.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0724 eq 005.png]]....................(8.19)
 
<br>
 
<br>
 
Thus, if a well is tested at several different stabilized rates and the stabilized flowing bottomhole pressure (BHP), ''p''<sub>''wf''</sub>, is measured at each rate (that is, if pseudosteady-state is attained at each rate), '''Eq. 8.19''' implies that a plot of test data should produce a straight line with slope ''J'' and intercepts ''q'' = 0 when ''p''<sub>''wf''</sub> = [[File:Vol5 page 0724 inline 002.png]] and [[File:Vol5 page 0724 inline 003.png]] when ''p''<sub>''wf''</sub> = 0. (See '''Fig. 8.2'''.) In practice, actual field data will fall below the theoretical straight line for pressures below the bubblepoint pressure of the oil because of increasing gas saturations and oil viscosities that increase the resistance to flow.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0724 Image 0001.png|'''Fig. 8.2 – Field data drop below idealized straight line at pressures below bubblepoint.'''
 
File:vol5 Page 0724 Image 0001.png|'''Fig. 8.2 – Field data drop below idealized straight line at pressures below bubblepoint.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Generalized Drainage Area Shapes ===
 
=== Generalized Drainage Area Shapes ===
  
'''Eq. 8.17''' is limited to a well centered in a circular drainage area. A similar equation models pseudosteady-state flow in more general reservoir shapes<ref name="r4" />:
+
'''Eq. 8.17''' is limited to a well centered in a circular drainage area. A similar equation models pseudosteady-state flow in more general reservoir shapes<ref name="r4">_</ref>:<br/><br/>[[File:Vol5 page 0725 eq 001.png|RTENOTITLE]]....................(8.20)<br/><br/>where ''A'' is the drainage area in square feet, and ''C''<sub>''A''</sub> is the dimensionless shape factor for a specific drainage-area shape and configuration. '''Table 8.A-1''' (Appendix) gives values of ''C''<sub>''A''</sub>.
<br>
 
<br>
 
[[File:Vol5 page 0725 eq 001.png]]....................(8.20)
 
<br>
 
<br>
 
where ''A'' is the drainage area in square feet, and ''C''<sub>''A''</sub> is the dimensionless shape factor for a specific drainage-area shape and configuration. '''Table 8.A-1''' (Appendix) gives values of ''C''<sub>''A''</sub>.  
 
  
The productivity index, ''J'', can be expressed for general drainage-area geometry as
+
The productivity index, ''J'', can be expressed for general drainage-area geometry as<br/><br/>[[File:Vol5 page 0725 eq 002.png|RTENOTITLE]]....................(8.21)<br/><br/>Other numerical constants tabulated in '''Table 8.A-1''' allow us to calculate the maximum elapsed time during which a reservoir is infinite-acting (so that the ''Ei''-function solution can be used), the time required for the for the pseudosteady-state solution to predict pressure drawdown within 1% accuracy, and time required for the pseudosteady-state solution to be exact. For a given reservoir geometry, the maximum time a reservoir is infinite acting can be determined using the entry in the column "Use Infinite System Solution With Less Than 1% Error for ''t''<sub>''DA''</sub> <." This ''t''<sub>''DA''</sub> is defined as 0.0002637''kt''/''ϕμc''<sub>''t''</sub>''A'', so this means that the time in hours is calculated from<br/><br/>[[File:Vol5 page 0725 eq 003.png|RTENOTITLE]]....................(8.22)<br/><br/>Time required for the pseudosteady-state equation to be accurate within 1% can be found from the entry in the column titled "Less Than 1% Error for t DA., " Finally, the time required for the pseudosteady-state equation to be exact is found in the entry in the column "Exact for ''t''<sub>''DA''</sub> >."<br/><br/>'''Figs. 8.3 and 8.4''' show the flow regimes that occur at various times. These figures show ''p''<sub>''wf''</sub> in a well flowing at constant rate, plotted as a function of time on both logarithmic and linear scales. In the transient region, the reservoir is infinite acting and is modeled by '''Eq. 8.9''', which implies that ''p''<sub>''wf''</sub> is a linear function of log ''t''. In the pseudosteady-state region, the reservoir is modeled by '''Eq. 8.20''' in the general case or '''Eqs. 8.15''' or '''8.13''' for the special case of a well centered in a cylindrical drainage area. '''Eq. 8.13''' shows a linear relationship between ''p''<sub>''wf''</sub> and ''t'' during pseudosteady-state flow. This linear relationship also exists in generalized reservoir geometries.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0725 eq 002.png]]....................(8.21)
 
<br>
 
<br>
 
Other numerical constants tabulated in '''Table 8.A-1''' allow us to calculate the maximum elapsed time during which a reservoir is infinite-acting (so that the ''Ei''-function solution can be used), the time required for the for the pseudosteady-state solution to predict pressure drawdown within 1% accuracy, and time required for the pseudosteady-state solution to be exact. For a given reservoir geometry, the maximum time a reservoir is infinite acting can be determined using the entry in the column "Use Infinite System Solution With Less Than 1% Error for ''t''<sub>''DA''</sub> <." This ''t''<sub>''DA''</sub> is defined as 0.0002637''kt''/''ϕμc''<sub>''t''</sub>''A'', so this means that the time in hours is calculated from
 
<br>
 
<br>
 
[[File:Vol5 page 0725 eq 003.png]]....................(8.22)
 
<br>
 
<br>
 
Time required for the pseudosteady-state equation to be accurate within 1% can be found from the entry in the column titled "Less Than 1% Error for t DA., " Finally, the time required for the pseudosteady-state equation to be exact is found in the entry in the column "Exact for ''t''<sub>''DA''</sub> >."  
 
<br>
 
<br>
 
'''Figs. 8.3 and 8.4''' show the flow regimes that occur at various times. These figures show ''p''<sub>''wf''</sub> in a well flowing at constant rate, plotted as a function of time on both logarithmic and linear scales. In the transient region, the reservoir is infinite acting and is modeled by '''Eq. 8.9''', which implies that ''p''<sub>''wf''</sub> is a linear function of log ''t''. In the pseudosteady-state region, the reservoir is modeled by '''Eq. 8.20''' in the general case or '''Eqs. 8.15''' or '''8.13''' for the special case of a well centered in a cylindrical drainage area. '''Eq. 8.13''' shows a linear relationship between ''p''<sub>''wf''</sub> and ''t'' during pseudosteady-state flow. This linear relationship also exists in generalized reservoir geometries.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0726 Image 0001.png|'''Fig. 8.3 – Flow regions on semilogarithmic coordinates.'''
 
File:vol5 Page 0726 Image 0001.png|'''Fig. 8.3 – Flow regions on semilogarithmic coordinates.'''
  
 
File:vol5 Page 0726 Image 0002.png|'''Fig. 8.4 – Flow regions on Cartesian coordinates.'''
 
File:vol5 Page 0726 Image 0002.png|'''Fig. 8.4 – Flow regions on Cartesian coordinates.'''
 
</gallery>
 
</gallery>
<br/>
 
  
At times between the end of the transient region and the beginning of the pseudosteady-state region, there is a transition region, sometimes called the late-transient region. This region is, for practical purposes, nonexistent for wells centered in circular, square, or hexagonal drainage areas, as '''Table 8.A-1''' indicates. However, for a well off-center in its drainage area, the late-transient region can span a significant time region, as '''Table 8.A-1''' also indicates.  
+
At times between the end of the transient region and the beginning of the pseudosteady-state region, there is a transition region, sometimes called the late-transient region. This region is, for practical purposes, nonexistent for wells centered in circular, square, or hexagonal drainage areas, as '''Table 8.A-1''' indicates. However, for a well off-center in its drainage area, the late-transient region can span a significant time region, as '''Table 8.A-1''' also indicates.
  
 
=== Steady-State Flow ===
 
=== Steady-State Flow ===
  
Pseudosteady-state flow describes production from a closed drainage area (one with no-flow outer boundaries, either permanent and caused by zero-permeability rock or "temporary" and caused by production from offset wells). In pseudosteady-state, reservoir pressure drops at the same rate with time at all points in the reservoir, including at the reservoir boundaries. Ideally, true steady-state flow can occur in the drainage area of a well, but only if pressure at the drainage boundaries of the well can be maintained constant while the well is producing at constant rate. While unlikely, steady-state flow is conceivable for wells with edgewater drive or in repeated flood patterns in a reservoir. The solution to the radial diffusivity equation is based on a constant-pressure outer boundary condition, instead of a no-flow outer boundary condition. The steady-state solution, applicable after boundary effects have been felt, is
+
Pseudosteady-state flow describes production from a closed drainage area (one with no-flow outer boundaries, either permanent and caused by zero-permeability rock or "temporary" and caused by production from offset wells). In pseudosteady-state, reservoir pressure drops at the same rate with time at all points in the reservoir, including at the reservoir boundaries. Ideally, true steady-state flow can occur in the drainage area of a well, but only if pressure at the drainage boundaries of the well can be maintained constant while the well is producing at constant rate. While unlikely, steady-state flow is conceivable for wells with edgewater drive or in repeated flood patterns in a reservoir. The solution to the radial diffusivity equation is based on a constant-pressure outer boundary condition, instead of a no-flow outer boundary condition. The steady-state solution, applicable after boundary effects have been felt, is<br/><br/>[[File:Vol5 page 0726 eq 001.png|RTENOTITLE]]....................(8.23)<br/>
<br>
 
<br>
 
[[File:Vol5 page 0726 eq 001.png]]....................(8.23)
 
<br>
 
<br>
 
  
 
=== Constant Pressure in the Well ===
 
=== Constant Pressure in the Well ===
  
Both the steady-state solution (constant pressure at the outer boundaries) and the pseudosteady-state solution (no-flow at the outer boundaries) assume constant rate production in the well. A well is actually more likely to be produced at something close to constant flowing BHP than constant rate. When pressure transients reach no-flow drainage area boundaries, the flow regime is not pseudosteady state; instead, it is more correctly called boundary-dominated flow. If the drainage boundaries are maintained at constant pressure, however, steady-state flow is achieved when the pressure transient reaches the reservoir boundaries.  
+
Both the steady-state solution (constant pressure at the outer boundaries) and the pseudosteady-state solution (no-flow at the outer boundaries) assume constant rate production in the well. A well is actually more likely to be produced at something close to constant flowing BHP than constant rate. When pressure transients reach no-flow drainage area boundaries, the flow regime is not pseudosteady state; instead, it is more correctly called boundary-dominated flow. If the drainage boundaries are maintained at constant pressure, however, steady-state flow is achieved when the pressure transient reaches the reservoir boundaries.<br/><br/>These different flow regimes are clarified with figures showing pressure distributions in the drainage area of wells with constant flow rate and constant-pressure outer boundaries ('''Fig. 8.5'''); constant BHP and constant-pressure outer boundaries (also '''Fig. 8.5'''); constant flow rate and no-flow outer boundaries ('''Fig. 8.6'''); and constant BHP and no-flow outer boundaries (also '''Fig. 8.6''').<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
These different flow regimes are clarified with figures showing pressure distributions in the drainage area of wells with constant flow rate and constant-pressure outer boundaries ('''Fig. 8.5'''); constant BHP and constant-pressure outer boundaries (also '''Fig. 8.5'''); constant flow rate and no-flow outer boundaries ('''Fig. 8.6'''); and constant BHP and no-flow outer boundaries (also '''Fig. 8.6''').  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0727 Image 0001.png|'''Fig. 8.5 – Flow regimes with constant well flow rate or constant well pressure and constant-pressure outer boundaries.'''
 
File:vol5 Page 0727 Image 0001.png|'''Fig. 8.5 – Flow regimes with constant well flow rate or constant well pressure and constant-pressure outer boundaries.'''
  
 
File:vol5 Page 0728 Image 0001.png|'''Fig. 8.6 – Flow regimes with constant well flow rate and no-flow outer boundaries.'''
 
File:vol5 Page 0728 Image 0001.png|'''Fig. 8.6 – Flow regimes with constant well flow rate and no-flow outer boundaries.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Wellbore Storage ===
 
=== Wellbore Storage ===
  
The'' Ei-''function solution to the diffusivity equation assumes constant flow rate in the reservoir, starting at time zero. In practice, only the rate at the surface can be controlled. Under ideal conditions, a constant surface rate can be maintained, but the first fluid produced will be fluid that was stored in the wellbore, and, at first, the flow rate from the reservoir into the wellbore will be zero. As the wellbore is unloaded, the reservoir rate approaches the surface rate ('''Fig. 8.7'''). Only as the reservoir and surface rates become approximately equal does the'' Ei-''function solution become valid. This wellbore unloading during flow tests is a special case of a general phenomenon called wellbore storage.  
+
The''Ei-''function solution to the diffusivity equation assumes constant flow rate in the reservoir, starting at time zero. In practice, only the rate at the surface can be controlled. Under ideal conditions, a constant surface rate can be maintained, but the first fluid produced will be fluid that was stored in the wellbore, and, at first, the flow rate from the reservoir into the wellbore will be zero. As the wellbore is unloaded, the reservoir rate approaches the surface rate ('''Fig. 8.7'''). Only as the reservoir and surface rates become approximately equal does the''Ei-''function solution become valid. This wellbore unloading during flow tests is a special case of a general phenomenon called wellbore storage.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0729 Image 0001.png|'''Fig. 8.7 – Reservoir rate approaches surface rate as the wellbore is unloaded.'''
 
File:vol5 Page 0729 Image 0001.png|'''Fig. 8.7 – Reservoir rate approaches surface rate as the wellbore is unloaded.'''
 
</gallery>
 
</gallery>
<br/>
 
  
For a pressure buildup test, the surface rate is zero starting at the instant of shut-in. However, fluid continues to flow into the wellbore from the reservoir because of existing pressure gradients. Idealized models of pressure buildup tests assume a reservoir rate of zero starting at the time of shut-in for the test. This assumption is obviously violated because of the afterflow into the wellbore. As the afterflow rate diminishes, the downhole rate approaches the surface rate (zero), and only as the afterflow rate approaches zero closely can the idealized models closely approximate actual well behavior ('''Fig. 8.8'''). Afterflow during buildup tests is another special case of wellbore storage.  
+
For a pressure buildup test, the surface rate is zero starting at the instant of shut-in. However, fluid continues to flow into the wellbore from the reservoir because of existing pressure gradients. Idealized models of pressure buildup tests assume a reservoir rate of zero starting at the time of shut-in for the test. This assumption is obviously violated because of the afterflow into the wellbore. As the afterflow rate diminishes, the downhole rate approaches the surface rate (zero), and only as the afterflow rate approaches zero closely can the idealized models closely approximate actual well behavior ('''Fig. 8.8'''). Afterflow during buildup tests is another special case of wellbore storage.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0730 Image 0001.png|'''Fig. 8.8 – Downhole rate approaches surface rate as afterflow rate diminishes.'''
 
File:vol5 Page 0730 Image 0001.png|'''Fig. 8.8 – Downhole rate approaches surface rate as afterflow rate diminishes.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The relationship between changes in bottomhole pressure and wellbore unloading or afterflow rates can be modeled with mass balances on the wellbore. There are two special cases of interest: a wellbore completely filled with a single-phase fluid ('''Fig. 8.9''', usually gas in practice) and a wellbore with a rising or falling liquid/gas interface in the well ('''Fig. 8.10''').  
+
The relationship between changes in bottomhole pressure and wellbore unloading or afterflow rates can be modeled with mass balances on the wellbore. There are two special cases of interest: a wellbore completely filled with a single-phase fluid ('''Fig. 8.9''', usually gas in practice) and a wellbore with a rising or falling liquid/gas interface in the well ('''Fig. 8.10''').<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0731 Image 0001.png|'''Fig. 8.9 – Schematic of wellbore containing single-phase liquid or gas.'''
 
File:vol5 Page 0731 Image 0001.png|'''Fig. 8.9 – Schematic of wellbore containing single-phase liquid or gas.'''
  
 
File:vol5 Page 0732 Image 0001.png|'''Fig. 8.10 – Schematic of wellbore with moving liquid/gas interface.'''
 
File:vol5 Page 0732 Image 0001.png|'''Fig. 8.10 – Schematic of wellbore with moving liquid/gas interface.'''
 
</gallery>
 
</gallery>
<br/>
 
  
For the wellbore filled with a single-phase fluid, <ref name="r4" />
+
For the wellbore filled with a single-phase fluid, <ref name="r4">_</ref><br/><br/>[[File:Vol5 page 0728 eq 001.png|RTENOTITLE]]....................(8.24)<br/><br/>For a well with a rising or falling liquid/gas interface, <ref name="r4">_</ref><br/><br/>[[File:Vol5 page 0728 eq 002.png|RTENOTITLE]]....................(8.25)<br/><br/>In most applications, ''p''<sub>''t''</sub> is assumed to be constant, a convenient but frequently inaccurate simplification. Both equations can be written in the general form<br/><br/>[[File:Vol5 page 0729 eq 001.png|RTENOTITLE]]....................(8.26)<br/><br/>where, for a fluid-filled wellbore,<br/><br/>[[File:Vol5 page 0729 eq 002.png|RTENOTITLE]]....................(8.27)<br/><br/>and, for a moving liquid/gas interface with unchanging surface pressure,<br/><br/>[[File:Vol5 page 0729 eq 003.png|RTENOTITLE]]....................(8.28)<br/><br/>''C'' is called the wellbore storage coefficient.<br/><br/>For special cases in which, at earliest times for a flowing well, all the production is coming from fluid stored in the wellbore and none is entering the wellbore from the formation (or, for a shut-in well, the rate of afterflow is equal to the rate before shut in), the integration of '''Eq. 8.26''' yields<br/><br/>[[File:Vol5 page 0729 eq 004.png|RTENOTITLE]]....................(8.29)<br/><br/>where Δ''p'' is the pressure change in the time because either the start of flow or shut in and Δ''t'' is the elapsed time. On a log-log plot of Δ''p'' vs. Δ''t'' during these early times, a straight line with a slope of unity will result. For any point on this unit slope line, the wellbore storage coefficient, ''C'', can be found from any point on the line (Δ''t'', Δ''p'') and '''Eq. 8.29''' ('''Fig. 8.11'''). Alternatively, the slope (''qB''/24''C'') of a plot of Δ''p'' vs. Δ''t'' on Cartesian coordinates also leads to an estimate of the wellbore-storage coefficient.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0728 eq 001.png]]....................(8.24)
 
<br>
 
<br>
 
For a well with a rising or falling liquid/gas interface, <ref name="r4" />
 
<br>
 
<br>
 
[[File:Vol5 page 0728 eq 002.png]]....................(8.25)
 
<br>
 
<br>
 
In most applications, ''p''<sub>''t''</sub> is assumed to be constant, a convenient but frequently inaccurate simplification. Both equations can be written in the general form  
 
<br>
 
<br>
 
[[File:Vol5 page 0729 eq 001.png]]....................(8.26)
 
<br>
 
<br>
 
where, for a fluid-filled wellbore,
 
<br>
 
<br>
 
[[File:Vol5 page 0729 eq 002.png]]....................(8.27)
 
<br>
 
<br>
 
and, for a moving liquid/gas interface with unchanging surface pressure,
 
<br>
 
<br>
 
[[File:Vol5 page 0729 eq 003.png]]....................(8.28)
 
<br>
 
<br>
 
''C'' is called the wellbore storage coefficient.  
 
<br>
 
<br>
 
For special cases in which, at earliest times for a flowing well, all the production is coming from fluid stored in the wellbore and none is entering the wellbore from the formation (or, for a shut-in well, the rate of afterflow is equal to the rate before shut in), the integration of '''Eq. 8.26''' yields
 
<br>
 
<br>
 
[[File:Vol5 page 0729 eq 004.png]]....................(8.29)
 
<br>
 
<br>
 
where Δ''p'' is the pressure change in the time because either the start of flow or shut in and Δ''t'' is the elapsed time. On a log-log plot of Δ''p'' vs. Δ''t'' during these early times, a straight line with a slope of unity will result. For any point on this unit slope line, the wellbore storage coefficient, ''C'', can be found from any point on the line (Δ''t'', Δ''p'') and '''Eq. 8.29''' ('''Fig. 8.11'''). Alternatively, the slope (''qB''/24''C'') of a plot of Δ''p'' vs. Δ''t'' on Cartesian coordinates also leads to an estimate of the wellbore-storage coefficient.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0733 Image 0001.png|'''Fig. 8.11 – Unit slope line on a log-log plot of Δ''p'' vs. Δ''t''.'''
 
File:vol5 Page 0733 Image 0001.png|'''Fig. 8.11 – Unit slope line on a log-log plot of Δ''p'' vs. Δ''t''.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Linear Flow ===
 
=== Linear Flow ===
  
Linear flow occurs in some reservoirs with long, highly conductive vertical fractures; in relatively long, relatively narrow reservoirs (channels, such as ancient stream beds); and near horizontal wells during certain times. For unsteady-state linear flow in an unbounded (infinite-acting) reservoir, <ref name="r4" />
+
Linear flow occurs in some reservoirs with long, highly conductive vertical fractures; in relatively long, relatively narrow reservoirs (channels, such as ancient stream beds); and near horizontal wells during certain times. For unsteady-state linear flow in an unbounded (infinite-acting) reservoir, <ref name="r4">_</ref><br/><br/>[[File:Vol5 page 0730 eq 001.png|RTENOTITLE]]....................(8.30)<br/>
<br>
 
<br>
 
[[File:Vol5 page 0730 eq 001.png]]....................(8.30)
 
<br>
 
<br>
 
  
 
=== Spherical Flow ===
 
=== Spherical Flow ===
  
Spherical flow occurs in wells with limited perforated intervals and into wireline formation test tools. The solution to the spherical/cylindrical, 1D form of the diffusivity equation, subject to the initial condition that pressure is uniform before production and the boundary conditions of constant flow rate and an infinitely large drainage area, is<ref name="r5" />
+
Spherical flow occurs in wells with limited perforated intervals and into wireline formation test tools. The solution to the spherical/cylindrical, 1D form of the diffusivity equation, subject to the initial condition that pressure is uniform before production and the boundary conditions of constant flow rate and an infinitely large drainage area, is<ref name="r5">_</ref><br/><br/>[[File:Vol5 page 0730 eq 002.png|RTENOTITLE]]....................(8.31)<br/><br/>where [[File:Vol5 page 0730 eq 003.png|RTENOTITLE]]....................(8.32)<br/><br/>and ''r''<sub>''sp''</sub> = the radius of the sphere into which flow converges.
<br>
 
<br>
 
[[File:Vol5 page 0730 eq 002.png]]....................(8.31)
 
<br>
 
<br>
 
where [[File:Vol5 page 0730 eq 003.png]]....................(8.32)
 
<br>
 
<br>
 
and ''r''<sub>''sp''</sub> = the radius of the sphere into which flow converges.  
 
  
 
=== Superposition ===
 
=== Superposition ===
  
The principle of superposition indicates that the total pressure at any point in a reservoir is the sum of the pressure drops at that point caused by flow in each of the wells in the reservoir. A simple illustration of this principle is the case of three wells in an infinite reservoir. Consider wells A, B, and C, that start to produce at times ''t''<sub>A</sub>, ''t''<sub>B</sub>, and ''t''<sub>C</sub> in an infinite-acting reservoir ('''Fig. 8.12'''). Application of the principle of superposition shows that<ref name="r4" />
+
The principle of superposition indicates that the total pressure at any point in a reservoir is the sum of the pressure drops at that point caused by flow in each of the wells in the reservoir. A simple illustration of this principle is the case of three wells in an infinite reservoir. Consider wells A, B, and C, that start to produce at times ''t''<sub>A</sub>, ''t''<sub>B</sub>, and ''t''<sub>C</sub> in an infinite-acting reservoir ('''Fig. 8.12'''). Application of the principle of superposition shows that<ref name="r4">_</ref><br/><br/>[[File:Vol5 page 0731 eq 001.png|RTENOTITLE]]....................(8.33)<br/><br/>For an infinite-acting reservoir, use the''Ei-''function solutions, including the logarithmic approximation at Well A:<br/><br/>[[File:Vol5 page 0731 eq 002.png|RTENOTITLE]]<br/>[[File:Vol5 page 0732 eq 001.png|RTENOTITLE]]....................(8.34)<br/><br/>where ''t''<sub>A</sub>, ''t''<sub>B</sub>, and ''t''<sub>C</sub> are times at which wells A, B, and C will begin to produce. The skin factor for Well A is included in '''Eq. 8.29'''. The skin factors for other wells are not, because skin factors for individual wells affect only pressures measured inside altered zones for those wells.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0731 eq 001.png]]....................(8.33)
 
<br>
 
<br>
 
For an infinite-acting reservoir, use the'' Ei-''function solutions, including the logarithmic approximation at Well A:
 
<br>
 
<br>
 
[[File:Vol5 page 0731 eq 002.png]]
 
<br>
 
[[File:Vol5 page 0732 eq 001.png]]....................(8.34)
 
<br>
 
<br>
 
where ''t''<sub>A</sub>, ''t''<sub>B</sub>, and ''t''<sub>C</sub> are times at which wells A, B, and C will begin to produce. The skin factor for Well A is included in '''Eq. 8.29'''. The skin factors for other wells are not, because skin factors for individual wells affect only pressures measured inside altered zones for those wells.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0733 Image 0002.png|'''Fig. 8.12 – Multiple-well system in infinite reservoir.'''
 
File:vol5 Page 0733 Image 0002.png|'''Fig. 8.12 – Multiple-well system in infinite reservoir.'''
 
</gallery>
 
</gallery>
<br/>
 
  
Next, consider the use of superposition to model the effects of boundaries in bounded reservoirs. Consider the well in '''Fig. 8.13''', a distance ''L'' from a single no-flow boundary (such as a sealing fault). Mathematically, this problem is identical to the problem of a well at distance 2''L'' from an "image" well; that is, a well that has the same production history as the actual well. The reason that the two-well system simulates the behavior of a well near a boundary is that a line equidistant between the two wells can be shown to be a no-flow boundary. That is, along this line the pressure gradient is zero, which means that there can be no flow. Thus, this problem is a simple problem of two wells in an infinite reservoir:
+
Next, consider the use of superposition to model the effects of boundaries in bounded reservoirs. Consider the well in '''Fig. 8.13''', a distance ''L'' from a single no-flow boundary (such as a sealing fault). Mathematically, this problem is identical to the problem of a well at distance 2''L'' from an "image" well; that is, a well that has the same production history as the actual well. The reason that the two-well system simulates the behavior of a well near a boundary is that a line equidistant between the two wells can be shown to be a no-flow boundary. That is, along this line the pressure gradient is zero, which means that there can be no flow. Thus, this problem is a simple problem of two wells in an infinite reservoir:<br/><br/>[[File:Vol5 page 0732 eq 002.png|RTENOTITLE]]....................(8.35)<br/><br/>The drawdown term of the image well does not include a skin factor.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0732 eq 002.png]]....................(8.35)
 
<br>
 
<br>
 
The drawdown term of the image well does not include a skin factor.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0734 Image 0001.png|'''Fig. 8.13 – Well near no-flow boundary illustrating use of imaging.'''
 
File:vol5 Page 0734 Image 0001.png|'''Fig. 8.13 – Well near no-flow boundary illustrating use of imaging.'''
 
</gallery>
 
</gallery>
<br/>
 
  
As examples, extend the imaging technique to model wells between boundaries intersecting at 90° ('''Fig. 8.14'''); wells between two parallel boundaries ('''Fig. 8.15'''); wells near single constant-pressure boundaries ('''Fig. 8.16'''); and wells at various locations in closed reservoirs ('''Fig. 8.17''').  
+
As examples, extend the imaging technique to model wells between boundaries intersecting at 90° ('''Fig. 8.14'''); wells between two parallel boundaries ('''Fig. 8.15'''); wells near single constant-pressure boundaries ('''Fig. 8.16'''); and wells at various locations in closed reservoirs ('''Fig. 8.17''').<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0735 Image 0001.png|'''Fig. 8.14 – Image-well locations for two intersecting no-flow boundaries.'''
 
File:vol5 Page 0735 Image 0001.png|'''Fig. 8.14 – Image-well locations for two intersecting no-flow boundaries.'''
  
Line 411: Line 137:
 
File:vol5 Page 0738 Image 0001.png|'''Fig. 8.17 – Image-well locations for well between parallel boundaries.'''
 
File:vol5 Page 0738 Image 0001.png|'''Fig. 8.17 – Image-well locations for well between parallel boundaries.'''
 
</gallery>
 
</gallery>
<br/>
 
  
One of the most frequently used applications of superposition is to model variable-rate production. Consider '''Fig. 8.18''', in which a well in an infinite-acting reservoir produces at rate ''q''<sub>1</sub> from time 0 to time ''t''<sub>1</sub>; ''q''<sub>2</sub> from ''t''<sub>1</sub> to ''t''<sub>2</sub>, and ''q''<sub>3</sub> for times greater than ''t''<sub>2</sub>. To model the total drawdown for ''t'' > ''t''<sub>2</sub>, add three drawdowns: the drawdown because of a well producing at rate ''q''<sub>1</sub> starting at time zero and continuing to produce to time ''t''; the drawdown because of a well producing at rate (''q''<sub>2</sub> – ''q''<sub>1</sub>), starting at time ''t''<sub>1</sub> and continuing to time ''t''; and the drawdown because of a well producing at rate (''q''<sub>3</sub> – ''q''<sub>2</sub>) starting at time ''t''<sub>2</sub> and continuing to time ''t''. The total drawdown is thus
+
One of the most frequently used applications of superposition is to model variable-rate production. Consider '''Fig. 8.18''', in which a well in an infinite-acting reservoir produces at rate ''q''<sub>1</sub> from time 0 to time ''t''<sub>1</sub>; ''q''<sub>2</sub> from ''t''<sub>1</sub> to ''t''<sub>2</sub>, and ''q''<sub>3</sub> for times greater than ''t''<sub>2</sub>. To model the total drawdown for ''t'' > ''t''<sub>2</sub>, add three drawdowns: the drawdown because of a well producing at rate ''q''<sub>1</sub> starting at time zero and continuing to produce to time ''t''; the drawdown because of a well producing at rate (''q''<sub>2</sub> – ''q''<sub>1</sub>), starting at time ''t''<sub>1</sub> and continuing to time ''t''; and the drawdown because of a well producing at rate (''q''<sub>3</sub> – ''q''<sub>2</sub>) starting at time ''t''<sub>2</sub> and continuing to time ''t''. The total drawdown is thus<br/><br/>[[File:Vol5 page 0733 eq 001.png|RTENOTITLE]]<br/>[[File:Vol5 page 0734 eq 001.png|RTENOTITLE]]....................(8.36)<br/><br/>Horner<ref name="r6">_</ref> proposed a convenient alternative to superposition to model the many changes in rate in the history of a typical well. With this approximation, the sequence of''Ei ''functions reflecting rate changes can be replaced with a single''Ei ''function that contains a single producing time and a single producing rate. The single rate is the most recent nonzero rate at which the well has produced, ''q''<sub>''n''</sub>. The single producing time, called ''t''<sub>''p''</sub>, is the ratio of cumulative production, ''N''<sub>''p''</sub>, to ''q''<sub>''n''</sub>.<br/><br/>[[File:Vol5 page 0734 eq 002.png|RTENOTITLE]]....................(8.37)<br/><br/>This approximation preserves the material balance in the drainage area of the well and properly gives greatest weight to most recent rate (as opposed to average rate), which dominates the pressure distribution near a well out to the radius of investigation achieved while the well was produced at rate ''q''<sub>''n''</sub>. The approximation is particularly useful for hand calculations. Given the widespread availability of computer software for analyzing flow and buildup tests on well, the use of more rigorous superposition to model variable-rate production histories is generally more appropriate.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0733 eq 001.png]]
 
<br>
 
[[File:Vol5 page 0734 eq 001.png]]....................(8.36)
 
<br>
 
<br>
 
Horner<ref name="r6" /> proposed a convenient alternative to superposition to model the many changes in rate in the history of a typical well. With this approximation, the sequence of'' Ei ''functions reflecting rate changes can be replaced with a single'' Ei ''function that contains a single producing time and a single producing rate. The single rate is the most recent nonzero rate at which the well has produced, ''q''<sub>''n''</sub>. The single producing time, called ''t''<sub>''p''</sub>, is the ratio of cumulative production, ''N''<sub>''p''</sub>, to ''q''<sub>''n''</sub>.
 
<br>
 
<br>
 
[[File:Vol5 page 0734 eq 002.png]]....................(8.37)
 
<br>
 
<br>
 
This approximation preserves the material balance in the drainage area of the well and properly gives greatest weight to most recent rate (as opposed to average rate), which dominates the pressure distribution near a well out to the radius of investigation achieved while the well was produced at rate ''q''<sub>''n''</sub>. The approximation is particularly useful for hand calculations. Given the widespread availability of computer software for analyzing flow and buildup tests on well, the use of more rigorous superposition to model variable-rate production histories is generally more appropriate.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0739 Image 0001.png|'''Fig. 8.18 – Production schedule for variable-rate well.'''
 
File:vol5 Page 0739 Image 0001.png|'''Fig. 8.18 – Production schedule for variable-rate well.'''
 
</gallery>
 
</gallery>
<br/>
 
  
 
=== Semilog Methods for Flow Tests ===
 
=== Semilog Methods for Flow Tests ===
  
The logarithmic approximation to the'' Ei-''function solution can be used as a basis for analysis of an ideal constant-rate flow test in a well. Written in terms of log<sub>10</sub>, this equation, which models the BHP for a well in a homogeneous-acting formation with an infinite-acting drainage area and, in absence of wellbore unloading, becomes
+
The logarithmic approximation to the''Ei-''function solution can be used as a basis for analysis of an ideal constant-rate flow test in a well. Written in terms of log<sub>10</sub>, this equation, which models the BHP for a well in a homogeneous-acting formation with an infinite-acting drainage area and, in absence of wellbore unloading, becomes<br/><br/>[[File:Vol5 page 0735 eq 001.png|RTENOTITLE]]....................(8.38)<br/><br/>This expression has the same form as the equation of a straight line, ''y'' = ''mx'' + ''b'', with the analogies<br/><br/>[[File:Vol5 page 0735 eq 002.png|RTENOTITLE]]....................(8.39)<br/><br/>[[File:Vol5 page 0735 eq 003.png|RTENOTITLE]]....................(8.40)<br/><br/>[[File:Vol5 page 0735 eq 004.png|RTENOTITLE]]....................(8.41)<br/><br/>and [[File:Vol5 page 0735 eq 005.png|RTENOTITLE]]....................(8.42)<br/><br/>These analogies suggest a graphical method of analysis. '''Eq. 8.38''' indicates that a plot of ''p''<sub>''wf''</sub> vs. log<sub>10</sub>(''t'') should be a straight line with slope m that will allow an estimate of effective permeability to the single liquid phase flowing. (See '''Fig. 8.19'''.)<br/><br/>[[File:Vol5 page 0736 eq 001.png|RTENOTITLE]]....................(8.43)<br/><br/>From the intercept, ''b'', at ''t'' = 1 hr [log<sub>10</sub>(1) = 0], ''p''<sub>1hr</sub>, calculate the skin factor.<br/><br/>[[File:Vol5 page 0736 eq 002.png|RTENOTITLE]]....................(8.44)<br/><br/>In these equations, the slope, ''m'', is given by<br/><br/>[[File:Vol5 page 0736 eq 003.png|RTENOTITLE]]....................(8.45)<br/><br/>'''Eq. 8.45''' indicates that ''m'' is most easily determined by choosing values of times ''t''<sub>1</sub> and ''t''<sub>2</sub> that differ by powers of 10 and is especially easy if ''t''<sub>1</sub> and ''t''<sub>2</sub> differ by one log cycle. The intercept, ''p''<sub>1hr</sub>, is the pressure at a time of 1 hour on the best straight line through the data. It may be necessary to extrapolate the straight line to a time of one hour to read the intercept.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0735 eq 001.png]]....................(8.38)
 
<br>
 
<br>
 
This expression has the same form as the equation of a straight line, ''y'' = ''mx'' + ''b'', with the analogies
 
<br>
 
<br>
 
[[File:Vol5 page 0735 eq 002.png]]....................(8.39)
 
<br>
 
<br>
 
[[File:Vol5 page 0735 eq 003.png]]....................(8.40)
 
<br>
 
<br>
 
[[File:Vol5 page 0735 eq 004.png]]....................(8.41)
 
<br>
 
<br>
 
and [[File:Vol5 page 0735 eq 005.png]]....................(8.42)
 
<br>
 
<br>
 
These analogies suggest a graphical method of analysis. '''Eq. 8.38''' indicates that a plot of ''p''<sub>''wf''</sub> vs. log<sub>10</sub>(''t'') should be a straight line with slope m that will allow an estimate of effective permeability to the single liquid phase flowing. (See '''Fig. 8.19'''.)
 
<br>
 
<br>
 
[[File:Vol5 page 0736 eq 001.png]]....................(8.43)
 
<br>
 
<br>
 
From the intercept, ''b'', at ''t'' = 1 hr [log<sub>10</sub>(1) = 0], ''p''<sub>1hr</sub>, calculate the skin factor.
 
<br>
 
<br>
 
[[File:Vol5 page 0736 eq 002.png]]....................(8.44)
 
<br>
 
<br>
 
In these equations, the slope, ''m'', is given by
 
<br>
 
<br>
 
[[File:Vol5 page 0736 eq 003.png]]....................(8.45)
 
<br>
 
<br>
 
'''Eq. 8.45''' indicates that ''m'' is most easily determined by choosing values of times ''t''<sub>1</sub> and ''t''<sub>2</sub> that differ by powers of 10 and is especially easy if ''t''<sub>1</sub> and ''t''<sub>2</sub> differ by one log cycle. The intercept, ''p''<sub>1hr</sub>, is the pressure at a time of 1 hour on the best straight line through the data. It may be necessary to extrapolate the straight line to a time of one hour to read the intercept.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0740 Image 0001.png|'''Fig. 8.19 – Straight line with slope ''m'' allows us to estimate effective permeability to the single liquid phase flowing.'''
 
File:vol5 Page 0740 Image 0001.png|'''Fig. 8.19 – Straight line with slope ''m'' allows us to estimate effective permeability to the single liquid phase flowing.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Semilog Methods for Pressure Buildup Test ===
 
=== Semilog Methods for Pressure Buildup Test ===
  
Consider the rate history for an idealized pressure test shown in '''Fig. 8.20'''. A well is produced at constant rate ''q'' for a time ''t''<sub>''p''</sub>, and then the well shut in (''q'' = 0) for a pressure buildup test. The rate history is modeled as the sum of two constant flow rate periods, one at rate ''q'', beginning at ''t'' = 0, and the other at rate –''q'', beginning at ''t'' = ''t''<sub>''p''</sub>, at which the time elapsed since shut-in, Δ''t'', is zero. Use the log approximation to the'' Ei-''function solution to model the drawdown, and sum them as '''Fig. 8.21''' shows. Represented mathematically, the superposition process is
+
Consider the rate history for an idealized pressure test shown in '''Fig. 8.20'''. A well is produced at constant rate ''q'' for a time ''t''<sub>''p''</sub>, and then the well shut in (''q'' = 0) for a pressure buildup test. The rate history is modeled as the sum of two constant flow rate periods, one at rate ''q'', beginning at ''t'' = 0, and the other at rate –''q'', beginning at ''t'' = ''t''<sub>''p''</sub>, at which the time elapsed since shut-in, Δ''t'', is zero. Use the log approximation to the''Ei-''function solution to model the drawdown, and sum them as '''Fig. 8.21''' shows. Represented mathematically, the superposition process is<br/><br/>[[File:Vol5 page 0737 eq 001.png|RTENOTITLE]]....................(8.46)<br/><br/>This can be simplified to<br/><br/>[[File:Vol5 page 0737 eq 002.png|RTENOTITLE]]....................(8.47)<br/><br/>Like the drawdown equation, '''Eq. 8.43''' can be interpreted as the equation of a straight line. The analogies are<br/><br/>[[File:Vol5 page 0737 eq 003.png|RTENOTITLE]]....................(8.48)<br/><br/>[[File:Vol5 page 0737 eq 004.png|RTENOTITLE]]....................(8.49)<br/><br/>[[File:Vol5 page 0738 eq 001.png|RTENOTITLE]]....................(8.41)<br/><br/>and [[File:Vol5 page 0738 eq 002.png|RTENOTITLE]]....................(8.50)<br/><br/>The group [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t''] is called the Horner time ratio (HTR) or sometimes simply the Horner time. Our simple model, which describes a buildup test in a homogeneous, infinite-acting reservoir, a well with one constant rate before shut in and without afterflow (wellbore storage), indicates that a graph of ''p''<sub>''ws''</sub> vs. the HTR should fall on a straight line. From the slope, ''m'', of this line, the permeability to the single-phase liquid flowing into the wellbore can be estimated. The intercept, ''b'', at log 10 [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t''] = 0 or [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t''] = 1 provides an estimate of original drainage area pressure, ''p''<sub>''i''</sub>.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0737 eq 001.png]]....................(8.46)
 
<br>
 
<br>
 
This can be simplified to
 
<br>
 
<br>
 
[[File:Vol5 page 0737 eq 002.png]]....................(8.47)
 
<br>
 
<br>
 
Like the drawdown equation, '''Eq. 8.43''' can be interpreted as the equation of a straight line. The analogies are
 
<br>
 
<br>
 
[[File:Vol5 page 0737 eq 003.png]]....................(8.48)
 
<br>
 
<br>
 
[[File:Vol5 page 0737 eq 004.png]]....................(8.49)
 
<br>
 
<br>
 
[[File:Vol5 page 0738 eq 001.png]]....................(8.41)
 
<br>
 
<br>
 
and [[File:Vol5 page 0738 eq 002.png]]....................(8.50)
 
<br>
 
<br>
 
The group [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t''] is called the Horner time ratio (HTR) or sometimes simply the Horner time. Our simple model, which describes a buildup test in a homogeneous, infinite-acting reservoir, a well with one constant rate before shut in and without afterflow (wellbore storage), indicates that a graph of ''p''<sub>''ws''</sub> vs. the HTR should fall on a straight line. From the slope, ''m'', of this line, the permeability to the single-phase liquid flowing into the wellbore can be estimated. The intercept, ''b'', at log 10 [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t''] = 0 or [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t''] = 1 provides an estimate of original drainage area pressure, ''p''<sub>''i''</sub>.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0740 Image 0002.png|'''Fig. 8.20 – Rate history for an idealized pressure buildup test.'''
 
File:vol5 Page 0740 Image 0002.png|'''Fig. 8.20 – Rate history for an idealized pressure buildup test.'''
  
 
File:vol5 Page 0741 Image 0001.png|'''Fig. 8.21 – Sum of ''Ei''-function solutions to model pressure buildup test.'''
 
File:vol5 Page 0741 Image 0001.png|'''Fig. 8.21 – Sum of ''Ei''-function solutions to model pressure buildup test.'''
 
</gallery>
 
</gallery>
<br/>
 
  
Obtain the slope, ''m'', from
+
Obtain the slope, ''m'', from<br/><br/>[[File:Vol5 page 0738 eq 003.png|RTENOTITLE]]....................(8.51)<br/><br/>where {[ (''t''<sub>''p''</sub> +Δ''t'')/Δ''t'']<sub>1</sub>, ''p''<sub>''ws''1</sub>} and {[ (''t''<sub>''p''</sub> +Δ''t'')/Δ''t'']<sub>2</sub>, ''p''<sub>''ws''2</sub>} are any two points on the straight-line ('''Fig. 8.22'''). Normally, choose [''t''<sub>''p''</sub> + Δ''t'')/Δ''t'']<sub>1</sub> and [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t'']<sub>2</sub> to be powers of 10.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0738 eq 003.png]]....................(8.51)
 
<br>
 
<br>
 
where {[ (''t''<sub>''p''</sub> +Δ''t'')/Δ''t'']<sub>1</sub>, ''p''<sub>''ws''1</sub>} and {[ (''t''<sub>''p''</sub> +Δ''t'')/Δ''t'']<sub>2</sub>, ''p''<sub>''ws''2</sub>} are any two points on the straight-line ('''Fig. 8.22'''). Normally, choose [''t''<sub>''p''</sub> + Δ''t'')/Δ''t'']<sub>1</sub> and [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t'']<sub>2</sub> to be powers of 10.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0741 Image 0001.png|'''Fig. 8.22 – Horner plot.'''
 
File:vol5 Page 0741 Image 0001.png|'''Fig. 8.22 – Horner plot.'''
 
</gallery>
 
</gallery>
<br/>
 
  
In '''Fig. 8.22''', which is called a Horner plot, the HTR on the horizontal axis decreases from left to right, so that shut-in time increases from left to right. In some Horner plots, the HTR increases from left to right; in that case shut-in time increases from right to left.  
+
In '''Fig. 8.22''', which is called a Horner plot, the HTR on the horizontal axis decreases from left to right, so that shut-in time increases from left to right. In some Horner plots, the HTR increases from left to right; in that case shut-in time increases from right to left.<br/><br/>Skin factor can be estimated from a pressure buildup test, even though the skin factor does not appear in the buildup equation, '''Eq. 8.47'''. Simultaneously solve the equation modeling the drawdown at the instant of shut in (at time ''t''<sub>''p''</sub>) with '''Eq. 8.47''', discard terms that are ordinarily negligible, and arrive at the result<br/><br/>[[File:Vol5 page 0739 eq 001.png|RTENOTITLE]]....................(8.52)<br/><br/>The radius-of-investigation concept is also useful for pressure buildup tests, as '''Fig. 8.23''' illustrates. The approximate position of the point at which the pressure has built up to a uniform level intersects the region in which the pressure is little affected by the shut-in is given by '''Eq. 8.11''', with elapsed time, ''t'', interpreted as shut-in time, Δt.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
Skin factor can be estimated from a pressure buildup test, even though the skin factor does not appear in the buildup equation, '''Eq. 8.47'''. Simultaneously solve the equation modeling the drawdown at the instant of shut in (at time ''t''<sub>''p''</sub>) with '''Eq. 8.47''', discard terms that are ordinarily negligible, and arrive at the result
 
<br>
 
<br>
 
[[File:Vol5 page 0739 eq 001.png]]....................(8.52)
 
<br>
 
<br>
 
The radius-of-investigation concept is also useful for pressure buildup tests, as '''Fig. 8.23''' illustrates. The approximate position of the point at which the pressure has built up to a uniform level intersects the region in which the pressure is little affected by the shut-in is given by '''Eq. 8.11''', with elapsed time, ''t'', interpreted as shut-in time, Δt.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0742 Image 0001.png|'''Fig. 8.23 – Radius of investigation in pressure buildup tests.'''
 
File:vol5 Page 0742 Image 0001.png|'''Fig. 8.23 – Radius of investigation in pressure buildup tests.'''
 
</gallery>
 
</gallery>
<br/>
+
</div></div><div class="toccolours mw-collapsible mw-collapsed">
 
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
 
 
 
 
== Type Curves ==
 
== Type Curves ==
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
<br>
+
<br/>Type curves provide a powerful method for analyzing pressure drawdown (flow) and buildup tests. Fundamentally, type curves are preplotted solutions to the flow equations, such as the diffusivity equation, for selected types of formations and selected initial and boundary conditions. Because of the way they are plotted (usually on logarithmic coordinates), it is convenient to compare actual field data plotted on the same coordinates to the type curves. The results of this comparison frequently include qualitative and quantitative descriptions of the formation and completion properties of the tested well.
Type curves provide a powerful method for analyzing pressure drawdown (flow) and buildup tests. Fundamentally, type curves are preplotted solutions to the flow equations, such as the diffusivity equation, for selected types of formations and selected initial and boundary conditions. Because of the way they are plotted (usually on logarithmic coordinates), it is convenient to compare actual field data plotted on the same coordinates to the type curves. The results of this comparison frequently include qualitative and quantitative descriptions of the formation and completion properties of the tested well.  
 
  
 
=== Dimensionless Variables ===
 
=== Dimensionless Variables ===
  
The solutions plotted on type curves are usually presented in terms of dimensionless variables. To review dimensionless variables, consider the'' Ei-''function solution to the flow equation, '''Eq. 8.2''', presented in terms of dimensional variables:
+
The solutions plotted on type curves are usually presented in terms of dimensionless variables. To review dimensionless variables, consider the''Ei-''function solution to the flow equation, '''Eq. 8.2''', presented in terms of dimensional variables:<br/><br/>[[File:Vol5 page 0740 eq 001.png|RTENOTITLE]]....................(8.2)<br/><br/>'''Eq. 8.2''' can be rewritten in terms of conventional definitions of dimensionless variables. (Variables that when the parameters are expressed in terms of the fundamental units of mass, length, and time, have no dimensions are sometimes said to have dimensions of zero.)<br/><br/>[[File:Vol5 page 0740 eq 002.png|RTENOTITLE]]....................(8.53)<br/><br/>In '''Eq. 8.53''', the definitions of the dimensionless variables are<br/><br/>[[File:Vol5 page 0741 eq 001.png|RTENOTITLE]]....................(8.54)<br/><br/>[[File:Vol5 page 0741 eq 002.png|RTENOTITLE]]....................(8.55)<br/><br/>and [[File:Vol5 page 0742 eq 001.png|RTENOTITLE]]....................(8.56)<br/><br/>The dimensionless form of '''Eq. 8.2''' has the advantage that this solution, ''p''<sub>''D''</sub>, to the diffusivity equation can be expressed in terms of a single variable, ''t''<sub>''D''</sub>, and single parameter, ''r''<sub>''D''</sub>. This leads to much simpler graphical or tabular presentation of the solution than would direct use of '''Eq. 8.2'''. Solutions to the diffusivity equation for more realistic reservoir models also include the dimensionless skin factor, ''s'', and wellbore storage coefficient, ''C''<sub>''D''</sub>, where<br/><br/>[[File:Vol5 page 0742 eq 002.png|RTENOTITLE]]....................(8.57)<br/>
<br>
 
<br>
 
[[File:Vol5 page 0740 eq 001.png]]....................(8.2)
 
<br>
 
<br>
 
'''Eq. 8.2''' can be rewritten in terms of conventional definitions of dimensionless variables. (Variables that when the parameters are expressed in terms of the fundamental units of mass, length, and time, have no dimensions are sometimes said to have dimensions of zero.)
 
<br>
 
<br>
 
[[File:Vol5 page 0740 eq 002.png]]....................(8.53)
 
<br>
 
<br>
 
In '''Eq. 8.53''', the definitions of the dimensionless variables are
 
<br>
 
<br>
 
[[File:Vol5 page 0741 eq 001.png]]....................(8.54)
 
<br>
 
<br>
 
[[File:Vol5 page 0741 eq 002.png]]....................(8.55)
 
<br>
 
<br>
 
and [[File:Vol5 page 0742 eq 001.png]]....................(8.56)
 
<br>
 
<br>
 
The dimensionless form of '''Eq. 8.2''' has the advantage that this solution, ''p''<sub>''D''</sub>, to the diffusivity equation can be expressed in terms of a single variable, ''t''<sub>''D''</sub>, and single parameter, ''r''<sub>''D''</sub>. This leads to much simpler graphical or tabular presentation of the solution than would direct use of '''Eq. 8.2'''. Solutions to the diffusivity equation for more realistic reservoir models also include the dimensionless skin factor, ''s'', and wellbore storage coefficient, ''C''<sub>''D''</sub>, where
 
<br>
 
<br>
 
[[File:Vol5 page 0742 eq 002.png]]....................(8.57)
 
<br>
 
<br>
 
  
 
=== Gringarten Type Curve ===
 
=== Gringarten Type Curve ===
  
Gringarten ''et al.''<ref name="r7" /> presented a type curve, commonly called the Gringarten type curve, that achieved widespread use. It is based on a solution to the radial diffusivity equation and the following assumptions: vertical well with constant production rate; infinite-acting, homogeneous-acting reservoir; single-phase, slightly compressible liquid flowing; infinitesimal skin factor (thin "membrane" at production face); and constant wellbore-storage coefficient. These assumptions indicate that the type curve was developed specifically for drawdown tests in undersaturated oil reservoirs. The type curve is also useful to analyze pressure buildup tests and for gas wells.  
+
Gringarten ''et al.''<ref name="r7">_</ref> presented a type curve, commonly called the Gringarten type curve, that achieved widespread use. It is based on a solution to the radial diffusivity equation and the following assumptions: vertical well with constant production rate; infinite-acting, homogeneous-acting reservoir; single-phase, slightly compressible liquid flowing; infinitesimal skin factor (thin "membrane" at production face); and constant wellbore-storage coefficient. These assumptions indicate that the type curve was developed specifically for drawdown tests in undersaturated oil reservoirs. The type curve is also useful to analyze pressure buildup tests and for gas wells.<br/><br/>In the Gringarten type curve, ''p''<sub>''D''</sub> is plotted vs. the time function ''t''<sub>''D''</sub>/''C''<sub>''D''</sub>, with a parameter ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> ('''Fig. 8.24'''). Each different value of ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> describes a pressure response with a shape different (in theory) from the responses for other values of the parameter. However, adjacent pairs of curves can be quite similar, and this fact can cause uncertainty when trying to match test data to the "uniquely correct" curve.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
In the Gringarten type curve, ''p''<sub>''D''</sub> is plotted vs. the time function ''t''<sub>''D''</sub>/''C''<sub>''D''</sub>, with a parameter ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> ('''Fig. 8.24'''). Each different value of ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> describes a pressure response with a shape different (in theory) from the responses for other values of the parameter. However, adjacent pairs of curves can be quite similar, and this fact can cause uncertainty when trying to match test data to the "uniquely correct" curve.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0743 Image 0001.png|'''Fig. 8.24 – Gringarten type curve with parameter ''C''<sub>''D''</sub>''e''<sup>2''s''</sup>.'''
 
File:vol5 Page 0743 Image 0001.png|'''Fig. 8.24 – Gringarten type curve with parameter ''C''<sub>''D''</sub>''e''<sup>2''s''</sup>.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Derivative Type Curve ===
 
=== Derivative Type Curve ===
  
The derivative type curve proposed by Bourdet ''et al.''<ref name="r8" /> eliminates the ambiguity in the Gringarten type curve. The "derivative" referred to in this type curve is the logarithmic derivative of the solution to the radial diffusivity equation presented on the Gringarten type curve. Two limiting forms of this solution help illustrate the nature of the derivative type curve. First, consider that part of a test response where the distorting effects of wellbore storage have vanished. This portion of the test is described by the logarithmic approximation to'' Ei-''function solution, '''Eq. 8.9''':
+
The derivative type curve proposed by Bourdet ''et al.''<ref name="r8">_</ref> eliminates the ambiguity in the Gringarten type curve. The "derivative" referred to in this type curve is the logarithmic derivative of the solution to the radial diffusivity equation presented on the Gringarten type curve. Two limiting forms of this solution help illustrate the nature of the derivative type curve. First, consider that part of a test response where the distorting effects of wellbore storage have vanished. This portion of the test is described by the logarithmic approximation to''Ei-''function solution, '''Eq. 8.9''':<br/><br/>[[File:Vol5 page 0743 eq 001.png|RTENOTITLE]]....................(8.9)<br/><br/>The derivative of (''p''<sub>''i''</sub> – ''p''<sub>''wf''</sub>) with respect to ln(''t''), expressed more simply as ''t∂''Δ''p''/''∂t'', is 70.6''qBμ''/''kh'', a constant. In terms of dimensionless variables, ''t''<sub>''D''</sub>(''∂p''<sub>''D''</sub>/''∂t''<sub>''D''</sub>) = 0.5. Thus, when the distorting effects of wellbore storage have disappeared, the pressure derivative will become constant in an infinite-acting reservoir, and, in terms of dimensionless variables, will have a value of 0.5.<br/><br/>When wellbore storage completely dominates the pressure response (all produced fluid comes from the wellbore, none from the formation),<br/><br/>[[File:Vol5 page 0743 eq 002.png|RTENOTITLE]]....................(8.29)<br/><br/>The derivative, ''t∂''Δ''p''/''∂t'', is ''qBt''/24''C'', the same as the pressure change itself. In terms of dimensionless variables, the derivative becomes<br/><br/>[[File:Vol5 page 0743 eq 003.png|RTENOTITLE]]....................(8.58)<br/><br/>The implication of '''Eq. 8.58''' is that, on logarithmic coordinates, graphs of ''p''<sub>''D''</sub> and ''t''<sub>''D''</sub>(''∂p''<sub>''D''</sub>/''∂t''<sub>''D''</sub>) vs. ''t''<sub>''D''</sub>/''C''<sub>''D''</sub> will coincide and will have slopes of unity.<br/><br/>For values of ''t''<sub>''D''</sub>(''∂p''<sub>''D''</sub>/''∂t''<sub>''D''</sub>) between the end of complete wellbore storage distortion and the start of infinite-acting radial flow, no simple solutions are available to guide us, but '''Fig. 8.25''' shows the derivatives, including those times. Note the unit slope lines at earliest times and the horizontal derivative at later times. The shapes of the derivative stems are much more distinctive than those for the pressure-change type curve.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0743 eq 001.png]]....................(8.9)
 
<br>
 
<br>
 
The derivative of (''p''<sub>''i''</sub> – ''p''<sub>''wf''</sub>) with respect to ln(''t''), expressed more simply as ''t∂''Δ''p''/''∂t'', is 70.6''qBμ''/''kh'', a constant. In terms of dimensionless variables, ''t''<sub>''D''</sub>(''∂p''<sub>''D''</sub>/''∂t''<sub>''D''</sub>) = 0.5. Thus, when the distorting effects of wellbore storage have disappeared, the pressure derivative will become constant in an infinite-acting reservoir, and, in terms of dimensionless variables, will have a value of 0.5.  
 
<br>
 
<br>
 
When wellbore storage completely dominates the pressure response (all produced fluid comes from the wellbore, none from the formation),
 
<br>
 
<br>
 
[[File:Vol5 page 0743 eq 002.png]]....................(8.29)
 
<br>
 
<br>
 
The derivative, ''t∂''Δ''p''/''∂t'', is ''qBt''/24''C'', the same as the pressure change itself. In terms of dimensionless variables, the derivative becomes
 
<br>
 
<br>
 
[[File:Vol5 page 0743 eq 003.png]]....................(8.58)
 
<br>
 
<br>
 
The implication of '''Eq. 8.58''' is that, on logarithmic coordinates, graphs of ''p''<sub>''D''</sub> and ''t''<sub>''D''</sub>(''∂p''<sub>''D''</sub>/''∂t''<sub>''D''</sub>) vs. ''t''<sub>''D''</sub>/''C''<sub>''D''</sub> will coincide and will have slopes of unity.  
 
<br>
 
<br>
 
For values of ''t''<sub>''D''</sub>(''∂p''<sub>''D''</sub>/''∂t''<sub>''D''</sub>) between the end of complete wellbore storage distortion and the start of infinite-acting radial flow, no simple solutions are available to guide us, but '''Fig. 8.25''' shows the derivatives, including those times. Note the unit slope lines at earliest times and the horizontal derivative at later times. The shapes of the derivative stems are much more distinctive than those for the pressure-change type curve.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0744 Image 0001.png|'''Fig. 8.25 – Bourdet’s derivative type curves.'''
 
File:vol5 Page 0744 Image 0001.png|'''Fig. 8.25 – Bourdet’s derivative type curves.'''
 
</gallery>
 
</gallery>
<br/>
 
  
For test analysis, we plot pressure change, ''p''<sub>''D''</sub>, and pressure derivative [''t''<sub>''D''</sub>(''∂p''<sub>''D''</sub>/''∂t''<sub>''D''</sub>)] on the same graph ('''Fig. 8.26'''). On this graph, a specific value of the parameter ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> refers to a pair of curves—one pressure-change curve and one pressure-derivative curve. Time regions can be defined conveniently on the basis of the combined pressure ('''Fig. 8.27''') and pressure derivative type curves.  
+
For test analysis, we plot pressure change, ''p''<sub>''D''</sub>, and pressure derivative [''t''<sub>''D''</sub>(''∂p''<sub>''D''</sub>/''∂t''<sub>''D''</sub>)] on the same graph ('''Fig. 8.26'''). On this graph, a specific value of the parameter ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> refers to a pair of curves—one pressure-change curve and one pressure-derivative curve. Time regions can be defined conveniently on the basis of the combined pressure ('''Fig. 8.27''') and pressure derivative type curves.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0744 Image 0002.png|'''Fig. 8.26 – Combined pressure change, pressure derivative type curves.'''
 
File:vol5 Page 0744 Image 0002.png|'''Fig. 8.26 – Combined pressure change, pressure derivative type curves.'''
  
 
File:vol5 Page 0745 Image 0001.png|'''Fig. 8.27- Combined pressure-change/pressure-derivative type curves define time regions conveniently.'''
 
File:vol5 Page 0745 Image 0001.png|'''Fig. 8.27- Combined pressure-change/pressure-derivative type curves define time regions conveniently.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The shape of the pressure- and pressure-derivative type curves provides a qualitative estimate of skin factor ('''Fig. 8.28'''). For a well with a large skin factor, the derivative rises to a maximum and then falls sharply before flattening out for the middle-time region (MTR). The pressure change curve rises along the unit-slope line and then flattens quickly. The pressure-change and pressure-derivative curves are separated by approximately two log cycles when wellbore storage (WBS) ends.  
+
The shape of the pressure- and pressure-derivative type curves provides a qualitative estimate of skin factor ('''Fig. 8.28'''). For a well with a large skin factor, the derivative rises to a maximum and then falls sharply before flattening out for the middle-time region (MTR). The pressure change curve rises along the unit-slope line and then flattens quickly. The pressure-change and pressure-derivative curves are separated by approximately two log cycles when wellbore storage (WBS) ends.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0745 Image 0002.png|'''Fig. 8.28 – Shape of the type curves provides a qualitative estimate of skin factor.'''
 
File:vol5 Page 0745 Image 0002.png|'''Fig. 8.28 – Shape of the type curves provides a qualitative estimate of skin factor.'''
 
</gallery>
 
</gallery>
<br/>
 
  
When the skin is near zero, the pressure derivative rises to a maximum and then falls only slightly before flattening for the MTR. The pressure change and pressure derivative are separated by approximately one log cycle when WBS ends. When the skin factor is negative, the pressure derivative approaches a horizontal line from below. The pressure change and pressure derivative curves leave the unit slope line at relatively early times and take a relatively long time to reach the MTR.  
+
When the skin is near zero, the pressure derivative rises to a maximum and then falls only slightly before flattening for the MTR. The pressure change and pressure derivative are separated by approximately one log cycle when WBS ends. When the skin factor is negative, the pressure derivative approaches a horizontal line from below. The pressure change and pressure derivative curves leave the unit slope line at relatively early times and take a relatively long time to reach the MTR.
  
 
=== Differences in Drawdown and Buildup Test Type Curves ===
 
=== Differences in Drawdown and Buildup Test Type Curves ===
  
The shapes of drawdown and buildup type curves are different, as '''Fig. 8.29''' illustrates. In this simplified case, in which wellbore storage distortion is absent, a well has produced for a dimensionless producing time, ''t''<sub>''pD''</sub>, of 10<sup>5</sup>, before shut-in. In the figure, note that, on a plot of ''p''<sub>''D''</sub> and ''p''<sub>''D''</sub>′ (the derivative) vs. ''t''<sub>''D''</sub> (dimensionless time since each test began), the shapes of the buildup and drawdown curves for infinite-acting radial flow coincide up to ''t''<sub>''D''</sub> = 10<sup>4</sup> and then begin to deviate. The buildup pressure-change curve is "flatter" than the drawdown curve at later times in an infinite-acting reservoir, and thus the slope of the buildup curve (the derivative) tends to deviate from the drawdown derivative. For many years, test analysts used a rule of thumb that buildup tests could be analyzed on a drawdown type curve only up to a maximum time of one-tenth the producing time before shut-in. That rule of thumb is appropriate for the conditions in '''Fig. 8.29'''.  
+
The shapes of drawdown and buildup type curves are different, as '''Fig. 8.29''' illustrates. In this simplified case, in which wellbore storage distortion is absent, a well has produced for a dimensionless producing time, ''t''<sub>''pD''</sub>, of 10<sup>5</sup>, before shut-in. In the figure, note that, on a plot of ''p''<sub>''D''</sub> and ''p''<sub>''D''</sub>′ (the derivative) vs. ''t''<sub>''D''</sub> (dimensionless time since each test began), the shapes of the buildup and drawdown curves for infinite-acting radial flow coincide up to ''t''<sub>''D''</sub> = 10<sup>4</sup> and then begin to deviate. The buildup pressure-change curve is "flatter" than the drawdown curve at later times in an infinite-acting reservoir, and thus the slope of the buildup curve (the derivative) tends to deviate from the drawdown derivative. For many years, test analysts used a rule of thumb that buildup tests could be analyzed on a drawdown type curve only up to a maximum time of one-tenth the producing time before shut-in. That rule of thumb is appropriate for the conditions in '''Fig. 8.29'''.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0746 Image 0001.png|'''Fig. 8.29 – Buildup and drawdown type curves have different shapes.'''
 
File:vol5 Page 0746 Image 0001.png|'''Fig. 8.29 – Buildup and drawdown type curves have different shapes.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Equivalent Drawdown Time ===
 
=== Equivalent Drawdown Time ===
  
Agarwal<ref name="r9" /> suggested a method of plotting pressure change data from a buildup test on a logarithmic graph that alters the shape so that it corresponds to that of a constant rate flow test during infinite-acting radial flow. The basis for Agarwal’s "equivalent time" is a combination of logarithmic approximations to'' Ei-''function solutions to the diffusivity equation. The equation modeling the drawdown at the instant of shut-in is
+
Agarwal<ref name="r9">_</ref> suggested a method of plotting pressure change data from a buildup test on a logarithmic graph that alters the shape so that it corresponds to that of a constant rate flow test during infinite-acting radial flow. The basis for Agarwal’s "equivalent time" is a combination of logarithmic approximations to''Ei-''function solutions to the diffusivity equation. The equation modeling the drawdown at the instant of shut-in is<br/><br/>[[File:Vol5 page 0746 eq 001.png|RTENOTITLE]]....................(8.59)<br/><br/>We model a buildup test with<br/><br/>[[File:Vol5 page 0746 eq 002.png|RTENOTITLE]]....................(8.60)<br/><br/>Combining '''Eqs. 8.59''' and '''8.60''' and simplifying,<br/><br/>[[File:Vol5 page 0746 eq 003.png|RTENOTITLE]]....................(8.61)<br/><br/>which can be rewritten as<br/><br/>[[File:Vol5 page 0747 eq 001.png|RTENOTITLE]]....................(8.62)<br/><br/>The forms of '''Eqs. 8.62''' and '''8.59''' are the same; thus '''Eq. 8.62''' is an "equivalent" drawdown equation, with the equivalent pressure change, (''p''<sub>''ws''</sub> – ''p''<sub>''wf''</sub>), a function of equivalent time, Δ''t''<sub>''e''</sub> = ''t''<sub>''p''</sub>Δ''t''/(''t''<sub>''p''</sub> + Δ''t''). The analogies between these equations suggest that, just as Δ''p'' = ''p''<sub>''i''</sub> − ''p''<sub>''wf''</sub> vs. ''t'' were plotted for drawdown tests, Δ''p'' = ''p''<sub>''ws''</sub> − ''p''<sub>''wf''</sub> vs. Δ''t''<sub>''e''</sub> can be plotted for buildup tests and achieve the same shapes on logarithmic graphs. However, the theoretical basis for this radial-equivalent time indicates that the equivalence exists only for infinite-acting radial flow and not for data influenced by wellbore storage or by effects of boundaries or other conditions that cause the flow pattern to deviate from radial. In practice, buildup test data for infinite-acting radial flow, including data distorted by wellbore storage, are transformed to the same shape as drawdown test data. However, data affected by boundaries or by linear flow (as in wells with hydraulic fractures) may not be transformed accurately.<br/><br/>Radial equivalent time has the properties<br/><br/>[[File:Vol5 page 0747 eq 002.png|RTENOTITLE]]....................(8.63)<br/>
<br>
 
<br>
 
[[File:Vol5 page 0746 eq 001.png]]....................(8.59)
 
<br>
 
<br>
 
We model a buildup test with
 
<br>
 
<br>
 
[[File:Vol5 page 0746 eq 002.png]]....................(8.60)
 
<br>
 
<br>
 
Combining '''Eqs. 8.59''' and '''8.60''' and simplifying,
 
<br>
 
<br>
 
[[File:Vol5 page 0746 eq 003.png]]....................(8.61)
 
<br>
 
<br>
 
which can be rewritten as
 
<br>
 
<br>
 
[[File:Vol5 page 0747 eq 001.png]]....................(8.62)
 
<br>
 
<br>
 
The forms of '''Eqs. 8.62''' and '''8.59''' are the same; thus '''Eq. 8.62''' is an "equivalent" drawdown equation, with the equivalent pressure change, (''p''<sub>''ws''</sub> – ''p''<sub>''wf''</sub>), a function of equivalent time, Δ''t''<sub>''e''</sub> = ''t''<sub>''p''</sub>Δ''t''/(''t''<sub>''p''</sub> + Δ''t''). The analogies between these equations suggest that, just as Δ''p'' = ''p''<sub>''i''</sub> − ''p''<sub>''wf''</sub> vs. ''t'' were plotted for drawdown tests, Δ''p'' = ''p''<sub>''ws''</sub> − ''p''<sub>''wf''</sub> vs. Δ''t''<sub>''e''</sub> can be plotted for buildup tests and achieve the same shapes on logarithmic graphs. However, the theoretical basis for this radial-equivalent time indicates that the equivalence exists only for infinite-acting radial flow and not for data influenced by wellbore storage or by effects of boundaries or other conditions that cause the flow pattern to deviate from radial. In practice, buildup test data for infinite-acting radial flow, including data distorted by wellbore storage, are transformed to the same shape as drawdown test data. However, data affected by boundaries or by linear flow (as in wells with hydraulic fractures) may not be transformed accurately.  
 
<br>
 
<br>
 
Radial equivalent time has the properties
 
<br>
 
<br>
 
[[File:Vol5 page 0747 eq 002.png]]....................(8.63)
 
<br>
 
<br>
 
  
 
=== Type-Curve Matching ===
 
=== Type-Curve Matching ===
  
 
The steps in type-curve matching for wells with infinite-acting radial flow are outlined here. Details vary for more complex reservoirs, but the general procedure is similar to that for infinite-acting reservoirs.
 
The steps in type-curve matching for wells with infinite-acting radial flow are outlined here. Details vary for more complex reservoirs, but the general procedure is similar to that for infinite-acting reservoirs.
<br>
 
* Plot field data on log-log coordinates with the same size log cycles as the type curve.
 
* Align the horizontal sections of the field data and the type curve.
 
* Align unit slope regions on the field data and the type curve.
 
* Select the value of ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> that best matches the field data.
 
* Select pressure and time match points (corresponding values of real and dimensionless variables from field data and type curve plots) from anywhere on the plot.
 
* Calculate permeability from the pressure match-point ratio,
 
*;[[File:Vol5 page 0747 eq 003.png]]....................(8.64)
 
*;
 
*;or [[File:Vol5 page 0747 eq 004.png]]....................(8.65)
 
* Calculate ''C''<sub>''D''</sub> from the time match-point ratio,
 
*;[[File:Vol5 page 0747 eq 005.png]]....................(8.66)
 
*;
 
*;or [[File:Vol5 page 0747 eq 006.png]]....................(8.67)
 
* Calculate ''s'' from the matching stem value, ''C''<sub>''D''</sub>''e''<sup>2''s''</sup>:
 
*;[[File:Vol5 page 0748 eq 001.png]]....................(8.68)
 
<br>
 
  
'''Fig. 8.30''' shows an example interpretation of match points. In practice, this matching and match-point interpretation procedure is done on the computer and monitor, and much of the process is transparent to the analyst.  
+
*Plot field data on log-log coordinates with the same size log cycles as the type curve.
<br/>
+
*Align the horizontal sections of the field data and the type curve.
<br/>
+
*Align unit slope regions on the field data and the type curve.
<gallery widths="300px" heights="200px">
+
*Select the value of ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> that best matches the field data.
 +
*Select pressure and time match points (corresponding values of real and dimensionless variables from field data and type curve plots) from anywhere on the plot.
 +
*Calculate permeability from the pressure match-point ratio,
 +
*;[[File:Vol5 page 0747 eq 003.png|RTENOTITLE]]....................(8.64)
 +
*;;or [[File:Vol5 page 0747 eq 004.png|RTENOTITLE]]....................(8.65)
 +
*Calculate ''C''<sub>''D''</sub> from the time match-point ratio,
 +
*;[[File:Vol5 page 0747 eq 005.png|RTENOTITLE]]....................(8.66)
 +
*;;or [[File:Vol5 page 0747 eq 006.png|RTENOTITLE]]....................(8.67)
 +
*Calculate ''s'' from the matching stem value, ''C''<sub>''D''</sub>''e''<sup>2''s''</sup>:;[[File:Vol5 page 0748 eq 001.png|RTENOTITLE]]....................(8.68)
 +
 
 +
 
 +
 
 +
'''Fig. 8.30''' shows an example interpretation of match points. In practice, this matching and match-point interpretation procedure is done on the computer and monitor, and much of the process is transparent to the analyst.<br/><br/><gallery widths="300px" heights="200px">
 
File:vol5 Page 0748 Image 0001.png|'''Fig. 8.30 – Example interpretation of match points.'''
 
File:vol5 Page 0748 Image 0001.png|'''Fig. 8.30 – Example interpretation of match points.'''
 
</gallery>
 
</gallery>
<br/>
+
</div></div><div class="toccolours mw-collapsible mw-collapsed">
 
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
 
 
== Damage and Stimulation ==
 
== Damage and Stimulation ==
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
=== Causes of Formation Damage ===
 
=== Causes of Formation Damage ===
  
The causes of formation damage that lead to positive skin factors include damage caused by drilling-fluid invasion, production, or injection.  
+
The causes of formation damage that lead to positive skin factors include damage caused by drilling-fluid invasion, production, or injection.<br/><br/>When mud filtrate invades the formation surrounding a borehole, it will generally remain in the formation even after the well is cased and perforated. This mud filtrate in the formation reduces the effective permeability to hydrocarbons near the wellbore. It may also cause clays in the formation to swell, reducing the absolute permeability of the formation. In addition, solid particles from the mud may enter the formation and reduce permeability at the formation face.<br/><br/>The production process may also reduce permeability and introduce a positive skin factor. For example, in an otherwise undersaturated oil reservoir, pressure near the well may be below the bubblepoint pressure, causing a free-gas saturation and reducing the effective permeability to oil. In a retrograde gas reservoir, the pressure near the wellbore may drop below the dewpoint and an immobile liquid phase may form and reduce the effective permeability to gas near the wellbore.<br/><br/>Injection can also cause damage. The water injected may be dirty; that is, it may contain fines that may plug the formation and reduce permeability. In other cases, the injected water may be incompatible with the formation water, causing solids to precipitate and plug the formation. In still other cases, the injected water may be incompatible with clays in the formation (e.g., fresh water can destabilize some clays, causing fines to migrate and plug the formation).
<br>
 
<br>
 
When mud filtrate invades the formation surrounding a borehole, it will generally remain in the formation even after the well is cased and perforated. This mud filtrate in the formation reduces the effective permeability to hydrocarbons near the wellbore. It may also cause clays in the formation to swell, reducing the absolute permeability of the formation. In addition, solid particles from the mud may enter the formation and reduce permeability at the formation face.  
 
<br>
 
<br>
 
The production process may also reduce permeability and introduce a positive skin factor. For example, in an otherwise undersaturated oil reservoir, pressure near the well may be below the bubblepoint pressure, causing a free-gas saturation and reducing the effective permeability to oil. In a retrograde gas reservoir, the pressure near the wellbore may drop below the dewpoint and an immobile liquid phase may form and reduce the effective permeability to gas near the wellbore.  
 
<br>
 
<br>
 
Injection can also cause damage. The water injected may be dirty; that is, it may contain fines that may plug the formation and reduce permeability. In other cases, the injected water may be incompatible with the formation water, causing solids to precipitate and plug the formation. In still other cases, the injected water may be incompatible with clays in the formation (e.g., fresh water can destabilize some clays, causing fines to migrate and plug the formation).  
 
  
 
=== Altered Zone and Skin Effect ===
 
=== Altered Zone and Skin Effect ===
  
A two-region reservoir model ('''Fig. 8.31''') is a convenient representation of a damaged well (and some stimulated wells with radially symmetric permeability alteration around the wellbore). In this model, the altered zone around the wellbore is assumed to have uniform permeability ''k''<sub>''s''</sub> out to a radius ''r''<sub>''s''</sub>, beyond which the formation permeability, ''k'', is unaltered.  
+
A two-region reservoir model ('''Fig. 8.31''') is a convenient representation of a damaged well (and some stimulated wells with radially symmetric permeability alteration around the wellbore). In this model, the altered zone around the wellbore is assumed to have uniform permeability ''k''<sub>''s''</sub> out to a radius ''r''<sub>''s''</sub>, beyond which the formation permeability, ''k'', is unaltered.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0749 Image 0001.png|'''Fig. 8.31 – Two-region reservoir model with altered zone around wellbore and unaltered formation permeability beyond.'''
 
File:vol5 Page 0749 Image 0001.png|'''Fig. 8.31 – Two-region reservoir model with altered zone around wellbore and unaltered formation permeability beyond.'''
 
</gallery>
 
</gallery>
<br/>
 
  
For a damaged well, the reduced permeability in the altered zone causes an additional pressure drop, Δ''p''<sub>''s''</sub> ('''Fig. 8.32'''). The dimensionless skin factor, ''s'', and the additional pressure drop across the altered zone are related by
+
For a damaged well, the reduced permeability in the altered zone causes an additional pressure drop, Δ''p''<sub>''s''</sub> ('''Fig. 8.32'''). The dimensionless skin factor, ''s'', and the additional pressure drop across the altered zone are related by<br/><br/>[[File:Vol5 page 0748 eq 002.png|RTENOTITLE]]....................(8.69)<br/><br/>For a well with a known skin factor, ''s'', '''Eq. 8.69''' provides a method of translating the somewhat abstract dimensionless skin factor into a more concrete characterization of the practical effect of damage or stimulation.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0748 eq 002.png]]....................(8.69)
 
<br>
 
<br>
 
For a well with a known skin factor, ''s'', '''Eq. 8.69''' provides a method of translating the somewhat abstract dimensionless skin factor into a more concrete characterization of the practical effect of damage or stimulation.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0749 Image 0002.png|'''Fig. 8.32 – Additional pressure drop in damaged well is caused by reduced permeability in altered zone.'''
 
File:vol5 Page 0749 Image 0002.png|'''Fig. 8.32 – Additional pressure drop in damaged well is caused by reduced permeability in altered zone.'''
 
</gallery>
 
</gallery>
<br/>
 
  
In a two-region reservoir model, the skin factor, ''s'', is related to the properties of the altered zone:
+
In a two-region reservoir model, the skin factor, ''s'', is related to the properties of the altered zone:<br/><br/>[[File:Vol5 page 0750 eq 001.png|RTENOTITLE]]....................(8.70)<br/><br/>Rearrange '''Eq. 8.70''' and solve for the permeability of the altered zone:<br/><br/>[[File:Vol5 page 0750 eq 002.png|RTENOTITLE]]....................(8.71)<br/><br/>Rearrangements of '''Eq. 8.70''' provide a second method of translating skin into a more concrete characterization of a well with altered permeability near the wellbore. If the depth of damage can be estimated for a well with a known skin factor, ''s'', the permeability of the altered zone can be estimated. Even if the depth of permeability alteration, ''r''<sub>''s''</sub>, is estimated '''Eq. 8.71''' can still provide a reasonable estimate of altered zone permeability because ''r''<sub>''s''</sub> appears in a logarithmic term. Alternatively, an estimate of the permeability reduction ratio (for example, from laboratory tests on cores) can produce an estimate of the depth of damage from another rearrangement of '''Eq. 8.70''',<br/><br/>[[File:Vol5 page 0750 eq 003.png|RTENOTITLE]]....................(8.72)<br/>
<br>
+
 
<br>
 
[[File:Vol5 page 0750 eq 001.png]]....................(8.70)
 
<br>
 
<br>
 
Rearrange '''Eq. 8.70''' and solve for the permeability of the altered zone:
 
<br>
 
<br>
 
[[File:Vol5 page 0750 eq 002.png]]....................(8.71)
 
<br>
 
<br>
 
Rearrangements of '''Eq. 8.70''' provide a second method of translating skin into a more concrete characterization of a well with altered permeability near the wellbore. If the depth of damage can be estimated for a well with a known skin factor, ''s'', the permeability of the altered zone can be estimated. Even if the depth of permeability alteration, ''r''<sub>''s''</sub>, is estimated '''Eq. 8.71''' can still provide a reasonable estimate of altered zone permeability because ''r''<sub>''s''</sub> appears in a logarithmic term. Alternatively, an estimate of the permeability reduction ratio (for example, from laboratory tests on cores) can produce an estimate of the depth of damage from another rearrangement of '''Eq. 8.70''',
 
<br>
 
<br>
 
[[File:Vol5 page 0750 eq 003.png]]....................(8.72)
 
<br>
 
<br>
 
 
=== Apparent Wellbore Radius ===
 
=== Apparent Wellbore Radius ===
  
A third method of translating skin to a more concrete characterization of near-well conditions is to calculate apparent or effective wellbore radius, ''r''<sub>''wa''</sub>. Apparent wellbore radius is defined as
+
A third method of translating skin to a more concrete characterization of near-well conditions is to calculate apparent or effective wellbore radius, ''r''<sub>''wa''</sub>. Apparent wellbore radius is defined as<br/><br/>[[File:Vol5 page 0750 eq 004.png|RTENOTITLE]]....................(8.73)<br/><br/>or [[File:Vol5 page 0750 eq 005.png|RTENOTITLE]]....................(8.74)<br/><br/>For a stimulated well, the pressure drawdown at the wellbore is the same as it would be in a formation with unaltered permeability but with wellbore radius equal to the apparent wellbore radius. This concept has value in some simulation applications. Note that ''r''<sub>''wa''</sub> can be calculated from the actual wellbore radius and skin factor.<br/><br/>'''Eqs. 8.73''' and '''8.74''' are also useful to illustrate the minimum (i.e., the most-negative possible) skin factor. This minimum skin, ''s''<sub>min</sub>, occurs when the apparent wellbore radius is equal to the drainage radius of the well:<br/><br/>[[File:Vol5 page 0750 eq 006.png|RTENOTITLE]]....................(8.75)<br/><br/>For a well with a circular drainage area of 40 acres for which ''r''<sub>''e''</sub> is 745 ft and a wellbore radius of 0.3 ft, the minimum skin (maximum stimulation) is ''s''<sub>min</sub> = - ln(''r''<sub>''e''</sub>/''r''<sub>''w''</sub>) = −(745/0.3) = −7.82. Such a skin implies increasing the permeability throughout the entire altered zone to infinity—clearly an idealistic "upper limit." More realistically, research<ref name="r10">_</ref> has shown that the half-length, ''L''<sub>''f''</sub>, of a highly conductive vertical fracture is related to ''r''<sub>''wa''</sub> by<br/><br/>[[File:Vol5 page 0751 eq 001.png|RTENOTITLE]]....................(8.76)<br/><br/>or [[File:Vol5 page 0751 eq 002.png|RTENOTITLE]]....................(8.77)<br/><br/>Thus, for ''L''<sub>''f''</sub> = ''r''<sub>''e''</sub> = 745 ft, ''s'' = −7.12 is a more realistic minimum (for the given drainage radius and wellbore radius).
<br>
 
<br>
 
[[File:Vol5 page 0750 eq 004.png]]....................(8.73)
 
<br>
 
<br>
 
or [[File:Vol5 page 0750 eq 005.png]]....................(8.74)
 
<br>
 
<br>
 
For a stimulated well, the pressure drawdown at the wellbore is the same as it would be in a formation with unaltered permeability but with wellbore radius equal to the apparent wellbore radius. This concept has value in some simulation applications. Note that ''r''<sub>''wa''</sub> can be calculated from the actual wellbore radius and skin factor.  
 
<br>
 
<br>
 
'''Eqs. 8.73''' and '''8.74''' are also useful to illustrate the minimum (i.e., the most-negative possible) skin factor. This minimum skin, ''s''<sub>min</sub>, occurs when the apparent wellbore radius is equal to the drainage radius of the well:
 
<br>
 
<br>
 
[[File:Vol5 page 0750 eq 006.png]]....................(8.75)
 
<br>
 
<br>
 
For a well with a circular drainage area of 40 acres for which ''r''<sub>''e''</sub> is 745 ft and a wellbore radius of 0.3 ft, the minimum skin (maximum stimulation) is ''s''<sub>min</sub> = - ln(''r''<sub>''e''</sub>/''r''<sub>''w''</sub>) = −(745/0.3) = −7.82. Such a skin implies increasing the permeability throughout the entire altered zone to infinity—clearly an idealistic "upper limit." More realistically, research<ref name="r10" /> has shown that the half-length, ''L''<sub>''f''</sub>, of a highly conductive vertical fracture is related to ''r''<sub>''wa''</sub> by
 
<br>
 
<br>
 
[[File:Vol5 page 0751 eq 001.png]]....................(8.76)
 
<br>
 
<br>
 
or [[File:Vol5 page 0751 eq 002.png]]....................(8.77)
 
<br>
 
<br>
 
Thus, for ''L''<sub>''f''</sub> = ''r''<sub>''e''</sub> = 745 ft, ''s'' = −7.12 is a more realistic minimum (for the given drainage radius and wellbore radius).  
 
  
 
=== Flow Efficiency ===
 
=== Flow Efficiency ===
  
A fourth way to characterize a well with nonzero skin is to calculate the flow efficiency of the well. Flow efficiency, ''E''<sub>''f''</sub>, is defined as the ratio of the actual productivity index of the well (including skin) to the ideal productivity index if the skin factor were zero. Because the productivity index is the ratio of stabilized flow rate to pressure drop required to sustain that stabilized rate,
+
A fourth way to characterize a well with nonzero skin is to calculate the flow efficiency of the well. Flow efficiency, ''E''<sub>''f''</sub>, is defined as the ratio of the actual productivity index of the well (including skin) to the ideal productivity index if the skin factor were zero. Because the productivity index is the ratio of stabilized flow rate to pressure drop required to sustain that stabilized rate,<br/><br/>[[File:Vol5 page 0751 eq 003.png|RTENOTITLE]]....................(8.78)<br/><br/>[[File:Vol5 page 0751 eq 004.png|RTENOTITLE]]....................(8.79)<br/><br/>and [[File:Vol5 page 0751 eq 005.png|RTENOTITLE]]....................(8.80)<br/><br/>For a well with neither damage nor stimulation, ''E''<sub>''f''</sub> = 1; for a damaged well, ''E''<sub>''f''</sub> < 1; and for a stimulated well, ''E''<sub>''f''</sub> > 1.
<br>
 
<br>
 
[[File:Vol5 page 0751 eq 003.png]]....................(8.78)
 
<br>
 
<br>
 
[[File:Vol5 page 0751 eq 004.png]]....................(8.79)
 
<br>
 
<br>
 
and [[File:Vol5 page 0751 eq 005.png]]....................(8.80)
 
<br>
 
<br>
 
For a well with neither damage nor stimulation, ''E''<sub>''f''</sub> = 1; for a damaged well, ''E''<sub>''f''</sub> < 1; and for a stimulated well, ''E''<sub>''f''</sub> > 1.  
 
  
 
=== Geometric Skin ===
 
=== Geometric Skin ===
  
When the area open to flow decreases, the pressure drop is greater than when the area is unchanged all the way to the formation face. Examples include flow converging to perforations ('''Fig. 8.33'''), partial penetration ('''Fig. 8.34'''), and an incompletely perforated interval ('''Fig. 8.35''').  
+
When the area open to flow decreases, the pressure drop is greater than when the area is unchanged all the way to the formation face. Examples include flow converging to perforations ('''Fig. 8.33'''), partial penetration ('''Fig. 8.34'''), and an incompletely perforated interval ('''Fig. 8.35''').<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0751 Image 0001.png|'''Fig. 8.33 – When perforation spacing is large, converging flow results in positive skin factor.'''
 
File:vol5 Page 0751 Image 0001.png|'''Fig. 8.33 – When perforation spacing is large, converging flow results in positive skin factor.'''
  
Line 858: Line 270:
 
File:vol5 Page 0752 Image 0002.png|'''Fig. 8.35 – In an incompletely penetrated well, only a portion of the total productive interval (''h'') is perforated, increasing pressure drop and skin.'''
 
File:vol5 Page 0752 Image 0002.png|'''Fig. 8.35 – In an incompletely penetrated well, only a portion of the total productive interval (''h'') is perforated, increasing pressure drop and skin.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.33''' illustrates flow converging into perforations. When the perforation spacing is too large, this converging flow results in a positive skin factor. The skin increases as vertical permeability decreases and increases as shot density decreases.  
+
'''Fig. 8.33''' illustrates flow converging into perforations. When the perforation spacing is too large, this converging flow results in a positive skin factor. The skin increases as vertical permeability decreases and increases as shot density decreases.<br/><br/>'''''Partial Penetration.''''' '''Fig. 8.34''' illustrates flow converging into an interval that is only partly penetrated by perforations. When a well is completed in only a fraction of the productive interval, the flow must converge through a smaller area, increasing the pressure drop near the well (compared to a fully completed interval). The additional pressure drop near the well results in a more positive skin. It increases as the vertical permeability decreases and as the perforated interval as a fraction of the total interval decreases. Formation damage (reduced permeability) near the completion face can significantly increase the additional pressure drop and thus the calculated skin factor.<br/><br/>'''''Incompletely Perforated Interval.''''' Partial penetration is a special case of an incompletely perforated interval ('''Fig. 8.35'''). In the general case, the well is perforated starting at a distance ''h''<sub>1</sub> from the top of the productive interval and has perforations extending over a distance, ''h''<sub>''p''</sub>, in an interval of total thickness, ''h''. The total skin for the well in this general situation is<br/><br/>[[File:Vol5 page 0753 eq 001.png|RTENOTITLE]]....................(8.81)<br/><br/>In '''Eq. 8.81''', ''s''<sub>''d''</sub> is the skin caused by formation damage, and s p is the skin resulting from an incompletely perforated interval. This equation is not valid for a stimulated well.<br/><br/>The skin factor for an incompletely perforated interval, ''s''<sub>''p''</sub>, can be quantified by<ref name="r11">_</ref><br/><br/>[[File:Vol5 page 0753 eq 002.png|RTENOTITLE]]....................(8.82)<br/><br/>where [[File:Vol5 page 0753 eq 003.png|RTENOTITLE]]....................(8.83)<br/><br/>[[File:Vol5 page 0753 eq 004.png|RTENOTITLE]]....................(8.84)<br/><br/>[[File:Vol5 page 0753 eq 005.png|RTENOTITLE]]....................(8.85)<br/><br/>[[File:Vol5 page 0753 eq 006.png|RTENOTITLE]]....................(8.86)<br/><br/>and [[File:Vol5 page 0753 eq 007.png|RTENOTITLE]]....................(8.87)<br/><br/>The most significant limitation in applying '''Eq. 8.82''' in practice is the difficulty in estimating accurately the vertical-to-horizontal-permeability ratio, ''k''<sub>''v''</sub>/''k''<sub>''h''</sub>. Fortunately, this ratio appears only in a logarithmic term in '''Eq. 8.82''', so errors will not seriously distort the calculated value of ''s''<sub>''p''</sub>.<br/><br/>'''''Deviated Well.''''' For a deviated well ('''Fig. 8.36'''), which penetrates the formation at an angle other than 90°, more surface is in contact with the formation. This introduces a negative skin factor, ''s''<sub>''θ''</sub>, which makes the total skin factor, ''s'', more negative.<br/><br/>[[File:Vol5 page 0753 eq 008.png|RTENOTITLE]]....................(8.88)<br/><br/>The effect increases as the vertical permeability increases and increases as the angle from the vertical, ''θ''<sub>''w''</sub>, increases. The deviated well skin factor, ''s''<sub>''θ''</sub>, is given by a correlation of simulated results<ref name="r12">_</ref> (valid for ''θ''<sub>''w''</sub> < 75°):<br/><br/>[[File:Vol5 page 0753 eq 009.png|RTENOTITLE]]....................(8.89)<br/><br/>where [[File:Vol5 page 0753 eq 010.png|RTENOTITLE]]....................(8.90)<br/><br/>and [[File:Vol5 page 0753 eq 011.png|RTENOTITLE]]....................(8.91)<br/><br/>'''''Gravel-Pack Skin.''''' When a well is gravel packed ('''Fig. 8.37'''), there is a pressure drop through the gravel pack within the perforations, given by<ref name="r13">_</ref><br/><br/>[[File:Vol5 page 0754 eq 001.png|RTENOTITLE]]....................(8.92)<br/><br/>where ''s''<sub>''gp''</sub> is the skin factor because of Darcy flow through the gravel pack; ''h'', the net pay thickness, ft; ''k''<sub>''gp''</sub>, the permeability of the gravel in the gravel pack, md; ''k'', the reservoir permeability, md; ''L''<sub>''g''</sub>, the length of the flow path through the gravel pack, ft; ''n'', the number of perforations open; and ''r''<sub>''p''</sub>, the radius of the perforation tunnel, ft. '''Eq. 8.92''' does not include the effects of non-Darcy flow, which may be extremely important in high-rate gas wells.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
'''''Partial Penetration.''''' '''Fig. 8.34''' illustrates flow converging into an interval that is only partly penetrated by perforations. When a well is completed in only a fraction of the productive interval, the flow must converge through a smaller area, increasing the pressure drop near the well (compared to a fully completed interval). The additional pressure drop near the well results in a more positive skin. It increases as the vertical permeability decreases and as the perforated interval as a fraction of the total interval decreases. Formation damage (reduced permeability) near the completion face can significantly increase the additional pressure drop and thus the calculated skin factor.  
 
<br>
 
<br>
 
'''''Incompletely Perforated Interval.''''' Partial penetration is a special case of an incompletely perforated interval ('''Fig. 8.35'''). In the general case, the well is perforated starting at a distance ''h''<sub>1</sub> from the top of the productive interval and has perforations extending over a distance, ''h''<sub>''p''</sub>, in an interval of total thickness, ''h''. The total skin for the well in this general situation is
 
<br>
 
<br>
 
[[File:Vol5 page 0753 eq 001.png]]....................(8.81)
 
<br>
 
<br>
 
In '''Eq. 8.81''', ''s''<sub>''d''</sub> is the skin caused by formation damage, and s p is the skin resulting from an incompletely perforated interval. This equation is not valid for a stimulated well.  
 
<br>
 
<br>
 
The skin factor for an incompletely perforated interval, ''s''<sub>''p''</sub>, can be quantified by<ref name="r11" />
 
<br>
 
<br>
 
[[File:Vol5 page 0753 eq 002.png]]....................(8.82)
 
<br>
 
<br>
 
where [[File:Vol5 page 0753 eq 003.png]]....................(8.83)
 
<br>
 
<br>
 
[[File:Vol5 page 0753 eq 004.png]]....................(8.84)
 
<br>
 
<br>
 
[[File:Vol5 page 0753 eq 005.png]]....................(8.85)
 
<br>
 
<br>
 
[[File:Vol5 page 0753 eq 006.png]]....................(8.86)
 
<br>
 
<br>
 
and [[File:Vol5 page 0753 eq 007.png]]....................(8.87)
 
<br>
 
<br>
 
The most significant limitation in applying '''Eq. 8.82''' in practice is the difficulty in estimating accurately the vertical-to-horizontal-permeability ratio, ''k''<sub>''v''</sub>/''k''<sub>''h''</sub>. Fortunately, this ratio appears only in a logarithmic term in '''Eq. 8.82''', so errors will not seriously distort the calculated value of ''s''<sub>''p''</sub>.  
 
<br>
 
<br>
 
'''''Deviated Well.''''' For a deviated well ('''Fig. 8.36'''), which penetrates the formation at an angle other than 90°, more surface is in contact with the formation. This introduces a negative skin factor, ''s''<sub>''θ''</sub>, which makes the total skin factor, ''s'', more negative.
 
<br>
 
<br>
 
[[File:Vol5 page 0753 eq 008.png]]....................(8.88)
 
<br>
 
<br>
 
The effect increases as the vertical permeability increases and increases as the angle from the vertical, ''θ''<sub>''w''</sub>, increases. The deviated well skin factor, ''s''<sub>''θ''</sub>, is given by a correlation of simulated results<ref name="r12" /> (valid for ''θ''<sub>''w''</sub> < 75°):
 
<br>
 
<br>
 
[[File:Vol5 page 0753 eq 009.png]]....................(8.89)
 
<br>
 
<br>
 
where [[File:Vol5 page 0753 eq 010.png]]....................(8.90)
 
<br>
 
<br>
 
and [[File:Vol5 page 0753 eq 011.png]]....................(8.91)
 
<br>
 
<br>
 
'''''Gravel-Pack Skin.''''' When a well is gravel packed ('''Fig. 8.37'''), there is a pressure drop through the gravel pack within the perforations, given by<ref name="r13" />
 
<br>
 
<br>
 
[[File:Vol5 page 0754 eq 001.png]]....................(8.92)
 
<br>
 
<br>
 
where ''s''<sub>''gp''</sub> is the skin factor because of Darcy flow through the gravel pack; ''h'', the net pay thickness, ft; ''k''<sub>''gp''</sub>, the permeability of the gravel in the gravel pack, md; ''k'', the reservoir permeability, md; ''L''<sub>''g''</sub>, the length of the flow path through the gravel pack, ft; ''n'', the number of perforations open; and ''r''<sub>''p''</sub>, the radius of the perforation tunnel, ft. '''Eq. 8.92''' does not include the effects of non-Darcy flow, which may be extremely important in high-rate gas wells.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0754 Image 0001.png|'''Fig. 8.36 – More surface of a deviated well is in contact with formation, introducing a negative skin factor.'''
 
File:vol5 Page 0754 Image 0001.png|'''Fig. 8.36 – More surface of a deviated well is in contact with formation, introducing a negative skin factor.'''
  
 
File:vol5 Page 0755 Image 0001.png|'''Fig. 8.37 – Pressure drop in a gravel-packed well includes the skin effect in perforations.'''
 
File:vol5 Page 0755 Image 0001.png|'''Fig. 8.37 – Pressure drop in a gravel-packed well includes the skin effect in perforations.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''''Completion Skin.''''' For a perforated well, any reduced permeability, ''k''<sub>''dp''</sub>, in the zone surrounding the perforations ('''Fig. 8.38''') introduces an additional pressure drop. The additional skin is<ref name="r14" />
+
'''''Completion Skin.''''' For a perforated well, any reduced permeability, ''k''<sub>''dp''</sub>, in the zone surrounding the perforations ('''Fig. 8.38''') introduces an additional pressure drop. The additional skin is<ref name="r14">_</ref><br/><br/>[[File:Vol5 page 0754 eq 002.png|RTENOTITLE]]....................(8.93)<br/><br/>and [[File:Vol5 page 0754 eq 003.png|RTENOTITLE]]....................(8.94)<br/><br/>where ''s''<sub>''p''</sub> is the geometric skin from flow converging to the perforations; ''s''<sub>''d''</sub>, the damage skin; ''s''<sub>''dp''</sub>, perforation damage skin; ''k''<sub>''d''</sub>, permeability of the damaged zone around the wellbore, md; ''k''<sub>''dp''</sub>, permeability of the damaged zone around perforation tunnels, md; ''k'', reservoir permeability, md; ''L''<sub>''p''</sub>, length of perforation tunnel, ft; ''n'', number of perforations; ''h'', formation thickness, ft; ''r''<sub>''d''</sub>, radius of the damaged zone around the wellbore, ft; ''r''<sub>''dp''</sub>, radius of the damages zone around the perforation tunnel, ft; ''r''<sub>''p''</sub>, radius of the perforation tunnel, ft; and ''r''<sub>''w''</sub>, wellbore radius, ft. '''Eq. 8.94''' does not include the effects of non-Darcy flow.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0754 eq 002.png]]....................(8.93)
 
<br>
 
<br>
 
and [[File:Vol5 page 0754 eq 003.png]]....................(8.94)
 
<br>
 
<br>
 
where ''s''<sub>''p''</sub> is the geometric skin from flow converging to the perforations; ''s''<sub>''d''</sub>, the damage skin; ''s''<sub>''dp''</sub>, perforation damage skin; ''k''<sub>''d''</sub>, permeability of the damaged zone around the wellbore, md; ''k''<sub>''dp''</sub>, permeability of the damaged zone around perforation tunnels, md; ''k'', reservoir permeability, md; ''L''<sub>''p''</sub>, length of perforation tunnel, ft; ''n'', number of perforations; ''h'', formation thickness, ft; ''r''<sub>''d''</sub>, radius of the damaged zone around the wellbore, ft; ''r''<sub>''dp''</sub>, radius of the damages zone around the perforation tunnel, ft; ''r''<sub>''p''</sub>, radius of the perforation tunnel, ft; and ''r''<sub>''w''</sub>, wellbore radius, ft. '''Eq. 8.94''' does not include the effects of non-Darcy flow.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0755 Image 0002.png|'''Fig. 8.38 – Reduced permeability in damaged zone surrounding perforations introduces additional pressure drop and skin.'''
 
File:vol5 Page 0755 Image 0002.png|'''Fig. 8.38 – Reduced permeability in damaged zone surrounding perforations introduces additional pressure drop and skin.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''''Hydraulically Fractured Wells.''''' Wells are frequently fractured hydraulically to improve their productivity, especially in low-permeability formations where fractures increase the effective drained area and in high-permeability formations where they penetrate near-well damage or promote sand control. These fractures, almost always vertical ('''Fig. 8.39'''), are high-conductivity paths between the reservoir and the wellbore. If the fracture conductivity is large enough relative to the formation permeability and fracture length, the pressure drop within the fracture will be negligible. This distributes the pressure drop caused by fluid influx into the wellbore over a much larger area, resulting in a negative skin factor, which is interpreted as a geometric skin.  
+
'''''Hydraulically Fractured Wells.''''' Wells are frequently fractured hydraulically to improve their productivity, especially in low-permeability formations where fractures increase the effective drained area and in high-permeability formations where they penetrate near-well damage or promote sand control. These fractures, almost always vertical ('''Fig. 8.39'''), are high-conductivity paths between the reservoir and the wellbore. If the fracture conductivity is large enough relative to the formation permeability and fracture length, the pressure drop within the fracture will be negligible. This distributes the pressure drop caused by fluid influx into the wellbore over a much larger area, resulting in a negative skin factor, which is interpreted as a geometric skin.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0756 Image 0001.png|'''Fig. 8.39 – Hydraulic fractures change the flow pattern in a reservoir and introduce a negative geometric skin effect.'''
 
File:vol5 Page 0756 Image 0001.png|'''Fig. 8.39 – Hydraulic fractures change the flow pattern in a reservoir and introduce a negative geometric skin effect.'''
 
</gallery>
 
</gallery>
<br/>
 
  
Dimensionless fracture conductivity, ''C''<sub>''r''</sub>, is defined by
+
Dimensionless fracture conductivity, ''C''<sub>''r''</sub>, is defined by<br/><br/>[[File:Vol5 page 0756 eq 001.png|RTENOTITLE]]....................(8.95)<br/><br/>where ''w''<sub>''f''</sub> is the fracture length, ft; ''k''<sub>''f''</sub>, the permeability of the proppant in the fracture; ''k'', the formation permeability, md; and ''L''<sub>''f''</sub>, the fracture half-length, ft. Pressure drop in the fracture is negligible for ''C''<sub>''r''</sub> > 100.<br/>
<br>
+
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<br>
 
[[File:Vol5 page 0756 eq 001.png]]....................(8.95)
 
<br>
 
<br>
 
where ''w''<sub>''f''</sub> is the fracture length, ft; ''k''<sub>''f''</sub>, the permeability of the proppant in the fracture; ''k'', the formation permeability, md; and ''L''<sub>''f''</sub>, the fracture half-length, ft. Pressure drop in the fracture is negligible for ''C''<sub>''r''</sub> > 100.  
 
<br>
 
<br>
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
 
 
== Modifications for Gases and Multiphase Flow ==
 
== Modifications for Gases and Multiphase Flow ==
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
=== Diffusitivity Equation for Gas Flow ===
 
=== Diffusitivity Equation for Gas Flow ===
  
The diffusivity equation for liquids, '''Eq. 8.1''',
+
The diffusivity equation for liquids, '''Eq. 8.1''',<br/><br/>[[File:Vol5 page 0756 eq 002.png|RTENOTITLE]]....................(8.1)<br/><br/>was derived from three principles: conservation of mass, the equation of state for slightly compressible liquids, and Darcy’s law. This form of the diffusivity equation is linear, which makes solutions (such as the''Ei-''function solution) much easier to find and which allows us to use superposition in time and space to develop solutions for complex flow geometries and for variable rate histories from simple, single-well solutions.
<br>
 
<br>
 
[[File:Vol5 page 0756 eq 002.png]]....................(8.1)
 
<br>
 
<br>
 
was derived from three principles: conservation of mass, the equation of state for slightly compressible liquids, and Darcy’s law. This form of the diffusivity equation is linear, which makes solutions (such as the'' Ei-''function solution) much easier to find and which allows us to use superposition in time and space to develop solutions for complex flow geometries and for variable rate histories from simple, single-well solutions.  
 
  
 
=== Pseudopressure ===
 
=== Pseudopressure ===
  
Other forms of the equation for flow of gases must be developed because the equation of state for a slightly compressible liquid will not be applicable. First, introducing the real gas law,
+
Other forms of the equation for flow of gases must be developed because the equation of state for a slightly compressible liquid will not be applicable. First, introducing the real gas law,<br/><br/>[[File:Vol5 page 0757 eq 001.png|RTENOTITLE]]....................(8.96)<br/><br/>to replace the slightly compressible equation of state results in a more complex, nonlinear partial differential equation. This equation can be partially linearized by introducing the pseudopressure transformation, <ref name="r15">_</ref><br/><br/>[[File:Vol5 page 0757 eq 002.png|RTENOTITLE]]....................(8.97)<br/><br/>where ''p''<sub>0</sub> is an arbitrary "base" pressure, frequently chosen to be zero psia. The resulting form of the diffusivity equation is<br/><br/>[[File:Vol5 page 0757 eq 003.png|RTENOTITLE]]....................(8.98)<br/><br/>'''Eq. 8.98''' has the same form as the diffusivity equation for slightly compressible liquids, with pressure replaced by pseudopressure, ''p''<sub>''p''</sub>. However, this equation is nonlinear because the product ''μc''<sub>''t''</sub> is a strong function of pressure. Fortunately, research has shown that the equation can be treated as linear, and the''Ei-''function is valid for gases if ''μc''<sub>''t''</sub> is evaluated at the pressure at the beginning of a flow period until the time when boundaries begin to have a significant influence on the pressure drop at the well; that is, as long as the reservoir is infinite-acting.
<br>
 
<br>
 
[[File:Vol5 page 0757 eq 001.png]]....................(8.96)
 
<br>
 
<br>
 
to replace the slightly compressible equation of state results in a more complex, nonlinear partial differential equation. This equation can be partially linearized by introducing the pseudopressure transformation, <ref name="r15" />
 
<br>
 
<br>
 
[[File:Vol5 page 0757 eq 002.png]]....................(8.97)
 
<br>
 
<br>
 
where ''p''<sub>0</sub> is an arbitrary "base" pressure, frequently chosen to be zero psia. The resulting form of the diffusivity equation is
 
<br>
 
<br>
 
[[File:Vol5 page 0757 eq 003.png]]....................(8.98)
 
<br>
 
<br>
 
'''Eq. 8.98''' has the same form as the diffusivity equation for slightly compressible liquids, with pressure replaced by pseudopressure, ''p''<sub>''p''</sub>. However, this equation is nonlinear because the product ''μc''<sub>''t''</sub> is a strong function of pressure. Fortunately, research has shown that the equation can be treated as linear, and the'' Ei-''function is valid for gases if ''μc''<sub>''t''</sub> is evaluated at the pressure at the beginning of a flow period until the time when boundaries begin to have a significant influence on the pressure drop at the well; that is, as long as the reservoir is infinite-acting.  
 
  
 
=== Pressure-Squared and Pressure Approximations ===
 
=== Pressure-Squared and Pressure Approximations ===
  
By assuming that the product ''μz'' is constant, then, from '''Eq. 8.97''', pseudopressure becomes
+
By assuming that the product ''μz'' is constant, then, from '''Eq. 8.97''', pseudopressure becomes<br/><br/>[[File:Vol5 page 0757 eq 004.png|RTENOTITLE]]....................(8.99)<br/><br/>and the diffusivity equation becomes<br/><br/>[[File:Vol5 page 0757 eq 005.png|RTENOTITLE]]....................(8.100)<br/><br/>The independent variable has become ''p''<sup>2</sup>, and, in terms of this variable, the''Ei-''function solution is valid when the assumption that ''μz'' is constant is valid. This is true (based on empirical evidence) even though '''Eq. 8.100''' is nonlinear (pressure-dependent ''μc''<sub>''t''</sub>), but it is valid only for an infinite-acting reservoir.<br/><br/>'''Fig. 8.40''' shows the range of validity of this assumption for a reservoir temperature of 200°F and several different gas gravities. The ''μz'' product is fairly constant at pressures below approximately 2,000 psia (the shaded area in the figure). Conclusions are similar at other temperatures from 100 to 300°F.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0757 eq 004.png]]....................(8.99)
 
<br>
 
<br>
 
and the diffusivity equation becomes
 
<br>
 
<br>
 
[[File:Vol5 page 0757 eq 005.png]]....................(8.100)
 
<br>
 
<br>
 
The independent variable has become ''p''<sup>2</sup>, and, in terms of this variable, the'' Ei-''function solution is valid when the assumption that ''μz'' is constant is valid. This is true (based on empirical evidence) even though '''Eq. 8.100''' is nonlinear (pressure-dependent ''μc''<sub>''t''</sub>), but it is valid only for an infinite-acting reservoir.  
 
<br>
 
<br>
 
'''Fig. 8.40''' shows the range of validity of this assumption for a reservoir temperature of 200°F and several different gas gravities. The ''μz'' product is fairly constant at pressures below approximately 2,000 psia (the shaded area in the figure). Conclusions are similar at other temperatures from 100 to 300°F.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0758 Image 0001.png|'''Fig. 8.40 - ''μz'' product is fairly constant at low pressures. Shaded area indicates pressure range in which ''μz'' is constant.'''
 
File:vol5 Page 0758 Image 0001.png|'''Fig. 8.40 - ''μz'' product is fairly constant at low pressures. Shaded area indicates pressure range in which ''μz'' is constant.'''
 
</gallery>
 
</gallery>
<br/>
 
  
By assuming that the group ''p''/''μz'' is constant, from '''Eq. 8.97''', pseudopressure becomes
+
By assuming that the group ''p''/''μz'' is constant, from '''Eq. 8.97''', pseudopressure becomes<br/><br/>[[File:Vol5 page 0757 eq 006.png|RTENOTITLE]]....................(8.101)<br/><br/>and the diffusivity equation becomes<br/><br/>[[File:Vol5 page 0758 eq 001.png|RTENOTITLE]]....................(8.102)<br/><br/>The independent variable has become ''p'', and, in terms of pressure, the''Ei-''function is valid (from empirical evidence) when the assumption that p/''μz'' is constant is valid. This is true even though '''Eq. 8.102''' is nonlinear (pressure-dependent ''μc''<sub>''t''</sub>) , but is valid only for an infinite-acting reservoir.<br/><br/>'''Fig. 8.41''' shows the range of validity of this assumption (shaded area in the figure) for a reservoir temperature of 200°F and several different gas gravities. The group ''p''/''μz'' is fairly constant at pressures above approximately 3,000 psia as it is at other temperatures from 100 to 300°F.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0757 eq 006.png]]....................(8.101)
 
<br>
 
<br>
 
and the diffusivity equation becomes
 
<br>
 
<br>
 
[[File:Vol5 page 0758 eq 001.png]]....................(8.102)
 
<br>
 
<br>
 
The independent variable has become ''p'', and, in terms of pressure, the'' Ei-''function is valid (from empirical evidence) when the assumption that p/''μz'' is constant is valid. This is true even though '''Eq. 8.102''' is nonlinear (pressure-dependent ''μc''<sub>''t''</sub>) , but is valid only for an infinite-acting reservoir.  
 
<br>
 
<br>
 
'''Fig. 8.41''' shows the range of validity of this assumption (shaded area in the figure) for a reservoir temperature of 200°F and several different gas gravities. The group ''p''/''μz'' is fairly constant at pressures above approximately 3,000 psia as it is at other temperatures from 100 to 300°F.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0758 Image 0002.png|'''Fig. 8.41 - ''p/μz'' product is fairly constant for high pressure (''p'' > 3,000 psia). Shaded area indicates pressure range in which ''μz'' is constant.'''
 
File:vol5 Page 0758 Image 0002.png|'''Fig. 8.41 - ''p/μz'' product is fairly constant for high pressure (''p'' > 3,000 psia). Shaded area indicates pressure range in which ''μz'' is constant.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The implication of these results is that the choice of variable for gas well-flow equations depends on the situation. The pressure-squared approximation is valid only for low pressures (''p'' < 2,000 psia), the pressure approximation is valid only for high pressures (''p'' > 3,000 psia), and the pseudopressure transformation is valid for all pressure ranges. For pressure transient test analysis using software, the pseudopressure is almost always the optimal variable to use. For hand analysis, only pressure or pressure-squared approaches are feasible.  
+
The implication of these results is that the choice of variable for gas well-flow equations depends on the situation. The pressure-squared approximation is valid only for low pressures (''p'' < 2,000 psia), the pressure approximation is valid only for high pressures (''p'' > 3,000 psia), and the pseudopressure transformation is valid for all pressure ranges. For pressure transient test analysis using software, the pseudopressure is almost always the optimal variable to use. For hand analysis, only pressure or pressure-squared approaches are feasible.
  
 
=== Pseudotime ===
 
=== Pseudotime ===
  
Although the diffusivity equation written for gas flow has the same form as the diffusivity equation for slightly compressible liquids, with pressure replaced by pseudopressure, it is a nonlinear equation because the product, ''μc''<sub>''t''</sub>, is strongly pressure dependent. In some cases, the remaining nonlinearity cannot be ignored. To solve this problem, Agarwal<ref name="r16" /> introduced the pseudotime transformation to further linearize the diffusivity equation for gas. (The linearization is not rigorous, but is adequate for many practical purposes. <ref name="r17" />) The definition of pseudotime is
+
Although the diffusivity equation written for gas flow has the same form as the diffusivity equation for slightly compressible liquids, with pressure replaced by pseudopressure, it is a nonlinear equation because the product, ''μc''<sub>''t''</sub>, is strongly pressure dependent. In some cases, the remaining nonlinearity cannot be ignored. To solve this problem, Agarwal<ref name="r16">_</ref> introduced the pseudotime transformation to further linearize the diffusivity equation for gas. (The linearization is not rigorous, but is adequate for many practical purposes. <ref name="r17">_</ref>) The definition of pseudotime is<br/><br/>[[File:Vol5 page 0759 eq 001.png|RTENOTITLE]]....................(8.103)<br/><br/>In terms of pseudotime, ''t''<sub>''ap''</sub>, the diffusivity equation becomes<br/><br/>[[File:Vol5 page 0759 eq 002.png|RTENOTITLE]]....................(8.104)<br/><br/>Subsequent studies<ref name="r18">_</ref> have shown that the pseudotime transformation is particularly useful for analysis of flow and buildup tests distorted by wellbore storage when using type curves designed to model flow of slightly compressible liquids.<br/><br/>Because the pressure in the integrand of '''Eq. 8.103''' is a function of position in the reservoir, it is not obvious where the pressure is to be evaluated. Empirical observations<ref name="r18">_</ref> indicate that the pressure should be evaluated at BHP during wellbore storage distortion for both buildup and flow tests. During the middle time region for buildup tests, it should be evaluated at BHP, and, for flow tests, at the average reservoir pressure at the start of the test. For flow tests in infinite-acting reservoirs, this is equivalent to using ordinary time as the independent variable.
<br>
 
<br>
 
[[File:Vol5 page 0759 eq 001.png]]....................(8.103)
 
<br>
 
<br>
 
In terms of pseudotime, ''t''<sub>''ap''</sub>, the diffusivity equation becomes
 
<br>
 
<br>
 
[[File:Vol5 page 0759 eq 002.png]]....................(8.104)
 
<br>
 
<br>
 
Subsequent studies<ref name="r18" /> have shown that the pseudotime transformation is particularly useful for analysis of flow and buildup tests distorted by wellbore storage when using type curves designed to model flow of slightly compressible liquids.  
 
<br>
 
<br>
 
Because the pressure in the integrand of '''Eq. 8.103''' is a function of position in the reservoir, it is not obvious where the pressure is to be evaluated. Empirical observations<ref name="r18" /> indicate that the pressure should be evaluated at BHP during wellbore storage distortion for both buildup and flow tests. During the middle time region for buildup tests, it should be evaluated at BHP, and, for flow tests, at the average reservoir pressure at the start of the test. For flow tests in infinite-acting reservoirs, this is equivalent to using ordinary time as the independent variable.  
 
  
 
=== Normalized Transformed Variables ===
 
=== Normalized Transformed Variables ===
  
The pseudopressure and pseudotime transformations provide excellent results when used as part of the analysis procedure for gas well tests. However, they are inconvenient for two reasons: the values of both variables will often be in the range of 10<sup>5</sup> to 10<sup>9</sup>, and they do not have units of actual pressure and time. Thus, the intuitive "feel" for the transformed variables is lost, and they may tend to be regarded as "black box" output—never helpful in test analysis. The use of pseudopressure and pseudotime require different test interpretation equations for oil wells than for gas wells.  
+
The pseudopressure and pseudotime transformations provide excellent results when used as part of the analysis procedure for gas well tests. However, they are inconvenient for two reasons: the values of both variables will often be in the range of 10<sup>5</sup> to 10<sup>9</sup>, and they do not have units of actual pressure and time. Thus, the intuitive "feel" for the transformed variables is lost, and they may tend to be regarded as "black box" output—never helpful in test analysis. The use of pseudopressure and pseudotime require different test interpretation equations for oil wells than for gas wells.<br/><br/>These difficulties are overcome by normalizing pseudopressure and pseudotime by multiplying them by constants<ref name="r19">_</ref>:<br/><br/>[[File:Vol5 page 0759 eq 003.png|RTENOTITLE]]....................(8.105)<br/><br/>and [[File:Vol5 page 0759 eq 004.png|RTENOTITLE]]....................(8.106)<br/><br/>This normalization, or multiplication by appropriate constants, gives the new variables the same units—and similar ranges—as pressure and time, respectively. With these transformations, the equations for analysis of gas wells in terms of normalized pseudopressure and pseudotime, which are called adjusted pressure and adjusted time, are obtained from the equations for analysis of oil well tests by simple substitution. Of course, the transformations require the computer. Commercial well-test analysis software often provides these transformations.<br/><br/>'''Table 8.1''' summarizes plotting methods and interpretation equations for oil well tests. It also presents information for gas well tests analyzed with ordinary pressure and time, adjusted pressure and time, pressure squared and time, and, finally, pseudopressure and time. The table includes a definition of ''p''<sub>DMBH</sub>, a dimensionless pressure defined by Matthews, Brons, and Hazebroek<ref name="r20">_</ref> that is useful in estimating current average drainage pressure. See this topic in '''Section 8.8'''.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
These difficulties are overcome by normalizing pseudopressure and pseudotime by multiplying them by constants<ref name="r19" />:
 
<br>
 
<br>
 
[[File:Vol5 page 0759 eq 003.png]]....................(8.105)
 
<br>
 
<br>
 
and [[File:Vol5 page 0759 eq 004.png]]....................(8.106)
 
<br>
 
<br>
 
This normalization, or multiplication by appropriate constants, gives the new variables the same units—and similar ranges—as pressure and time, respectively. With these transformations, the equations for analysis of gas wells in terms of normalized pseudopressure and pseudotime, which are called adjusted pressure and adjusted time, are obtained from the equations for analysis of oil well tests by simple substitution. Of course, the transformations require the computer. Commercial well-test analysis software often provides these transformations.  
 
<br>
 
<br>
 
'''Table 8.1''' summarizes plotting methods and interpretation equations for oil well tests. It also presents information for gas well tests analyzed with ordinary pressure and time, adjusted pressure and time, pressure squared and time, and, finally, pseudopressure and time. The table includes a definition of ''p''<sub>DMBH</sub>, a dimensionless pressure defined by Matthews, Brons, and Hazebroek<ref name="r20" /> that is useful in estimating current average drainage pressure. See this topic in '''Section 8.8'''.
 
<br>
 
<br>
 
<gallery widths=300px heights=200px>
 
 
File:Vol5 Page 0760 Image 0001.png|'''Table 8.1'''
 
File:Vol5 Page 0760 Image 0001.png|'''Table 8.1'''
  
 
File:Vol5 Page 0761 Image 0001.png|'''Table 8.1'''
 
File:Vol5 Page 0761 Image 0001.png|'''Table 8.1'''
 
</gallery>
 
</gallery>
<br>
 
  
In '''Table 8.1''', the HTR for gas well buildup tests is best estimated to be simply (''t''<sub>''p''</sub> + Δ''t''<sub>''a''</sub>)/Δ''t''<sub>''a''</sub>. This conclusion is based on the findings of Spivey and Lee. <ref name="r18" /> Thus, when using adjusted pressure and time, the HTR is calculated using the actual producing time,''t''<sub>''p''</sub>.  
+
In '''Table 8.1''', the HTR for gas well buildup tests is best estimated to be simply (''t''<sub>''p''</sub> + Δ''t''<sub>''a''</sub>)/Δ''t''<sub>''a''</sub>. This conclusion is based on the findings of Spivey and Lee. <ref name="r18">_</ref> Thus, when using adjusted pressure and time, the HTR is calculated using the actual producing time,''t''<sub>''p''</sub>.
  
 
=== Non-Darcy Flow ===
 
=== Non-Darcy Flow ===
  
The flow equations shown to this point assume that Darcy’s law is an appropriate model for gas flow into wells. However, as the flow velocity and Reynolds number near the well increase, the result is a transition from laminar and turbulent flow and then to turbulent flow. This transitional (and possibly turbulent) flow is called non-Darcy (non-laminar) flow. The high velocities at which the flow is transitional occur in the immediate vicinity of the well, and the additional pressure drop caused by this transitional flow is similar to a zone of altered permeability that is characterized with a skin factor. In the case of non-Darcy flow, however, the additional "skin effect" caused by the deviations from Darcy’s law is rate dependent.  
+
The flow equations shown to this point assume that Darcy’s law is an appropriate model for gas flow into wells. However, as the flow velocity and Reynolds number near the well increase, the result is a transition from laminar and turbulent flow and then to turbulent flow. This transitional (and possibly turbulent) flow is called non-Darcy (non-laminar) flow. The high velocities at which the flow is transitional occur in the immediate vicinity of the well, and the additional pressure drop caused by this transitional flow is similar to a zone of altered permeability that is characterized with a skin factor. In the case of non-Darcy flow, however, the additional "skin effect" caused by the deviations from Darcy’s law is rate dependent.<br/><br/>An adequate model for the apparent skin factor, s′, determined from a flow or buildup test is<br/><br/>[[File:Vol5 page 0761 eq 001.png|RTENOTITLE]]....................(8.107)<br/><br/>In '''Eq. 8.107''', ''s'' is the "true" skin because of damage or stimulation; ''D'' is a non-Darcy flow coefficient (assumed constant), with units of D/Mscf; and ''q''<sub>''g''</sub> is the gas flow rate with units of Mscf/D. The absolute value of the gas rate is used because the contribution to the skin is positive regardless of whether the gas well is a producer or an injector.<br/><br/>The true skin for a gas well cannot be obtained from information in a single test conducted at constant rate (including a buildup test following constant-rate production). However, skin calculated from tests conducted at several different rates (for example, associated with a multipoint deliverability test on a well) can be used to determine the true skin and the non-Darcy flow coefficient. '''Fig. 8.42''' illustrates the process for a well tested at three different rates, with an apparent skin factor determined at each rate.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
An adequate model for the apparent skin factor, s′, determined from a flow or buildup test is
 
<br>
 
<br>
 
[[File:Vol5 page 0761 eq 001.png]]....................(8.107)
 
<br>
 
<br>
 
In '''Eq. 8.107''', ''s'' is the "true" skin because of damage or stimulation; ''D'' is a non-Darcy flow coefficient (assumed constant), with units of D/Mscf; and ''q''<sub>''g''</sub> is the gas flow rate with units of Mscf/D. The absolute value of the gas rate is used because the contribution to the skin is positive regardless of whether the gas well is a producer or an injector.  
 
<br>
 
<br>
 
The true skin for a gas well cannot be obtained from information in a single test conducted at constant rate (including a buildup test following constant-rate production). However, skin calculated from tests conducted at several different rates (for example, associated with a multipoint deliverability test on a well) can be used to determine the true skin and the non-Darcy flow coefficient. '''Fig. 8.42''' illustrates the process for a well tested at three different rates, with an apparent skin factor determined at each rate.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0762 Image 0001.png|'''Fig. 8.42 – Process to obtain true skin from rate-dependent skin factor.'''
 
File:vol5 Page 0762 Image 0001.png|'''Fig. 8.42 – Process to obtain true skin from rate-dependent skin factor.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The apparent skin factor extrapolated to zero rate is the true skin (in this case, 3.4), and the slope of the curve is the non-Darcy flow coefficient, ''D'' (in this case, 5.1×10<sup>–4</sup> D/Mscf). When this method is used, take care to ensure that the permeabilities obtained from the different tests are the same; otherwise, the skin factors will be inconsistent and erroneous.  
+
The apparent skin factor extrapolated to zero rate is the true skin (in this case, 3.4), and the slope of the curve is the non-Darcy flow coefficient, ''D'' (in this case, 5.1×10<sup>–4</sup> D/Mscf). When this method is used, take care to ensure that the permeabilities obtained from the different tests are the same; otherwise, the skin factors will be inconsistent and erroneous.<br/><br/>Often, only one test is available. In this case, the non-Darcy flow coefficient, ''D'', can be estimated from<ref name="r3">_</ref><br/><br/>[[File:Vol5 page 0762 eq 001.png|RTENOTITLE]]....................(8.108)<br/><br/>The turbulence parameter, ''β'', can be estimated from<ref name="r21">_</ref><br/><br/>[[File:Vol5 page 0762 eq 002.png|RTENOTITLE]]....................(8.109)<br/><br/>The correlation represented by '''Eq. 8.109''' will provide only a crude estimate of the turbulence parameter, ''β''. Further, the correlation assumes that the non-Darcy flow occurs in the formation near the wellbore rather than through the perforations. In a gravel-packed well, the most significant additional pressure drop caused by non-Darcy flow may occur in the perforation channels through the casing.
<br>
 
<br>
 
Often, only one test is available. In this case, the non-Darcy flow coefficient, ''D'', can be estimated from<ref name="r3" />
 
<br>
 
<br>
 
[[File:Vol5 page 0762 eq 001.png]]....................(8.108)
 
<br>
 
<br>
 
The turbulence parameter, ''β'', can be estimated from<ref name="r21" />
 
<br>
 
<br>
 
[[File:Vol5 page 0762 eq 002.png]]....................(8.109)
 
<br>
 
<br>
 
The correlation represented by '''Eq. 8.109''' will provide only a crude estimate of the turbulence parameter, ''β''. Further, the correlation assumes that the non-Darcy flow occurs in the formation near the wellbore rather than through the perforations. In a gravel-packed well, the most significant additional pressure drop caused by non-Darcy flow may occur in the perforation channels through the casing.  
 
  
 
=== Multiphase Flow ===
 
=== Multiphase Flow ===
  
The equations modeling flow in reservoirs can be modified to include multiphase flow. Perrine<ref name="r22" /> suggested simple and easily applied modifications and Martin<ref name="r23" /> gave them a theoretical basis. These modifications are based on the simplifying assumption that the saturation gradients in the drainage area of the tested well are small. Thus, as examples, the modifications may lead to reasonable approximations for solution-gas drive reservoirs and are inappropriate for water-drive reservoirs with a water bank (and saturation discontinuity) in the drainage area of the tested well. The Perrine-Martin modification for constant-rate flow in an infinite-acting reservoir is
+
The equations modeling flow in reservoirs can be modified to include multiphase flow. Perrine<ref name="r22">_</ref> suggested simple and easily applied modifications and Martin<ref name="r23">_</ref> gave them a theoretical basis. These modifications are based on the simplifying assumption that the saturation gradients in the drainage area of the tested well are small. Thus, as examples, the modifications may lead to reasonable approximations for solution-gas drive reservoirs and are inappropriate for water-drive reservoirs with a water bank (and saturation discontinuity) in the drainage area of the tested well. The Perrine-Martin modification for constant-rate flow in an infinite-acting reservoir is<br/><br/>[[File:Vol5 page 0763 eq 001.png|RTENOTITLE]]....................(8.110)<br/><br/>and the Horner equation modeling a buildup test in an infinite-acting reservoir becomes<br/><br/>[[File:Vol5 page 0763 eq 002.png|RTENOTITLE]]....................(8.111)<br/><br/>In '''Eqs. 8.110''' and '''8.111''', ''q''<sub>''Rt''</sub> represents the total reservoir flow rate (RB/D) and is given by<br/><br/>[[File:Vol5 page 0763 eq 003.png|RTENOTITLE]]....................(8.112)<br/><br/>and ''λ''<sub>''t''</sub> represents the total mobility, given by<br/><br/>[[File:Vol5 page 0763 eq 004.png|RTENOTITLE]]....................(8.113)<br/><br/>The total mobility, ''λ''<sub>''t''</sub>, can be determined from a pressure buildup test run on a well that produces two or three phases simultaneously. Because '''Eq. 8.111''' implies that ''λ''<sub>''t''</sub> is related to the slope, ''m'', of a Horner plot of ''p''<sub>''ws''</sub> vs. log(''t''<sub>''p''</sub>+ Δ''t'')/Δ''t'' by<br/><br/>[[File:Vol5 page 0763 eq 005.png|RTENOTITLE]]....................(8.114)<br/><br/>The slope, ''m'', of a plot of ''p''<sub>''wf''</sub> vs. log(''t'') data from a constant-rate flow test has the same interpretation. Perrine<ref name="r22">_</ref> also showed that the permeability to each phase flowing can be estimated from the relations<br/><br/>[[File:Vol5 page 0763 eq 006.png|RTENOTITLE]]....................(8.115)<br/><br/>[[File:Vol5 page 0763 eq 007.png|RTENOTITLE]]....................(8.116)<br/><br/>and [[File:Vol5 page 0763 eq 008.png|RTENOTITLE]]....................(8.117)<br/><br/>The quantity (''q''<sub>''g''</sub> – ''q''<sub>''o''</sub>''R''<sub>''s''</sub>/1,000)''B''<sub>''g''</sub> in '''Eqs. 8.112''' and '''8.116''' is the free-gas flow rate in the reservoir; that is, the difference in the total gas rate, ''q''<sub>''g''</sub>, and the dissolved gas rate, ''q''<sub>''o''</sub>''R''<sub>''s''</sub>/1,000. Skin factor for multiphase flow test analysis using semilog plots is calculated from<br/><br/>[[File:Vol5 page 0763 eq 009.png|RTENOTITLE]]....................(8.118)<br/><br/>For analysis of tests using type curves, note that the pressure match point on a type curve is related to total and individual phase mobilities and rates by<br/><br/>[[File:Vol5 page 0764 eq 001.png|RTENOTITLE]]....................(8.119)<br/><br/>and the time match point is related to the dimensionless storage coefficient by<br/><br/>[[File:Vol5 page 0764 eq 002.png|RTENOTITLE]]....................(8.120)<br/><br/>The practical implication of '''Eqs. 8.119''' and '''8.120''' is that total mobility and individual phase permeability are determined from the pressure-match point on a type-curve match. The dimensionless storage coefficient is determined from the time-match point resulting in the calculation of skin factor from<br/><br/>[[File:Vol5 page 0764 eq 003.png|RTENOTITLE]]....................(8.121)<br/><br/>just as for single-phase flow. When the conditions for applicability of the Perrine-Martin approximations (small saturation gradients in the drainage area of the tested well) are not satisfied, use of a reservoir simulator for test analysis is an appropriate alternative.<br/>
<br>
+
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<br>
 
[[File:Vol5 page 0763 eq 001.png]]....................(8.110)
 
<br>
 
<br>
 
and the Horner equation modeling a buildup test in an infinite-acting reservoir becomes
 
<br>
 
<br>
 
[[File:Vol5 page 0763 eq 002.png]]....................(8.111)
 
<br>
 
<br>
 
In '''Eqs. 8.110''' and '''8.111''', ''q''<sub>''Rt''</sub> represents the total reservoir flow rate (RB/D) and is given by
 
<br>
 
<br>
 
[[File:Vol5 page 0763 eq 003.png]]....................(8.112)
 
<br>
 
<br>
 
and ''λ''<sub>''t''</sub> represents the total mobility, given by
 
<br>
 
<br>
 
[[File:Vol5 page 0763 eq 004.png]]....................(8.113)
 
<br>
 
<br>
 
The total mobility, ''λ''<sub>''t''</sub>, can be determined from a pressure buildup test run on a well that produces two or three phases simultaneously. Because '''Eq. 8.111''' implies that ''λ''<sub>''t''</sub> is related to the slope, ''m'', of a Horner plot of ''p''<sub>''ws''</sub> vs. log(''t''<sub>''p''''' '''</sub>+ Δ''t'')/Δ''t'' by
 
<br>
 
<br>
 
[[File:Vol5 page 0763 eq 005.png]]....................(8.114)
 
<br>
 
<br>
 
The slope, ''m'', of a plot of ''p''<sub>''wf''</sub> vs. log(''t'') data from a constant-rate flow test has the same interpretation. Perrine<ref name="r22" /> also showed that the permeability to each phase flowing can be estimated from the relations
 
<br>
 
<br>
 
[[File:Vol5 page 0763 eq 006.png]]....................(8.115)
 
<br>
 
<br>
 
[[File:Vol5 page 0763 eq 007.png]]....................(8.116)
 
<br>
 
<br>
 
and [[File:Vol5 page 0763 eq 008.png]]....................(8.117)
 
<br>
 
<br>
 
The quantity (''q''<sub>''g''</sub> – ''q''<sub>''o''</sub>''R''<sub>''s''</sub>/1,000)''B''<sub>''g''</sub> in '''Eqs. 8.112''' and '''8.116''' is the free-gas flow rate in the reservoir; that is, the difference in the total gas rate, ''q''<sub>''g''</sub>, and the dissolved gas rate, ''q''<sub>''o''</sub>''R''<sub>''s''</sub>/1,000. Skin factor for multiphase flow test analysis using semilog plots is calculated from
 
<br>
 
<br>
 
[[File:Vol5 page 0763 eq 009.png]]....................(8.118)
 
<br>
 
<br>
 
For analysis of tests using type curves, note that the pressure match point on a type curve is related to total and individual phase mobilities and rates by
 
<br>
 
<br>
 
[[File:Vol5 page 0764 eq 001.png]]....................(8.119)
 
<br>
 
<br>
 
and the time match point is related to the dimensionless storage coefficient by
 
<br>
 
<br>
 
[[File:Vol5 page 0764 eq 002.png]]....................(8.120)
 
<br>
 
<br>
 
The practical implication of '''Eqs. 8.119''' and '''8.120''' is that total mobility and individual phase permeability are determined from the pressure-match point on a type-curve match. The dimensionless storage coefficient is determined from the time-match point resulting in the calculation of skin factor from
 
<br>
 
<br>
 
[[File:Vol5 page 0764 eq 003.png]]....................(8.121)
 
<br>
 
<br>
 
just as for single-phase flow. When the conditions for applicability of the Perrine-Martin approximations (small saturation gradients in the drainage area of the tested well) are not satisfied, use of a reservoir simulator for test analysis is an appropriate alternative.  
 
<br>
 
<br>
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
 
 
== Diagnostic Plot ==
 
== Diagnostic Plot ==
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
=== Introduction ===
 
=== Introduction ===
  
The diagnostic plot is a log-log plot of the pressure change and pressure derivative (vertical axis) from a pressure transient test vs. elapsed time (horizontal axis). '''Fig. 8.43''' shows an example. The diagnostic plot can be divided into three time regions: early, middle, and late. At the earliest times on a plot (the early-time region), wellbore and near-wellbore effects dominate. These effects include wellbore storage, formation damage, partial penetration, phase redistribution, and stimulation (hydraulic fractures or acidization). At intermediate times (the middle-time region), a reservoir will ordinarily be infinite acting. For a homogeneous reservoir, the pressure derivative will be horizontal during this time region. Data in this region lead to the most accurate estimates of formation permeability. At the latest times in a test (the late-time region), boundary effects dominate curve shapes. The types of boundaries that may affect the pressure response include sealing faults, closed reservoirs, and gas/water, gas/oil, and oil/water contacts. Several common flow regimes and the diagnostic plots associated with these flow regimes are discussed in the remainder of '''Section 8.6'''.  
+
The diagnostic plot is a log-log plot of the pressure change and pressure derivative (vertical axis) from a pressure transient test vs. elapsed time (horizontal axis). '''Fig. 8.43''' shows an example. The diagnostic plot can be divided into three time regions: early, middle, and late. At the earliest times on a plot (the early-time region), wellbore and near-wellbore effects dominate. These effects include wellbore storage, formation damage, partial penetration, phase redistribution, and stimulation (hydraulic fractures or acidization). At intermediate times (the middle-time region), a reservoir will ordinarily be infinite acting. For a homogeneous reservoir, the pressure derivative will be horizontal during this time region. Data in this region lead to the most accurate estimates of formation permeability. At the latest times in a test (the late-time region), boundary effects dominate curve shapes. The types of boundaries that may affect the pressure response include sealing faults, closed reservoirs, and gas/water, gas/oil, and oil/water contacts. Several common flow regimes and the diagnostic plots associated with these flow regimes are discussed in the remainder of '''Section 8.6'''.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0765 Image 0001.png|'''Fig. 8.43 – Diagnostic plot showing time regions.'''
 
File:vol5 Page 0765 Image 0001.png|'''Fig. 8.43 – Diagnostic plot showing time regions.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Volumetric Behavior ===
 
=== Volumetric Behavior ===
  
Volumetric behavior is defined as that pressure response time dominated by the wellbore, reservoir, or part of the reservoir acting like a uniform-pressure "tank" with fluid entering or leaving the tank. The most common example of volumetric behavior is wellbore storage, which dominates during the early-time region. The "tank" is the wellbore, in which the pressure is uniform. Fluid either leaves this tank (earliest times in a flow test, before the reservoir begins to respond) or enters the tank (earliest times in a buildup test). Another example is pseudosteady-state (boundary-dominated) flow in a closed reservoir during constant-rate production. In this case, the reservoir is the tank; pressure is changing at the same rate throughout (although it is not the same at all points), and fluid is leaving the reservoir through the producing well. As a final example, in a test the reservoir may behave like a tank with recharge (fluid influx) entering from a secondary source of pressure support, such as a large supply of hydrocarbons in a lower-permeability medium in pressure communication with the reservoir being tested.  
+
Volumetric behavior is defined as that pressure response time dominated by the wellbore, reservoir, or part of the reservoir acting like a uniform-pressure "tank" with fluid entering or leaving the tank. The most common example of volumetric behavior is wellbore storage, which dominates during the early-time region. The "tank" is the wellbore, in which the pressure is uniform. Fluid either leaves this tank (earliest times in a flow test, before the reservoir begins to respond) or enters the tank (earliest times in a buildup test). Another example is pseudosteady-state (boundary-dominated) flow in a closed reservoir during constant-rate production. In this case, the reservoir is the tank; pressure is changing at the same rate throughout (although it is not the same at all points), and fluid is leaving the reservoir through the producing well. As a final example, in a test the reservoir may behave like a tank with recharge (fluid influx) entering from a secondary source of pressure support, such as a large supply of hydrocarbons in a lower-permeability medium in pressure communication with the reservoir being tested.<br/><br/>The equation modeling wellbore storage (derived from a mass balance on the wellbore) is<br/><br/>[[File:Vol5 page 0765 eq 001.png|RTENOTITLE]]....................(8.29)<br/><br/>The equation modeling pseudosteady-state flow in a cylindrical drainage area is<br/><br/>[[File:Vol5 page 0765 eq 002.png|RTENOTITLE]]....................(8.18)<br/><br/>The general form is<br/><br/>[[File:Vol5 page 0765 eq 003.png|RTENOTITLE]]....................(8.122)<br/><br/>The derivative of the general form is<br/><br/>[[File:Vol5 page 0765 eq 004.png|RTENOTITLE]]....................(8.123)<br/><br/>The implication is that the derivative plot will have unit slope (up one log cycle as it moves over one log cycle) on log-log coordinates, and the pressure change plot will approach unity at long times when b v is not equal to zero ('''Fig. 8.44'''). In wellbore storage, ''b''<sub>''v''</sub> is zero, and the derivative and pressure change plots will lie on top of one another. During pseudosteady-state flow or recharge, the pressure change and pressure derivative plots will not coincide.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
The equation modeling wellbore storage (derived from a mass balance on the wellbore) is
 
<br>
 
<br>
 
[[File:Vol5 page 0765 eq 001.png]]....................(8.29)
 
<br>
 
<br>
 
The equation modeling pseudosteady-state flow in a cylindrical drainage area is
 
<br>
 
<br>
 
[[File:Vol5 page 0765 eq 002.png]]....................(8.18)
 
<br>
 
<br>
 
The general form is
 
<br>
 
<br>
 
[[File:Vol5 page 0765 eq 003.png]]....................(8.122)
 
<br>
 
<br>
 
The derivative of the general form is
 
<br>
 
<br>
 
[[File:Vol5 page 0765 eq 004.png]]....................(8.123)
 
<br>
 
<br>
 
The implication is that the derivative plot will have unit slope (up one log cycle as it moves over one log cycle) on log-log coordinates, and the pressure change plot will approach unity at long times when b v is not equal to zero ('''Fig. 8.44'''). In wellbore storage, ''b''<sub>''v''</sub> is zero, and the derivative and pressure change plots will lie on top of one another. During pseudosteady-state flow or recharge, the pressure change and pressure derivative plots will not coincide.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0766 Image 0001.png|'''Fig. 8.44 – Volumetric flow produces derivative with unit slope.'''
 
File:vol5 Page 0766 Image 0001.png|'''Fig. 8.44 – Volumetric flow produces derivative with unit slope.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Radial Flow ===
 
=== Radial Flow ===
  
Infinite-acting radial flow is common in reservoirs, and data in the radial flow regime can be used to estimate formation permeability and skin factor. Common situations in which radial flow occurs include flow into vertical wells after wellbore storage distortion has ceased and before boundary effects, hydraulically fractured wells after the transient has moved well beyond the tips of the fracture, horizontal wells before the transient has reached the top and bottom of the productive interval, and horizontal wells after the transient has moved beyond the ends of the wellbore.  
+
Infinite-acting radial flow is common in reservoirs, and data in the radial flow regime can be used to estimate formation permeability and skin factor. Common situations in which radial flow occurs include flow into vertical wells after wellbore storage distortion has ceased and before boundary effects, hydraulically fractured wells after the transient has moved well beyond the tips of the fracture, horizontal wells before the transient has reached the top and bottom of the productive interval, and horizontal wells after the transient has moved beyond the ends of the wellbore.<br/><br/>The equation used to model radial flow for a well producing at constant rate is the familiar logarithmic approximation to the line-source solution,<br/><br/>[[File:Vol5 page 0766 eq 001.png|RTENOTITLE]]....................(8.124)<br/><br/>Equations modeling radial flow have the general form<br/><br/>[[File:Vol5 page 0766 eq 002.png|RTENOTITLE]]....................(8.125)<br/><br/>with derivative<br/><br/>[[File:Vol5 page 0766 eq 003.png|RTENOTITLE]]....................(8.126)<br/><br/>On the diagnostic plot ('''Fig. 8.45'''), radial flow is indicated by a horizontal derivative.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
The equation used to model radial flow for a well producing at constant rate is the familiar logarithmic approximation to the line-source solution,
 
<br>
 
<br>
 
[[File:Vol5 page 0766 eq 001.png]]....................(8.124)
 
<br>
 
<br>
 
Equations modeling radial flow have the general form
 
<br>
 
<br>
 
[[File:Vol5 page 0766 eq 002.png]]....................(8.125)
 
<br>
 
<br>
 
with derivative
 
<br>
 
<br>
 
[[File:Vol5 page 0766 eq 003.png]]....................(8.126)
 
<br>
 
<br>
 
On the diagnostic plot ('''Fig. 8.45'''), radial flow is indicated by a horizontal derivative.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0767 Image 0001.png|'''Fig. 8.45 – Radial flow appears as a horizontal derivative on the diagnostic plot.'''
 
File:vol5 Page 0767 Image 0001.png|'''Fig. 8.45 – Radial flow appears as a horizontal derivative on the diagnostic plot.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Linear Flow ===
 
=== Linear Flow ===
  
Linear flow is also common and occurs in channel reservoirs, hydraulically fractured wells, and horizontal wells. Data from linear flow regimes can be used to estimate channel width or fracture half-length if an estimate of permeability is available. In horizontal wells, an estimate of permeability perpendicular to the well can be made if the productive well length open to flow is known.  
+
Linear flow is also common and occurs in channel reservoirs, hydraulically fractured wells, and horizontal wells. Data from linear flow regimes can be used to estimate channel width or fracture half-length if an estimate of permeability is available. In horizontal wells, an estimate of permeability perpendicular to the well can be made if the productive well length open to flow is known.<br/><br/>An equation that models linear flow in a channel reservoir of width ''w'' is<br/><br/>[[File:Vol5 page 0767 eq 001.png|RTENOTITLE]]....................(8.127)<br/><br/>For a hydraulically fractured well with fracture half-length ''L''<sub>''f''</sub>,<br/><br/>[[File:Vol5 page 0767 eq 002.png|RTENOTITLE]]....................(8.128)<br/><br/>The general form is<br/><br/>[[File:Vol5 page 0767 eq 003.png|RTENOTITLE]]....................(8.129)<br/><br/>The derivative is<br/><br/>[[File:Vol5 page 0767 eq 004.png|RTENOTITLE]]....................(8.130)<br/><br/>Linear flow on the diagnostic plot is indicated when a derivative follows a half-slope line—that is, a line that moves up vertically by one log cycle for each two cycles of horizontal movement ('''Fig. 8.46'''). The pressure change may or may not also follow a half-slope line. In a hydraulically fractured well, the pressure change will follow a half-slope line unless the fracture is damaged. In a channel reservoir, a hydraulically fractured well with damage, or a horizontal well, the pressure change will approach the half-slope line from above.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
An equation that models linear flow in a channel reservoir of width ''w'' is
 
<br>
 
<br>
 
[[File:Vol5 page 0767 eq 001.png]]....................(8.127)
 
<br>
 
<br>
 
For a hydraulically fractured well with fracture half-length ''L''<sub>''f''</sub>,
 
<br>
 
<br>
 
[[File:Vol5 page 0767 eq 002.png]]....................(8.128)
 
<br>
 
<br>
 
The general form is
 
<br>
 
<br>
 
[[File:Vol5 page 0767 eq 003.png]]....................(8.129)
 
<br>
 
<br>
 
The derivative is
 
<br>
 
<br>
 
[[File:Vol5 page 0767 eq 004.png]]....................(8.130)
 
<br>
 
<br>
 
Linear flow on the diagnostic plot is indicated when a derivative follows a half-slope line—that is, a line that moves up vertically by one log cycle for each two cycles of horizontal movement ('''Fig. 8.46'''). The pressure change may or may not also follow a half-slope line. In a hydraulically fractured well, the pressure change will follow a half-slope line unless the fracture is damaged. In a channel reservoir, a hydraulically fractured well with damage, or a horizontal well, the pressure change will approach the half-slope line from above.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0768 Image 0001.png|'''Fig. 8.46 – Linear flow derivative follows a half-slope line on a diagnostic plot.'''
 
File:vol5 Page 0768 Image 0001.png|'''Fig. 8.46 – Linear flow derivative follows a half-slope line on a diagnostic plot.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Bilinear Flow ===
 
=== Bilinear Flow ===
  
Bilinear flow occurs primarily in wells with low-conductivity hydraulic fractures. Flow is linear within the fracture to the well, and also linear (normal to fracture flow) from the formation into the fracture. Estimates of fracture conductivity, ''w''<sub>''f''</sub>''k''<sub>''f''</sub>, can be made with data from this flow regime when estimates of formation permeability are available.  
+
Bilinear flow occurs primarily in wells with low-conductivity hydraulic fractures. Flow is linear within the fracture to the well, and also linear (normal to fracture flow) from the formation into the fracture. Estimates of fracture conductivity, ''w''<sub>''f''</sub>''k''<sub>''f''</sub>, can be made with data from this flow regime when estimates of formation permeability are available.<br/><br/>For a hydraulically fractured well, an equation that models bilinear flow is<br/><br/>[[File:Vol5 page 0768 eq 001.png|RTENOTITLE]]....................(8.131)<br/><br/>The general form is<br/><br/>[[File:Vol5 page 0768 eq 002.png|RTENOTITLE]]....................(8.132)<br/><br/>The derivative is<br/><br/>[[File:Vol5 page 0768 eq 003.png|RTENOTITLE]]....................(8.133)<br/><br/>Bilinear flow derivatives plot as a quarter-slope line on the diagnostic plot ('''Fig. 8.47'''). The quarter-slope line moves up one log cycle as it moves over four log cycles. The pressure change does not necessarily follow a quarter-slope line. In a damaged, hydraulically fractured well, the pressure change curve will approach the quarter-slope line from above; in an undamaged hydraulically fractured well (Δ''p''<sub>''s''</sub> = 0), the pressure change will typically follow the quarter-slope line when the effects of wellbore storage have ended.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
For a hydraulically fractured well, an equation that models bilinear flow is
 
<br>
 
<br>
 
[[File:Vol5 page 0768 eq 001.png]]....................(8.131)
 
<br>
 
<br>
 
The general form is
 
<br>
 
<br>
 
[[File:Vol5 page 0768 eq 002.png]]....................(8.132)
 
<br>
 
<br>
 
The derivative is
 
<br>
 
<br>
 
[[File:Vol5 page 0768 eq 003.png]]....................(8.133)
 
<br>
 
<br>
 
Bilinear flow derivatives plot as a quarter-slope line on the diagnostic plot ('''Fig. 8.47'''). The quarter-slope line moves up one log cycle as it moves over four log cycles. The pressure change does not necessarily follow a quarter-slope line. In a damaged, hydraulically fractured well, the pressure change curve will approach the quarter-slope line from above; in an undamaged hydraulically fractured well (Δ''p''<sub>''s''</sub> = 0), the pressure change will typically follow the quarter-slope line when the effects of wellbore storage have ended.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0768 Image 0002.png|'''Fig. 8.47 – Bilinear flow derivative follows a quarter-slope line on the diagnostic plot.'''
 
File:vol5 Page 0768 Image 0002.png|'''Fig. 8.47 – Bilinear flow derivative follows a quarter-slope line on the diagnostic plot.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Spherical Flow ===
 
=== Spherical Flow ===
  
The flow pattern is spherical when the pressure transient can propagate freely in three dimensions and converge into a "point." This can occur for wells that penetrate only a short distance into the formation (actually hemispherical flow), wells that have only a limited number of perforations open to flow, horizontal wells with inflow over only short intervals, and during wireline formation tests. Data in the spherical-flow regime can be used to estimate the mean permeability,
+
The flow pattern is spherical when the pressure transient can propagate freely in three dimensions and converge into a "point." This can occur for wells that penetrate only a short distance into the formation (actually hemispherical flow), wells that have only a limited number of perforations open to flow, horizontal wells with inflow over only short intervals, and during wireline formation tests. Data in the spherical-flow regime can be used to estimate the mean permeability,<br/><br/>[[File:Vol5 page 0769 eq 001.png|RTENOTITLE]]....................(8.134)<br/><br/>An equation that models spherical flow is<br/><br/>[[File:Vol5 page 0769 eq 002.png|RTENOTITLE]]....................(8.31)<br/><br/>where [[File:Vol5 page 0769 eq 003.png|RTENOTITLE]]....................(8.32)<br/><br/>and ''r''<sub>''sp''</sub> is the radius of the sphere into which flow converges. The general form is<br/><br/>[[File:Vol5 page 0769 eq 004.png|RTENOTITLE]]....................(8.135)<br/><br/>and the derivative is<br/><br/>[[File:Vol5 page 0769 eq 005.png|RTENOTITLE]]....................(8.136)<br/><br/>Spherical flow on the diagnostic plot produces a derivative line with a slope of −1/2. The pressure change during spherical flow approaches a horizontal line from below, and never exhibits a straight line with the same slope as the derivative ('''Fig. 8.48'''). Spherical flow can occur during either buildup or drawdown tests.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0769 eq 001.png]]....................(8.134)
 
<br>
 
<br>
 
An equation that models spherical flow is
 
<br>
 
<br>
 
[[File:Vol5 page 0769 eq 002.png]]....................(8.31)
 
<br>
 
<br>
 
where [[File:Vol5 page 0769 eq 003.png]]....................(8.32)
 
<br>
 
<br>
 
and ''r''<sub>''sp''</sub> is the radius of the sphere into which flow converges. The general form is
 
<br>
 
<br>
 
[[File:Vol5 page 0769 eq 004.png]]....................(8.135)
 
<br>
 
<br>
 
and the derivative is
 
<br>
 
<br>
 
[[File:Vol5 page 0769 eq 005.png]]....................(8.136)
 
<br>
 
<br>
 
Spherical flow on the diagnostic plot produces a derivative line with a slope of −1/2. The pressure change during spherical flow approaches a horizontal line from below, and never exhibits a straight line with the same slope as the derivative ('''Fig. 8.48'''). Spherical flow can occur during either buildup or drawdown tests.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0769 Image 0001.png|'''Fig. 8.48 – Spherical flow derivative has slow of – 1/2.'''
 
File:vol5 Page 0769 Image 0001.png|'''Fig. 8.48 – Spherical flow derivative has slow of – 1/2.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Flow Regimes on the Diagnostic Plot ===
 
=== Flow Regimes on the Diagnostic Plot ===
  
A major application of the diagnostic plot is the potential that it provides in identifying the flow regimes that appear in a logical sequence during a buildup or flow test. For example, consider '''Fig. 8.49'''. At early times, the unit slope line on both derivative and pressure change, indicating wellbore storage. Later, a derivative with a slope of −1/2, indicating possible spherical flow, followed by a horizontal derivative, indicating infinite-acting radial flow. Boundary effects, including a unit-slope line, follow, indicating possible recharge of the reservoir pressure.  
+
A major application of the diagnostic plot is the potential that it provides in identifying the flow regimes that appear in a logical sequence during a buildup or flow test. For example, consider '''Fig. 8.49'''. At early times, the unit slope line on both derivative and pressure change, indicating wellbore storage. Later, a derivative with a slope of −1/2, indicating possible spherical flow, followed by a horizontal derivative, indicating infinite-acting radial flow. Boundary effects, including a unit-slope line, follow, indicating possible recharge of the reservoir pressure.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0770 Image 0001.png|'''Fig. 8.49 – Well test diagnostic plot indicating several flow regimes.'''
 
File:vol5 Page 0770 Image 0001.png|'''Fig. 8.49 – Well test diagnostic plot indicating several flow regimes.'''
 
</gallery>
 
</gallery>
<br/>
+
</div></div><div class="toccolours mw-collapsible mw-collapsed">
 
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
 
 
== Behavior of Bounded Reservoirs ==
 
== Behavior of Bounded Reservoirs ==
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
=== Introduction ===
 
=== Introduction ===
  
Reservoir boundaries have significant influences on the shape of the diagnostic plot. The effects of boundaries appear following the middle-time region (infinite-acting radial flow) in a test. Recognizing the influence of boundaries can be as important as analyzing the test quantitatively. However, a problem in recognition is that many reservoir models may produce similar pressure responses. The model selected to interpret the test quantitatively must be consistent with geological and geophysical interpretations. Once the proper reservoir model has been determined, test analysis may be relatively straight-forward type-curve matching or regression analysis using modern well-test analysis software.  
+
Reservoir boundaries have significant influences on the shape of the diagnostic plot. The effects of boundaries appear following the middle-time region (infinite-acting radial flow) in a test. Recognizing the influence of boundaries can be as important as analyzing the test quantitatively. However, a problem in recognition is that many reservoir models may produce similar pressure responses. The model selected to interpret the test quantitatively must be consistent with geological and geophysical interpretations. Once the proper reservoir model has been determined, test analysis may be relatively straight-forward type-curve matching or regression analysis using modern well-test analysis software.<br/><br/>The shapes of the diagnostic plots for a buildup test and a drawdown test are essentially identical during the early- and middle-time regions for most tests. However, boundary effects can cause quite different shapes for a given reservoir model at late times in buildup and drawdown tests. This problem is augmented by the common use of "equivalent time" functions to analyze buildup tests on drawdown type curves. (There are different equivalent time functions for radial flow, linear flow, and bilinear flow, as discussed in more detail in the section on analysis of hydraulically fractured wells.)<br/><br/>Basically, equivalent time functions apply rigorously only to situations where either the producing time and the shut-in time both lie within the middle-time region or, as is commonly the case, the shut-in time is much less than the producing time before shut in.<br/><br/>To further complicate matters for buildup test analysis, the shape of the derivative curve depends on how the derivative is calculated and plotted. The derivative of pressure change may be taken with respect to the logarithm of either shut-in time or equivalent time. The derivative may then be plotted vs. either of these time functions, and the shape differs for each plotting function. Some pressure transient test analysis software allows the user a choice in the time function used in taking the derivative and another choice in time plotting function; for other software, the time functions used are "hard-wired." The results can be bewildering.
<br>
 
<br>
 
The shapes of the diagnostic plots for a buildup test and a drawdown test are essentially identical during the early- and middle-time regions for most tests. However, boundary effects can cause quite different shapes for a given reservoir model at late times in buildup and drawdown tests. This problem is augmented by the common use of "equivalent time" functions to analyze buildup tests on drawdown type curves. (There are different equivalent time functions for radial flow, linear flow, and bilinear flow, as discussed in more detail in the section on analysis of hydraulically fractured wells.)  
 
<br>
 
<br>
 
Basically, equivalent time functions apply rigorously only to situations where either the producing time and the shut-in time both lie within the middle-time region or, as is commonly the case, the shut-in time is much less than the producing time before shut in.  
 
<br>
 
<br>
 
To further complicate matters for buildup test analysis, the shape of the derivative curve depends on how the derivative is calculated and plotted. The derivative of pressure change may be taken with respect to the logarithm of either shut-in time or equivalent time. The derivative may then be plotted vs. either of these time functions, and the shape differs for each plotting function. Some pressure transient test analysis software allows the user a choice in the time function used in taking the derivative and another choice in time plotting function; for other software, the time functions used are "hard-wired." The results can be bewildering.  
 
  
 
=== Well in an Infinite-Acting Reservoir ===
 
=== Well in an Infinite-Acting Reservoir ===
  
Infinite-acting, radial flow reservoirs were described in the previous section. '''Figs. 8.50 and 8.51''' show their diagnostic curves. For these plots, the derivative was taken with respect to shut-in time and derivative and pressure change curves are plotted vs. shut-in time. Both pressure and time are in terms of dimensionless variables. Wellbore storage distortion is not included in any of the diagnostic plots in this section.  
+
Infinite-acting, radial flow reservoirs were described in the previous section. '''Figs. 8.50 and 8.51''' show their diagnostic curves. For these plots, the derivative was taken with respect to shut-in time and derivative and pressure change curves are plotted vs. shut-in time. Both pressure and time are in terms of dimensionless variables. Wellbore storage distortion is not included in any of the diagnostic plots in this section.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0771 Image 0001.png|'''Fig. 8.50 – Diagnostic plot for constant-rate drawdown test in an infinite-acting reservoir.'''
 
File:vol5 Page 0771 Image 0001.png|'''Fig. 8.50 – Diagnostic plot for constant-rate drawdown test in an infinite-acting reservoir.'''
  
 
File:vol5 Page 0771 Image 0001.png|'''Fig. 8.51 – Diagnostic plot for buildup test.'''
 
File:vol5 Page 0771 Image 0001.png|'''Fig. 8.51 – Diagnostic plot for buildup test.'''
 
</gallery>
 
</gallery>
<br/>
 
  
Notice the significant difference in the shapes of both the derivative and pressure change curves for buildup and drawdown tests, with the pressure change curves flattening for buildup tests and the derivatives moving downward with an ultimate slope of –1. The time at which the flattening of the pressure change curve (and corresponding downward movement of the derivative) becomes apparent is a function of the producing time before shut-in. The longer the producing time, the longer the flattening is delayed and the longer the time the buildup diagnostic plot is essentially identical to the drawdown diagnostic plot.  
+
Notice the significant difference in the shapes of both the derivative and pressure change curves for buildup and drawdown tests, with the pressure change curves flattening for buildup tests and the derivatives moving downward with an ultimate slope of –1. The time at which the flattening of the pressure change curve (and corresponding downward movement of the derivative) becomes apparent is a function of the producing time before shut-in. The longer the producing time, the longer the flattening is delayed and the longer the time the buildup diagnostic plot is essentially identical to the drawdown diagnostic plot.<br/><br/>'''Fig. 8.52''' is the diagnostic plot that results when the derivative is taken with respect to radial equivalent time and the time-plotting function is radial equivalent time. The drawdown and buildup curves appear to be identical for all times. However, the radial equivalent time has a maximum value of the producing time before shut-in, so, for the buildup plots, the curves terminate at these maximum values of the time plotting function, and all points "stack up" at these limiting values of the plotting function. Our conclusion is that radial equivalent time is more satisfactory as a variable for taking the derivative and as a plotting function for an infinite-acting reservoir because the shape of the diagnostic plot is the same as for a constant-rate drawdown test.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
'''Fig. 8.52''' is the diagnostic plot that results when the derivative is taken with respect to radial equivalent time and the time-plotting function is radial equivalent time. The drawdown and buildup curves appear to be identical for all times. However, the radial equivalent time has a maximum value of the producing time before shut-in, so, for the buildup plots, the curves terminate at these maximum values of the time plotting function, and all points "stack up" at these limiting values of the plotting function. Our conclusion is that radial equivalent time is more satisfactory as a variable for taking the derivative and as a plotting function for an infinite-acting reservoir because the shape of the diagnostic plot is the same as for a constant-rate drawdown test.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0772 Image 0001.png|'''Fig. 8.52 – Diagnostic plot when derivative is taken with respect to and plotted against radial equivalent time.'''
 
File:vol5 Page 0772 Image 0001.png|'''Fig. 8.52 – Diagnostic plot when derivative is taken with respect to and plotted against radial equivalent time.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Linear No-Flow Boundary ===
 
=== Linear No-Flow Boundary ===
  
When a well is near a single no-flow boundary ('''Fig. 8.53''') or, as a practical matter, much closer to one boundary than to any other, and when sufficient time has elapsed for the boundary to have an influence on the pressure response during the test, the characteristic diagnostic plot, as '''Fig. 8.54''' shows, results for a constant-rate drawdown test. (Wellbore storage may distort some of the earlier data on this diagnostic plot.) The derivative will double in value over approximately 1 2/3 log cycles (from 0.5 to 1.0 on a plot of dimensionless variables). Similar responses occur in naturally fractured reservoirs with transient flow from the matrix to the fractures.  
+
When a well is near a single no-flow boundary ('''Fig. 8.53''') or, as a practical matter, much closer to one boundary than to any other, and when sufficient time has elapsed for the boundary to have an influence on the pressure response during the test, the characteristic diagnostic plot, as '''Fig. 8.54''' shows, results for a constant-rate drawdown test. (Wellbore storage may distort some of the earlier data on this diagnostic plot.) The derivative will double in value over approximately 1 2/3 log cycles (from 0.5 to 1.0 on a plot of dimensionless variables). Similar responses occur in naturally fractured reservoirs with transient flow from the matrix to the fractures.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0772 Image 0002.png|'''Fig. 8.53 – Well near a single no-flow boundary.'''
 
File:vol5 Page 0772 Image 0002.png|'''Fig. 8.53 – Well near a single no-flow boundary.'''
  
 
File:vol5 Page 0773 Image 0001.png|'''Fig. 8.54 – Diagnostic plot for constant-rate drawdown test influenced by a single no-flow boundary.'''
 
File:vol5 Page 0773 Image 0001.png|'''Fig. 8.54 – Diagnostic plot for constant-rate drawdown test influenced by a single no-flow boundary.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.55''' is the diagnostic plot for a buildup test with the derivative taken with respect to shut-in time and plotted vs. shut-in time. (Wellbore storage may distort some of the earlier data on this plot.) The longer the producing time before shut-in, the more nearly the shape of the diagnostic plot for a buildup test resembles the diagnostic plot for a drawdown test. The derivative has a slope of –1 for shut-in times much longer than producing time before shut-in.  
+
'''Fig. 8.55''' is the diagnostic plot for a buildup test with the derivative taken with respect to shut-in time and plotted vs. shut-in time. (Wellbore storage may distort some of the earlier data on this plot.) The longer the producing time before shut-in, the more nearly the shape of the diagnostic plot for a buildup test resembles the diagnostic plot for a drawdown test. The derivative has a slope of –1 for shut-in times much longer than producing time before shut-in.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0773 Image 0002.png|'''Fig. 8.55 – Diagnostic plot for buildup test with derivative taken with respect to shut-in time. Long producing time before shut-in produces curve resembling plot for drawdown test.'''
 
File:vol5 Page 0773 Image 0002.png|'''Fig. 8.55 – Diagnostic plot for buildup test with derivative taken with respect to shut-in time. Long producing time before shut-in produces curve resembling plot for drawdown test.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.56''' is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. Derivatives double over a small fraction of a log cycle for short producing times and, in general, the shapes of the diagnostic plots for buildup tests are similar to drawdown diagnostic plots only for longer producing times before shut-in.  
+
'''Fig. 8.56''' is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. Derivatives double over a small fraction of a log cycle for short producing times and, in general, the shapes of the diagnostic plots for buildup tests are similar to drawdown diagnostic plots only for longer producing times before shut-in.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0774 Image 0001.png|'''Fig. 8.56 – Diagnostic plot for buildup test with derivative taken with respect to and plotted against equivalent time shows that derivative doubles over small fraction of log cycle for short producing times.'''
 
File:vol5 Page 0774 Image 0001.png|'''Fig. 8.56 – Diagnostic plot for buildup test with derivative taken with respect to and plotted against equivalent time shows that derivative doubles over small fraction of log cycle for short producing times.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.57''' is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. In this case, the diagnostic plot is similar to the drawdown response, but the plots are not identical. Notice that the derivative doubles over approximately 1 2/3 log cycle. This procedure for taking the derivative and preparing the diagnostic plot is the most satisfactory of the alternatives considered.  
+
'''Fig. 8.57''' is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. In this case, the diagnostic plot is similar to the drawdown response, but the plots are not identical. Notice that the derivative doubles over approximately 1 2/3 log cycle. This procedure for taking the derivative and preparing the diagnostic plot is the most satisfactory of the alternatives considered.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0774 Image 0002.png|'''Fig. 8.57 – Diagnostic plot for buildup test with derivative taken with respect to radial equivalent time and plotted vs. shut-in time resembles drawdown response.'''
 
File:vol5 Page 0774 Image 0002.png|'''Fig. 8.57 – Diagnostic plot for buildup test with derivative taken with respect to radial equivalent time and plotted vs. shut-in time resembles drawdown response.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Linear Constant-Pressure Boundary ===
 
=== Linear Constant-Pressure Boundary ===
  
When a well is much nearer a single boundary (similar to '''Fig. 8.53''') but with a constant-pressure at that boundary and boundary effects are encountered during the test, the diagnostic plot shown in '''Fig. 8.58''' will result in a constant-rate drawdown test. (Wellbore storage effects could also occur early in the test.) The derivative has a slope of –1 at late times on the diagnostic plot.  
+
When a well is much nearer a single boundary (similar to '''Fig. 8.53''') but with a constant-pressure at that boundary and boundary effects are encountered during the test, the diagnostic plot shown in '''Fig. 8.58''' will result in a constant-rate drawdown test. (Wellbore storage effects could also occur early in the test.) The derivative has a slope of –1 at late times on the diagnostic plot.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0775 Image 0001.png|'''Fig. 8.58 – Derivative has a slope of -1 for a well located near a single, constant-pressure boundary.'''
 
File:vol5 Page 0775 Image 0001.png|'''Fig. 8.58 – Derivative has a slope of -1 for a well located near a single, constant-pressure boundary.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.59''' is the diagnostic plot for a buildup test, with derivative taken with respect to shut-in time and plotted vs. shut-in time. This diagnostic plot is identical to the drawdown plot if steady state was achieved during the flow period preceding the buildup test. For other cases, with shorter producing times, the derivative has a slope steeper than the drawdown slope of –1.  
+
'''Fig. 8.59''' is the diagnostic plot for a buildup test, with derivative taken with respect to shut-in time and plotted vs. shut-in time. This diagnostic plot is identical to the drawdown plot if steady state was achieved during the flow period preceding the buildup test. For other cases, with shorter producing times, the derivative has a slope steeper than the drawdown slope of –1.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0775 Image 0002.png|'''Fig. 8.59 – Diagnostic plot for buildup test with derivative taken with respect to and plotted vs. shut-in time is similar to drawdown plot if steady state achieved during flow period preceding buildup test.'''
 
File:vol5 Page 0775 Image 0002.png|'''Fig. 8.59 – Diagnostic plot for buildup test with derivative taken with respect to and plotted vs. shut-in time is similar to drawdown plot if steady state achieved during flow period preceding buildup test.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.60''' is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. For short producing times, the derivative falls precipitously.  
+
'''Fig. 8.60''' is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. For short producing times, the derivative falls precipitously.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0776 Image 0001.png|'''Fig. 8.60 – Diagnostic plot for buildup test with derivative taken with respect to and plotted against equivalent time shows that derivative falls precipitously for short producing times.'''
 
File:vol5 Page 0776 Image 0001.png|'''Fig. 8.60 – Diagnostic plot for buildup test with derivative taken with respect to and plotted against equivalent time shows that derivative falls precipitously for short producing times.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.61''' is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. The shapes of the diagnostic plots are similar to, but not identical to, the drawdown diagnostic plot for all producing times before shut-in. The diagnostic plot prepared in this way is the most satisfactory of the alternatives considered.  
+
'''Fig. 8.61''' is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. The shapes of the diagnostic plots are similar to, but not identical to, the drawdown diagnostic plot for all producing times before shut-in. The diagnostic plot prepared in this way is the most satisfactory of the alternatives considered.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0776 Image 0002.png|'''Fig. 8.61 – Diagnostic plot for buildup test with derivative taken with respect to and plotted against shut-in time resembles drawdown diagnostic plot for all producing times before shut-in.'''
 
File:vol5 Page 0776 Image 0002.png|'''Fig. 8.61 – Diagnostic plot for buildup test with derivative taken with respect to and plotted against shut-in time resembles drawdown diagnostic plot for all producing times before shut-in.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Well in a Channel ===
 
=== Well in a Channel ===
  
When a well is between two parallel no-flow boundaries and the pressure transient encounters both during a test long before the ends of the reservoir influence the test data, the diagnostic plot in '''Fig. 8.62''' results for a constant-rate drawdown test. Before the boundary effects, with characteristic derivative slope of 1/2, wellbore storage, radial flow (or hemiradial flow if the well is much nearer one boundary than the other) will usually appear on the diagnostic plot. Diagnostic plots with similar shapes occur for a well between two sealing faults, a hydraulically fractured well with a high-conductivity fracture, and a horizontal well during early linear flow.  
+
When a well is between two parallel no-flow boundaries and the pressure transient encounters both during a test long before the ends of the reservoir influence the test data, the diagnostic plot in '''Fig. 8.62''' results for a constant-rate drawdown test. Before the boundary effects, with characteristic derivative slope of 1/2, wellbore storage, radial flow (or hemiradial flow if the well is much nearer one boundary than the other) will usually appear on the diagnostic plot. Diagnostic plots with similar shapes occur for a well between two sealing faults, a hydraulically fractured well with a high-conductivity fracture, and a horizontal well during early linear flow.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0777 Image 0001.png|'''Fig. 8.62 – Constant-rate drawdown diagnostic plot, well between two no-flow boundaries, characteristic derivative slope of 1/2.'''
 
File:vol5 Page 0777 Image 0001.png|'''Fig. 8.62 – Constant-rate drawdown diagnostic plot, well between two no-flow boundaries, characteristic derivative slope of 1/2.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.63''' is the diagnostic plot for a buildup test, with derivative taken with respect to shut-in time and plotted vs. shut-in time. The longer the producing time before shut-in, the more similar the curve shape is to the drawdown-test diagnostic plot. The derivative has a slope of –1/2 when shut-in time is much larger than producing time.  
+
'''Fig. 8.63''' is the diagnostic plot for a buildup test, with derivative taken with respect to shut-in time and plotted vs. shut-in time. The longer the producing time before shut-in, the more similar the curve shape is to the drawdown-test diagnostic plot. The derivative has a slope of –1/2 when shut-in time is much larger than producing time.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0777 Image 0002.png|'''Fig. 8.63 – Diagnostic plot for buildup test in well between two boundaries with derivative taken with respect to shut-in time. Longer producing time results in curve more similar to drawdown-test diagnostic plot. (Derivative has a slope of – 1/2 when shut-in time is much longer than producing time.)'''
 
File:vol5 Page 0777 Image 0002.png|'''Fig. 8.63 – Diagnostic plot for buildup test in well between two boundaries with derivative taken with respect to shut-in time. Longer producing time results in curve more similar to drawdown-test diagnostic plot. (Derivative has a slope of – 1/2 when shut-in time is much longer than producing time.)'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.64''' is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. This plot is not particularly useful for test analysis. However, linear equivalent time produces a more useful diagnostic plot as long as channel ends do not affect the pressure response.  
+
'''Fig. 8.64''' is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. This plot is not particularly useful for test analysis. However, linear equivalent time produces a more useful diagnostic plot as long as channel ends do not affect the pressure response.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0778 Image 0001.png|'''Fig. 8.64 – Diagnostic plot for buildup test with derivative taken with respect to and plotted against equivalent time is not particularly useful for test analysis.'''
 
File:vol5 Page 0778 Image 0001.png|'''Fig. 8.64 – Diagnostic plot for buildup test with derivative taken with respect to and plotted against equivalent time is not particularly useful for test analysis.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.65''' is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. The derivative is similar to, but not identical to, the drawdown response. This method is the most useful for test analysis among the alternatives discussed.  
+
'''Fig. 8.65''' is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. The derivative is similar to, but not identical to, the drawdown response. This method is the most useful for test analysis among the alternatives discussed.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0778 Image 0002.png|'''Fig. 8.65 – Diagnostic plot for buildup test in well between two flow barriers. Derivative taken with respect to equivalent time and plotted against shut-in time resembles drawdown response.'''
 
File:vol5 Page 0778 Image 0002.png|'''Fig. 8.65 – Diagnostic plot for buildup test in well between two flow barriers. Derivative taken with respect to equivalent time and plotted against shut-in time resembles drawdown response.'''
 
</gallery>
 
</gallery>
<br/>
+
</div></div><div class="toccolours mw-collapsible mw-collapsed">
 
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
 
 
== Estimating Average Reservoir Pressure ==
 
== Estimating Average Reservoir Pressure ==
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
<br>
+
<br/>Two different method types, one using data from the middle-time region and the second using data from the late-time region (LTR), are commonly applied in estimating average reservoir pressure. The middle-time region methods are the Matthews-Brons-Hazebroek (MBH) method<ref name="r20">_</ref> and the Ramey-Cobb method. <ref name="r24">_</ref> The LTR methods are the modified Muskat method<ref name="r25">_</ref> and the Arps-Smith method. <ref name="r26">_</ref>
Two different method types, one using data from the middle-time region and the second using data from the late-time region (LTR), are commonly applied in estimating average reservoir pressure. The middle-time region methods are the Matthews-Brons-Hazebroek (MBH) method<ref name="r20" /> and the Ramey-Cobb method. <ref name="r24" /> The LTR methods are the modified Muskat method<ref name="r25" /> and the Arps-Smith method. <ref name="r26" />  
 
  
 
=== Middle-Time Region Methods ===
 
=== Middle-Time Region Methods ===
  
The MTR methods are based on extrapolation of the middle-time region and the correction of the extrapolated pressure. The advantage of these methods is that they use pressure data only from the middle-time region, which means they require relatively short tests. The disadvantages are the need for accurate fluid property estimates, a known drainage area shape and size, and the location of the well within the drainage area.  
+
The MTR methods are based on extrapolation of the middle-time region and the correction of the extrapolated pressure. The advantage of these methods is that they use pressure data only from the middle-time region, which means they require relatively short tests. The disadvantages are the need for accurate fluid property estimates, a known drainage area shape and size, and the location of the well within the drainage area.<br/><br/>'''''Drainage Area Shapes.''''' The MTR methods depend on the shape of the drainage area. Matthews-Brons-Hazebroek<ref name="r20">_</ref> developed a series of curves that model buildup tests in many shapes. As a matter of interest, these graphs were generated using image wells to simulate boundaries.<br/><br/>'''Figs. 8.66 through 8.68''' illustrate representative dimensionless pressures as calculated by the MBH method. '''Fig. 8.66''' is a plot of dimensionless pressure as defined by the MBH method plotted against dimensionless producing time calculated using the drainage area. Dimensionless pressure is defined as<br/><br/>[[File:Vol5 page 0777 eq 001.png|RTENOTITLE]]....................(8.137)<br/><br/>and dimensionless time is<br/><br/>[[File:Vol5 page 0778 eq 001.png|RTENOTITLE]]....................(8.138)<br/><br/>In '''Eq. 8.137''', ''p''* = the extrapolated pressure at a HTR of unity, [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]] = the current average drainage area pressure, and ''m'' = the slope of the MTR straight line on a Horner plot. In '''Eq. 8.138''', ''t''<sub>''p''</sub> = the producing time before shut-in, and ''A'' = the well’s drainage area expressed in square feet.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
'''''Drainage Area Shapes.''''' The MTR methods depend on the shape of the drainage area. Matthews-Brons-Hazebroek<ref name="r20" /> developed a series of curves that model buildup tests in many shapes. As a matter of interest, these graphs were generated using image wells to simulate boundaries.  
 
<br>
 
<br>
 
'''Figs. 8.66 through 8.68''' illustrate representative dimensionless pressures as calculated by the MBH method. '''Fig. 8.66''' is a plot of dimensionless pressure as defined by the MBH method plotted against dimensionless producing time calculated using the drainage area. Dimensionless pressure is defined as
 
<br>
 
<br>
 
[[File:Vol5 page 0777 eq 001.png]]....................(8.137)
 
<br>
 
<br>
 
and dimensionless time is
 
<br>
 
<br>
 
[[File:Vol5 page 0778 eq 001.png]]....................(8.138)
 
<br>
 
<br>
 
In '''Eq. 8.137''', ''p''* = the extrapolated pressure at a HTR of unity, [[File:Vol5 page 0781 inline 001.png]] = the current average drainage area pressure, and ''m'' = the slope of the MTR straight line on a Horner plot. In '''Eq. 8.138''', ''t''<sub>''p''</sub> = the producing time before shut-in, and ''A'' = the well’s drainage area expressed in square feet.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0779 Image 0001.png|'''Fig. 8.66 – MBH pressures for wells in square drainage area.'''
 
File:vol5 Page 0779 Image 0001.png|'''Fig. 8.66 – MBH pressures for wells in square drainage area.'''
  
Line 1,598: Line 484:
 
File:vol5 Page 0780 Image 0001.png|'''Fig. 8.68 – MBH pressures for wells located in a 4x1 rectangular reservoir.'''
 
File:vol5 Page 0780 Image 0001.png|'''Fig. 8.68 – MBH pressures for wells located in a 4x1 rectangular reservoir.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The four different curves in '''Fig. 8.66''' represent four different locations of a well within a square drainage area. On this plot of dimensionless pressure on a linear scale vs. dimensionless time on a logarithmic scale, these curves eventually become straight lines. For example, for a well centered in a square drainage area, the line becomes straight at a dimensionless time of approximately 0.2. The time at which the line becomes straight is an indication that a well has reached pseudosteady-state flow at that dimensionless time.  
+
The four different curves in '''Fig. 8.66''' represent four different locations of a well within a square drainage area. On this plot of dimensionless pressure on a linear scale vs. dimensionless time on a logarithmic scale, these curves eventually become straight lines. For example, for a well centered in a square drainage area, the line becomes straight at a dimensionless time of approximately 0.2. The time at which the line becomes straight is an indication that a well has reached pseudosteady-state flow at that dimensionless time.<br/><br/>'''Fig. 8.67''' shows the Matthews-Brons-Hazebroek correlations for a well in the various positions in a 2×1 rectangle. The wells eventually reach pseudosteady state and the lines become straight, but, in general, the time to reach pseudosteady state is longer for the 2×1 rectangle than it was for the square rectangle. Furthermore, the farther the well is off center within the drainage area, the longer the time required to reach pseudosteady state. The difference is on the order of one full log cycle between the case in which the well is centered in the drainage area and that for a well most off-centered in the drainage area, which is the lowest curve on this plot. '''Fig. 8.68''' shows the MBH pressures for wells in various positions in a 4×1 rectangle. Matthews-Brons-Hazebroek generated many similar graphs for other drainage-area shapes.<br/><br/>'''''Example of the Matthews-Brons-Hazebroek Method.''''' This method will be applied to a well in a reservoir with the following properties: t p = 482 hours, ''ϕ'' = 0.15, ''μ'' = 0.25 cp, ''c''<sub>''t''</sub> = 1.615 × 10<sup>–5</sup>, and ''A'' = 1,500 × 3,000 ft (a 2 × 1 reservoir, well centered).<br/><br/>First, plot well shut-in pressure against the HTR on semilog coordinates. In '''Fig. 8.69''', which is an ordinary HTR plot, the wellbore storage affects the data at large values of HTR, followed by the straight-line middle-time region, in turn followed by a deviation of the curve as it begins moving toward a fully built-up pressure.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
'''Fig. 8.67''' shows the Matthews-Brons-Hazebroek correlations for a well in the various positions in a 2×1 rectangle. The wells eventually reach pseudosteady state and the lines become straight, but, in general, the time to reach pseudosteady state is longer for the 2×1 rectangle than it was for the square rectangle. Furthermore, the farther the well is off center within the drainage area, the longer the time required to reach pseudosteady state. The difference is on the order of one full log cycle between the case in which the well is centered in the drainage area and that for a well most off-centered in the drainage area, which is the lowest curve on this plot. '''Fig. 8.68''' shows the MBH pressures for wells in various positions in a 4×1 rectangle. Matthews-Brons-Hazebroek generated many similar graphs for other drainage-area shapes.  
 
<br>
 
<br>
 
'''''Example of the Matthews-Brons-Hazebroek Method.''''' This method will be applied to a well in a reservoir with the following properties: t p = 482 hours, ''ϕ'' = 0.15, ''μ'' = 0.25 cp, ''c''<sub>''t''</sub> = 1.615 × 10<sup>–5</sup>, and ''A'' = 1,500 × 3,000 ft (a 2 × 1 reservoir, well centered).  
 
<br>
 
<br>
 
First, plot well shut-in pressure against the HTR on semilog coordinates. In '''Fig. 8.69''', which is an ordinary HTR plot, the wellbore storage affects the data at large values of HTR, followed by the straight-line middle-time region, in turn followed by a deviation of the curve as it begins moving toward a fully built-up pressure.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0780 Image 0002.png|'''Fig. 8.69 – Horner plot yields slope, Horner time ratio, and extrapolated pressure.'''
 
File:vol5 Page 0780 Image 0002.png|'''Fig. 8.69 – Horner plot yields slope, Horner time ratio, and extrapolated pressure.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The MTR straight line on this Horner graph is extrapolated to a HTR of 1 to determine ''p''*. In this case, ''p''* = 2,689.4 psi. From the slope of the semilog straight line, 26.7 psi/cycle, we calculate ''k'' = 7.5 md.  
+
The MTR straight line on this Horner graph is extrapolated to a HTR of 1 to determine ''p''*. In this case, ''p''* = 2,689.4 psi. From the slope of the semilog straight line, 26.7 psi/cycle, we calculate ''k'' = 7.5 md.<br/><br/>Next, calculate the dimensionless producing time, ''t''<sub>''p''AD</sub>, with '''Eq. 8.138'''.<br/><br/>[[File:Vol5 page 0780 eq 001.png|RTENOTITLE]]<br/><br/>To calculate dimensionless production time, use the same producing time used in preparing the Horner graph. If the actual producing time is quite long, replace it with the time required to reach pseudosteady state, but remember to use the same producing time in the HTR and in calculating the dimensionless time for the MBH function. The time to reach pseudosteady state is determined by observing the appropriate MBH graph and finding when the dimensionless pressure vs. time becomes a straight line.<br/><br/>The next step is to select the appropriate MBH chart for the drainage area shape and well location being evaluated. Because the example well is centered in a 2×1 rectangle, choose '''Fig. 8.67'''. On this chart, enter the graph at a dimensionless producing time of 0.35, as illustrated in '''Fig. 8.70''', and read across to find the dimensionless pressure, ''p''<sub>MBHD</sub>, which has a value of 2.05.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
Next, calculate the dimensionless producing time, ''t''<sub>''p''AD</sub>, with '''Eq. 8.138'''.
 
<br>
 
<br>
 
[[File:Vol5 page 0780 eq 001.png]]
 
<br>
 
<br>
 
To calculate dimensionless production time, use the same producing time used in preparing the Horner graph. If the actual producing time is quite long, replace it with the time required to reach pseudosteady state, but remember to use the same producing time in the HTR and in calculating the dimensionless time for the MBH function. The time to reach pseudosteady state is determined by observing the appropriate MBH graph and finding when the dimensionless pressure vs. time becomes a straight line.  
 
<br>
 
<br>
 
The next step is to select the appropriate MBH chart for the drainage area shape and well location being evaluated. Because the example well is centered in a 2×1 rectangle, choose '''Fig. 8.67'''. On this chart, enter the graph at a dimensionless producing time of 0.35, as illustrated in '''Fig. 8.70''', and read across to find the dimensionless pressure, ''p''<sub>MBHD</sub>, which has a value of 2.05.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0781 Image 0001.png|'''Fig. 8.70 – Entering chart at producing time yields MBH pressure.'''
 
File:vol5 Page 0781 Image 0001.png|'''Fig. 8.70 – Entering chart at producing time yields MBH pressure.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The next step is to calculate the average reservoir pressure, [[File:Vol5 page 0781 inline 001.png]]. From rearrangement of '''Eq. 8.137''',
+
The next step is to calculate the average reservoir pressure, [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]]. From rearrangement of '''Eq. 8.137''',<br/><br/>[[File:Vol5 page 0781 eq 001.png|RTENOTITLE]]....................(8.139)<br/><br/>In this case, the extrapolated ''p''* = 2,689.4 psi, the slope of the MTR = 26.7, and the dimensionless pressure= 2.05. Thus,<br/><br/>[[File:Vol5 page 0781 eq 002.png|RTENOTITLE]]<br/><br/>'''''Ramey-Cobb Method.''''' The Ramey-Cobb method<ref name="r24">_</ref> also uses information from a Horner plot of buildup test data. After determining permeability from the Horner plot, dimensionless producing time, ''t''<sub>''p''AD</sub>, can be calculated.<br/><br/>The third step differs from the MBH method in that the Dietz shape factors, ''C''<sub>''A''</sub>, from '''Table 8.A-1''' for the drainage-area shape and well location that best describes the tested well are used. (For the physical significance of the shape factor, see Ramey and Cobb. <ref name="r24">_</ref>) For the example well, the drainage area is a 2 × 1 rectangle, and the shape factor is 21.8369. Ramey and Cobb found a relationship between shape factor and the HTR at which the pressure on the MTR is current average drainage area pressure, [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]]. The relationship is<br/><br/>[[File:Vol5 page 0781 eq 003.png|RTENOTITLE]]....................(8.140)<br/><br/>In the example test, the dimensionless producing time is 0.35, so the HTR that corresponds to the average reservoir pressure is 7.63.<br/><br/>[[File:Vol5 page 0782 eq 001.png|RTENOTITLE]]<br/><br/>Enter the Horner plot at a HTR of 7.63, read up to the extrapolated MTR straight line, then read across to the vertical axis. The resulting average reservoir pressure is 2,665.8 ('''Fig. 8.71'''). The result, for practical purposes, is identical to the result obtained using the MBH method.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0781 eq 001.png]]....................(8.139)
 
<br>
 
<br>
 
In this case, the extrapolated ''p''* = 2,689.4 psi, the slope of the MTR = 26.7, and the dimensionless pressure= 2.05. Thus,
 
<br>
 
<br>
 
[[File:Vol5 page 0781 eq 002.png]]
 
<br>
 
<br>
 
'''''Ramey-Cobb Method.''''' The Ramey-Cobb method<ref name="r24" /> also uses information from a Horner plot of buildup test data. After determining permeability from the Horner plot, dimensionless producing time, ''t''<sub>''p''AD</sub>, can be calculated.  
 
<br>
 
<br>
 
The third step differs from the MBH method in that the Dietz shape factors, ''C''<sub>''A''</sub>, from '''Table 8.A-1''' for the drainage-area shape and well location that best describes the tested well are used. (For the physical significance of the shape factor, see Ramey and Cobb. <ref name="r24" />) For the example well, the drainage area is a 2 × 1 rectangle, and the shape factor is 21.8369. Ramey and Cobb found a relationship between shape factor and the HTR at which the pressure on the MTR is current average drainage area pressure, [[File:Vol5 page 0781 inline 001.png]]. The relationship is
 
<br>
 
<br>
 
[[File:Vol5 page 0781 eq 003.png]]....................(8.140)
 
<br>
 
<br>
 
In the example test, the dimensionless producing time is 0.35, so the HTR that corresponds to the average reservoir pressure is 7.63.
 
<br>
 
<br>
 
[[File:Vol5 page 0782 eq 001.png]]
 
<br>
 
<br>
 
Enter the Horner plot at a HTR of 7.63, read up to the extrapolated MTR straight line, then read across to the vertical axis. The resulting average reservoir pressure is 2,665.8 ('''Fig. 8.71'''). The result, for practical purposes, is identical to the result obtained using the MBH method.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0782 Image 0001.png|'''Fig. 8.71 – Ramey-Cobb results replicate those of MBH method.'''
 
File:vol5 Page 0782 Image 0001.png|'''Fig. 8.71 – Ramey-Cobb results replicate those of MBH method.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The MBH and Ramey-Cobb methods use only data in the MTR. Once enough data is available to identify the MTR, the test can be terminated, which reduces test costs. The disadvantages of these methods are the need to know the drainage area size, shape, location of the well within that drainage area, and an accurate measurement of fluid properties. In the MBH method, the well can be in transient flow at the time of shut-in, but in the Ramey-Cobb method, the well must have reached pseudosteady state before shut-in. Results for the two methods should be identical, because they are based on the same theory. When it is applicable (pseudosteady state before shut-in), the Ramey-Cobb method is preferred because it is easier to apply.  
+
The MBH and Ramey-Cobb methods use only data in the MTR. Once enough data is available to identify the MTR, the test can be terminated, which reduces test costs. The disadvantages of these methods are the need to know the drainage area size, shape, location of the well within that drainage area, and an accurate measurement of fluid properties. In the MBH method, the well can be in transient flow at the time of shut-in, but in the Ramey-Cobb method, the well must have reached pseudosteady state before shut-in. Results for the two methods should be identical, because they are based on the same theory. When it is applicable (pseudosteady state before shut-in), the Ramey-Cobb method is preferred because it is easier to apply.
  
 
=== Late-Time Region Methods ===
 
=== Late-Time Region Methods ===
  
Methods using LTR data are based on extrapolation of the post-middle-time region data trend. The advantages of these methods are that they require neither accurate fluid property estimates nor the drainage area size and shape. They do require that the well be reasonably centered within its drainage area. The disadvantage is that they require the post-middle-time region transient data. Thus, they require longer and more expensive shut-in tests to provide the data required for analysis.  
+
Methods using LTR data are based on extrapolation of the post-middle-time region data trend. The advantages of these methods are that they require neither accurate fluid property estimates nor the drainage area size and shape. They do require that the well be reasonably centered within its drainage area. The disadvantage is that they require the post-middle-time region transient data. Thus, they require longer and more expensive shut-in tests to provide the data required for analysis.<br/><br/>'''Modified Muskat Method.''' The modified Muskat method is based on the theoretical observation first published by Larsen<ref name="r25">_</ref> that, for late-time data (after boundary effects have appeared), the difference between current average reservoir pressure, [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]], and shut-in BHP, ''p''<sub>''ws''</sub>, declines exponentially. In equation form,<br/><br/>[[File:Vol5 page 0782 eq 002.png|RTENOTITLE]]....................(8.141)<br/><br/>or [[File:Vol5 page 0783 eq 001.png|RTENOTITLE]]....................(8.142)<br/><br/>'''Eq. 8.142''' leads to a procedure for estimating average drainage-area pressure, [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]]. This method requires a trial-and-error approach. To select data suitable for analysis with this method, use the diagnostic plot to determine the start of boundary effects. Then assume a value for [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]], and plot log[[File:Vol5 page 0783 inline 001.png|RTENOTITLE]] vs. time. If the curve is concave downward, the assumed pressure is too low; if the curve is concave upward, the assumed pressure is too high. Try different values for [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]] until the graph is a straight line, as predicted by theory.<br/><br/>On the example ('''Fig. 8.72'''), once the data begin to fall on a straight line, they tend to remain on that straight line. Shown are curves for assumed values of [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]] = 5,600; 5,575; and 5,560. On the first curve, for [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]] = 5,600, the final data points are trending above the straight line. For the lower curve, with [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]] = 5,560, the last few data points are trending below the straight line. For the assumed value [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]] = 5,575, all of the data points fall on a straight line making this assumption the right estimate of [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]]. The advantage to this method is that it is very easy to apply. It works best with a well reasonably centered within a drainage area.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
'''Modified Muskat Method.''' The modified Muskat method is based on the theoretical observation first published by Larsen<ref name="r25" /> that, for late-time data (after boundary effects have appeared), the difference between current average reservoir pressure, [[File:Vol5 page 0781 inline 001.png]], and shut-in BHP, ''p''<sub>''ws''</sub>, declines exponentially. In equation form,
 
<br>
 
<br>
 
[[File:Vol5 page 0782 eq 002.png]]....................(8.141)
 
<br>
 
<br>
 
or
 
[[File:Vol5 page 0783 eq 001.png]]....................(8.142)
 
<br>
 
<br>
 
'''Eq. 8.142''' leads to a procedure for estimating average drainage-area pressure, [[File:Vol5 page 0781 inline 001.png]]. This method requires a trial-and-error approach. To select data suitable for analysis with this method, use the diagnostic plot to determine the start of boundary effects. Then assume a value for [[File:Vol5 page 0781 inline 001.png]], and plot log[[File:Vol5 page 0783 inline 001.png]] vs. time. If the curve is concave downward, the assumed pressure is too low; if the curve is concave upward, the assumed pressure is too high. Try different values for [[File:Vol5 page 0781 inline 001.png]] until the graph is a straight line, as predicted by theory.  
 
<br>
 
<br>
 
On the example ('''Fig. 8.72'''), once the data begin to fall on a straight line, they tend to remain on that straight line. Shown are curves for assumed values of [[File:Vol5 page 0781 inline 001.png]] = 5,600; 5,575; and 5,560. On the first curve, for [[File:Vol5 page 0781 inline 001.png]] = 5,600, the final data points are trending above the straight line. For the lower curve, with [[File:Vol5 page 0781 inline 001.png]] = 5,560, the last few data points are trending below the straight line. For the assumed value [[File:Vol5 page 0781 inline 001.png]] = 5,575, all of the data points fall on a straight line making this assumption the right estimate of [[File:Vol5 page 0781 inline 001.png]]. The advantage to this method is that it is very easy to apply. It works best with a well reasonably centered within a drainage area.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0783 Image 0001.png|'''Fig. 8.72 – Trial-and-error plot identifies best value of average pressure.'''
 
File:vol5 Page 0783 Image 0001.png|'''Fig. 8.72 – Trial-and-error plot identifies best value of average pressure.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The weaknesses of this method are that it is more sensitive to estimates that are too low rather than to estimates that are too high and that it is not easily automated and, therefore, not as widely incorporated into well-test analysis software as some other methods.  
+
The weaknesses of this method are that it is more sensitive to estimates that are too low rather than to estimates that are too high and that it is not easily automated and, therefore, not as widely incorporated into well-test analysis software as some other methods.<br/><br/>'''''Arps-Smith Method.''''' This is an alternative method for analyzing LTR data. <ref name="r26">_</ref> The theoretical basis for this originally empirical method is also '''Eq. 8.141'''. Differentiating '''Eq. 8.141''' with respect to time,<br/><br/>[[File:Vol5 page 0783 eq 002.png|RTENOTITLE]]....................(8.143)<br/><br/>To apply this method, plot the change in BHP with time, ''dp''<sub>''ws''</sub>/''dt'' vs. ''p''<sub>''ws''</sub>, on Cartesian coordinates. On such a plot, data for the LTR should fall on a straight line, and extrapolation of that line to ''dp''<sub>''ws''</sub>/''dt'' = 0 provides an estimate of the average drainage area pressure, [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]].<br/><br/>In '''Fig. 8.73''', the final points from an example test fall on a straight line. Extrapolating the straight line to the horizontal axis gives the average pressure at the intercept. For this example, which is the same test illustrated with the modified Muskat method, the average pressure is 5,575 psi, which is the same value found with the Muskat method.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
'''''Arps-Smith Method.''''' This is an alternative method for analyzing LTR data. <ref name="r26" /> The theoretical basis for this originally empirical method is also '''Eq. 8.141'''. Differentiating '''Eq. 8.141''' with respect to time,
 
<br>
 
<br>
 
[[File:Vol5 page 0783 eq 002.png]]....................(8.143)
 
<br>
 
<br>
 
To apply this method, plot the change in BHP with time, ''dp''<sub>''ws''</sub>/''dt'' vs. ''p''<sub>''ws''</sub>, on Cartesian coordinates. On such a plot, data for the LTR should fall on a straight line, and extrapolation of that line to ''dp''<sub>''ws''</sub>/''dt'' = 0 provides an estimate of the average drainage area pressure, [[File:Vol5 page 0781 inline 001.png]].  
 
<br>
 
<br>
 
In '''Fig. 8.73''', the final points from an example test fall on a straight line. Extrapolating the straight line to the horizontal axis gives the average pressure at the intercept. For this example, which is the same test illustrated with the modified Muskat method, the average pressure is 5,575 psi, which is the same value found with the Muskat method.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0784 Image 0001.png|'''Fig. 8.73 – Extrapolation of straight line to ''x''-axis yields average pressure.'''
 
File:vol5 Page 0784 Image 0001.png|'''Fig. 8.73 – Extrapolation of straight line to ''x''-axis yields average pressure.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The advantages of the Arps-Smith method are that it is simple to apply and easily automated (which means that it is easily implemented into well-test analysis software or into spreadsheets). The disadvantages are that it requires data in the LTR, which means that it requires longer, more expensive tests. It assumes that shut-in pressure approaches average pressure exponentially, which is most nearly true for wells centered in the drainage area, and it requires numerical differentiation of pressure with respect to time, which tends to magnify any noise that may be present in the data.  
+
The advantages of the Arps-Smith method are that it is simple to apply and easily automated (which means that it is easily implemented into well-test analysis software or into spreadsheets). The disadvantages are that it requires data in the LTR, which means that it requires longer, more expensive tests. It assumes that shut-in pressure approaches average pressure exponentially, which is most nearly true for wells centered in the drainage area, and it requires numerical differentiation of pressure with respect to time, which tends to magnify any noise that may be present in the data.<br/><br/>The modified Muskat and Arps-Smith methods actually apply for shut-in times in the range,<br/><br/>[[File:Vol5 page 0784 eq 001.png|RTENOTITLE]]....................(8.144)<br/><br/>In '''Fig. 8.74''', the data points with darker dots are on the type curve for the derivative. These are the data in the range for which the modified Muskat and Arps-Smith methods work.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
The modified Muskat and Arps-Smith methods actually apply for shut-in times in the range,
 
<br>
 
<br>
 
[[File:Vol5 page 0784 eq 001.png]]....................(8.144)
 
<br>
 
<br>
 
In '''Fig. 8.74''', the data points with darker dots are on the type curve for the derivative. These are the data in the range for which the modified Muskat and Arps-Smith methods work.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0784 Image 0002.png|'''Fig. 8.74 – Darker dots represent data in range of Muskat, Arps-Smith methods.'''
 
File:vol5 Page 0784 Image 0002.png|'''Fig. 8.74 – Darker dots represent data in range of Muskat, Arps-Smith methods.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''Fig. 8.74''' illustrates one of the disadvantages of these two methods. Many other reservoir models will exhibit similar diagnostic plots, but data like that shown with the dark dots in this figure will not extrapolate to the correct average reservoir drainage area pressure. Examples of these other cases include dual-porosity reservoirs during the early transition from fracture flow to total system flow, layered reservoirs, and composite reservoirs with an inner zone mobility much lower than the outer zone mobility.  
+
'''Fig. 8.74''' illustrates one of the disadvantages of these two methods. Many other reservoir models will exhibit similar diagnostic plots, but data like that shown with the dark dots in this figure will not extrapolate to the correct average reservoir drainage area pressure. Examples of these other cases include dual-porosity reservoirs during the early transition from fracture flow to total system flow, layered reservoirs, and composite reservoirs with an inner zone mobility much lower than the outer zone mobility.<br/>
<br>
+
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<br>
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
 
 
== Hydraulically Fractured Wells ==
 
== Hydraulically Fractured Wells ==
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
<br>
+
<br/>Many wells—particularly gas wells in low-permeability formations—require hydraulic fracturing to be commercially viable. Interpretation of pressure-transient data in hydraulically fractured wells is important for evaluating the success of fracture treatments and predicting the future performance of fractured wells. This section includes graphical techniques for analyzing post-fracture pressure transient tests after identifying several flow patterns that are characteristic of hydraulically fractured wells. Often, identification of specific flow patterns can aid in well test analysis.
Many wells—particularly gas wells in low-permeability formations—require hydraulic fracturing to be commercially viable. Interpretation of pressure-transient data in hydraulically fractured wells is important for evaluating the success of fracture treatments and predicting the future performance of fractured wells. This section includes graphical techniques for analyzing post-fracture pressure transient tests after identifying several flow patterns that are characteristic of hydraulically fractured wells. Often, identification of specific flow patterns can aid in well test analysis.  
 
  
 
=== Flow Patterns in Hydraulically Fractured Wells ===
 
=== Flow Patterns in Hydraulically Fractured Wells ===
  
Five distinct flow patterns ('''Fig. 8.75''') occur in the fracture and formation around a hydraulically fractured well. <ref name="r27" /> Successive flow patterns, which often are separated by transition periods, include fracture linear, bilinear, formation linear, elliptical, and pseudoradial flow. Fracture linear flow ('''Fig. 8.75a''') is very short-lived and may be masked by wellbore-storage effects. During this flow period, most of the fluid entering the wellbore comes from fluid expansion in the fracture, and the flow pattern is essentially linear.  
+
Five distinct flow patterns ('''Fig. 8.75''') occur in the fracture and formation around a hydraulically fractured well. <ref name="r27">_</ref> Successive flow patterns, which often are separated by transition periods, include fracture linear, bilinear, formation linear, elliptical, and pseudoradial flow. Fracture linear flow ('''Fig. 8.75a''') is very short-lived and may be masked by wellbore-storage effects. During this flow period, most of the fluid entering the wellbore comes from fluid expansion in the fracture, and the flow pattern is essentially linear.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0786 Image 0001.png|'''Fig. 8.75 - Flow periods in a vertically fractured well.<ref name="r27" />'''
 
File:vol5 Page 0786 Image 0001.png|'''Fig. 8.75 - Flow periods in a vertically fractured well.<ref name="r27" />'''
 
</gallery>
 
</gallery>
<br/>
 
  
Because of its extremely short duration, the fracture linear flow period often is of no practical use in well test analysis. The duration of the fracture linear flow period is estimated by<ref name="r27" />
+
Because of its extremely short duration, the fracture linear flow period often is of no practical use in well test analysis. The duration of the fracture linear flow period is estimated by<ref name="r27">_</ref><br/><br/>[[File:Vol5 page 0785 eq 001.png|RTENOTITLE]]....................(8.145)<br/><br/>where ''t''<sub>''LfD''</sub> is dimensionless time in terms of fracture half-length,<br/><br/>[[File:Vol5 page 0785 eq 002.png|RTENOTITLE]]....................(8.146)<br/><br/>The dimensionless fracture conductivity, ''C''<sub>''r''</sub>, is<br/><br/>[[File:Vol5 page 0785 eq 003.png|RTENOTITLE]]....................(8.147)<br/><br/>and ''η''<sub>''fD''</sub> is dimensionless hydraulic diffusivity defined by<br/><br/>[[File:Vol5 page 0785 eq 004.png|RTENOTITLE]]....................(8.148)<br/><br/>Bilinear flow ('''Fig. 8.75b''') evolves only in finite-conductivity fractures as fluid in the surrounding formation flows linearly into the fracture and before fracture tip effects begin to influence well behavior. Fractures are considered to be finite conductivity when ''C''<sub>''r''</sub> < 100. Most of the fluid entering the wellbore during this flow period comes from the formation. During the bilinear flow period, BHP, ''p''<sub>''wf''</sub>, is a linear function of t<sub>1/4</sub> on Cartesian coordinates.<br/><br/>A log-log plot of (''p''<sub>''i''</sub> – ''p''<sub>''wf''</sub>) as a function of time exhibits a slope of 1/4 unless the fracture is damaged. The pressure derivative also has a slope of 1/4 during this same time period. The duration of bilinear flow depends on dimensionless fracture conductivity and is given by '''Eqs. 8.149a''' through '''8.149c'''<ref name="r27">_</ref> for a range of dimensionless times and fracture conductivities:<br/><br/>[[File:Vol5 page 0786 eq 001.png|RTENOTITLE]]....................(8.149a)<br/><br/>[[File:Vol5 page 0786 eq 002.png|RTENOTITLE]]....................(8.149b)<br/><br/>and [[File:Vol5 page 0786 eq 003.png|RTENOTITLE]]....................(8.149c)<br/><br/>Formation linear flow ('''Fig. 8.75c''') occurs only in high-conductivity (''C''<sub>''r''</sub> ≥ 100) fractures. This period continues to a dimensionless time of ''t''<sub>''LfD''</sub> ≅ 0.016. The transition from fracture linear flow to formation linear flow is complete by a time of ''t''<sub>''LfD''</sub> = 10<sup>–4</sup>. On Cartesian coordinates, ''p''<sub>''wf''</sub> is a linear function of t<sup>1/2</sup>, and a log-log plot of (''p''<sub>''i''</sub> – ''p''<sub>''wf''</sub>) has a slope of 1/2 unless the fracture is damaged. The pressure derivative plot exhibits a slope of 1/2. Elliptical flow ('''Fig. 8.75d''') is a transitional flow period that occurs between a linear or near-linear flow pattern at early times and a radial or near—radial flow pattern at late times.<br/><br/>Pseudoradial flow ('''Fig. 8.75e''') occurs with fractures of all conductivities. After a sufficiently long flow period, the fracture appears to the reservoir as an expanded wellbore (consistent with the effective wellbore radius concept suggested by Prats ''et al.''<ref name="r10">_</ref>). At this time, the drainage pattern can be considered as a circle for practical purposes. (The larger the fracture conductivity, the later the development of an essentially radial drainage pattern.) If the fracture length is large relative to the drainage area, then boundary effects distort or entirely mask the pseudoradial flow regime. Pseudoradial flow begins at ''t''<sub>''LfD''</sub> ≅ 3 for high-conductivity fractures (''C''<sub>''r''</sub> ≥ 100) and at slightly smaller values of ''t''<sub>''LfD''</sub> for lower values of ''C''<sub>''r''</sub>.<br/><br/>These flow patterns also appear in pressure-buildup tests and occur at approximately the same dimensionless times as in flow tests. The physical interpretation is that the pressure has built up to an essentially uniform value throughout a particular region at a given time during a buildup test. For example, at a given time during bilinear or formation linear flow, pressure has built up to a uniform level throughout an approximately rectangular region around the fracture. At a later time during elliptical flow, pressure has built up to a uniform level throughout an approximately elliptical region centered at the wellbore. At a given time during pseudoradial flow, pressure has built up to a uniform level throughout an approximately circular region centered at the wellbore. The area of the region and the pressure level within that area increase with increasing shut-in time. '''Example 8.1''' illustrates how to estimate the duration of flow periods for hydraulically fractured wells.
<br>
+
 
<br>
+
----
[[File:Vol5 page 0785 eq 001.png]]....................(8.145)
+
 
<br>
+
<br/>'''Example 8.1: Estimating Duration of Flow Periods in a Hydraulically Fractured Well''' For each case, estimate the end of the linear flow period and the time at which pseudoradial flow period begins. Assume that pseudoradial flow begins when ''t''<sub>''LfD''</sub> = 3. '''Table 8.2''' gives the data for each case.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
where ''t''<sub>''LfD''</sub> is dimensionless time in terms of fracture half-length,
 
<br>
 
<br>
 
[[File:Vol5 page 0785 eq 002.png]]....................(8.146)
 
<br>
 
<br>
 
The dimensionless fracture conductivity, ''C''<sub>''r''</sub>, is
 
<br>
 
<br>
 
[[File:Vol5 page 0785 eq 003.png]]....................(8.147)
 
<br>
 
<br>
 
and ''η''<sub>''fD''</sub> is dimensionless hydraulic diffusivity defined by
 
<br>
 
<br>
 
[[File:Vol5 page 0785 eq 004.png]]....................(8.148)
 
<br>
 
<br>
 
Bilinear flow ('''Fig. 8.75b''') evolves only in finite-conductivity fractures as fluid in the surrounding formation flows linearly into the fracture and before fracture tip effects begin to influence well behavior. Fractures are considered to be finite conductivity when ''C''<sub>''r''</sub> < 100. Most of the fluid entering the wellbore during this flow period comes from the formation. During the bilinear flow period, BHP, ''p''<sub>''wf''</sub>, is a linear function of t<sub>1/4</sub> on Cartesian coordinates.  
 
<br>
 
<br>
 
A log-log plot of (''p''<sub>''i''</sub> – ''p''<sub>''wf''</sub>) as a function of time exhibits a slope of 1/4 unless the fracture is damaged. The pressure derivative also has a slope of 1/4 during this same time period. The duration of bilinear flow depends on dimensionless fracture conductivity and is given by '''Eqs. 8.149a''' through '''8.149c'''<ref name="r27" /> for a range of dimensionless times and fracture conductivities:
 
<br>
 
<br>
 
[[File:Vol5 page 0786 eq 001.png]]....................(8.149a)
 
<br>
 
<br>
 
[[File:Vol5 page 0786 eq 002.png]]....................(8.149b)
 
<br>
 
<br>
 
and [[File:Vol5 page 0786 eq 003.png]]....................(8.149c)
 
<br>
 
<br>
 
Formation linear flow ('''Fig. 8.75c''') occurs only in high-conductivity (''C''<sub>''r''</sub> ≥ 100) fractures. This period continues to a dimensionless time of ''t''<sub>''LfD''</sub> ≅ 0.016. The transition from fracture linear flow to formation linear flow is complete by a time of ''t''<sub>''LfD''</sub> = 10<sup>–4</sup>. On Cartesian coordinates, ''p''<sub>''wf''</sub> is a linear function of t<sup>1/2</sup>, and a log-log plot of (''p''<sub>''i''</sub> – ''p''<sub>''wf''</sub>) has a slope of 1/2 unless the fracture is damaged. The pressure derivative plot exhibits a slope of 1/2. Elliptical flow ('''Fig. 8.75d''') is a transitional flow period that occurs between a linear or near-linear flow pattern at early times and a radial or near—radial flow pattern at late times.  
 
<br>
 
<br>
 
Pseudoradial flow ('''Fig. 8.75e''') occurs with fractures of all conductivities. After a sufficiently long flow period, the fracture appears to the reservoir as an expanded wellbore (consistent with the effective wellbore radius concept suggested by Prats ''et al.''<ref name="r10" />). At this time, the drainage pattern can be considered as a circle for practical purposes. (The larger the fracture conductivity, the later the development of an essentially radial drainage pattern.) If the fracture length is large relative to the drainage area, then boundary effects distort or entirely mask the pseudoradial flow regime. Pseudoradial flow begins at ''t''<sub>''LfD''</sub> ≅ 3 for high-conductivity fractures (''C''<sub>''r''</sub> ≥ 100) and at slightly smaller values of ''t''<sub>''LfD''</sub> for lower values of ''C''<sub>''r''</sub>.  
 
<br>
 
<br>
 
These flow patterns also appear in pressure-buildup tests and occur at approximately the same dimensionless times as in flow tests. The physical interpretation is that the pressure has built up to an essentially uniform value throughout a particular region at a given time during a buildup test. For example, at a given time during bilinear or formation linear flow, pressure has built up to a uniform level throughout an approximately rectangular region around the fracture. At a later time during elliptical flow, pressure has built up to a uniform level throughout an approximately elliptical region centered at the wellbore. At a given time during pseudoradial flow, pressure has built up to a uniform level throughout an approximately circular region centered at the wellbore. The area of the region and the pressure level within that area increase with increasing shut-in time. '''Example 8.1''' illustrates how to estimate the duration of flow periods for hydraulically fractured wells.
 
<br>
 
----  
 
<br>
 
'''Example 8.1: Estimating Duration of Flow Periods in a Hydraulically Fractured Well'''
 
For each case, estimate the end of the linear flow period and the time at which pseudoradial flow period begins. Assume that pseudoradial flow begins when ''t''<sub>''LfD''</sub> = 3. '''Table 8.2''' gives the data for each case.
 
<br>
 
<br>
 
<gallery widths=300px heights=200px>
 
 
File:Vol5 Page 0787 Image 0001.png|'''Table 8.2'''
 
File:Vol5 Page 0787 Image 0001.png|'''Table 8.2'''
 
</gallery>
 
</gallery>
<br>
 
  
''Solution''. The end of the linear flow regime occurs at a dimensionless time of ''t''<sub>''LfD''</sub> ≅ 0.016 or, using '''Eq. 8.146''',
+
''Solution''. The end of the linear flow regime occurs at a dimensionless time of ''t''<sub>''LfD''</sub> ≅ 0.016 or, using '''Eq. 8.146''',<br/><br/>[[File:Vol5 page 0787 eq 001.png|RTENOTITLE]]<br/><br/>Similarly, the time to reach pseudoradial flow is ''t''<sub>''LfD''</sub> ≅ 3, or<br/><br/>[[File:Vol5 page 0787 eq 002.png|RTENOTITLE]]<br/><br/>'''Table 8.2''' summarizes the results.
<br>
+
 
<br>
 
[[File:Vol5 page 0787 eq 001.png]]
 
<br>
 
<br>
 
Similarly, the time to reach pseudoradial flow is ''t''<sub>''LfD''</sub> ≅ 3, or
 
<br>
 
<br>
 
[[File:Vol5 page 0787 eq 002.png]]
 
<br>
 
<br>
 
'''Table 8.2''' summarizes the results.
 
<br>
 
 
----
 
----
<br>
+
 
 +
 
  
 
=== Flow Geometry and Depth of Investigation of a Vertically Fractured Well ===
 
=== Flow Geometry and Depth of Investigation of a Vertically Fractured Well ===
  
Fluid flow in a vertically fractured well has been described using elliptical geometry. <ref name="r28" /> The equation for an ellipse with its major axis along the ''x''-axis and minor axis along the ''y''-axis is
+
Fluid flow in a vertically fractured well has been described using elliptical geometry. <ref name="r28">_</ref> The equation for an ellipse with its major axis along the ''x''-axis and minor axis along the ''y''-axis is<br/><br/>[[File:Vol5 page 0788 eq 001.png|RTENOTITLE]]....................(8.150)<br/><br/>where the endpoints of the major and minor axes are (±''a''<sub>''f''</sub>, 0) and (0, ±''b''<sub>''f''</sub>), respectively. The foci of the ellipse are ±''c''<sub>''f''</sub> where ''c''<sub>''f''</sub><sup>2</sup> = ''a''<sub>''f''</sub><sup>2</sup> – ''b''<sub>''f''</sub><sup>2</sup>. In terms of a well with a single vertical fracture with two wings of equal length, ''L''<sub>''f''</sub>, the relation becomes ''L''<sub>''f''</sub><sup>2</sup> = ''a''<sub>''f''</sub><sup>2</sup> – ''b''<sub>''f''</sub><sup>2</sup>, where ''L''<sub>''f''</sub> is the focal length of the ellipse. '''Fig. 8.76''' shows the elliptical geometry of a vertically fractured well.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0788 eq 001.png]]....................(8.150)
 
<br>
 
<br>
 
where the endpoints of the major and minor axes are (±''a''<sub>''f''</sub>, 0) and (0, ±''b''<sub>''f''</sub>), respectively. The foci of the ellipse are ±''c''<sub>''f''</sub> where ''c''<sub>''f''</sub><sup>2</sup> = ''a''<sub>''f''</sub><sup>2</sup> – ''b''<sub>''f''</sub><sup>2</sup>. In terms of a well with a single vertical fracture with two wings of equal length, ''L''<sub>''f''</sub>, the relation becomes ''L''<sub>''f''</sub><sup>2</sup> = ''a''<sub>''f''</sub><sup>2</sup> – ''b''<sub>''f''</sub><sup>2</sup>, where ''L''<sub>''f''</sub> is the focal length of the ellipse. '''Fig. 8.76''' shows the elliptical geometry of a vertically fractured well.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0788 Image 0001.png|'''Fig. 8.76 – Elliptical flow pattern around a vertically fractured well.'''
 
File:vol5 Page 0788 Image 0001.png|'''Fig. 8.76 – Elliptical flow pattern around a vertically fractured well.'''
 
</gallery>
 
</gallery>
<br/>
 
  
Hale and Evers<ref name="r28" /> defined a depth of investigation for a vertically fractured well. Their definition is based on a definition of dimensionless time at a distance ''b''<sub>''f''</sub>, the length of the minor axis:
+
Hale and Evers<ref name="r28">_</ref> defined a depth of investigation for a vertically fractured well. Their definition is based on a definition of dimensionless time at a distance ''b''<sub>''f''</sub>, the length of the minor axis:<br/><br/>[[File:Vol5 page 0788 eq 002.png|RTENOTITLE]]....................(8.151)<br/><br/>Solving for the length of the minor axis,<br/><br/>[[File:Vol5 page 0788 eq 003.png|RTENOTITLE]]....................(8.152)<br/><br/>Assuming that pseudosteady-state flow exists out to distance, ''b''<sub>''f''</sub>, at dimensionless time ''t''<sub>''bD''</sub> = 1/''π'' as in linear systems, '''Eq. 8.152''' becomes<br/><br/>[[File:Vol5 page 0788 eq 004.png|RTENOTITLE]]....................(8.153)<br/><br/>which represents the depth of investigation in a direction perpendicular to the fracture at time, ''t'', for a vertically fractured well. In gas wells, the terms ''μ'' and ''c''<sub>''t''</sub> should be [[File:Vol5 page 0789 inline 001.png|RTENOTITLE]] and [[File:Vol5 page 0789 inline 002.png|RTENOTITLE]], evaluated at average drainage-area pressure, [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]].<br/><br/>The elliptical pattern of the propagating pressure transient can be fully described in terms of the lengths of the major axis, ''a''<sub>''f''</sub>, the minor axis, ''b''<sub>''f''</sub>, and the focus, ''L''<sub>''f''</sub>. Using the estimate of ''b''<sub>''f''</sub> from '''Eq. 8.153''' and an estimate of ''L''<sub>''f''</sub> obtained by one of the methods described in sections that follow, the length of the major axis can be estimated from<br/><br/>[[File:Vol5 page 0789 eq 001.png|RTENOTITLE]]....................(8.154)<br/><br/>Given values of ''a''<sub>''f''</sub> and ''b''<sub>''f''</sub>, the depth of investigation at a particular time, ''t'', in any direction from the fracture can be calculated using '''Eq. 8.150'''. Furthermore, the area, ''A'', enclosed by the ellipse at time, t (the area of the reservoir sampled by the pressure transient), is given by<br/><br/>[[File:Vol5 page 0789 eq 002.png|RTENOTITLE]]....................(8.155)<br/><br/>The coefficient 0.0002878 in '''Eq. 8.153''' is strictly correct only for highly conductive fractures (''C''<sub>''r''</sub> ≥ 100). As ''C''<sub>''r''</sub> becomes smaller, the ratio ''a''<sub>''f''</sub>/''b''<sub>''f''</sub> also becomes smaller. The lower bound of ''a''<sub>''f''</sub>/''b''<sub>''f''</sub> is 1 (a circle) as ''C''<sub>''r''</sub> approaches 0.
<br>
 
<br>
 
[[File:Vol5 page 0788 eq 002.png]]....................(8.151)
 
<br>
 
<br>
 
Solving for the length of the minor axis,
 
<br>
 
<br>
 
[[File:Vol5 page 0788 eq 003.png]]....................(8.152)
 
<br>
 
<br>
 
Assuming that pseudosteady-state flow exists out to distance, ''b''<sub>''f''</sub>, at dimensionless time ''t''<sub>''bD''</sub> = 1/''π'' as in linear systems, '''Eq. 8.152''' becomes
 
<br>
 
<br>
 
[[File:Vol5 page 0788 eq 004.png]]....................(8.153)
 
<br>
 
<br>
 
which represents the depth of investigation in a direction perpendicular to the fracture at time, ''t'', for a vertically fractured well. In gas wells, the terms ''μ'' and ''c''<sub>''t''</sub> should be [[File:Vol5 page 0789 inline 001.png]] and [[File:Vol5 page 0789 inline 002.png]], evaluated at average drainage-area pressure, [[File:Vol5 page 0781 inline 001.png]].  
 
<br>
 
<br>
 
The elliptical pattern of the propagating pressure transient can be fully described in terms of the lengths of the major axis, ''a''<sub>''f''</sub>, the minor axis, ''b''<sub>''f''</sub>, and the focus, ''L''<sub>''f''</sub>. Using the estimate of ''b''<sub>''f''</sub> from '''Eq. 8.153''' and an estimate of ''L''<sub>''f''</sub> obtained by one of the methods described in sections that follow, the length of the major axis can be estimated from
 
<br>
 
<br>
 
[[File:Vol5 page 0789 eq 001.png]]....................(8.154)
 
<br>
 
<br>
 
Given values of ''a''<sub>''f''</sub> and ''b''<sub>''f''</sub>, the depth of investigation at a particular time, ''t'', in any direction from the fracture can be calculated using '''Eq. 8.150'''. Furthermore, the area, ''A'', enclosed by the ellipse at time, t (the area of the reservoir sampled by the pressure transient), is given by
 
<br>
 
<br>
 
[[File:Vol5 page 0789 eq 002.png]]....................(8.155)
 
<br>
 
<br>
 
The coefficient 0.0002878 in '''Eq. 8.153''' is strictly correct only for highly conductive fractures (''C''<sub>''r''</sub> ≥ 100). As ''C''<sub>''r''</sub> becomes smaller, the ratio ''a''<sub>''f''</sub>/''b''<sub>''f''</sub> also becomes smaller. The lower bound of ''a''<sub>''f''</sub>/''b''<sub>''f''</sub> is 1 (a circle) as ''C''<sub>''r''</sub> approaches 0.  
 
  
 
=== Fracture Damage ===
 
=== Fracture Damage ===
  
Two major types of fracture damage are frequent: choked fracture damage and fracture-face damage. The choked-fracture damage means that the fracture has a reduced permeability in the immediate vicinity of the wellbore ('''Fig. 8.77'''). In this case, ''k''<sub>''f''</sub> is used for the permeability in the propped portion of the fracture farther along the wellbore, and ''k''<sub>''fs''</sub> for reduced permeability near the wellbore, out to a length, ''L''<sub>''s''</sub>, in the fracture.  
+
Two major types of fracture damage are frequent: choked fracture damage and fracture-face damage. The choked-fracture damage means that the fracture has a reduced permeability in the immediate vicinity of the wellbore ('''Fig. 8.77'''). In this case, ''k''<sub>''f''</sub> is used for the permeability in the propped portion of the fracture farther along the wellbore, and ''k''<sub>''fs''</sub> for reduced permeability near the wellbore, out to a length, ''L''<sub>''s''</sub>, in the fracture.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0789 Image 0001.png|'''Fig. 8.77 – Permeability differs between propped and near-wellbore portions of fracture.'''
 
File:vol5 Page 0789 Image 0001.png|'''Fig. 8.77 – Permeability differs between propped and near-wellbore portions of fracture.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The choked-fracture skin factor, ''s''<sub>''f''</sub>, is<ref name="r29" />
+
The choked-fracture skin factor, ''s''<sub>''f''</sub>, is<ref name="r29">_</ref><br/><br/>[[File:Vol5 page 0789 eq 003.png|RTENOTITLE]]....................(8.156)<br/><br/>Fracture face damage in a hydraulically fractured well ('''Fig. 8.78''') is a permeability reduction around the edges of the fracture, usually caused by invasion of the fracture fluid into the formation or an adverse reaction with the fracturing fluid. The equation for fracture face skin is<ref name="r29">_</ref><br/><br/>[[File:Vol5 page 0789 eq 004.png|RTENOTITLE]]....................(8.157)<br/><br/><br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0789 eq 003.png]]....................(8.156)
 
<br>
 
<br>
 
Fracture face damage in a hydraulically fractured well ('''Fig. 8.78''') is a permeability reduction around the edges of the fracture, usually caused by invasion of the fracture fluid into the formation or an adverse reaction with the fracturing fluid. The equation for fracture face skin is<ref name="r29" />
 
<br>
 
<br>
 
[[File:Vol5 page 0789 eq 004.png]]....................(8.157)
 
<br>
 
<br>
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0790 Image 0001.png|'''Fig. 8.78 – Permeability reduction around edges of fracture represents fracture face damage.'''
 
File:vol5 Page 0790 Image 0001.png|'''Fig. 8.78 – Permeability reduction around edges of fracture represents fracture face damage.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Specialized Methods for Post-Fracture Well-Test Analysis ===
 
=== Specialized Methods for Post-Fracture Well-Test Analysis ===
  
Generally, the objectives of post-fracture pressure-transient test analysis are to assess the success of the fracture treatment and to estimate the fracture half-length, fracture conductivity, and formation permeability. Three specialized methods of analyzing these post-fracture transient tests are included in this section: pseudoradial flow, bilinear flow, and linear flow.  
+
Generally, the objectives of post-fracture pressure-transient test analysis are to assess the success of the fracture treatment and to estimate the fracture half-length, fracture conductivity, and formation permeability. Three specialized methods of analyzing these post-fracture transient tests are included in this section: pseudoradial flow, bilinear flow, and linear flow.<br/><br/>'''''Bilinear Flow Method.''''' The bilinear flow method<ref name="r30">_</ref> applies to test data obtained during the bilinear flow regime in wells with finite-conductivity vertical fractures. Bilinear flow is indicated by a quarter-slope line on a log-log graph of pressure derivative vs. ''t'' or Δ''t''<sub>''e''</sub>.<br/><br/>During bilinear flow,<br/><br/>[[File:Vol5 page 0790 eq 001.png|RTENOTITLE]]....................(8.158)<br/><br/>and [[File:Vol5 page 0790 eq 002.png|RTENOTITLE]]....................(8.159)<br/><br/>The following procedure is recommended for analyzing test data obtained in the bilinear flow regime (that is, data in the time range with quarter slope on the diagnostic plot). In Step 1, note the use of "bilinear equivalent time," Δ''t''<sub>''Be''</sub>. Radial equivalent time is rigorously correct as a plotting function only for infinite-acting radial flow.
<br>
+
 
<br>
+
#For a constant-rate flow test, plot ''p''<sub>''wf''</sub> vs. ''t''<sup>1/4</sup> on Cartesian coordinates. For a buildup test, plot ''p''<sub>''ws''</sub> vs. Δ''t''<sub>''Be''</sub><sup>1/4</sup>, where
'''''Bilinear Flow Method.''''' The bilinear flow method<ref name="r30" /> applies to test data obtained during the bilinear flow regime in wells with finite-conductivity vertical fractures. Bilinear flow is indicated by a quarter-slope line on a log-log graph of pressure derivative vs. ''t'' or Δ''t''<sub>''e''</sub>.  
+
#:[[File:Vol5 page 0790 eq 003.png|RTENOTITLE]]....................(8.160)
<br>
 
<br>
 
During bilinear flow,
 
<br>
 
<br>
 
[[File:Vol5 page 0790 eq 001.png]]....................(8.158)
 
<br>
 
<br>
 
and [[File:Vol5 page 0790 eq 002.png]]....................(8.159)
 
<br>
 
<br>
 
The following procedure is recommended for analyzing test data obtained in the bilinear flow regime (that is, data in the time range with quarter slope on the diagnostic plot). In Step 1, note the use of "bilinear equivalent time," Δ''t''<sub>''Be''</sub>. Radial equivalent time is rigorously correct as a plotting function only for infinite-acting radial flow.
 
<br>
 
#For a constant-rate flow test, plot ''p''<sub>''wf''</sub> vs. ''t''<sup>1/4</sup> on Cartesian coordinates. For a buildup test, plot ''p''<sub>''ws''</sub> vs. Δ''t''<sub>''Be''</sub><sup>1/4</sup>, where  
 
#:[[File:Vol5 page 0790 eq 003.png]]....................(8.160)
 
 
#Determine the slope, ''m''<sub>''B''</sub>, of the straight line region of the plot.
 
#Determine the slope, ''m''<sub>''B''</sub>, of the straight line region of the plot.
 
#Determine the pressure extrapolated to time zero, ''p''<sub>''o''</sub>, and the fracture skin, ''s''<sub>''f''</sub>, from
 
#Determine the pressure extrapolated to time zero, ''p''<sub>''o''</sub>, and the fracture skin, ''s''<sub>''f''</sub>, from
#:[[File:Vol5 page 0790 eq 004.png]]....................(8.161)
+
#:[[File:Vol5 page 0790 eq 004.png|RTENOTITLE]]....................(8.161)
 
#:for drawdown and buildup tests, respectively.
 
#:for drawdown and buildup tests, respectively.
#From independent knowledge of ''k'' (for example, from a prefracture well test), estimate the fracture conductivity, ''w''<sub>''f''</sub>''k''<sub>''f''</sub>, using ''m''<sub>''B''</sub> and the relationship  
+
#From independent knowledge of ''k'' (for example, from a prefracture well test), estimate the fracture conductivity, ''w''<sub>''f''</sub>''k''<sub>''f''</sub>, using ''m''<sub>''B''</sub> and the relationship
#:[[File:Vol5 page 0790 eq 005.png]]....................(8.162)
+
#:[[File:Vol5 page 0790 eq 005.png|RTENOTITLE]]....................(8.162)
#:where [[File:Vol5 page 0789 inline 001.png]] and [[File:Vol5 page 0789 inline 002.png]], evaluated at [[File:Vol5 page 0781 inline 001.png]], are used for a gas well test.  
+
#:where [[File:Vol5 page 0789 inline 001.png|RTENOTITLE]] and [[File:Vol5 page 0789 inline 002.png|RTENOTITLE]], evaluated at [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]], are used for a gas well test.
<br>
+
 
 +
 
  
 
'''Fig. 8.79''' is an example of bilinear flow analysis. The bilinear flow analysis method has the following important limitations.
 
'''Fig. 8.79''' is an example of bilinear flow analysis. The bilinear flow analysis method has the following important limitations.
<br>
+
 
* No estimate of fracture half-length, ''L''<sub>''f''</sub>.
+
*No estimate of fracture half-length, ''L''<sub>''f''</sub>.
* In wells with low-conductivity fractures, wellbore storage frequently distorts early test data for a sufficient length of time so that the quarter-slope line characteristic of bilinear flow may not appear on a log-log plot of test data.
+
*In wells with low-conductivity fractures, wellbore storage frequently distorts early test data for a sufficient length of time so that the quarter-slope line characteristic of bilinear flow may not appear on a log-log plot of test data.
* An independent estimate of ''k'' is required. This suggests that prefracture well tests should be conducted before fracturing the well, thus obtaining independent estimates of formation properties.
+
*An independent estimate of ''k'' is required. This suggests that prefracture well tests should be conducted before fracturing the well, thus obtaining independent estimates of formation properties.
<br/>
+
 
<br/>
+
<br/><br/><gallery widths="300px" heights="200px">
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0791 Image 0001.png|'''Fig. 8.79 – Bilinear flow analysis.'''
 
File:vol5 Page 0791 Image 0001.png|'''Fig. 8.79 – Bilinear flow analysis.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''''Linear Flow Method.''''' The linear flow method<ref name="r30" /> applies to test data obtained during formation linear flow in wells with high-conductivity fractures (''C''<sub>''r''</sub> ≥ 100). After wellbore storage effects have ended, formation linear flow occurs up to a dimensionless time of ''t''<sub>''LfD''</sub> = 0.016, which means that a log-log plot of pressure derivative against time will have a slope of one-half. The plot of pressure change vs. time, however, will have a half-slope only if the fracture skin is zero. The pressure and pressure derivative are
+
'''''Linear Flow Method.''''' The linear flow method<ref name="r30">_</ref> applies to test data obtained during formation linear flow in wells with high-conductivity fractures (''C''<sub>''r''</sub> ≥ 100). After wellbore storage effects have ended, formation linear flow occurs up to a dimensionless time of ''t''<sub>''LfD''</sub> = 0.016, which means that a log-log plot of pressure derivative against time will have a slope of one-half. The plot of pressure change vs. time, however, will have a half-slope only if the fracture skin is zero. The pressure and pressure derivative are<br/><br/>[[File:Vol5 page 0791 eq 001.png|RTENOTITLE]]....................(8.163)<br/><br/>and [[File:Vol5 page 0791 eq 002.png|RTENOTITLE]]....................(8.164)<br/><br/>so that<br/><br/>[[File:Vol5 page 0791 eq 003.png|RTENOTITLE]]....................(8.165)<br/><br/>which indicates that a log-log plot of the derivative against time will have a slope of one-half. Radial equivalent time applies rigorously only for radial flow in an infinite-acting reservoir. When linear flow is the flow pattern occurring at both times (''t''<sub>''p''</sub> + Δ''t'') and Δ''t'', a more useful equivalent time function is the linear equivalent time, Δ''t''<sub>''eL''</sub>.<br/><br/>[[File:Vol5 page 0792 eq 001.png|RTENOTITLE]]....................(8.166)<br/><br/>Test conditions in which linear flow occurs at both (''t''<sub>''p''</sub> + Δ''t'') and Δ''t'' are rare, and, consequently, '''Eq. 8.166''' is not necessarily rigorously correct for well-test analysis. Fortunately, when ''t''<sub>''p''</sub> >> Δ''t''<sub>max</sub>, Δ''t''<sub>''eL''</sub> ≈ Δ''t''. '''Fig. 8.80''' is an example of a plot used in linear flow analysis.
<br>
 
<br>
 
[[File:Vol5 page 0791 eq 001.png]]....................(8.163)
 
<br>
 
<br>
 
and [[File:Vol5 page 0791 eq 002.png]]....................(8.164)
 
<br>
 
<br>
 
so that
 
<br>
 
<br>
 
[[File:Vol5 page 0791 eq 003.png]]....................(8.165)
 
<br>
 
<br>
 
which indicates that a log-log plot of the derivative against time will have a slope of one-half. Radial equivalent time applies rigorously only for radial flow in an infinite-acting reservoir. When linear flow is the flow pattern occurring at both times (''t''<sub>''p''</sub> + Δ''t'') and Δ''t'', a more useful equivalent time function is the linear equivalent time, Δ''t''<sub>''eL''</sub>.
 
<br>
 
<br>
 
[[File:Vol5 page 0792 eq 001.png]]....................(8.166)
 
<br>
 
<br>
 
Test conditions in which linear flow occurs at both (''t''<sub>''p''</sub> + Δ''t'') and Δ''t'' are rare, and, consequently, '''Eq. 8.166''' is not necessarily rigorously correct for well-test analysis. Fortunately, when ''t''<sub>''p''</sub> >> Δ''t''<sub>max</sub>, Δ''t''<sub>''eL''</sub> ≈ Δ''t''. '''Fig. 8.80''' is an example of a plot used in linear flow analysis.  
 
  
 
The linear flow analysis method also has limitations.
 
The linear flow analysis method also has limitations.
<br>
+
 
* The method applies only for fractures with high conductivities. Strictly speaking, linear flow occurs for the condition of uniform flux into a fracture (same flow rate from the formation per unit cross-sectional area of the fracture at all points along the fracture) rather than for infinite fracture conductivity. Therefore, only very early test data (''t''<sub>''Lf D''</sub> ≤ 0.016) exhibit linear flow in a high-conductivity fracture.
+
*The method applies only for fractures with high conductivities. Strictly speaking, linear flow occurs for the condition of uniform flux into a fracture (same flow rate from the formation per unit cross-sectional area of the fracture at all points along the fracture) rather than for infinite fracture conductivity. Therefore, only very early test data (''t''<sub>''Lf D''</sub> ≤ 0.016) exhibit linear flow in a high-conductivity fracture.
* Some or all of these early data may be distorted by wellbore storage, further limiting the amount of linear-flow data available for analysis.
+
*Some or all of these early data may be distorted by wellbore storage, further limiting the amount of linear-flow data available for analysis.
* Estimating fracture half-length requires an independent estimate of permeability, ''k'', which suggests the need for a prefracture well test.
+
*Estimating fracture half-length requires an independent estimate of permeability, ''k'', which suggests the need for a prefracture well test.
<br/>
+
 
<br/>
+
<br/><br/><gallery widths="300px" heights="200px">
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0792 Image 0001.png|'''Fig. 8.80 – Plot used in linear flow analysis.'''
 
File:vol5 Page 0792 Image 0001.png|'''Fig. 8.80 – Plot used in linear flow analysis.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''''Pseudoradial Flow Method.''''' The pseudoradial flow method applies when a short, highly conductive fracture is created in a high-permeability formation, so that pseudoradial flow develops in a short time. The time required to achieve pseudoradial flow for an infinitely conductive fracture (''C''<sub>''r''</sub> ≥ 100) in either a flow test or a pressure buildup test is estimated by
+
'''''Pseudoradial Flow Method.''''' The pseudoradial flow method applies when a short, highly conductive fracture is created in a high-permeability formation, so that pseudoradial flow develops in a short time. The time required to achieve pseudoradial flow for an infinitely conductive fracture (''C''<sub>''r''</sub> ≥ 100) in either a flow test or a pressure buildup test is estimated by<br/><br/>[[File:Vol5 page 0792 eq 002.png|RTENOTITLE]]....................(8.167)<br/><br/>The beginning of pseudoradial flow is characterized by the flattening of the pressure derivative on a log-log plot and by the start of a straight line on a semilog plot. Hence, when the pseudoradial flow regime is reached, conventional semilog analysis can be used to calculate permeability and skin factor. For a highly conductive fracture, skin factor is related to fracture half-length by<ref name="r10">_</ref><br/><br/>[[File:Vol5 page 0792 eq 003.png|RTENOTITLE]]....................(8.168)<br/><br/>'''Fig. 8.81''' shows an example.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0792 eq 002.png]]....................(8.167)
 
<br>
 
<br>
 
The beginning of pseudoradial flow is characterized by the flattening of the pressure derivative on a log-log plot and by the start of a straight line on a semilog plot. Hence, when the pseudoradial flow regime is reached, conventional semilog analysis can be used to calculate permeability and skin factor. For a highly conductive fracture, skin factor is related to fracture half-length by<ref name="r10" />
 
<br>
 
<br>
 
[[File:Vol5 page 0792 eq 003.png]]....................(8.168)
 
<br>
 
<br>
 
'''Fig. 8.81''' shows an example.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0793 Image 0001.png|'''Fig. 8.81 – Pseudoradial flow analysis.'''
 
File:vol5 Page 0793 Image 0001.png|'''Fig. 8.81 – Pseudoradial flow analysis.'''
 
</gallery>
 
</gallery>
<br/>
 
  
 
A recommended procedure for analyzing test data from the pseudoradial flow regime is as follows.
 
A recommended procedure for analyzing test data from the pseudoradial flow regime is as follows.
<br>
 
* For a drawdown test, plot ''p''<sub>''wf''</sub> vs. log ''t''. For a buildup test, plot ''p''<sub>''ws''</sub> vs. the HTR.
 
* Determine the position and slope, ''m'', of the semilog straight line and the intercept, ''p''<sub>1hr</sub> on the line.
 
* Using ''m'', calculate values of ''k'' and ''s'' (or ''s''′ for a gas well).
 
* Calculate the fracture half-length, ''L''<sub>''f''</sub>, using '''Eq. 8.168'''.
 
<br>
 
  
The pseudoradial flow method has the following limitations that seldom make it applicable in practice. <ref name="r30" />
+
*For a drawdown test, plot ''p''<sub>''wf''</sub> vs. log ''t''. For a buildup test, plot ''p''<sub>''ws''</sub> vs. the HTR.
<br>
+
*Determine the position and slope, ''m'', of the semilog straight line and the intercept, ''p''<sub>1hr</sub> on the line.
* The conditions that are most favorable for the occurrence of pseudoradial flow are short, highly conductive fractures in high-permeability formations. These formations, however, are rarely fractured. The most common application of hydraulic fractures—wells with long fractures in low-permeability formations—require impractically long test times to reach pseudoradial flow.
+
*Using ''m'', calculate values of ''k'' and ''s'' (or ''s''′ for a gas well).
* For gas wells, the apparent skin factor, ''s''′, calculated from test data is often affected by non-Darcy flow.
+
*Calculate the fracture half-length, ''L''<sub>''f''</sub>, using '''Eq. 8.168'''.
* The pseudoradial method applies only to highly conductive (''C''<sub>''r''</sub> ≥ 100) fractures. For lower conductivity fractures, fracture lengths calculated using the skin factor ('''Eq. 8.168''') will be too low.
+
 
<br>
+
 
 +
 
 +
The pseudoradial flow method has the following limitations that seldom make it applicable in practice. <ref name="r30">_</ref>
 +
 
 +
*The conditions that are most favorable for the occurrence of pseudoradial flow are short, highly conductive fractures in high-permeability formations. These formations, however, are rarely fractured. The most common application of hydraulic fractures—wells with long fractures in low-permeability formations—require impractically long test times to reach pseudoradial flow.
 +
*For gas wells, the apparent skin factor, ''s''′, calculated from test data is often affected by non-Darcy flow.
 +
*The pseudoradial method applies only to highly conductive (''C''<sub>''r''</sub> ≥ 100) fractures. For lower conductivity fractures, fracture lengths calculated using the skin factor ('''Eq. 8.168''') will be too low.
 +
 
 +
 
  
 
=== Using Type Curves for Hydraulically Fractured Wells ===
 
=== Using Type Curves for Hydraulically Fractured Wells ===
  
Type curves are the most common method of analyzing hydraulically fractured wells. The independent variable for most type curves for analyzing hydraulically fractured wells is the dimensionless time based on hydraulic fracture half-length, ''t''<sub>''Lf D''</sub>. The dependent variable is usually the dimensionless pressure, ''p''<sub>''D''</sub>.  
+
Type curves are the most common method of analyzing hydraulically fractured wells. The independent variable for most type curves for analyzing hydraulically fractured wells is the dimensionless time based on hydraulic fracture half-length, ''t''<sub>''Lf D''</sub>. The dependent variable is usually the dimensionless pressure, ''p''<sub>''D''</sub>.<br/><br/>For type curves used for manual type-curve matching, most vary only one parameter. The Cinco type curve<ref name="r27">_</ref> is obtained for zero ''C''<sub>''Lf D''</sub> and ''s''<sub>''f''</sub>&nbsp;; the only parameter is dimensionless fracture conductivity, ''C''<sub>''r''</sub> or ''F''<sub>''cD''</sub> (where ''F''<sub>''cD''</sub> = ''πc''<sub>''r''</sub>). The choked-fracture skin is analyzed by assuming ''C''<sub>''Lf D''</sub> and infinite ''C''<sub>''r''</sub> with single parameter ''s''<sub>''f''</sub>. The wellbore-storage type curve<ref name="r31">_</ref> sets ''s''<sub>''f''</sub> to 0 and ''C''<sub>''r''</sub> (''F''<sub>''cD''</sub>) to infinity and varies the coefficient ''C''<sub>''Lf D''</sub>.<br/><br/>When using type curves in commercial software, the computer can set any two of the three parameters to fixed values (other than their limiting values) and vary the third parameter to obtain the matching stems.<br/><br/>'''''Procedures for Analyzing Fractured Wells With Type Curves.''''' The following steps outline the procedure for analyzing fractured wells with type curves.
<br>
+
 
<br>
+
*Graph field data pressure change and pressure derivatives.
For type curves used for manual type-curve matching, most vary only one parameter. The Cinco type curve<ref name="r27" /> is obtained for zero ''C''<sub>''Lf D''</sub> and ''s''<sub>''f''</sub> ; the only parameter is dimensionless fracture conductivity, ''C''<sub>''r''</sub> or ''F''<sub>''cD''</sub> (where ''F''<sub>''cD''</sub> = ''πc''<sub>''r''</sub>). The choked-fracture skin is analyzed by assuming ''C''<sub>''Lf D''</sub> and infinite ''C''<sub>''r''</sub> with single parameter ''s''<sub>''f''</sub>. The wellbore-storage type curve<ref name="r31" /> sets ''s''<sub>''f''</sub> to 0 and ''C''<sub>''r''</sub> (''F''<sub>''cD''</sub>) to infinity and varies the coefficient ''C''<sub>''Lf D''</sub>.  
+
*Match field data to the appropriate type curve.
<br>
+
*Find the match point and matching stem.
<br>
+
*Calculate the formation permeability from the pressure match point.
When using type curves in commercial software, the computer can set any two of the three parameters to fixed values (other than their limiting values) and vary the third parameter to obtain the matching stems.  
+
*Calculate ''L''<sub>''f''</sub> from the time match point.
<br>
+
*Interpret the matching stem value appropriate for a given type curve. For one type curve, this can be ''w''<sub>''f''</sub>''k''<sub>''f''</sub>, which will provide an estimate of fracture conductivity. For another, it can be ''s''<sub>''f''</sub>, the choked-fracture skin, or, for a third, it can be ''C'', the wellbore-storage coefficient.
<br>
+
 
'''''Procedures for Analyzing Fractured Wells With Type Curves.''''' The following steps outline the procedure for analyzing fractured wells with type curves.
+
 
<br>
 
* Graph field data pressure change and pressure derivatives.
 
* Match field data to the appropriate type curve.
 
* Find the match point and matching stem.
 
* Calculate the formation permeability from the pressure match point.
 
* Calculate ''L''<sub>''f''</sub> from the time match point.
 
* Interpret the matching stem value appropriate for a given type curve. For one type curve, this can be ''w''<sub>''f</sub>k''<sub>''f''</sub>, which will provide an estimate of fracture conductivity. For another, it can be ''s''<sub>''f''</sub>, the choked-fracture skin, or, for a third, it can be ''C'', the wellbore-storage coefficient.
 
<br>
 
  
To interpret the match points for a test with unknown permeability, use '''Eqs. 8.169''' and '''8.170'''. The formation permeability, ''k'', is determined from the pressure match point; that is, the relationship between the pressure derivative and pressure change found at a match point given by
+
To interpret the match points for a test with unknown permeability, use '''Eqs. 8.169''' and '''8.170'''. The formation permeability, ''k'', is determined from the pressure match point; that is, the relationship between the pressure derivative and pressure change found at a match point given by<br/><br/>[[File:Vol5 page 0794 eq 001.png|RTENOTITLE]]....................(8.169)<br/><br/>From the time match point, calculate the fracture half-length:<br/><br/>[[File:Vol5 page 0794 eq 002.png|RTENOTITLE]]....................(8.170)<br/><br/>Matching can be ambiguous for hydraulically fractured wells; the data can appear to match equally well in several different positions. The ambiguity can be reduced or eliminated if a prefracture permeability is determined, and the post-fracture test data forced to match the permeability.<br/><br/>'''''Type Curves Used for Analysis in Fractured Wells.''''' The Cinco type curve ('''Fig. 8.82'''),<ref name="r27">_</ref> assumes that ''C''<sub>''Lf D''</sub> = 0 and ''s''<sub>''f''</sub> = 0. The type-curve stems on this curve are obtained by varying values of ''C''<sub>''r''</sub> or ''F''<sub>''cD''</sub>. With the Cinco type curve, the fracture conductivity, ''w''<sub>''f''</sub>''k''<sub>''f''</sub>, can be determined from the matching parameter:<br/><br/>[[File:Vol5 page 0794 eq 003.png|RTENOTITLE]]....................(8.171)<br/><br/>''Choked-Fracture Type Curve.'' '''Fig. 8.83''' shows the choked-fracture type curve. <ref name="r29">_</ref> The choked-fracture type curve is generated with wellbore-storage coefficient, ''C''<sub>''Lf D''</sub>, of zero and infinite fracture conductivity, ''C''<sub>''r''</sub>. On this type curve, the stems represent different values of the fracture skin, ''s''<sub>''f''</sub>. The fracture skin, ''s''<sub>''f''</sub>, can be used to find the additional pressure drop from<br/><br/>[[File:Vol5 page 0794 eq 004.png|RTENOTITLE]]....................(8.172)<br/><br/>''Wellbore-Storage Type Curve.'' The wellbore-storage type curve ('''Fig. 8.84''') takes into account the possibility of wellbore storage. The wellbore-storage type assumes ''s''<sub>''f''</sub> = 0 and ''C''<sub>''r''</sub> = ∞. To interpret a best-fitting stem for this type curve, use the following:<br/><br/>[[File:Vol5 page 0795 eq 001.png|RTENOTITLE]]....................(8.173)<br/><br/><br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0794 eq 001.png]]....................(8.169)
 
<br>
 
<br>
 
From the time match point, calculate the fracture half-length:
 
<br>
 
<br>
 
[[File:Vol5 page 0794 eq 002.png]]....................(8.170)
 
<br>
 
<br>
 
Matching can be ambiguous for hydraulically fractured wells; the data can appear to match equally well in several different positions. The ambiguity can be reduced or eliminated if a prefracture permeability is determined, and the post-fracture test data forced to match the permeability.  
 
<br>
 
<br>
 
'''''Type Curves Used for Analysis in Fractured Wells.''''' The Cinco type curve ('''Fig. 8.82'''),<ref name="r27" /> assumes that ''C''<sub>''Lf D''</sub> = 0 and ''s''<sub>''f''</sub> = 0. The type-curve stems on this curve are obtained by varying values of ''C''<sub>''r''</sub> or ''F''<sub>''cD''</sub>. With the Cinco type curve, the fracture conductivity, ''w''<sub>''f</sub>k''<sub>''f''</sub>, can be determined from the matching parameter:
 
<br>
 
<br>
 
[[File:Vol5 page 0794 eq 003.png]]....................(8.171)
 
<br>
 
<br>
 
''Choked-Fracture Type Curve.'' '''Fig. 8.83''' shows the choked-fracture type curve. <ref name="r29" /> The choked-fracture type curve is generated with wellbore-storage coefficient, ''C''<sub>''Lf D''</sub>, of zero and infinite fracture conductivity, ''C''<sub>''r''</sub>. On this type curve, the stems represent different values of the fracture skin, ''s''<sub>''f''</sub>. The fracture skin, ''s''<sub>''f''</sub>, can be used to find the additional pressure drop from
 
<br>
 
<br>
 
[[File:Vol5 page 0794 eq 004.png]]....................(8.172)
 
<br>
 
<br>
 
''Wellbore-Storage Type Curve.'' The wellbore-storage type curve ('''Fig. 8.84''') takes into account the possibility of wellbore storage. The wellbore-storage type assumes ''s''<sub>''f''</sub> = 0 and ''C''<sub>''r''</sub> = ∞. To interpret a best-fitting stem for this type curve, use the following:
 
<br>
 
<br>
 
[[File:Vol5 page 0795 eq 001.png]]....................(8.173)
 
<br>
 
<br>
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0795 Image 0001.png|'''Fig. 8.82 – Cinco type curve.'''
 
File:vol5 Page 0795 Image 0001.png|'''Fig. 8.82 – Cinco type curve.'''
  
Line 2,088: Line 638:
 
File:vol5 Page 0796 Image 0001.png|'''Fig. 8.84 – Wellbore-storage type curve.'''
 
File:vol5 Page 0796 Image 0001.png|'''Fig. 8.84 – Wellbore-storage type curve.'''
 
</gallery>
 
</gallery>
<br/>
 
  
 
=== Limitations of Type-Curve Analysis in Hydraulically Fractured Wells ===
 
=== Limitations of Type-Curve Analysis in Hydraulically Fractured Wells ===
  
Although it is the most common methodology for analyzing hydraulically fractured well, type-curve analysis still has some limitations.  
+
Although it is the most common methodology for analyzing hydraulically fractured well, type-curve analysis still has some limitations.<br/><br/>First, type-curves for analysis of hydraulically fractured wells are usually based on solutions for constant-rate drawdown tests. For buildup tests, shut-in time itself may possibly be used as a plotting function in those cases in which producing time is much greater than the shut-in time. Equivalent time can be used in some cases, but equivalent time has different definitions depending on the flow regime: radial, linear, and bilinear flow. Another possibility is to use a "superposition" type curve, which depends on the specific durations of flow and buildup periods. Superposition type curves can be readily generated with computer software.<br/><br/>Another problem with type curves is that they may ignore important behavior. The type curve that takes into account wellbore storage does not consider a variable wellbore storage coefficient. This can be caused by phase redistribution in the wellbore, for example. The widely available type curves that have been discussed do not include boundary effects. With gas wells, the probability of non-Darcy flow is high, but available type curves don’t take this into account.<br/><br/>An independent estimate of permeability may also be needed. A number of different type curves or a variety of stems on a given type curve may seem to match test data equally well. To remove this ambiguity, the best solution is to have an independent estimate of permeability.<br/>
<br>
+
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<br>
 
First, type-curves for analysis of hydraulically fractured wells are usually based on solutions for constant-rate drawdown tests. For buildup tests, shut-in time itself may possibly be used as a plotting function in those cases in which producing time is much greater than the shut-in time. Equivalent time can be used in some cases, but equivalent time has different definitions depending on the flow regime: radial, linear, and bilinear flow. Another possibility is to use a "superposition" type curve, which depends on the specific durations of flow and buildup periods. Superposition type curves can be readily generated with computer software.  
 
<br>
 
<br>
 
Another problem with type curves is that they may ignore important behavior. The type curve that takes into account wellbore storage does not consider a variable wellbore storage coefficient. This can be caused by phase redistribution in the wellbore, for example. The widely available type curves that have been discussed do not include boundary effects. With gas wells, the probability of non-Darcy flow is high, but available type curves don’t take this into account.  
 
<br>
 
<br>
 
An independent estimate of permeability may also be needed. A number of different type curves or a variety of stems on a given type curve may seem to match test data equally well. To remove this ambiguity, the best solution is to have an independent estimate of permeability.  
 
<br>
 
<br>
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
 
 
 
 
== Naturally Fractured Reservoirs ==
 
== Naturally Fractured Reservoirs ==
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
<br>
+
<br/>This section focuses on interpretation of well test data from wells completed in naturally fractured reservoirs. Because of the presence of two distinct types of porous media, the assumption of homogeneous behavior (discussed in previous sections) is no longer valid in naturally fractured reservoirs. This section includes two naturally fractured reservoir models, the physics governing fluid flow in these reservoirs and semilog and type-curve analysis techniques for well tests in these reservoirs.
This section focuses on interpretation of well test data from wells completed in naturally fractured reservoirs. Because of the presence of two distinct types of porous media, the assumption of homogeneous behavior (discussed in previous sections) is no longer valid in naturally fractured reservoirs. This section includes two naturally fractured reservoir models, the physics governing fluid flow in these reservoirs and semilog and type-curve analysis techniques for well tests in these reservoirs.  
 
  
 
=== Naturally Fractured Reservoir Models ===
 
=== Naturally Fractured Reservoir Models ===
  
Naturally fractured reservoirs are characterized by the presence of two distinct types of porous media: matrix and fracture. Because of the different fluid storage and conductivity characteristics of the matrix and fractures, these reservoirs often are called dual-porosity reservoirs. '''Fig. 8.85''' illustrates a naturally fractured reservoir composed of a rock matrix surrounded by an irregular system of vugs and natural fractures. Fortunately, it has been observed that a real, heterogeneous, naturally fractured reservoir has a characteristic behavior that can be interpreted using an equivalent, homogeneous dual-porosity model such as that shown in the idealized sketch.  
+
Naturally fractured reservoirs are characterized by the presence of two distinct types of porous media: matrix and fracture. Because of the different fluid storage and conductivity characteristics of the matrix and fractures, these reservoirs often are called dual-porosity reservoirs. '''Fig. 8.85''' illustrates a naturally fractured reservoir composed of a rock matrix surrounded by an irregular system of vugs and natural fractures. Fortunately, it has been observed that a real, heterogeneous, naturally fractured reservoir has a characteristic behavior that can be interpreted using an equivalent, homogeneous dual-porosity model such as that shown in the idealized sketch.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0797 Image 0001.png|'''Fig. 8.85 – Actual (a) and idealized (b) dual-porosity reservoir model.<ref name="r32" />'''
 
File:vol5 Page 0797 Image 0001.png|'''Fig. 8.85 – Actual (a) and idealized (b) dual-porosity reservoir model.<ref name="r32" />'''
 
</gallery>
 
</gallery>
<br/>
 
  
Several models have been proposed to represent the pressure behavior in a naturally fractured reservoir. These models differ conceptually only in the assumptions made to describe fluid flow in the matrix. Most dual-porosity models assume that production from the naturally fractured system comes from the matrix, to the fracture, and then to the wellbore (i.e., that the matrix does not produce directly into the wellbore). Furthermore, the models assume that the matrix has low permeability but large storage capacity relative to the natural fracture system, while the fractures have high permeability but low storage capacity relative to the natural fracture system. Warren and Root<ref name="r32" /> introduced two dual-porosity parameters, in addition to the usual single-porosity parameters, which can be used to describe dual-porosity reservoirs.  
+
Several models have been proposed to represent the pressure behavior in a naturally fractured reservoir. These models differ conceptually only in the assumptions made to describe fluid flow in the matrix. Most dual-porosity models assume that production from the naturally fractured system comes from the matrix, to the fracture, and then to the wellbore (i.e., that the matrix does not produce directly into the wellbore). Furthermore, the models assume that the matrix has low permeability but large storage capacity relative to the natural fracture system, while the fractures have high permeability but low storage capacity relative to the natural fracture system. Warren and Root<ref name="r32">_</ref> introduced two dual-porosity parameters, in addition to the usual single-porosity parameters, which can be used to describe dual-porosity reservoirs.<br/><br/>Interporosity flow is the fluid exchange between the two media (the matrix and fractures) constituting a dual-porosity system. Warren and Root<ref name="r32">_</ref> defined the interporosity flow coefficient, ''λ'', as<br/><br/>[[File:Vol5 page 0797 eq 001.png|RTENOTITLE]]....................(8.174)<br/><br/>where ''k''<sub>''m''</sub> is the permeability of the matrix, k<sub>f</sub> is the permeability of the natural fractures, and α is the parameter characteristic of the system geometry.<br/><br/>The interporosity flow coefficient is a measure of how easily fluid flows from the matrix to the fractures. The parameter α is defined by<ref name="r33">_</ref><br/><br/>[[File:Vol5 page 0797 eq 002.png|RTENOTITLE]]....................(8.175)<br/><br/>where ''L'' is a characteristic dimension of a matrix block and ''j'' is the number of normal sets of planes limiting the less-permeable medium (''j'' = 1, 2, 3). For example, ''j'' = 3 in the idealized reservoir cube model in '''Fig. 8.85'''. On the other hand, for the multilayered or "slab" model shown in '''Fig. 8.86''', <ref name="r34">_</ref> ''j'' = 1. For the slab model, letting ''L'' = ''h''<sub>''m''</sub> (the thickness of an individual matrix block), ''λ'' becomes<br/><br/>[[File:Vol5 page 0797 eq 003.png|RTENOTITLE]]....................(8.176)<br/><br/>The storativity ratio, <ref name="r33">_</ref> ''ω'', is defined by<br/><br/>[[File:Vol5 page 0797 eq 004.png|RTENOTITLE]]....................(8.177)<br/><br/>where ''V'' is the ratio of the total volume of one medium to the bulk volume of the total system and ''ϕ'' is the ratio of the pore volume of one medium to the total volume of that medium. Subscripts ''f'' and ''f'' + ''m'' refer to the fracture and to the total system (fractures plus matrix), respectively. Consequently, the storativity ratio is a measure of the relative fracture storage capacity in the reservoir.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
Interporosity flow is the fluid exchange between the two media (the matrix and fractures) constituting a dual-porosity system. Warren and Root<ref name="r32" /> defined the interporosity flow coefficient, ''λ'', as
 
<br>
 
<br>
 
[[File:Vol5 page 0797 eq 001.png]]....................(8.174)
 
<br>
 
<br>
 
where ''k''<sub>''m''</sub> is the permeability of the matrix, k<sub>f</sub> is the permeability of the natural fractures, and α is the parameter characteristic of the system geometry.  
 
<br>
 
<br>
 
The interporosity flow coefficient is a measure of how easily fluid flows from the matrix to the fractures. The parameter α is defined by<ref name="r33" />
 
<br>
 
<br>
 
[[File:Vol5 page 0797 eq 002.png]]....................(8.175)
 
<br>
 
<br>
 
where ''L'' is a characteristic dimension of a matrix block and ''j'' is the number of normal sets of planes limiting the less-permeable medium (''j'' = 1, 2, 3). For example, ''j'' = 3 in the idealized reservoir cube model in '''Fig. 8.85'''. On the other hand, for the multilayered or "slab" model shown in '''Fig. 8.86''', <ref name="r34" /> ''j'' = 1. For the slab model, letting ''L'' = ''h''<sub>''m''</sub> (the thickness of an individual matrix block), ''λ'' becomes
 
<br>
 
<br>
 
[[File:Vol5 page 0797 eq 003.png]]....................(8.176)
 
<br>
 
<br>
 
The storativity ratio, <ref name="r33" /> ''ω'', is defined by
 
<br>
 
<br>
 
[[File:Vol5 page 0797 eq 004.png]]....................(8.177)
 
<br>
 
<br>
 
where ''V'' is the ratio of the total volume of one medium to the bulk volume of the total system and ''ϕ'' is the ratio of the pore volume of one medium to the total volume of that medium. Subscripts ''f'' and ''f'' + ''m'' refer to the fracture and to the total system (fractures plus matrix), respectively. Consequently, the storativity ratio is a measure of the relative fracture storage capacity in the reservoir.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0798 Image 0001.png|'''Fig. 8.86 – Schematic reservoir with rectangular matrix elements.<ref name="r34" />'''
 
File:vol5 Page 0798 Image 0001.png|'''Fig. 8.86 – Schematic reservoir with rectangular matrix elements.<ref name="r34" />'''
 
</gallery>
 
</gallery>
<br/>
 
  
Many models have been developed for naturally fractures reservoirs. Two common models, pseudosteady-state and transient flow, that describe flow in the less-permeable matrix are presented here. Pseudosteady-state flow was assumed by Warren and Root<ref name="r32" /> and Barenblatt ''et al.''<ref name="r35" />; others, notably deSwaan, <ref name="r36" /> assumed transient flow in the matrix. Intuition suggests that, in a low-permeability matrix, very long times should be required to reach pseudosteady-state and that transient matrix flow should dominate; however, test analysis suggests that pseudosteady-state flow is quite common. A possible explanation of this apparent inconsistency is that matrix flow is almost always transient but can exhibit a behavior much like pseudosteady-state, if there is a significant impediment to flow from the less-permeable medium to the more-permeable one (such as low-permeability solution deposits on the faces of fractures).  
+
Many models have been developed for naturally fractures reservoirs. Two common models, pseudosteady-state and transient flow, that describe flow in the less-permeable matrix are presented here. Pseudosteady-state flow was assumed by Warren and Root<ref name="r32">_</ref> and Barenblatt ''et al.''<ref name="r35">_</ref>; others, notably deSwaan, <ref name="r36">_</ref> assumed transient flow in the matrix. Intuition suggests that, in a low-permeability matrix, very long times should be required to reach pseudosteady-state and that transient matrix flow should dominate; however, test analysis suggests that pseudosteady-state flow is quite common. A possible explanation of this apparent inconsistency is that matrix flow is almost always transient but can exhibit a behavior much like pseudosteady-state, if there is a significant impediment to flow from the less-permeable medium to the more-permeable one (such as low-permeability solution deposits on the faces of fractures).
  
 
=== Pseudosteady-State Matrix Flow Model ===
 
=== Pseudosteady-State Matrix Flow Model ===
  
The pseudosteady-state flow model assumes that, at a given time, the pressure in the matrix is decreasing at the same rate at all points and, thus, flow from the matrix to the fracture is proportional to the difference between matrix pressure and pressure in the adjacent fracture. Specifically, this model, which does not allow unsteady-state pressure gradients within the matrix, assumes that pseudosteady-state flow conditions are present from the beginning of flow.  
+
The pseudosteady-state flow model assumes that, at a given time, the pressure in the matrix is decreasing at the same rate at all points and, thus, flow from the matrix to the fracture is proportional to the difference between matrix pressure and pressure in the adjacent fracture. Specifically, this model, which does not allow unsteady-state pressure gradients within the matrix, assumes that pseudosteady-state flow conditions are present from the beginning of flow.<br/><br/>Because it assumes a pressure distribution in the matrix that would be reached only after what could be a considerable flow period, the pseudosteady-state flow model obviously is oversimplified. Again, this model seems to match a surprising number of field tests. One possible reason is that damage to the face of the matrix could cause the flow from matrix to fracture to be controlled by a sort of choke (the thin, low-permeability, damaged zone) and, therefore, is proportional to pressure differences upstream and downstream of the choke. In the next two sections, semilog and type-curve analysis techniques are presented for well tests in naturally fractured reservoirs exhibiting pseudosteady-state flow characteristics.<br/><br/>'''''Semilog Analysis Technique.''''' The pseudosteady-state matrix flow solution developed by Warren and Root<ref name="r32">_</ref> predicts that, on a semilog graph of test data, two parallel straight lines will develop. '''Fig. 8.87''' shows this characteristic pressure response.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
Because it assumes a pressure distribution in the matrix that would be reached only after what could be a considerable flow period, the pseudosteady-state flow model obviously is oversimplified. Again, this model seems to match a surprising number of field tests. One possible reason is that damage to the face of the matrix could cause the flow from matrix to fracture to be controlled by a sort of choke (the thin, low-permeability, damaged zone) and, therefore, is proportional to pressure differences upstream and downstream of the choke. In the next two sections, semilog and type-curve analysis techniques are presented for well tests in naturally fractured reservoirs exhibiting pseudosteady-state flow characteristics.  
 
<br>
 
<br>
 
'''''Semilog Analysis Technique.''''' The pseudosteady-state matrix flow solution developed by Warren and Root<ref name="r32" /> predicts that, on a semilog graph of test data, two parallel straight lines will develop. '''Fig. 8.87''' shows this characteristic pressure response.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0799 Image 0001.png|'''Fig. 8.87 - Characteristic pressure response of flow test exhibiting pseudosteady-state matrix flow.'''
 
File:vol5 Page 0799 Image 0001.png|'''Fig. 8.87 - Characteristic pressure response of flow test exhibiting pseudosteady-state matrix flow.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The initial straight line reflects flow in the fracture system only. At this time, the formation is behaving like a homogeneous formation with fluid flow originating only from the fracture system with no contribution from the matrix. Consequently, the slope of the initial semilog straight line is proportional to the permeability-thickness product of the natural fracture system, just as it is for any homogeneous system. Following a discrete pressure drop in the fracture system, the fluid in the matrix begins to flow into the fracture, and a rather flat transition region appears.  
+
The initial straight line reflects flow in the fracture system only. At this time, the formation is behaving like a homogeneous formation with fluid flow originating only from the fracture system with no contribution from the matrix. Consequently, the slope of the initial semilog straight line is proportional to the permeability-thickness product of the natural fracture system, just as it is for any homogeneous system. Following a discrete pressure drop in the fracture system, the fluid in the matrix begins to flow into the fracture, and a rather flat transition region appears.<br/><br/>Finally, the matrix and the fracture each reach an equilibrium condition, and a second straight line appears. At this time, the reservoir again is behaving like a homogeneous system, but now the system consists of both the matrix and the fractures. The slope of the second semilog straight line is proportional to the total permeability-thickness product of the matrix/fracture system. Because the permeability of the fractures is much greater than that of the matrix, the slope of the second line is almost identical to that of the initial line.<br/><br/>Similar shapes are predicted for pressure buildup tests ('''Fig. 8.88'''). The lower curve, A, represents the ideal buildup test plot predicted by Warren and Root. <ref name="r32">_</ref> The shape of a semilog plot of test data from a naturally fractured reservoir is almost never the same as that predicted by Warren and Root’s model. Wellbore storage almost always obscures the initial straight line and often obscures part of the transition region between the straight lines. The upper curve, ''B'', in '''Fig. 8.88''' shows a more common pressure response.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
Finally, the matrix and the fracture each reach an equilibrium condition, and a second straight line appears. At this time, the reservoir again is behaving like a homogeneous system, but now the system consists of both the matrix and the fractures. The slope of the second semilog straight line is proportional to the total permeability-thickness product of the matrix/fracture system. Because the permeability of the fractures is much greater than that of the matrix, the slope of the second line is almost identical to that of the initial line.  
 
<br>
 
<br>
 
Similar shapes are predicted for pressure buildup tests ('''Fig. 8.88'''). The lower curve, A, represents the ideal buildup test plot predicted by Warren and Root. <ref name="r32" /> The shape of a semilog plot of test data from a naturally fractured reservoir is almost never the same as that predicted by Warren and Root’s model. Wellbore storage almost always obscures the initial straight line and often obscures part of the transition region between the straight lines. The upper curve, ''B'', in '''Fig. 8.88''' shows a more common pressure response.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0799 Image 0002.png|'''Fig. 8.88 – Characteristic buildup pressure response predicted by the Warren and Root<ref name="r32" /> pseudosteady-state model.'''
 
File:vol5 Page 0799 Image 0002.png|'''Fig. 8.88 – Characteristic buildup pressure response predicted by the Warren and Root<ref name="r32" /> pseudosteady-state model.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The reservoir permeability-thickness product, ''kh'' [actually the ''kh'' of the fractures, or (''kh'')<sub>''f''</sub>, because (''kh'') m is usually negligible], can be obtained from the slope, ''m'', of the two semilog straight lines. Storativity, ''ω'', can be determined from their vertical displacement, ''δp''. The interporosity flow coefficient, ''λ'', can be obtained from the time of intersection of a horizontal line, drawn through the middle of the transition curve, with either the first or second semilog straight line. <ref name="r33" />  
+
The reservoir permeability-thickness product, ''kh'' [actually the ''kh'' of the fractures, or (''kh'')<sub>''f''</sub>, because (''kh'') m is usually negligible], can be obtained from the slope, ''m'', of the two semilog straight lines. Storativity, ''ω'', can be determined from their vertical displacement, ''δp''. The interporosity flow coefficient, ''λ'', can be obtained from the time of intersection of a horizontal line, drawn through the middle of the transition curve, with either the first or second semilog straight line. <ref name="r33">_</ref><br/><br/>When semilog analysis is possible (i.e., when the correct semilog straight line can be identified), the following procedure is recommended for semilog analysis of buildup or drawdown test data from wells completed in naturally fractured reservoirs. Although presented in variables for slightly compressible fluids (liquids), the same procedure is applicable to gas well tests when the appropriate variables are used.<br/>
<br>
+
 
<br>
+
*From the slope of the initial straight line (if present) or final straight line (more likely to be present), determine the permeability-thickness product, ''kh''. In either case, the slope, ''m'', is related to the total ''kh'' of the system, which is essentially all in the fractures. The permeability-thickness product is given by
When semilog analysis is possible (i.e., when the correct semilog straight line can be identified), the following procedure is recommended for semilog analysis of buildup or drawdown test data from wells completed in naturally fractured reservoirs. Although presented in variables for slightly compressible fluids (liquids), the same procedure is applicable to gas well tests when the appropriate variables are used.
+
*:[[File:Vol5 page 0800 eq 001.png|RTENOTITLE]]....................(8.178)
<br>
+
*:where [[File:Vol5 page 0800 inline 001.png|RTENOTITLE]] is equal to (''kh'')<sub>''f''</sub>/''h''. Strictly speaking, the slope of the second straight line is related to [(''kh'')<sub>''f''</sub> + (''kh'')<sub>''m''</sub> ], but (''kh'')<sub>''m''</sub> ordinarily is negligible compared to (''kh'')<sub>''f''</sub>.
<br>
+
*If both initial and final straight lines can be identified (or the position of the initial line can at least be approximated) and the pressure difference, ''δp'', established, then the storativity ratio, ''ω'' is calculated from
* From the slope of the initial straight line (if present) or final straight line (more likely to be present), determine the permeability-thickness product, ''kh''. In either case, the slope, ''m'', is related to the total ''kh'' of the system, which is essentially all in the fractures. The permeability-thickness product is given by  
+
*:[[File:Vol5 page 0800 eq 002.png|RTENOTITLE]]....................(8.179)
*:[[File:Vol5 page 0800 eq 001.png]]....................(8.178)
+
 
*:where [[File:Vol5 page 0800 inline 001.png]] is equal to (''kh'')<sub>''f ''</sub>/''h''. Strictly speaking, the slope of the second straight line is related to [(''kh'')<sub>''f''</sub> + (''kh'')<sub>''m''</sub> ], but (''kh'')<sub>''m''</sub> ordinarily is negligible compared to (''kh'')<sub>''f''</sub>.  
+
 
* If both initial and final straight lines can be identified (or the position of the initial line can at least be approximated) and the pressure difference, ''δp'', established, then the storativity ratio, ''ω'' is calculated from  
+
 
*:[[File:Vol5 page 0800 eq 002.png]]....................(8.179)
+
If the times of intersection of a horizontal line drawn through the midpoint of the transition data with the first and second semilog straight lines are denoted by ''t''<sub>l</sub> and ''t''<sub>2</sub>, respectively, the storativity ratio may also be calculated from<br/><br/>[[File:Vol5 page 0800 eq 003.png|RTENOTITLE]]....................(8.180)<br/><br/>For a buildup test, where the times of intersection of a horizontal line drawn through the midpoint of the transition data with the first and second semilog straight lines are denoted by [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t'']<sub>1</sub> and [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t'']<sub>2</sub>, respectively, the storativity ratio may be calculated from<br/><br/>[[File:Vol5 page 0800 eq 004.png|RTENOTITLE]]....................(8.181)<br/><br/>The interporosity flow coefficient, ''λ'', is calculated<ref name="r33">_</ref> for a drawdown test by<br/><br/>[[File:Vol5 page 0800 eq 005.png|RTENOTITLE]]....................(8.182)<br/><br/>or for a buildup test by<br/><br/>[[File:Vol5 page 0801 eq 001.png|RTENOTITLE]]....................(8.183)<br/><br/>where ''γ'' = 1.781.<br/><br/>The terms (''ϕV'')<sub>''m''</sub> and (''c''<sub>''t''</sub>)<sub>''m''</sub> in '''Eq. 8.183''' are obtained by conventional methods. A porosity log usually reads only the matrix porosity (not the fracture porosity) and thus gives ''ϕ''<sub>''m''</sub>, while (''c''<sub>''t''</sub>)<sub>''m''</sub> is the sum of ''c''<sub>''o''</sub>''S''<sub>''o''</sub>, ''c''<sub>''g''</sub>''S''<sub>''g''</sub>, ''c''<sub>''w''</sub>''S''<sub>''w''</sub>, and ''c''<sub>''f''</sub>. ''V''<sub>''m''</sub> usually can be assumed to be essentially 1.0. From the definition of ''ω'' in '''Eq. 8.177''',<br/><br/>[[File:Vol5 page 0801 eq 002.png|RTENOTITLE]]....................(8.184)<br/><br/>The second semilog straight line should be extrapolated to ''p''<sub>1hr</sub>, and the skin factor is<br/><br/>[[File:Vol5 page 0801 eq 003.png|RTENOTITLE]]....................(8.185)<br/><br/>where Δ''p''<sub>1hr</sub> is equal to (''p''<sub>''i''</sub> – ''p''<sub>1 hr</sub>) for a drawdown test or [''p''<sub>1 hr</sub> - ''p''<sub>''wf''</sub>(Δ''t''=0)] for a buildup test.<br/>
<br>
+
 
 +
*The second semilog straight line should be extrapolated to ''p''* ('''Fig. 8.89'''). From ''p''*, [[File:Vol5 page 0781 inline 001.png|RTENOTITLE]] can be found using conventional methods (such as the Matthew-Brons-Hazebroek ''p''* method).
  
If the times of intersection of a horizontal line drawn through the midpoint of the transition data with the first and second semilog straight lines are denoted by ''t''<sub>l</sub> and ''t''<sub>2</sub>, respectively, the storativity ratio may also be calculated from
+
<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0800 eq 003.png]]....................(8.180)
 
<br>
 
<br>
 
For a buildup test, where the times of intersection of a horizontal line drawn through the midpoint of the transition data with the first and second semilog straight lines are denoted by [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t'']<sub>1</sub> and [(''t''<sub>''p''</sub> + Δ''t'')/Δ''t'']<sub>2</sub>, respectively, the storativity ratio may be calculated from
 
<br>
 
<br>
 
[[File:Vol5 page 0800 eq 004.png]]....................(8.181)
 
<br>
 
<br>
 
The interporosity flow coefficient, ''λ'', is calculated<ref name="r33" /> for a drawdown test by
 
<br>
 
<br>
 
[[File:Vol5 page 0800 eq 005.png]]....................(8.182)
 
<br>
 
<br>
 
or for a buildup test by
 
<br>
 
<br>
 
[[File:Vol5 page 0801 eq 001.png]]....................(8.183)
 
<br>
 
<br>
 
where ''γ'' = 1.781.
 
<br>
 
<br>
 
The terms (''ϕV'')<sub>''m''</sub> and (''c''<sub>''t''</sub>)<sub>''m''</sub> in '''Eq. 8.183''' are obtained by conventional methods. A porosity log usually reads only the matrix porosity (not the fracture porosity) and thus gives ''ϕ''<sub>''m''</sub>, while (''c''<sub>''t''</sub>)<sub>''m''</sub> is the sum of ''c''<sub>''o''</sub>''S''<sub>''o''</sub>, ''c''<sub>''g''</sub>''S''<sub>''g''</sub>, ''c''<sub>''w''</sub>''S''<sub>''w''</sub>, and ''c''<sub>''f''</sub>. ''V''<sub>''m''</sub> usually can be assumed to be essentially 1.0. From the definition of ''ω'' in '''Eq. 8.177''',
 
<br>
 
<br>
 
[[File:Vol5 page 0801 eq 002.png]]....................(8.184)
 
<br>
 
<br>
 
The second semilog straight line should be extrapolated to ''p''<sub>1hr</sub>, and the skin factor is
 
<br>
 
<br>
 
[[File:Vol5 page 0801 eq 003.png]]....................(8.185)
 
<br>
 
<br>
 
where Δ''p''<sub>1hr</sub> is equal to (''p''<sub>''i''</sub> – ''p''<sub>1 hr</sub>) for a drawdown test or [''p''<sub>1 hr</sub> - ''p''<sub>''wf''</sub>(Δ''t''=0)] for a buildup test.
 
<br>
 
<br>
 
* The second semilog straight line should be extrapolated to ''p''* ('''Fig. 8.89'''). From ''p''*, [[File:Vol5 page 0781 inline 001.png]] can be found using conventional methods (such as the Matthew-Brons-Hazebroek ''p''* method).
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0801 Image 0001.png|'''Fig. 8.89 – Extrapolation of second semilog straight line to ''p''*.'''
 
File:vol5 Page 0801 Image 0001.png|'''Fig. 8.89 – Extrapolation of second semilog straight line to ''p''*.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''''Type Curve Analysis Technique.''''' Particularly because of wellbore-storage distortion, type curves are quite useful for identifying and analyzing dual-porosity systems. '''Fig. 8.90''' shows an example of the Bourdet ''et al.''<ref name="r37" /> type curves developed for pseudosteady-state matrix flow. Initially, test data follow a curve for some value of ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> where ''C''<sub>''D''</sub> is the dimensionless wellbore storage coefficient. In '''Fig. 8.90''', the earliest data for the well follow the curve for ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> = 1. The data then deviate from the early fit and follow a transition curve characterized by the parameter ''λe''<sup>-2''s''</sup>. In '''Fig. 8.90''', the data follow the curve for ''λe''<sup>–2''s''</sup> = 3×10<sup>–4</sup>. When equilibrium is reached between the matrix and fracture systems, the data then follow another ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> curve. In the example, the later data follow the ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> = 0.1 curve.  
+
'''''Type Curve Analysis Technique.''''' Particularly because of wellbore-storage distortion, type curves are quite useful for identifying and analyzing dual-porosity systems. '''Fig. 8.90''' shows an example of the Bourdet ''et al.''<ref name="r37">_</ref> type curves developed for pseudosteady-state matrix flow. Initially, test data follow a curve for some value of ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> where ''C''<sub>''D''</sub> is the dimensionless wellbore storage coefficient. In '''Fig. 8.90''', the earliest data for the well follow the curve for ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> = 1. The data then deviate from the early fit and follow a transition curve characterized by the parameter ''λe''<sup>-2''s''</sup>. In '''Fig. 8.90''', the data follow the curve for ''λe''<sup>–2''s''</sup> = 3×10<sup>–4</sup>. When equilibrium is reached between the matrix and fracture systems, the data then follow another ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> curve. In the example, the later data follow the ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> = 0.1 curve.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0802 Image 0001.png|'''Fig. 8.90 – Type curves for pseudosteady-state matrix flow. (After Bourdet ''et al''.<ref name="r37" />)'''
 
File:vol5 Page 0802 Image 0001.png|'''Fig. 8.90 – Type curves for pseudosteady-state matrix flow. (After Bourdet ''et al''.<ref name="r37" />)'''
 
</gallery>
 
</gallery>
<br/>
 
  
At earliest times, the reservoir is behaving like a homogeneous reservoir with all fluid originating from the fracture system. During intermediate times, there is a transition region as the matrix begins to produce into the fractures. At later times, the system again is behaving like a homogeneous system with both matrix and fractures contributing to fluid production.  
+
At earliest times, the reservoir is behaving like a homogeneous reservoir with all fluid originating from the fracture system. During intermediate times, there is a transition region as the matrix begins to produce into the fractures. At later times, the system again is behaving like a homogeneous system with both matrix and fractures contributing to fluid production.<br/><br/>'''Fig. 8.91''' illustrates the derivative type curves for a formation with pseudosteady-state matrix flow. <ref name="r37">_</ref> The most notable feature, characteristic of naturally fractured reservoirs, is the dip below the homogeneous reservoir curve. The curves dipping downward are characterized by a parameter ''λC''<sub>''D''</sub>/''ω'' (1 − ''ω''), while the curves returning to the homogeneous reservoir curves are characterized by the parameter ''λC''<sub>''D''</sub>/''ω'' (1 − ''ω''). Test data that follow this pattern on the derivative type curve can reasonably be interpreted as identifying a dual-porosity reservoir with pseudosteady-state matrix flow (a theory that needs to be confirmed with geological information and reservoir performance). Pressure and pressure derivative type curves can be used together for analysis of a dual-porosity reservoir. The pressure derivative data are especially useful for identifying the dual-porosity behavior. Manual type-curve analysis for well in naturally fractured reservoirs is tedious, and the interpretation involved is difficult. Most current analysis uses commercial software.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
'''Fig. 8.91''' illustrates the derivative type curves for a formation with pseudosteady-state matrix flow. <ref name="r37" /> The most notable feature, characteristic of naturally fractured reservoirs, is the dip below the homogeneous reservoir curve. The curves dipping downward are characterized by a parameter ''λC''<sub>''D''</sub>/''ω'' (1 − ''ω''), while the curves returning to the homogeneous reservoir curves are characterized by the parameter ''λC''<sub>''D''</sub>/''ω'' (1 − ''ω''). Test data that follow this pattern on the derivative type curve can reasonably be interpreted as identifying a dual-porosity reservoir with pseudosteady-state matrix flow (a theory that needs to be confirmed with geological information and reservoir performance). Pressure and pressure derivative type curves can be used together for analysis of a dual-porosity reservoir. The pressure derivative data are especially useful for identifying the dual-porosity behavior. Manual type-curve analysis for well in naturally fractured reservoirs is tedious, and the interpretation involved is difficult. Most current analysis uses commercial software.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0803 Image 0001.png|'''Fig. 8.91 – Derivative type curves for a pseudosteady-state matrix flow. (After Bourdey ''et al''.<ref name="r37" />)'''
 
File:vol5 Page 0803 Image 0001.png|'''Fig. 8.91 – Derivative type curves for a pseudosteady-state matrix flow. (After Bourdey ''et al''.<ref name="r37" />)'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''''Transient Matrix Flow Model.''''' The more probable flow regime in the matrix is unsteady-state or transient flow; that is, flow in which an increasing pressure drawdown starts at the matrix/fracture interface and moves further into the matrix with increasing time. Only at late times should pseudosteady-state flow be achieved, although a matrix with a thin, low-permeability damaged zone at the fracture face may behave as predicted by the pseudosteady-state matrix flow model even though the flow in the matrix is actually unsteady-state.  
+
'''''Transient Matrix Flow Model.''''' The more probable flow regime in the matrix is unsteady-state or transient flow; that is, flow in which an increasing pressure drawdown starts at the matrix/fracture interface and moves further into the matrix with increasing time. Only at late times should pseudosteady-state flow be achieved, although a matrix with a thin, low-permeability damaged zone at the fracture face may behave as predicted by the pseudosteady-state matrix flow model even though the flow in the matrix is actually unsteady-state.<br/><br/>A semilog graph of test data for a formation with transient matrix flow has a characteristic shape different from that for pseudosteady-state flow in the matrix. Three distinct flow regimes have been identified that are characteristic of dual-porosity reservoir behavior with transient matrix flow. '''Fig. 8.92''' illustrates these flow regimes on a semilog graph as regimes 1, 2, and 3.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
A semilog graph of test data for a formation with transient matrix flow has a characteristic shape different from that for pseudosteady-state flow in the matrix. Three distinct flow regimes have been identified that are characteristic of dual-porosity reservoir behavior with transient matrix flow. '''Fig. 8.92''' illustrates these flow regimes on a semilog graph as regimes 1, 2, and 3.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0803 Image 0002.png|'''Fig. 8.92 – Flow regimes in a dual-porosity reservoir with transient matrix flow.<ref name="r34" />'''
 
File:vol5 Page 0803 Image 0002.png|'''Fig. 8.92 – Flow regimes in a dual-porosity reservoir with transient matrix flow.<ref name="r34" />'''
 
</gallery>
 
</gallery>
<br/>
 
  
Flow regime 1 occurs at early times during which all production comes from the fractures. Flow regime 2 occurs when production from the matrix into the fracture begins and continues until the matrix-to-fracture transfer reaches equilibrium. This equilibrium point marks the beginning of flow regime 3, during which total system flow, from matrix to fracture to wellbore, is dominant. The same three flow regimes appear when there is pseudosteady-state matrix flow. The duration and shape of the transition flow regimes, however, is considerably different for the two matrix flow models.  
+
Flow regime 1 occurs at early times during which all production comes from the fractures. Flow regime 2 occurs when production from the matrix into the fracture begins and continues until the matrix-to-fracture transfer reaches equilibrium. This equilibrium point marks the beginning of flow regime 3, during which total system flow, from matrix to fracture to wellbore, is dominant. The same three flow regimes appear when there is pseudosteady-state matrix flow. The duration and shape of the transition flow regimes, however, is considerably different for the two matrix flow models.<br/><br/>Serra ''et al.''<ref name="r34">_</ref> observed that pressures from each of these flow regimes will plot as straight lines on conventional semilog graphs. Flow regimes 1 and 3, which correspond to the classical early- and late-time semilog straight-line periods, respectively, have the same slope. Flow regime 2 is an intermediate transitional period between the first and third flow regimes. The semilog straight line of flow regime 2 has a slope of approximately one-half that of flow regimes 1 and 3. If all or any two of these regimes can be identified, then a complete analysis is possible using semilog methods alone. Certain nonideal conditions, however, may make this analysis difficult to apply.<br/><br/>Flow regime 1 often is distorted or obscured by wellbore storage, which often makes this flow regime difficult to identify. Flow regime 2, the transition, also may be obscured by wellbore storage. Flow regime 3 sometimes requires a long flow period followed by a long shut-in time to be observed, especially in formations with low permeability. Furthermore, boundary effects may appear before flow regime 3 is fully developed.<br/><br/>'''''Semilog Analysis Techniques.''''' Serra ''et al.''<ref name="r34">_</ref> presented a semilog method for analyzing well test data in dual-porosity reservoirs exhibiting transient matrix flow ('''Fig. 8.92'''). They found that the existence of the transition region, flow regime 2, and either flow regime 1 or flow regime 3 is sufficient to obtain a complete analysis of drawdown or buildup test data. Further, they assumed unsteady-state flow in the matrix, no wellbore storage, and rectangular matrix-block geometry, as '''Fig. 8.86''' shows. The rectangular matrix-block geometry is adequate, although different assumed geometries can lead to slightly different interpretation results.<br/><br/>The major weakness of the Serra ''et al.'' method is that it assumes no wellbore storage. In many cases, flow regimes 1 and 2 are partially or even totally obscured by wellbore storage, making analysis by the Serra ''et al.'' method impossible or difficult. Despite this limitation, the Serra ''et al.'' method has great practical value when used in conjunction with type-curve methods. These calculations of the Serra ''et al.'' method apply to both buildup and drawdown test data and are applicable for well test analysis of slightly compressible liquids and gas well tests.<br/><br/>'''''Type Curve Analysis Technique.''''' Bourdet ''et al.''<ref name="r37">_</ref> presented type curves for analyzing well tests in dual-porosity reservoirs including the effects of wellbore storage and unsteady-state flow in the matrix. The type curves are useful supplements to the Serra ''et al.'' semilog analysis. '''Fig. 8.93''' gives an example of the pressure and pressure derivative type curves for transient matrix flow. Early (fracture-dominated) data are fit by a ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> value indicative of homogeneous behavior. Data in the transition region are fit by curves characterized by a parameter ''β''′. Finally, data in the homogeneous-acting, fracture-plus-matrix flow regime are fit by another ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> curve.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
Serra ''et al.''<ref name="r34" /> observed that pressures from each of these flow regimes will plot as straight lines on conventional semilog graphs. Flow regimes 1 and 3, which correspond to the classical early- and late-time semilog straight-line periods, respectively, have the same slope. Flow regime 2 is an intermediate transitional period between the first and third flow regimes. The semilog straight line of flow regime 2 has a slope of approximately one-half that of flow regimes 1 and 3. If all or any two of these regimes can be identified, then a complete analysis is possible using semilog methods alone. Certain nonideal conditions, however, may make this analysis difficult to apply.  
 
<br>
 
<br>
 
Flow regime 1 often is distorted or obscured by wellbore storage, which often makes this flow regime difficult to identify. Flow regime 2, the transition, also may be obscured by wellbore storage. Flow regime 3 sometimes requires a long flow period followed by a long shut-in time to be observed, especially in formations with low permeability. Furthermore, boundary effects may appear before flow regime 3 is fully developed.  
 
<br>
 
<br>
 
'''''Semilog Analysis Techniques.''''' Serra ''et al.''<ref name="r34" /> presented a semilog method for analyzing well test data in dual-porosity reservoirs exhibiting transient matrix flow ('''Fig. 8.92'''). They found that the existence of the transition region, flow regime 2, and either flow regime 1 or flow regime 3 is sufficient to obtain a complete analysis of drawdown or buildup test data. Further, they assumed unsteady-state flow in the matrix, no wellbore storage, and rectangular matrix-block geometry, as '''Fig. 8.86''' shows. The rectangular matrix-block geometry is adequate, although different assumed geometries can lead to slightly different interpretation results.  
 
<br>
 
<br>
 
The major weakness of the Serra ''et al.'' method is that it assumes no wellbore storage. In many cases, flow regimes 1 and 2 are partially or even totally obscured by wellbore storage, making analysis by the Serra ''et al.'' method impossible or difficult. Despite this limitation, the Serra ''et al.'' method has great practical value when used in conjunction with type-curve methods. These calculations of the Serra ''et al.'' method apply to both buildup and drawdown test data and are applicable for well test analysis of slightly compressible liquids and gas well tests.  
 
<br>
 
<br>
 
'''''Type Curve Analysis Technique.''''' Bourdet ''et al.''<ref name="r37" /> presented type curves for analyzing well tests in dual-porosity reservoirs including the effects of wellbore storage and unsteady-state flow in the matrix. The type curves are useful supplements to the Serra ''et al.'' semilog analysis. '''Fig. 8.93''' gives an example of the pressure and pressure derivative type curves for transient matrix flow. Early (fracture-dominated) data are fit by a ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> value indicative of homogeneous behavior. Data in the transition region are fit by curves characterized by a parameter ''β''′. Finally, data in the homogeneous-acting, fracture-plus-matrix flow regime are fit by another ''C''<sub>''D''</sub>''e''<sup>2''s''</sup> curve.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0804 Image 0001.png|'''Fig. 8.93 – Type curves for transient matrix flow.<ref name="r37" />'''
 
File:vol5 Page 0804 Image 0001.png|'''Fig. 8.93 – Type curves for transient matrix flow.<ref name="r37" />'''
 
</gallery>
 
</gallery>
<br/>
 
  
On the derivative type curve, early data also are fit by a derivative curve reflecting homogeneous behavior. '''Fig. 8.94''' shows an actual example. If wellbore-storage distortion ceases before the transition region begins (which did not happen in the example but is possible in other cases), the derivative data will be horizontal and should be aligned with the (''t''<sub>''D''</sub>/''C''<sub>''D''</sub>)''p''<sub>''D''</sub>′ = 0.5 curve. However, if the transition region is present (recall that its semilog slope is half that of the middle-time straight line), the derivative curve will flatten and should be aligned with the (''t''<sub>''D''</sub>/''C''<sub>''D''</sub>)''p''<sub>''D''</sub>′ = 0.25 curve as shown in this example. The homogeneous (fracture-plus-matrix) data should, after wellbore distortion has ceased and before boundary effects have appeared, be horizontal on the derivative type curve and should be aligned with the (''t''<sub>''D''</sub>/''C''<sub>''D''</sub>)''p''<sub>''D''</sub>′ = 0.5 curve as this example shows.  
+
On the derivative type curve, early data also are fit by a derivative curve reflecting homogeneous behavior. '''Fig. 8.94''' shows an actual example. If wellbore-storage distortion ceases before the transition region begins (which did not happen in the example but is possible in other cases), the derivative data will be horizontal and should be aligned with the (''t''<sub>''D''</sub>/''C''<sub>''D''</sub>)''p''<sub>''D''</sub>′ = 0.5 curve. However, if the transition region is present (recall that its semilog slope is half that of the middle-time straight line), the derivative curve will flatten and should be aligned with the (''t''<sub>''D''</sub>/''C''<sub>''D''</sub>)''p''<sub>''D''</sub>′ = 0.25 curve as shown in this example. The homogeneous (fracture-plus-matrix) data should, after wellbore distortion has ceased and before boundary effects have appeared, be horizontal on the derivative type curve and should be aligned with the (''t''<sub>''D''</sub>/''C''<sub>''D''</sub>)''p''<sub>''D''</sub>′ = 0.5 curve as this example shows.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0805 Image 0001.png|'''Fig. 8.94 – Example of using type curves for transient matrix flow.<ref name="r37" />'''
 
File:vol5 Page 0805 Image 0001.png|'''Fig. 8.94 – Example of using type curves for transient matrix flow.<ref name="r37" />'''
 
</gallery>
 
</gallery>
<br/>
 
 
Manual type-curve matching is tedious and difficult, especially with the interpolation involved. Analysis ordinarily uses commercially available software to analyze these kinds of tests after the reservoir model has been identified.
 
<br>
 
<br>
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
 
  
 +
Manual type-curve matching is tedious and difficult, especially with the interpolation involved. Analysis ordinarily uses commercially available software to analyze these kinds of tests after the reservoir model has been identified.<br/>
 +
</div></div><div class="toccolours mw-collapsible mw-collapsed">
 
== Horizontal Well Analysis ==
 
== Horizontal Well Analysis ==
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
<br>
+
<br/>Productivity estimates in horizontal wells are subject to more uncertainty than comparable estimates in vertical wells. Further, it is much more difficult to interpret well test data because of 3D flow geometry. The radial symmetry usually present in a vertical well does not exist. Several flow regimes can potentially occur and need to be considered in analyzing test data from horizontal wells. Wellbore storage effects can be much more significant and partial penetration and end effects commonly complicate interpretation.<br/><br/>In vertical wells, variables such as average permeability, net vertical thickness, and skin are used. Horizontal wells need more detail. Not only is vertical thickness important, but the horizontal dimensions of the reservoir, relative to the horizontal wellbore, need to be known.
Productivity estimates in horizontal wells are subject to more uncertainty than comparable estimates in vertical wells. Further, it is much more difficult to interpret well test data because of 3D flow geometry. The radial symmetry usually present in a vertical well does not exist. Several flow regimes can potentially occur and need to be considered in analyzing test data from horizontal wells. Wellbore storage effects can be much more significant and partial penetration and end effects commonly complicate interpretation.  
 
<br>
 
<br>
 
In vertical wells, variables such as average permeability, net vertical thickness, and skin are used. Horizontal wells need more detail. Not only is vertical thickness important, but the horizontal dimensions of the reservoir, relative to the horizontal wellbore, need to be known.  
 
  
 
=== Steps in Evaluating Horizontal Well-Test Data ===
 
=== Steps in Evaluating Horizontal Well-Test Data ===
  
There are three basic steps in evaluating pressure-transient data from a horizontal well. First, identify the specific flow regimes in the test data. Second, apply the proper analytical and graphical procedures to the data. Finally, evaluate the uniqueness and sensitivity of the results to properties derived from analysis or simply assumed.  
+
There are three basic steps in evaluating pressure-transient data from a horizontal well. First, identify the specific flow regimes in the test data. Second, apply the proper analytical and graphical procedures to the data. Finally, evaluate the uniqueness and sensitivity of the results to properties derived from analysis or simply assumed.<br/><br/>'''''Identify Flow Regimes.''''' As discussed in previous sections, evaluation of data from a vertical wellbore will generally center on a single flow regime, such as infinite-acting radial flow, known as the MTR. However, a pressure-transient test in a horizontal well can involve as many as five major and distinct regimes that need to be identified. These regimes may or may not occur in a given test and may or may not be obscured by wellbore storage effects.<br/><br/>'''''Apply the Proper Procedures.''''' Each flow regime can be modeled by an equation that can be used to estimate important reservoir properties. At best, only groups of analytical parameters can be determined directly from equations. It is imperative that the proper analytical and graphical procedures be applied to the data. In many cases, when solving for specific parameters, the application of these analytical expressions may involve a complex iterative procedure.<br/><br/>'''''Evaluate Uniqueness and Sensitivity.''''' Experience indicates that results of horizontal well test analysis are seldom unique, so it is important that the uniqueness and sensitivity of the results to assumed properties be evaluated. Simulation of the test using properties that have been determined from the test can confirm that at least the analysis is consistent with the test data. A simulator can also determine whether other sets of formation properties will also lead to a fit of the data.
<br>
 
<br>
 
'''''Identify Flow Regimes.''''' As discussed in previous sections, evaluation of data from a vertical wellbore will generally center on a single flow regime, such as infinite-acting radial flow, known as the MTR. However, a pressure-transient test in a horizontal well can involve as many as five major and distinct regimes that need to be identified. These regimes may or may not occur in a given test and may or may not be obscured by wellbore storage effects.  
 
<br>
 
<br>
 
'''''Apply the Proper Procedures.''''' Each flow regime can be modeled by an equation that can be used to estimate important reservoir properties. At best, only groups of analytical parameters can be determined directly from equations. It is imperative that the proper analytical and graphical procedures be applied to the data. In many cases, when solving for specific parameters, the application of these analytical expressions may involve a complex iterative procedure.  
 
<br>
 
<br>
 
'''''Evaluate Uniqueness and Sensitivity.''''' Experience indicates that results of horizontal well test analysis are seldom unique, so it is important that the uniqueness and sensitivity of the results to assumed properties be evaluated. Simulation of the test using properties that have been determined from the test can confirm that at least the analysis is consistent with the test data. A simulator can also determine whether other sets of formation properties will also lead to a fit of the data.  
 
  
 
=== Horizontal Well Flow Regimes ===
 
=== Horizontal Well Flow Regimes ===
  
Different formation properties can be calculated from the data in each of the five different flow regimes. Any flow regime may be absent from a plot of test data because of geometry, wellbore storage, or other factors. Nor does the fact that they can appear mean that they do appear. The five different flow regimes that can occur are early radial, hemi-radial, early linear, late pseudoradial, and late linear.  
+
Different formation properties can be calculated from the data in each of the five different flow regimes. Any flow regime may be absent from a plot of test data because of geometry, wellbore storage, or other factors. Nor does the fact that they can appear mean that they do appear. The five different flow regimes that can occur are early radial, hemi-radial, early linear, late pseudoradial, and late linear.<br/><br/>'''Fig. 8.95''' shows a horizontal well with length, ''L''<sub>''w''</sub>, within a reservoir that is assumed to be a rectangular parallelepiped or a "box reservoir" drainage area. In this discussion, it is assumed that the axes of the coordinate system coincide with the direction of principal permeability and the well produces over its entire length, ''L''<sub>''w''</sub>.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
'''Fig. 8.95''' shows a horizontal well with length, ''L''<sub>''w''</sub>, within a reservoir that is assumed to be a rectangular parallelepiped or a "box reservoir" drainage area. In this discussion, it is assumed that the axes of the coordinate system coincide with the direction of principal permeability and the well produces over its entire length, ''L''<sub>''w''</sub>.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0806 Image 0001.png|'''Fig. 8.95 – Well and reservoir geometry and nomenclature for a horizontal well.'''
 
File:vol5 Page 0806 Image 0001.png|'''Fig. 8.95 – Well and reservoir geometry and nomenclature for a horizontal well.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The axes for this box are the usual ''x''-, ''y''-, and ''z''-axes. Notice that the x-axis is measured along the bottom edge of the reservoir, going from left to right in the direction perpendicular to the well. The ''y''-axis lies along the axis from front to back of the reservoir, parallel to the wellbore. The ''z''-axis is oriented in the direction of reservoir thickness.  
+
The axes for this box are the usual ''x''-, ''y''-, and ''z''-axes. Notice that the x-axis is measured along the bottom edge of the reservoir, going from left to right in the direction perpendicular to the well. The ''y''-axis lies along the axis from front to back of the reservoir, parallel to the wellbore. The ''z''-axis is oriented in the direction of reservoir thickness.<br/><br/>The total width of the reservoir perpendicular to the wellbore is ''a''<sub>''H''</sub>, the total length in a direction parallel to the wellbore is ''b''<sub>''H''</sub>, and the total height of the reservoir is the net pay thickness, ''h''. Notice the parameters for the distance from the well to the various borders. Along the axis of the well, the shortest distance from the end of the well to a boundary is ''d''<sub>''y''</sub>, and the longest distance from the other end of the well to the boundary is ''D''<sub>''y''</sub>. In the vertical direction, the shortest distance to a vertical boundary is ''d''<sub>''z''</sub>, and the longest distance to a vertical boundary is ''D''<sub>''z''</sub>.<br/><br/>'''''Characteristics of Flow Regimes.''''' Consider a well producing at a constant rate. The early radial flow regime occurs before the area drained or the pressure transient caused by this production encounters either of the vertical boundaries of the reservoir. '''Fig. 8.96''' shows a radial flow pattern penetrating out into the reservoir. Actually, however, this flow pattern is likely to be elliptical, moving further into the reservoir at a given time in the higher-permeability ''x''-direction than in the lower-permeability ''z''-direction. This phenomenon causes no significant complications in our analysis.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
The total width of the reservoir perpendicular to the wellbore is ''a''<sub>''H''</sub>, the total length in a direction parallel to the wellbore is ''b''<sub>''H''</sub>, and the total height of the reservoir is the net pay thickness, ''h''. Notice the parameters for the distance from the well to the various borders. Along the axis of the well, the shortest distance from the end of the well to a boundary is ''d''<sub>''y''</sub>, and the longest distance from the other end of the well to the boundary is ''D''<sub>''y''</sub>. In the vertical direction, the shortest distance to a vertical boundary is ''d''<sub>''z''</sub>, and the longest distance to a vertical boundary is ''D''<sub>''z''</sub>.  
 
<br>
 
<br>
 
'''''Characteristics of Flow Regimes.''''' Consider a well producing at a constant rate. The early radial flow regime occurs before the area drained or the pressure transient caused by this production encounters either of the vertical boundaries of the reservoir. '''Fig. 8.96''' shows a radial flow pattern penetrating out into the reservoir. Actually, however, this flow pattern is likely to be elliptical, moving further into the reservoir at a given time in the higher-permeability ''x''-direction than in the lower-permeability ''z''-direction. This phenomenon causes no significant complications in our analysis.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0807 Image 0001.png|'''Fig. 8.96 – Early-radial flow is not influenced by reservoir boundaries.'''
 
File:vol5 Page 0807 Image 0001.png|'''Fig. 8.96 – Early-radial flow is not influenced by reservoir boundaries.'''
 
</gallery>
 
</gallery>
<br/>
 
  
When the wellbore is much nearer one vertical boundary than the other, another flow regime, called half-radial or hemiradial flow ('''Fig. 8.97''') may exist. Hemiradial flow can occur immediately following the early radial flow regime, if the well is much nearer one of the vertical boundaries than the other. Eventually, the area affected by the production will include the entire thickness of the reservoir. When that happens, a linear flow pattern may develop, as '''Fig. 8.98''' shows.  
+
When the wellbore is much nearer one vertical boundary than the other, another flow regime, called half-radial or hemiradial flow ('''Fig. 8.97''') may exist. Hemiradial flow can occur immediately following the early radial flow regime, if the well is much nearer one of the vertical boundaries than the other. Eventually, the area affected by the production will include the entire thickness of the reservoir. When that happens, a linear flow pattern may develop, as '''Fig. 8.98''' shows.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0807 Image 0002.png|'''Fig. 8.97 – Hemiradial flow occurs when effects of one vertical boundary are felt before the effects of the other.'''
 
File:vol5 Page 0807 Image 0002.png|'''Fig. 8.97 – Hemiradial flow occurs when effects of one vertical boundary are felt before the effects of the other.'''
  
 
File:vol5 Page 0808 Image 0001.png|'''Fig. 8.98 – Early-linear flow may develop after both vertical boundaries are encountered.'''
 
File:vol5 Page 0808 Image 0001.png|'''Fig. 8.98 – Early-linear flow may develop after both vertical boundaries are encountered.'''
 
</gallery>
 
</gallery>
<br/>
 
  
Eventually, flow will begin to come into the wellbore from beyond the ends of the well. Until these end effects become important, early linear flow continues. Once end effects become important a transition period is followed by a later pseudoradial flow regime, as '''Fig. 8.99''' illustrates.  
+
Eventually, flow will begin to come into the wellbore from beyond the ends of the well. Until these end effects become important, early linear flow continues. Once end effects become important a transition period is followed by a later pseudoradial flow regime, as '''Fig. 8.99''' illustrates.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0808 Image 0002.png|'''Fig. 8.99 – Late pseudoradial flow begins after flow enters wellbore from beyond ends of well.'''
 
File:vol5 Page 0808 Image 0002.png|'''Fig. 8.99 – Late pseudoradial flow begins after flow enters wellbore from beyond ends of well.'''
 
</gallery>
 
</gallery>
<br/>
 
  
This flow regime continues until the area affected by the production reaches one of the sides of the reservoir. Once the area affected is the entire width of the reservoir (that is, the pressure transient has reached both sides of the reservoir), then the late-linear flow regime begins ('''Fig. 8.100''').  
+
This flow regime continues until the area affected by the production reaches one of the sides of the reservoir. Once the area affected is the entire width of the reservoir (that is, the pressure transient has reached both sides of the reservoir), then the late-linear flow regime begins ('''Fig. 8.100''').<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0808 Image 0003.png|'''Fig. 8.100 – Late-linear flow includes flow from the drainage volume perpendicular to the well.'''
 
File:vol5 Page 0808 Image 0003.png|'''Fig. 8.100 – Late-linear flow includes flow from the drainage volume perpendicular to the well.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Identifying Flow Regimes in Horizontal Wells ===
 
=== Identifying Flow Regimes in Horizontal Wells ===
  
All of these flow regimes in a test can be identified on a diagnostic log-log plot of the pressure change, Δ''p'', and pressure derivative, ''p''′, against the logarithm of time ('''Fig. 8.101''').  
+
All of these flow regimes in a test can be identified on a diagnostic log-log plot of the pressure change, Δ''p'', and pressure derivative, ''p''′, against the logarithm of time ('''Fig. 8.101''').<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0809 Image 0001.png|'''Fig. 8.101 – Idealized diagnostic plot identifies all normally expected flow regimes.'''
 
File:vol5 Page 0809 Image 0001.png|'''Fig. 8.101 – Idealized diagnostic plot identifies all normally expected flow regimes.'''
 
</gallery>
 
</gallery>
<br/>
 
  
A unit-slope line appears during wellbore storage; a horizontal derivative during early radial flow, and then, later, in pseudoradial flow; and a half-slope line in early-linear flow and then in late-linear flow. (These half-slope lines appear on the derivative but not on the pressure-change curves.) This does not imply that all these flow regimes will appear in any given test; in fact, that would be rare. But these are the shapes that identify the flow regimes that may appear in the test being analyzed.  
+
A unit-slope line appears during wellbore storage; a horizontal derivative during early radial flow, and then, later, in pseudoradial flow; and a half-slope line in early-linear flow and then in late-linear flow. (These half-slope lines appear on the derivative but not on the pressure-change curves.) This does not imply that all these flow regimes will appear in any given test; in fact, that would be rare. But these are the shapes that identify the flow regimes that may appear in the test being analyzed.<br/><br/>The shapes that may appear in a drawdown test (which is the basis of '''Fig. 8.101''') may not appear in a buildup test because of the complex superposition of flow regimes. For example, a test would have to be in linear flow both at time (''t''<sub>''p''</sub> + Δ''t'') and at time Δ t to ensure appearance of a derivative with half-slope; this is highly unlikely. The best way to solve the problem is to ensure that a buildup test on a horizontal well is run with a producing time, ''t''<sub>''p''</sub>, much greater than the maximum shut-in time in the test (that is, ''t''<sub>''p''</sub> > 10 Δ''t'' max ).<br/><br/>'''Table 8.A-2''' (see Appendix) summarizes the working equations for permeability, skin, and start and end of each of the recognized flow regimes. Different investigators have found different equations for start and end of various flow regimes, especially linear flow regimes. This is partly because of a difference of assumptions about flow into the wellbore. Uniform flux or infinite conductivity models are common; neither is rigorously applicable in practice. <ref name="r38">_</ref> In this section, the equations for duration of flow regimes derived by Odeh and Babu<ref name="r39">_</ref> are used. This model assumes uniform flux into the wellbore.<br/><br/>'''''Early-Radial Flow.''''' Early-radial flow is similar to the radial flow period in a vertical well ('''Fig. 8.96'''). The governing equation for this flow regime is<br/><br/>[[File:Vol5 page 0809 eq 001.png|RTENOTITLE]]....................(8.186)<br/><br/>Data for this period may be masked by wellbore storage effects, but, when present, they may be analyzed on a semilog plot.<br/><br/>The early-radial flow regime may in theory start at time zero, in absence of wellbore storage effects. The end of the early-radial flow regime may occur when the transient reaches a vertical boundary or when flow comes from beyond the end of the wellbore. The end of the period is the smaller of these two values.<br/><br/>'''Eq. 8.187'''<ref name="r39">_</ref> says that the period must end when the transient reaches the nearest boundary, ''d''<sub>''z''</sub>, from the well. This equation includes the permeability in the vertical direction:<br/><br/>[[File:Vol5 page 0810 eq 001.png|RTENOTITLE]]....................(8.187)<br/><br/>The radial flow regime may also end when flow from beyond the end of the wellbore becomes important. '''Eq. 8.188''' gives the time by<br/><br/>[[File:Vol5 page 0810 eq 002.png|RTENOTITLE]]....................(8.188)<br/><br/>''L''<sub>''w''</sub> is the completed length of the well, and k y is the permeability in the direction parallel to the wellbore. The actual end is the lesser of the two times calculated from '''Eqs. 8.187''' and '''8.188'''. It is helpful to check the expected duration of the early-radial flow regime after estimating the parameters necessary to make these calculations.<br/><br/>'''Eq. 8.186''' suggests that possible radial flow on the diagnostic plot be identified and then bottomhole flowing pressure be plotted against time during the appropriate time range on semilog coordinates. The slope of the straight line that results is<br/><br/>[[File:Vol5 page 0810 eq 003.png|RTENOTITLE]]....................(8.189)<br/><br/>The group [[File:Vol5 page 0810 inline 001.png|RTENOTITLE]] can be found from the slope, ''m''<sub>erf</sub>:<br/><br/>[[File:Vol5 page 0810 eq 004.png|RTENOTITLE]]....................(8.190)<br/><br/>Effective completed length of the well must be known to make this calculation. This is not necessarily the same as the perforated or completed length of the well. Some sections of the well may not produce at all.<br/><br/>The equation for calculating the altered permeability skin, ''s''<sub>''d''</sub>, for early-radial flow is<br/><br/>[[File:Vol5 page 0810 eq 005.png|RTENOTITLE]]....................(8.191)<br/><br/>When analyzing a buildup test rather than a constant-rate flow test, plot the HTR or equivalent time on the horizontal axis of the semilog plot, and then plot shut-in or equivalent time on the vertical axis. Note that this plotting is correct only if (''t''<sub>''p''</sub> + Δ''t'') and Δ''t'' appear in this time period simultaneously; that is, radial flow must exist at both time (''t''<sub>''p''</sub> + Δ''t'') and time Δ''t''. This is unlikely because radial-flow regime may exist at time Δ''t'', but a different flow regime is likely at time (''t''<sub>''p''</sub> + Δ''t'').
<br>
+
 
<br>
 
The shapes that may appear in a drawdown test (which is the basis of '''Fig. 8.101''') may not appear in a buildup test because of the complex superposition of flow regimes. For example, a test would have to be in linear flow both at time (''t''<sub>''p''</sub> + Δ''t'') and at time Δ t to ensure appearance of a derivative with half-slope; this is highly unlikely. The best way to solve the problem is to ensure that a buildup test on a horizontal well is run with a producing time, ''t''<sub>''p''</sub>, much greater than the maximum shut-in time in the test (that is, ''t''<sub>''p''</sub> > 10 Δ''t'' max ).  
 
<br>
 
<br>
 
'''Table 8.A-2''' (see Appendix) summarizes the working equations for permeability, skin, and start and end of each of the recognized flow regimes. Different investigators have found different equations for start and end of various flow regimes, especially linear flow regimes. This is partly because of a difference of assumptions about flow into the wellbore. Uniform flux or infinite conductivity models are common; neither is rigorously applicable in practice. <ref name="r38" /> In this section, the equations for duration of flow regimes derived by Odeh and Babu<ref name="r39" /> are used. This model assumes uniform flux into the wellbore.  
 
<br>
 
<br>
 
'''''Early-Radial Flow.''''' Early-radial flow is similar to the radial flow period in a vertical well ('''Fig. 8.96'''). The governing equation for this flow regime is
 
<br>
 
<br>
 
[[File:Vol5 page 0809 eq 001.png]]....................(8.186)
 
<br>
 
<br>
 
Data for this period may be masked by wellbore storage effects, but, when present, they may be analyzed on a semilog plot.  
 
<br>
 
<br>
 
The early-radial flow regime may in theory start at time zero, in absence of wellbore storage effects. The end of the early-radial flow regime may occur when the transient reaches a vertical boundary or when flow comes from beyond the end of the wellbore. The end of the period is the smaller of these two values.  
 
<br>
 
<br>
 
'''Eq. 8.187'''<ref name="r39" /> says that the period must end when the transient reaches the nearest boundary, ''d''<sub>''z''</sub>, from the well. This equation includes the permeability in the vertical direction:
 
<br>
 
<br>
 
[[File:Vol5 page 0810 eq 001.png]]....................(8.187)
 
<br>
 
<br>
 
The radial flow regime may also end when flow from beyond the end of the wellbore becomes important. '''Eq. 8.188''' gives the time by
 
<br>
 
<br>
 
[[File:Vol5 page 0810 eq 002.png]]....................(8.188)
 
<br>
 
<br>
 
''L''<sub>''w''</sub> is the completed length of the well, and k y is the permeability in the direction parallel to the wellbore. The actual end is the lesser of the two times calculated from '''Eqs. 8.187''' and '''8.188'''. It is helpful to check the expected duration of the early-radial flow regime after estimating the parameters necessary to make these calculations.  
 
<br>
 
<br>
 
'''Eq. 8.186''' suggests that possible radial flow on the diagnostic plot be identified and then bottomhole flowing pressure be plotted against time during the appropriate time range on semilog coordinates. The slope of the straight line that results is
 
<br>
 
<br>
 
[[File:Vol5 page 0810 eq 003.png]]....................(8.189)
 
<br>
 
<br>
 
The group [[File:Vol5 page 0810 inline 001.png]] can be found from the slope, ''m''<sub>erf</sub>:
 
<br>
 
<br>
 
[[File:Vol5 page 0810 eq 004.png]]....................(8.190)
 
<br>
 
<br>
 
Effective completed length of the well must be known to make this calculation. This is not necessarily the same as the perforated or completed length of the well. Some sections of the well may not produce at all.  
 
<br>
 
<br>
 
The equation for calculating the altered permeability skin, ''s''<sub>''d''</sub>, for early-radial flow is
 
<br>
 
<br>
 
[[File:Vol5 page 0810 eq 005.png]]....................(8.191)
 
<br>
 
<br>
 
When analyzing a buildup test rather than a constant-rate flow test, plot the HTR or equivalent time on the horizontal axis of the semilog plot, and then plot shut-in or equivalent time on the vertical axis. Note that this plotting is correct only if (''t''<sub>''p''</sub> + Δ''t'') and Δ''t'' appear in this time period simultaneously; that is, radial flow must exist at both time (''t''<sub>''p''</sub> + Δ''t'') and time Δ''t''. This is unlikely because radial-flow regime may exist at time Δ''t'', but a different flow regime is likely at time (''t''<sub>''p''</sub> + Δ''t'').  
 
<br>
 
 
----
 
----
<br>
+
 
'''''Example 8.2: Well Erf-1'''''
+
<br/>'''''Example 8.2: Well Erf-1''''' For drawdown test data from Well Erf-1, <ref name="r39">_</ref> the diagnostic plot indicates the data from approximately 0.24 to 24 hours may be in early-radial flow. The following information is available for this well: ''q'' = 800 STB/D, ''μ'' = 1 cp, ''B'' = 1.25 RB/STB, ''r''<sub>''w''</sub> = 0.25 ft, ''ϕ'' = 0.2, ''c''<sub>''t''</sub> = 15×10 –6 psi<sup>–1</sup>, centered in box-shaped drainage area, ''h'' = 200 ft, ''b''<sub>''H''</sub> = 4,000 ft, and ''a''<sub>''H''</sub> = 2,000 ft, ''L''<sub>''w''</sub> =1,000 ft, and, from analysis of data in early linear flow regime, ''k''<sub>''x''</sub> = 200 md. '''Table 8.3''' shows the pressure change data for 0.24 to 24 hours.<br/><br/><gallery widths="300px" heights="200px">
For drawdown test data from Well Erf-1, <ref name="r39" /> the diagnostic plot indicates the data from approximately 0.24 to 24 hours may be in early-radial flow. The following information is available for this well: ''q'' = 800 STB/D, ''μ'' = 1 cp, ''B'' = 1.25 RB/STB, ''r''<sub>''w''</sub> = 0.25 ft, ''ϕ'' = 0.2, ''c''<sub>''t''</sub> = 15×10 –6 psi<sup>–1</sup>, centered in box-shaped drainage area, ''h'' = 200 ft, ''b''<sub>''H''</sub> = 4,000 ft, and ''a''<sub>''H''</sub> = 2,000 ft, ''L''<sub>''w''</sub> =1,000 ft, and, from analysis of data in early linear flow regime, ''k''<sub>''x''</sub> = 200 md. '''Table 8.3''' shows the pressure change data for 0.24 to 24 hours.  
 
<br>
 
<br>
 
<gallery widths=300px heights=200px>
 
 
File:Vol5 Page 0812 Image 0002.png|'''Table 8.3'''
 
File:Vol5 Page 0812 Image 0002.png|'''Table 8.3'''
 
</gallery>
 
</gallery>
<br>
 
  
Plot (''p''<sub>''i''</sub> − ''p''<sub>''wf''</sub> ) = Δ''p'' vs. ''t'' on semilog coordinates ('''Fig. 8.102'''). The plot results in a straight line with a slope of 8 psi/cycle. In '''Fig. 8.102''', at ''t'' = 2.4 hours, the points begin to deviate from the straight line, as expected from calculations for flow regime duration that follow. The pressure change at 1 hour is 39 psia. Using the slope of 8 and '''Eq. 8.190''',
+
Plot (''p''<sub>''i''</sub> − ''p''<sub>''wf''</sub> ) = Δ''p'' vs. ''t'' on semilog coordinates ('''Fig. 8.102'''). The plot results in a straight line with a slope of 8 psi/cycle. In '''Fig. 8.102''', at ''t'' = 2.4 hours, the points begin to deviate from the straight line, as expected from calculations for flow regime duration that follow. The pressure change at 1 hour is 39 psia. Using the slope of 8 and '''Eq. 8.190''',<br/><br/>[[File:Vol5 page 0811 eq 001.png|RTENOTITLE]]<br/><br/>Thus, because ''k''<sub>''x''</sub> = 200 md, ''k''<sub>''z''</sub> = 2 md. Using the value of 39 for Δ''p''<sub>1''hr''</sub> from '''Fig. 8.102''', skin from '''Eq. 8.191''' is<br/><br/>[[File:Vol5 page 0811 eq 002.png|RTENOTITLE]]<br/><br/>The start of the early-radial flow regime is controlled by wellbore storage, which appears to have vanished at times earlier than 0.24 hours in this example. The end of the early radial flow regime is expected at the lesser of the two values derived from '''Eqs. 8.187''' and '''8.188'''. For a centered well, ''d''<sub>''z''</sub> = ''h''/2 = 100 ft, and '''Eq. 8.187''' gives<br/><br/>[[File:Vol5 page 0811 eq 003.png|RTENOTITLE]]<br/><br/>Assuming ''k''<sub>''y''</sub> = ''k''<sub>''x''</sub> = 200 md, '''Eq. 8.188''' gives<br/><br/>[[File:Vol5 page 0811 eq 004.png|RTENOTITLE]]<br/><br/>Thus, expect the early-radial flow regime to end at approximately 1.875 hours, which is the smaller value and is consistent with observed test data.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0811 eq 001.png]]
 
<br>
 
<br>
 
Thus, because ''k''<sub>''x''</sub> = 200 md, ''k''<sub>''z''</sub> = 2 md. Using the value of 39 for Δ''p''<sub>1''hr''</sub> from '''Fig. 8.102''', skin from '''Eq. 8.191''' is
 
<br>
 
<br>
 
[[File:Vol5 page 0811 eq 002.png]]
 
<br>
 
<br>
 
The start of the early-radial flow regime is controlled by wellbore storage, which appears to have vanished at times earlier than 0.24 hours in this example. The end of the early radial flow regime is expected at the lesser of the two values derived from '''Eqs. 8.187''' and '''8.188'''. For a centered well, ''d''<sub>''z''</sub> = ''h''/2 = 100 ft, and '''Eq. 8.187''' gives
 
<br>
 
<br>
 
[[File:Vol5 page 0811 eq 003.png]]
 
<br>
 
<br>
 
Assuming ''k''<sub>''y''</sub> = ''k''<sub>''x''</sub> = 200 md, '''Eq. 8.188''' gives
 
<br>
 
<br>
 
[[File:Vol5 page 0811 eq 004.png]]
 
<br>
 
<br>
 
Thus, expect the early-radial flow regime to end at approximately 1.875 hours, which is the smaller value and is consistent with observed test data.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0812 Image 0001.png|'''Fig. 8.102 – Early-radial flow is indicated by semilog straight line for well Erf-1.'''
 
File:vol5 Page 0812 Image 0001.png|'''Fig. 8.102 – Early-radial flow is indicated by semilog straight line for well Erf-1.'''
</gallery>
+
</gallery><br/>
<br/>
+
 
<br>
 
 
----
 
----
<br>
+
 
 +
 
  
 
=== Hemiradial Flow ===
 
=== Hemiradial Flow ===
  
The hemiradial flow period ('''Fig. 8.97''') will occur only when the well is close to one of the vertical boundaries (either the upper or the lower boundaries) and is analogous to a vertical well near a fault. The governing equation is<ref name="r38" />
+
The hemiradial flow period ('''Fig. 8.97''') will occur only when the well is close to one of the vertical boundaries (either the upper or the lower boundaries) and is analogous to a vertical well near a fault. The governing equation is<ref name="r38">_</ref><br/><br/>[[File:Vol5 page 0812 eq 001.png|RTENOTITLE]]....................(8.192)<br/><br/>A horizontal derivative on the diagnostic plot identifies hemiradial flow. If data appear to fall into this flow regime, a straight line on a semilog plot would provide more confidence that radial flow has been identified. Consistency checks in the analysis coupled with a well survey will be required to distinguish hemiradial flow from early radial flow.<br/><br/>The time range in which the analysis for hemiradial flow is valid begins after the closest vertical boundary, ''d''<sub>''z''</sub>, affects the data and before the farthest boundary, ''D''<sub>''z''</sub>, affects them. In the absence of wellbore storage, the start of hemiradial flow is given by<br/><br/>[[File:Vol5 page 0812 eq 002.png|RTENOTITLE]]....................(8.193)<br/><br/>Note that the start of the hemiradial flow regime involves the shortest distance to a vertical boundary and the permeability in the vertical direction. However, wellbore storage will most likely determine the actual start of hemiradial flow.<br/><br/>The end of hemiradial flow occurs when pressure is affected by the farther vertical boundary or flow from beyond the ends of the wellbore, whichever occurs first. It is the smaller of the times calculated using '''Eqs. 8.194''' and '''8.195'''. If the hemiradial flow regime ends when pressure reaches the farthest vertical boundary, it depends on the distance, ''D''<sub>''z''</sub>, and the vertical permeability, ''k''<sub>''z''</sub>:<br/><br/>[[File:Vol5 page 0813 eq 001.png|RTENOTITLE]]....................(8.194)<br/><br/>When the appearance of end effects—flow from beyond the ends of the wellbore—causes the end of the hemiradial flow regime to appear, the end of the flow regime occurs when<br/><br/>[[File:Vol5 page 0813 eq 002.png|RTENOTITLE]]....................(8.195)<br/><br/>The completed length of the well, ''L''<sub>''w''</sub>, and the permeability ''k''<sub>''y''</sub>, parallel to the wellbore appear in this equation. These parameters determine when enough flow has come from beyond the ends of the wellbore to distort the radial flow pattern that appeared earlier.<br/><br/>[[File:Vol5 page 0813 eq 003.png|RTENOTITLE]]....................(8.196)<br/><br/>gives the slope of the semilog straight line for semiradial flow, ''m''<sub>hrf</sub>. The multiplier, 325.2, is twice the multiplier for early-radial flow. The equation to estimate the damage skin factor is also similar to that for radial flow but has a multiplier that differs by a factor of two.<br/><br/>[[File:Vol5 page 0813 eq 004.png|RTENOTITLE]]....................(8.197)<br/><br/>The equations relating slope and permeability and the equation for skin are similar in a buildup test to those for a drawdown test. The pressure change in the equation for skin is [''p''<sub>1hr</sub> − ''p''<sub>''wf''</sub> (Δ''t'' = 0)]. Semilog plots of buildup test data from the hemiradial flow regime cannot be analyzed rigorously using data from a Horner plot unless the pressure data at (''t''<sub>''p''</sub> + Δ''t'') and at time Δ''t'' are simultaneously in this flow regime. As a practical matter, the hemiradial flow regime is likely to appear clearly in the buildup test only when the producing time is much greater that the shut-in time.
<br>
 
<br>
 
[[File:Vol5 page 0812 eq 001.png]]....................(8.192)
 
<br>
 
<br>
 
A horizontal derivative on the diagnostic plot identifies hemiradial flow. If data appear to fall into this flow regime, a straight line on a semilog plot would provide more confidence that radial flow has been identified. Consistency checks in the analysis coupled with a well survey will be required to distinguish hemiradial flow from early radial flow.  
 
<br>
 
<br>
 
The time range in which the analysis for hemiradial flow is valid begins after the closest vertical boundary, ''d''<sub>''z''</sub>, affects the data and before the farthest boundary, ''D''<sub>''z''</sub>, affects them. In the absence of wellbore storage, the start of hemiradial flow is given by
 
<br>
 
<br>
 
[[File:Vol5 page 0812 eq 002.png]]....................(8.193)
 
<br>
 
<br>
 
Note that the start of the hemiradial flow regime involves the shortest distance to a vertical boundary and the permeability in the vertical direction. However, wellbore storage will most likely determine the actual start of hemiradial flow.  
 
<br>
 
<br>
 
The end of hemiradial flow occurs when pressure is affected by the farther vertical boundary or flow from beyond the ends of the wellbore, whichever occurs first. It is the smaller of the times calculated using '''Eqs. 8.194''' and '''8.195'''. If the hemiradial flow regime ends when pressure reaches the farthest vertical boundary, it depends on the distance, ''D''<sub>''z''</sub>, and the vertical permeability, ''k''<sub>''z''</sub>:
 
<br>
 
<br>
 
[[File:Vol5 page 0813 eq 001.png]]....................(8.194)
 
<br>
 
<br>
 
When the appearance of end effects—flow from beyond the ends of the wellbore—causes the end of the hemiradial flow regime to appear, the end of the flow regime occurs when
 
<br>
 
<br>
 
[[File:Vol5 page 0813 eq 002.png]]....................(8.195)
 
<br>
 
<br>
 
The completed length of the well, ''L''<sub>''w''</sub>, and the permeability ''k''<sub>''y''</sub>, parallel to the wellbore appear in this equation. These parameters determine when enough flow has come from beyond the ends of the wellbore to distort the radial flow pattern that appeared earlier.
 
<br>
 
<br>
 
[[File:Vol5 page 0813 eq 003.png]]....................(8.196)
 
<br>
 
<br>
 
gives the slope of the semilog straight line for semiradial flow, ''m''<sub>hrf</sub>. The multiplier, 325.2, is twice the multiplier for early-radial flow. The equation to estimate the damage skin factor is also similar to that for radial flow but has a multiplier that differs by a factor of two.
 
<br>
 
<br>
 
[[File:Vol5 page 0813 eq 004.png]]....................(8.197)
 
<br>
 
<br>
 
The equations relating slope and permeability and the equation for skin are similar in a buildup test to those for a drawdown test. The pressure change in the equation for skin is [''p''<sub>1hr</sub> − ''p''<sub>''wf''</sub> (Δ''t'' = 0)]. Semilog plots of buildup test data from the hemiradial flow regime cannot be analyzed rigorously using data from a Horner plot unless the pressure data at (''t''<sub>''p''</sub> + Δ''t'') and at time Δ''t'' are simultaneously in this flow regime. As a practical matter, the hemiradial flow regime is likely to appear clearly in the buildup test only when the producing time is much greater that the shut-in time.  
 
  
 
=== Early Linear Flow ===
 
=== Early Linear Flow ===
  
The governing equation for early-linear flow is<ref name="r38" />
+
The governing equation for early-linear flow is<ref name="r38">_</ref><br/><br/>[[File:Vol5 page 0813 eq 005.png|RTENOTITLE]]....................(8.198)<br/><br/>The "convergence skin," ''s''<sub>''c''</sub>, is discussed later in this section. The start of the early-linear flow regime ('''Fig. 8.98''') depends on the farthest distance to a vertical boundary, ''D''<sub>''z''</sub>, and the vertical permeability, ''k''<sub>''z''</sub>.<ref name="r39">_</ref><br/><br/>[[File:Vol5 page 0814 eq 001.png|RTENOTITLE]]....................(8.199)<br/><br/>Not until flow reaches that farthest vertical boundary can a linear flow pattern begin toward the well. This flow period ends when fluids flow from beyond the ends of the wellbore. Thus,<br/><br/>[[File:Vol5 page 0814 eq 002.png|RTENOTITLE]]....................(8.200)<br/><br/>Notice that the end depends on the effective completed length of the well, ''L''<sub>''w''</sub>, and on the permeability in the direction parallel to the well. This is the time in which end effects—flow beyond the ends of the well—begin to significantly distort the linear flow pattern.<br/><br/>The early-linear flow regime is identified in a drawdown test with a half-slope for the derivative. (Because of the skin effect, the pressure change curve on the diagnostic plot will only approach a half-slope asymptotically.) For data identified as being in this flow regime, plot pressure against the square root of time.<br/><br/>The slope of the straight line on such a plot, ''m''<sub>elf</sub>, can be used to estimate the square-root of ''k''<sub>''x''</sub>, the horizontal permeability perpendicular to the well:<br/><br/>[[File:Vol5 page 0814 eq 003.png|RTENOTITLE]]....................(8.201)<br/><br/>To calculate the damage skin,<br/><br/>[[File:Vol5 page 0814 eq 004.png|RTENOTITLE]]....................(8.202)<br/><br/>This equation includes a convergence skin, ''s''<sub>''c''</sub>, which is<ref name="r39">_</ref><br/><br/>[[File:Vol5 page 0814 eq 005.png|RTENOTITLE]]....................(8.203)<br/><br/>This convergence skin is an additional pressure drop that acts like a skin effect caused by flow moving from throughout the entire formation until it converges down to the small wellbore in the middle of the formation ('''Fig. 8.103'''). This convergence skin is defined in terms of the ratio of the permeability in the ''x''-direction, which is perpendicular to the wellbore, to the vertical permeability. It also involves the distance to the nearest vertical boundary, ''d''<sub>''z''</sub>, and the net pay thickness, ''h''.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0813 eq 005.png]]....................(8.198)
 
<br>
 
<br>
 
The "convergence skin," ''s''<sub>''c''</sub>, is discussed later in this section. The start of the early-linear flow regime ('''Fig. 8.98''') depends on the farthest distance to a vertical boundary, ''D''<sub>''z''</sub>, and the vertical permeability, ''k''<sub>''z''</sub>.<ref name="r39" />
 
<br>
 
<br>
 
[[File:Vol5 page 0814 eq 001.png]]....................(8.199)
 
<br>
 
<br>
 
Not until flow reaches that farthest vertical boundary can a linear flow pattern begin toward the well. This flow period ends when fluids flow from beyond the ends of the wellbore. Thus,
 
<br>
 
<br>
 
[[File:Vol5 page 0814 eq 002.png]]....................(8.200)
 
<br>
 
<br>
 
Notice that the end depends on the effective completed length of the well, ''L''<sub>''w''</sub>, and on the permeability in the direction parallel to the well. This is the time in which end effects—flow beyond the ends of the well—begin to significantly distort the linear flow pattern.  
 
<br>
 
<br>
 
The early-linear flow regime is identified in a drawdown test with a half-slope for the derivative. (Because of the skin effect, the pressure change curve on the diagnostic plot will only approach a half-slope asymptotically.) For data identified as being in this flow regime, plot pressure against the square root of time.  
 
<br>
 
<br>
 
The slope of the straight line on such a plot, ''m''<sub>elf</sub>, can be used to estimate the square-root of ''k''<sub>''x''</sub>, the horizontal permeability perpendicular to the well:
 
<br>
 
<br>
 
[[File:Vol5 page 0814 eq 003.png]]....................(8.201)
 
<br>
 
<br>
 
To calculate the damage skin,
 
<br>
 
<br>
 
[[File:Vol5 page 0814 eq 004.png]]....................(8.202)
 
<br>
 
<br>
 
This equation includes a convergence skin, ''s''<sub>''c''</sub>, which is<ref name="r39" />
 
<br>
 
<br>
 
[[File:Vol5 page 0814 eq 005.png]]....................(8.203)
 
<br>
 
<br>
 
This convergence skin is an additional pressure drop that acts like a skin effect caused by flow moving from throughout the entire formation until it converges down to the small wellbore in the middle of the formation ('''Fig. 8.103'''). This convergence skin is defined in terms of the ratio of the permeability in the ''x''-direction, which is perpendicular to the wellbore, to the vertical permeability. It also involves the distance to the nearest vertical boundary, ''d''<sub>''z''</sub>, and the net pay thickness, ''h''.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0815 Image 0001.png|'''Fig. 8.103 – Convergence skin accounts for increased pressure drop as flow deviates from the full vertical thickness of the reservoir into a radial pattern as it enters the well.'''
 
File:vol5 Page 0815 Image 0001.png|'''Fig. 8.103 – Convergence skin accounts for increased pressure drop as flow deviates from the full vertical thickness of the reservoir into a radial pattern as it enters the well.'''
 
</gallery>
 
</gallery>
<br/>
 
  
Kuchuk<ref name="r40" /> derived a different equation for convergence skin. As a practical matter, the Odeh-Babu and Kuchuk equations lead to the same result. When there has been a single rate preceding shut-in during early-linear flow, the buildup pressure is plotted against [[File:Vol5 page 0814 inline 001.png]] on a Cartesian plot, which is sometimes called a tandem root plot. The permeability, ''k''<sub>''x''</sub>, is calculated from the slope, ''m''<sub>elf</sub>, of the plot and '''Eq. 8.201'''. ''k''<sub>''x''</sub> has the same relationship to the slope that existed in a drawdown test. Skin for this flow regime is calculated with '''Eqs. 8.202''' and '''8.203'''.  
+
Kuchuk<ref name="r40">_</ref> derived a different equation for convergence skin. As a practical matter, the Odeh-Babu and Kuchuk equations lead to the same result. When there has been a single rate preceding shut-in during early-linear flow, the buildup pressure is plotted against [[File:Vol5 page 0814 inline 001.png|RTENOTITLE]] on a Cartesian plot, which is sometimes called a tandem root plot. The permeability, ''k''<sub>''x''</sub>, is calculated from the slope, ''m''<sub>elf</sub>, of the plot and '''Eq. 8.201'''. ''k''<sub>''x''</sub> has the same relationship to the slope that existed in a drawdown test. Skin for this flow regime is calculated with '''Eqs. 8.202''' and '''8.203'''.<br/><br/>Plots of buildup data from the early-linear flow regime cannot be analyzed rigorously with a plot of ''p''<sub>''ws''</sub> vs. [[File:Vol5 page 0814 inline 001.png|RTENOTITLE]] (that is, the tandem-root plot) unless data at (''t''<sub>''p''</sub> + Δ''t'') and Δ''t'' are simultaneously in this flow regime—highly unlikely—or unless ''t''<sub>''p''</sub> is much greater than Δ''t'', in which case simply plot ''p''<sub>''ws''</sub> vs. [[File:Vol5 page 0815 inline 001.png|RTENOTITLE]]. Little error results from ignoring the (''t''<sub>''p''</sub> + Δ''t'') term, which is essentially constant.
<br>
+
 
<br>
 
Plots of buildup data from the early-linear flow regime cannot be analyzed rigorously with a plot of ''p''<sub>''ws''</sub> vs. [[File:Vol5 page 0814 inline 001.png]] (that is, the tandem-root plot) unless data at (''t''<sub>''p''</sub> + Δ''t'') and Δ''t'' are simultaneously in this flow regime—highly unlikely—or unless ''t''<sub>''p''</sub> is much greater than Δ''t'', in which case simply plot ''p''<sub>''ws''</sub> vs. [[File:Vol5 page 0815 inline 001.png]]. Little error results from ignoring the (''t''<sub>''p''</sub> + Δ''t'') term, which is essentially constant.  
 
<br>
 
 
----
 
----
<br>
+
 
'''Example 8.3: Well Elf-2'''
+
<br/>'''Example 8.3: Well Elf-2''' The diagnostic plot for a drawdown test from Well Elf-2<ref name="r39">_</ref> indicates data in the early-linear flow regime because the derivative has a half-slope. The following data apply to this well: ''q'' = 800 STB/D; ''μ'' = 1 cp; ''B'' = 1.25 RB/STB; ''r''<sub>''w''</sub> = 0.25 ft; ''ϕ'' = 0.2; ''c''<sub>''t''</sub> = 15×10<sup>–6</sup> psi<sup>–1</sup>; centered in box-shaped drainage area 100 ft thick, 4,000 ft long, 4,000 ft wide; ''L''<sub>''w''</sub> = 2,500 ft; and, from early radial-flow regime data, ''k''<sub>''x''</sub>''k''<sub>''z''</sub> = 8,000 md<sup>2</sup>. In addition, '''Table 8.4''' shows the pressure-change data for this well.<br/><br/><gallery widths="300px" heights="200px">
The diagnostic plot for a drawdown test from Well Elf-2<ref name="r39" /> indicates data in the early-linear flow regime because the derivative has a half-slope. The following data apply to this well: ''q'' = 800 STB/D; ''μ'' = 1 cp; ''B'' = 1.25 RB/STB; ''r''<sub>''w''</sub> = 0.25 ft; ''ϕ'' = 0.2; ''c''<sub>''t''</sub> = 15×10<sup>–6</sup> psi<sup>–1</sup>; centered in box-shaped drainage area 100 ft thick, 4,000 ft long, 4,000 ft wide; ''L''<sub>''w''</sub> = 2,500 ft; and, from early radial-flow regime data, ''k''<sub>''x''</sub>''k''<sub>''z''</sub> = 8,000 md<sup>2</sup>. In addition, '''Table 8.4''' shows the pressure-change data for this well.  
 
<br>
 
<br>
 
<gallery widths=300px heights=200px>
 
 
File:Vol5 Page 0817 Image 0002.png|'''Table 8.4'''
 
File:Vol5 Page 0817 Image 0002.png|'''Table 8.4'''
 
</gallery>
 
</gallery>
<br>
 
  
A plot of pressure change vs. the square root of time ('''Fig. 8.104''') indicates early linear flow. The final point on the straight line is at a time of approximately 24 hours, but there may have been some deviation from the straight line by this time. From the slope of the straight line, ''m''<sub>elf</sub> = 0.934 psi/hr<sup>1/2</sup> and '''Eq. 8.201''', calculate the permeability in the horizontal plane perpendicular to the wellbore.
+
A plot of pressure change vs. the square root of time ('''Fig. 8.104''') indicates early linear flow. The final point on the straight line is at a time of approximately 24 hours, but there may have been some deviation from the straight line by this time. From the slope of the straight line, ''m''<sub>elf</sub> = 0.934 psi/hr<sup>1/2</sup> and '''Eq. 8.201''', calculate the permeability in the horizontal plane perpendicular to the wellbore.<br/><br/>[[File:Vol5 page 0815 eq 001.png|RTENOTITLE]]<br/><br/>or ''k''<sub>''x''</sub> = (20.1)<sup>2</sup> = 404 md.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0815 eq 001.png]]
 
<br>
 
<br>
 
or ''k''<sub>''x''</sub> = (20.1)<sup>2</sup> = 404 md.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0817 Image 0001.png|'''Fig. 8.104 – Early-linear flow is indicated by straight line on square-root-of-time plot for well Elf-2.'''
 
File:vol5 Page 0817 Image 0001.png|'''Fig. 8.104 – Early-linear flow is indicated by straight line on square-root-of-time plot for well Elf-2.'''
 
</gallery>
 
</gallery>
<br/>
 
  
Analysis of data from the early radial flow regime indicated that ''k''<sub>''x''</sub>''k''<sub>''z''</sub> is 8,000 md<sup>2</sup>; thus, ''k''<sub>''z''</sub> is approximately 19.8 md. To calculate ''s''<sub>''d''</sub>, use '''Eqs. 8.202''' and '''8.203''', noting that the value for (''p''<sub>''i''</sub> − ''p''<sub>''wf''</sub> ) = 3.1 at ''t'' = 0.
+
Analysis of data from the early radial flow regime indicated that ''k''<sub>''x''</sub>''k''<sub>''z''</sub> is 8,000 md<sup>2</sup>; thus, ''k''<sub>''z''</sub> is approximately 19.8 md. To calculate ''s''<sub>''d''</sub>, use '''Eqs. 8.202''' and '''8.203''', noting that the value for (''p''<sub>''i''</sub> − ''p''<sub>''wf''</sub> ) = 3.1 at ''t'' = 0.<br/><br/>[[File:Vol5 page 0815 eq 002.png|RTENOTITLE]] [[File:Vol5 page 0816 eq 001.png|RTENOTITLE]]<br/><br/>Then, ''s''<sub>''d''</sub> = 4.91 – 4.91 = 0.<br/><br/>Check the expected time range for the early-linear flow regime.<br/><br/>Using ''D''<sub>''z''</sub> = ''h''/2 = 50 ft and ''k''<sub>''z''</sub> = 20 md, calculate the beginning of linear flow with '''Eq. 8.199''':<br/><br/>[[File:Vol5 page 0816 eq 002.png|RTENOTITLE]]<br/><br/>Assuming ''k''<sub>''y''</sub> = ''k''<sub>''x''</sub> at 400 md, use '''Eq. 8.200''' to find the end of early linear flow.<br/><br/>[[File:Vol5 page 0816 eq 003.png|RTENOTITLE]]<br/><br/>These limits are reasonably consistent with the time range analyzed assuming early linear flow.
<br>
+
 
<br>
 
[[File:Vol5 page 0815 eq 002.png]]
 
[[File:Vol5 page 0816 eq 001.png]]
 
<br>
 
<br>
 
Then, ''s''<sub>''d''</sub> = 4.91 – 4.91 = 0.  
 
<br>
 
<br>
 
Check the expected time range for the early-linear flow regime.  
 
<br>
 
<br>
 
Using ''D''<sub>''z''</sub> = ''h''/2 = 50 ft and ''k''<sub>''z''</sub> = 20 md, calculate the beginning of linear flow with '''Eq. 8.199''':
 
<br>
 
<br>
 
[[File:Vol5 page 0816 eq 002.png]]
 
<br>
 
<br>
 
Assuming ''k''<sub>''y''</sub> = ''k''<sub>''x''</sub> at 400 md, use '''Eq. 8.200''' to find the end of early linear flow.
 
<br>
 
<br>
 
[[File:Vol5 page 0816 eq 003.png]]
 
<br>
 
<br>
 
These limits are reasonably consistent with the time range analyzed assuming early linear flow.  
 
<br>
 
 
----
 
----
<br>
+
 
 +
 
  
 
=== Late Pseudoradial ===
 
=== Late Pseudoradial ===
  
The governing equation for late-pseudoradial flow is<ref name="r38" />
+
The governing equation for late-pseudoradial flow is<ref name="r38">_</ref><br/><br/>[[File:Vol5 page 0816 eq 004.png|RTENOTITLE]]....................(8.204)<br/><br/>The late-pseudoradial flow period occurs only if <ref name="r39">_</ref><br/><br/>[[File:Vol5 page 0816 eq 005.png|RTENOTITLE]]....................(8.205)<br/><br/>Here, ''b''<sub>''H''</sub> is the dimension of the reservoir parallel to the wellbore. As long as the completed length of the well is relatively short compared with the length of the drainage area late-pseudoradial flow can occur.<br/><br/>The start of this flow period occurs when fluid flows from well beyond the ends of the wellbore ('''Fig. 8.99'''). It is approximated with<ref name="r39">_</ref><br/><br/>[[File:Vol5 page 0816 eq 006.png|RTENOTITLE]]....................(8.206)<br/><br/>This starting time depends on the completed length of the well, ''L''<sub>''w''</sub>, and on the permeability in the direction of the well, ''k''<sub>''y''</sub>. The end of this period, like others in this section, is approximated by the minimum of the results of two calculations. The first,<br/><br/>[[File:Vol5 page 0817 eq 001.png|RTENOTITLE]]....................(8.207)<br/><br/>depends on ''d''<sub>''y''</sub> and the length of the wellbore along with ''k''<sub>''y''</sub>, the permeability in the direction parallel to the wellbore. This is the time at which horizontal boundary effects first appear.<br/><br/>The other equation gives a time at which the radial-flow pattern begins to be distorted depending on the shortest distance, ''d''<sub>''x''</sub>, from the well to a boundary perpendicular to the wellbore and on ''k''<sub>''x''</sub>, the permeability in that direction.<br/><br/>[[File:Vol5 page 0818 eq 001.png|RTENOTITLE]]....................(8.208)<br/><br/>Whenever boundary effects first appear, whether in a direction that is parallel to the well or perpendicular to the axis of the well, the late-pseudoradial flow period will end.<br/><br/>The diagnostic plot helps identify the late-pseudoradial flow regime with the characteristic horizontal derivative. For data in the appropriate time range, prepare a semilog plot of pressure against time for a drawdown test. The slope of this plot will be ''m''<sub>prf</sub> and the relationship between that slope and the square root of ''k''<sub>''x''</sub>''k''<sub>''y''</sub>, or the permeabilities in the horizontal plane, is given by<br/><br/>[[File:Vol5 page 0818 eq 002.png|RTENOTITLE]]....................(8.209)<br/><br/>The skin equation is similar in form to those seen before:<br/><br/>[[File:Vol5 page 0818 eq 003.png|RTENOTITLE]]....................(8.210)<br/><br/>Again, the total skin depends on Δ''p''<sub>1hr</sub>. The convergence skin ('''Eq. 8.203''') is subtracted from the "total" skin to determine the damage skin.<br/><br/>For a buildup test preceded by production at a single rate, plot pressure against the HTR on a semilog graph. Permeability is calculated from '''Eq. 8.209''', the same as for a drawdown test. The skin equation is basically the same as for a drawdown test, except that the Δ''p''<sub>1hr</sub> is now ''p''<sub>1hr</sub> − ''p''<sub>''wf''</sub>. To obtain ''p''*, the extrapolated pressure, extrapolate the semilog straight line to a HTR of unity.<br/><br/>Semilog plots of buildup test data from the late-pseudoradial flow regime cannot be analyzed rigorously using a Horner plot unless pressures at (''t''<sub>''p''</sub> + Δ''t'') and at time Δ''t'' are simultaneously in the pseudoradial flow regime, which is highly unlikely. However, little error appears if the producing time before shut-in is much greater than the maximum shut-in time achieved in the buildup test.
<br>
+
 
<br>
 
[[File:Vol5 page 0816 eq 004.png]]....................(8.204)
 
<br>
 
<br>
 
The late-pseudoradial flow period occurs only if <ref name="r39" />
 
<br>
 
<br>
 
[[File:Vol5 page 0816 eq 005.png]]....................(8.205)
 
<br>
 
<br>
 
Here, ''b''<sub>''H''</sub> is the dimension of the reservoir parallel to the wellbore. As long as the completed length of the well is relatively short compared with the length of the drainage area late-pseudoradial flow can occur.  
 
<br>
 
<br>
 
The start of this flow period occurs when fluid flows from well beyond the ends of the wellbore ('''Fig. 8.99'''). It is approximated with<ref name="r39" />
 
<br>
 
<br>
 
[[File:Vol5 page 0816 eq 006.png]]....................(8.206)
 
<br>
 
<br>
 
This starting time depends on the completed length of the well, ''L''<sub>''w''</sub>, and on the permeability in the direction of the well, ''k''<sub>''y''</sub>. The end of this period, like others in this section, is approximated by the minimum of the results of two calculations. The first,
 
<br>
 
<br>
 
[[File:Vol5 page 0817 eq 001.png]]....................(8.207)
 
<br>
 
<br>
 
depends on ''d''<sub>''y''</sub> and the length of the wellbore along with ''k''<sub>''y''</sub>, the permeability in the direction parallel to the wellbore. This is the time at which horizontal boundary effects first appear.  
 
<br>
 
<br>
 
The other equation gives a time at which the radial-flow pattern begins to be distorted depending on the shortest distance, ''d''<sub>''x''</sub>, from the well to a boundary perpendicular to the wellbore and on ''k''<sub>''x''</sub>, the permeability in that direction.
 
<br>
 
<br>
 
[[File:Vol5 page 0818 eq 001.png]]....................(8.208)
 
<br>
 
<br>
 
Whenever boundary effects first appear, whether in a direction that is parallel to the well or perpendicular to the axis of the well, the late-pseudoradial flow period will end.  
 
<br>
 
<br>
 
The diagnostic plot helps identify the late-pseudoradial flow regime with the characteristic horizontal derivative. For data in the appropriate time range, prepare a semilog plot of pressure against time for a drawdown test. The slope of this plot will be ''m''<sub>prf</sub> and the relationship between that slope and the square root of ''k''<sub>''x''</sub>''k''<sub>''y''</sub>, or the permeabilities in the horizontal plane, is given by
 
<br>
 
<br>
 
[[File:Vol5 page 0818 eq 002.png]]....................(8.209)
 
<br>
 
<br>
 
The skin equation is similar in form to those seen before:
 
<br>
 
<br>
 
[[File:Vol5 page 0818 eq 003.png]]....................(8.210)
 
<br>
 
<br>
 
Again, the total skin depends on Δ''p''<sub>1hr</sub>. The convergence skin ('''Eq. 8.203''') is subtracted from the "total" skin to determine the damage skin.  
 
<br>
 
<br>
 
For a buildup test preceded by production at a single rate, plot pressure against the HTR on a semilog graph. Permeability is calculated from '''Eq. 8.209''', the same as for a drawdown test. The skin equation is basically the same as for a drawdown test, except that the Δ''p''<sub>1hr</sub> is now ''p''<sub>1hr</sub> − ''p''<sub>''wf''</sub>. To obtain ''p''*, the extrapolated pressure, extrapolate the semilog straight line to a HTR of unity.  
 
<br>
 
<br>
 
Semilog plots of buildup test data from the late-pseudoradial flow regime cannot be analyzed rigorously using a Horner plot unless pressures at (''t''<sub>''p''</sub> + Δ''t'') and at time Δ''t'' are simultaneously in the pseudoradial flow regime, which is highly unlikely. However, little error appears if the producing time before shut-in is much greater than the maximum shut-in time achieved in the buildup test.  
 
<br>
 
 
----
 
----
<br>
+
 
'''''Example 8.4: Well Prf-3'''''
+
<br/>'''''Example 8.4: Well Prf-3''''' The diagnostic plot suggests that a constant-rate drawdown test from Well Prf-3<ref name="r39">_</ref> includes data in the late-pseudoradial period. The following data are available from the test: ''q'' = 800 STB/D, ''μ'' = 1 cp, ''B'' = 1.25 RB/STB, ''r''<sub>''w''</sub> = 0.25 ft, ''ϕ'' = 0.2, ''c''<sub>''t''</sub> = 15×10<sup>–6</sup> psi<sup>–1</sup>, ''h'' = 150 ft, ''L''<sub>''w''</sub> = 900 ft, ''a''<sub>''H''</sub> = 5,280 ft, ''b''<sub>''H''</sub> = 5,280 ft, well centered in drainage volume, ''k''<sub>''x''</sub> = 100 md (from analysis of early linear flow), ''k''<sub>''x''</sub>''k''<sub>''z''</sub> = 1,000 md<sup>2</sup>, and ''k''<sub>''z''</sub> = 10 md (from analysis of early radial flow). '''Table 8.5''' gives the pressure change, Δ''p'' = ''p''<font size="4"><sub>''i''</sub></font> – ''p''<sub>''wf''</sub> vs. time.<br/><br/><gallery widths="300px" heights="200px">
The diagnostic plot suggests that a constant-rate drawdown test from Well Prf-3<ref name="r39" /> includes data in the late-pseudoradial period. The following data are available from the test: ''q'' = 800 STB/D, ''μ'' = 1 cp, ''B'' = 1.25 RB/STB, ''r''<sub>''w''</sub> = 0.25 ft, ''ϕ'' = 0.2, ''c''<sub>''t''</sub> = 15×10<sup>–6</sup> psi<sup>–1</sup>, ''h'' = 150 ft, ''L''<sub>''w''</sub> = 900 ft, ''a''<sub>''H''</sub> = 5,280 ft, ''b''<sub>''H''</sub> = 5,280 ft, well centered in drainage volume, ''k''<sub>''x''</sub> = 100 md (from analysis of early linear flow), ''k''<sub>''x''</sub>''k''<sub>''z''</sub> = 1,000 md<sup>2</sup>, and ''k''<sub>''z''</sub> = 10 md (from analysis of early radial flow). '''Table 8.5''' gives the pressure change, Δ''p'' = ''p''<font size= "4"><sub>''i''</sub></font> – ''p''<sub>''wf''</sub> vs. time.  
 
<br>
 
<br>
 
<gallery widths=300px heights=200px>
 
 
File:Vol5 Page 0820 Image 0002.png|'''Table 8.5'''
 
File:Vol5 Page 0820 Image 0002.png|'''Table 8.5'''
 
</gallery>
 
</gallery>
<br>
 
  
A plot of pressure change vs. the logarithm of time ('''Fig. 8.105''') confirms pseudoradial flow. A straight line fits all the data from 192 to 432 hours; the slope of the line, ''m''<sub>prf</sub>, is 15.3 psi/cycle, and Δ''p''<sub>1 hr</sub> = 18.94 psi (extrapolated). Then, from '''Eq. 8.209''',
+
A plot of pressure change vs. the logarithm of time ('''Fig. 8.105''') confirms pseudoradial flow. A straight line fits all the data from 192 to 432 hours; the slope of the line, ''m''<sub>prf</sub>, is 15.3 psi/cycle, and Δ''p''<sub>1 hr</sub> = 18.94 psi (extrapolated). Then, from '''Eq. 8.209''',<br/><br/>[[File:Vol5 page 0818 eq 004.png|RTENOTITLE]]<br/><br/>Thus, [[File:Vol5 page 0818 eq 005.png|RTENOTITLE]]<br/><br/>From Eqs. 8.210 and 8.203 ,<br/><br/>[[File:Vol5 page 0819 eq 001.png|RTENOTITLE]]<br/><br/>Here,<br/><br/>[[File:Vol5 page 0819 eq 002.png|RTENOTITLE]]<br/><br/>The pseudoradial flow regime should start at the time given by '''Eq. 8.206''':<br/><br/>[[File:Vol5 page 0819 eq 003.png|RTENOTITLE]]<br/><br/>It should end at a time given by the lesser of values from '''Eqs. 8.207''' and '''8.208'''. From '''Eq. 8.207''',<br/><br/>[[File:Vol5 page 0819 eq 004.png|RTENOTITLE]]<br/><br/>where ''d<sub>y</sub>'' = 1/2(5,200-900) = 2,190ft for this centered well. From '''Eq. 8.208''',<br/><br/>[[File:Vol5 page 0819 eq 005.png|RTENOTITLE]]<br/><br/>The smaller of these two values is 344 hours, which is thus the expected end of pseudoradial flow. The data on the figure that lie on the straight line show the time range from 192 to 432 hours, which is generally consistent with the expected duration of the flow regime.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
[[File:Vol5 page 0818 eq 004.png]]
 
<br>
 
<br>
 
Thus, [[File:Vol5 page 0818 eq 005.png]]
 
<br>
 
<br>
 
From Eqs. 8.210 and 8.203 ,
 
<br>
 
<br>
 
[[File:Vol5 page 0819 eq 001.png]]
 
<br>
 
<br>
 
Here,
 
<br>
 
<br>
 
[[File:Vol5 page 0819 eq 002.png]]
 
<br>
 
<br>
 
The pseudoradial flow regime should start at the time given by '''Eq. 8.206''':
 
<br>
 
<br>
 
[[File:Vol5 page 0819 eq 003.png]]
 
<br>
 
<br>
 
It should end at a time given by the lesser of values from '''Eqs. 8.207''' and '''8.208'''. From '''Eq. 8.207''',
 
<br>
 
<br>
 
[[File:Vol5 page 0819 eq 004.png]]
 
<br>
 
<br>
 
where ''d<sub>y</sub>'' = 1/2(5,200-900) = 2,190ft for this centered well. From '''Eq. 8.208''',
 
<br>
 
<br>
 
[[File:Vol5 page 0819 eq 005.png]]
 
<br>
 
<br>
 
The smaller of these two values is 344 hours, which is thus the expected end of pseudoradial flow. The data on the figure that lie on the straight line show the time range from 192 to 432 hours, which is generally consistent with the expected duration of the flow regime.  
 
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0820 Image 0001.png|'''Fig. 8.105 – Pseudoradial flow indicated by semilog straight line for well Prf-3.'''
 
File:vol5 Page 0820 Image 0001.png|'''Fig. 8.105 – Pseudoradial flow indicated by semilog straight line for well Prf-3.'''
</gallery>
+
</gallery><br/>
<br/>
+
 
<br>
 
 
----
 
----
<br>
+
 
 +
 
  
 
=== Late-Linear Flow ===
 
=== Late-Linear Flow ===
  
The governing equation for late-linear flow is<ref name="r38" /><ref name="r39" />
+
The governing equation for late-linear flow is<ref name="r38">_</ref><ref name="r39">_</ref><br/><br/>[[File:Vol5 page 0819 eq 006.png|RTENOTITLE]]....................(8.211)<br/><br/>The late-linear flow regime starts after the pressure transient has reached the boundaries in the ''z''- and ''y''-directions, and the flow behavior with regard to these directions has become pseudosteady state, as '''Fig. 8.100''' shows.<br/><br/>The start of this time period is the maximum of two equations. <ref name="r39">_</ref> The first depends on the time to reach the boundary, ''D''<sub>''y''</sub>, beyond the end of the horizontal well. It also depends on the permeability, ''k''<sub>''y''</sub>, in the direction parallel to the wellbore.<br/><br/>[[File:Vol5 page 0820 eq 001.png|RTENOTITLE]]....................(8.212)<br/><br/>Another requirement for the start of the late-linear flow regime is the time to reach the maximum vertical distance, ''D''<sub>''z''</sub>, divided by the vertical permeability:<br/><br/>[[File:Vol5 page 0820 eq 002.png|RTENOTITLE]]....................(8.213)<br/><br/>Usually, the start of the late-linear flow regime is dictated by the time to reach the boundaries in the ''y''-direction. The end of this period is given by the equation<br/><br/>[[File:Vol5 page 0821 eq 001.png|RTENOTITLE]]....................(8.214)<br/><br/>The end of the late-linear flow regime depends on reaching the nearest boundary in the direction perpendicular to the wellbore, which is the distance, ''d''<sub>''x''</sub>, away, and on the permeability ''k''<sub>''x''</sub> in that direction.<br/><br/>Identify the late-linear flow regime by a half-slope on the derivative in the diagnostic plot of drawdown test data. (The pressure change may approach a half-slope asymptotically.) For data that appear to be in this flow regime, plot pressure against the square root of time. From the slope ''m''<sub>llf</sub> of the plot, estimate permeability in the ''x''-direction from<br/><br/>[[File:Vol5 page 0821 eq 002.png|RTENOTITLE]]....................(8.215)<br/><br/>Alternatively, if ''k''<sub>''x''</sub> is known from an early-linear flow regime, estimate ''b''<sub>''H''</sub>, the length of the drainage area, from<br/><br/>[[File:Vol5 page 0821 eq 003.png|RTENOTITLE]]....................(8.216)<br/><br/>This late-linear flow regime is the only period that provides the data to calculate the total skin, ''s'', including the partial-penetration skin, ''s''<sub>''p''</sub>, and the convergence skin, ''s''<sub>''c''</sub>. To calculate the damage skin, ''s''<sub>''d''</sub>, use<br/><br/>[[File:Vol5 page 0821 eq 004.png|RTENOTITLE]]....................(8.217)<br/><br/>The total skin depends on Δ''p''<sub>''t''</sub> = 0. Subtracting the partial penetration skin, ''s''<sub>''p''</sub>, and the convergence skin, ''s''<sub>''c''</sub>, from the total skin yields the damage skin.<br/><br/>The partial-penetration skin is a complex function that is calculated with Eqs. A-25 through A-35 in '''Table 8.A-2'''. For a buildup test, plot pressure against the HTR. From the slope, ''m''<sub>llf</sub>, calculate ''k''<sub>''x''</sub> with '''Eq. 8.215''', exactly the same as for drawdown tests. Or, if ''k''<sub>''x''</sub> is known, estimate the length, ''b''<sub>''H''</sub>, of the drainage area with '''Eq. 8.216'''. Calculate the damage skin, ''s''<sub>''d''</sub>, from a pressure buildup test from '''Eq. 8.217''', where Δ''p''<sub>''t''</sub> = 0 = (''p''<sub>''t''</sub> = 0)<sub>ext</sub> – ''p''<sub>''wf''(''t'' = 0)</sub>.<br/><br/>Note that the same difficulty arises in using superposition to find plotting functions plots of buildup data from the late-linear flow regime as existed with the previous flow regimes. Pressures at both time (''t''<sub>''p''</sub> + Δ''t'') and time Δ''t'' must be in the late linear flow regime for a tandem-root plot to be valid. However, if ''t''<sub>''p''</sub> >> Δ''t''<sub>max</sub>, there is little error.
<br>
+
 
<br>
 
[[File:Vol5 page 0819 eq 006.png]]....................(8.211)
 
<br>
 
<br>
 
The late-linear flow regime starts after the pressure transient has reached the boundaries in the ''z''- and ''y''-directions, and the flow behavior with regard to these directions has become pseudosteady state, as '''Fig. 8.100''' shows.  
 
<br>
 
<br>
 
The start of this time period is the maximum of two equations. <ref name="r39" /> The first depends on the time to reach the boundary, ''D''<sub>''y''</sub>, beyond the end of the horizontal well. It also depends on the permeability, ''k''<sub>''y''</sub>, in the direction parallel to the wellbore.
 
<br>
 
<br>
 
[[File:Vol5 page 0820 eq 001.png]]....................(8.212)
 
<br>
 
<br>
 
Another requirement for the start of the late-linear flow regime is the time to reach the maximum vertical distance, ''D''<sub>''z''</sub>, divided by the vertical permeability:
 
<br>
 
<br>
 
[[File:Vol5 page 0820 eq 002.png]]....................(8.213)
 
<br>
 
<br>
 
Usually, the start of the late-linear flow regime is dictated by the time to reach the boundaries in the ''y''-direction. The end of this period is given by the equation
 
<br>
 
<br>
 
[[File:Vol5 page 0821 eq 001.png]]....................(8.214)
 
<br>
 
<br>
 
The end of the late-linear flow regime depends on reaching the nearest boundary in the direction perpendicular to the wellbore, which is the distance, ''d''<sub>''x''</sub>, away, and on the permeability ''k''<sub>''x''</sub> in that direction.  
 
<br>
 
<br>
 
Identify the late-linear flow regime by a half-slope on the derivative in the diagnostic plot of drawdown test data. (The pressure change may approach a half-slope asymptotically.) For data that appear to be in this flow regime, plot pressure against the square root of time. From the slope ''m''<sub>llf</sub> of the plot, estimate permeability in the ''x''-direction from
 
<br>
 
<br>
 
[[File:Vol5 page 0821 eq 002.png]]....................(8.215)
 
<br>
 
<br>
 
Alternatively, if ''k''<sub>''x''</sub> is known from an early-linear flow regime, estimate ''b''<sub>''H''</sub>, the length of the drainage area, from
 
<br>
 
<br>
 
[[File:Vol5 page 0821 eq 003.png]]....................(8.216)
 
<br>
 
<br>
 
This late-linear flow regime is the only period that provides the data to calculate the total skin, ''s'', including the partial-penetration skin, ''s''<sub>''p''</sub>, and the convergence skin, ''s''<sub>''c''</sub>. To calculate the damage skin, ''s''<sub>''d''</sub>, use
 
<br>
 
<br>
 
[[File:Vol5 page 0821 eq 004.png]]....................(8.217)
 
<br>
 
<br>
 
The total skin depends on Δ''p''<sub>''t''</sub> = 0. Subtracting the partial penetration skin, ''s''<sub>''p''</sub>, and the convergence skin, ''s''<sub>''c''</sub>, from the total skin yields the damage skin.  
 
<br>
 
<br>
 
The partial-penetration skin is a complex function that is calculated with Eqs. A-25 through A-35 in '''Table 8.A-2'''. For a buildup test, plot pressure against the HTR. From the slope, ''m''<sub>llf</sub>, calculate ''k''<sub>''x''</sub> with '''Eq. 8.215''', exactly the same as for drawdown tests. Or, if ''k''<sub>''x''</sub> is known, estimate the length, ''b''<sub>''H''</sub>, of the drainage area with '''Eq. 8.216'''. Calculate the damage skin, ''s''<sub>''d''</sub>, from a pressure buildup test from '''Eq. 8.217''', where Δ''p''<sub>''t''</sub> = 0 = (''p''<sub>''t''</sub> = 0)<sub>ext</sub> – ''p''<sub>''wf''(''t'' = 0)</sub>.  
 
<br>
 
<br>
 
Note that the same difficulty arises in using superposition to find plotting functions plots of buildup data from the late-linear flow regime as existed with the previous flow regimes. Pressures at both time (''t''<sub>''p''</sub> + Δ''t'') and time Δ''t'' must be in the late linear flow regime for a tandem-root plot to be valid. However, if ''t''<sub>''p''</sub> >> Δ''t''<sub>max</sub>, there is little error.  
 
<br>
 
 
----
 
----
<br>
+
 
'''Example 8.5: Well Llf-4'''
+
<br/>'''Example 8.5: Well Llf-4''' The diagnostic plot for a drawdown test from Well Llf-4<ref name="r39">_</ref> appears to include data in the late-linear flow regime (derivative with half-slope). The following data applies to this well: ''q'' = 800 STB/D, ''μ'' = 1 cp, ''B'' = 1.25 RB/STB, ''r''<sub>''w''</sub> = 0.25 ft, ''ϕ'' = 0.2, ''c''<sub>''t''</sub> = 15 × 10<sup>–6</sup> psi<sup>–1</sup> , ''h'' = 150 ft, ''L''<sub>''w''</sub> = 1,000 ft, ''b''<sub>''H''</sub> = 2,000 ft (well centered), ''a''<sub>''H''</sub> = 6,968 ft (well centered), ''D''<sub>''z''</sub> = 85 ft, ''d''<sub>''z''</sub> = 65 ft, ''k''<sub>''x''</sub>''k''<sub>''z''</sub> = 1,000 md<sup>2</sup> (from analysis of early-radial flow), and ''k''<sub>''x''</sub>''k''<sub>''y''</sub> = 5,000 md<sup>2</sup> (from analysis of pseudoradial flow). '''Table 8.6''' gives pressure change, Δ''p'' = ''p''<sub>''i''</sub> – ''p''<sub>''wf''</sub> , data vs. time.<br/><br/><gallery widths="300px" heights="200px">
The diagnostic plot for a drawdown test from Well Llf-4<ref name="r39" /> appears to include data in the late-linear flow regime (derivative with half-slope). The following data applies to this well: ''q'' = 800 STB/D, ''μ'' = 1 cp, ''B'' = 1.25 RB/STB, ''r''<sub>''w''</sub> = 0.25 ft, ''ϕ'' = 0.2, ''c''<sub>''t''</sub> = 15 × 10<sup>–6</sup> psi<sup>–1</sup> , ''h'' = 150 ft, ''L''<sub>''w''</sub> = 1,000 ft, ''b''<sub>''H''</sub> = 2,000 ft (well centered), ''a''<sub>''H''</sub> = 6,968 ft (well centered), ''D''<sub>''z''</sub> = 85 ft, ''d''<sub>''z''</sub> = 65 ft, ''k''<sub>''x''</sub>''k''<sub>''z''</sub> = 1,000 md<sup>2</sup> (from analysis of early-radial flow), and ''k''<sub>''x''</sub>''k''<sub>''y''</sub> = 5,000 md<sup>2</sup> (from analysis of pseudoradial flow). '''Table 8.6''' gives pressure change, Δ''p'' = ''p''<sub>''i''</sub> – ''p''<sub>''wf''</sub> , data vs. time.  
 
<br>
 
<br>
 
<gallery widths=300px heights=200px>
 
 
File:Vol5 Page 0825 Image 0002.png|'''Table 8.6'''
 
File:Vol5 Page 0825 Image 0002.png|'''Table 8.6'''
 
</gallery>
 
</gallery>
<br>
 
  
'''Fig. 8.106''' is a plot of pressure change vs. the square root of time. The straight line on this plot for the entire time range (60 to 240 hours) confirms late-linear flow for this range. The slope of the line is 1.56 psi/hr<sup>1/2</sup>, and the intercept is Δ''p''<sub>''t'' = 0</sub> =28.4 psi.  
+
'''Fig. 8.106''' is a plot of pressure change vs. the square root of time. The straight line on this plot for the entire time range (60 to 240 hours) confirms late-linear flow for this range. The slope of the line is 1.56 psi/hr<sup>1/2</sup>, and the intercept is Δ''p''<sub>''t'' = 0</sub> =28.4 psi.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0825 Image 0001.png|'''Fig. 8.106 – Later linear flow indicated by straight line on square-root-of-time plot for well Llf-4.'''
 
File:vol5 Page 0825 Image 0001.png|'''Fig. 8.106 – Later linear flow indicated by straight line on square-root-of-time plot for well Llf-4.'''
 
</gallery>
 
</gallery>
<br/>
 
  
From '''Eq. 8.215''',
+
From '''Eq. 8.215''',<br/><br/>[[File:Vol5 page 0822 eq 001.png|RTENOTITLE]]<br/><br/>Then, ''k''<sub>''x''</sub> = 100 md. Because ''k''<sub>''k''</sub>''k''<sub>''z''</sub> = 100 md<sup>2</sup>, ''k''<sub>''z''</sub> = 10 md. Also, because ''k''<sub>''x''</sub>''k''<sub>''y''</sub> = 5,000 md<sup>2</sup>, ''k''<sub>''y''</sub> =50 md. From '''Eq. 8.217''',<br/><br/>[[File:Vol5 page 0822 eq 002.png|RTENOTITLE]]<br/><br/>From '''Eq. 8.203''',<br/><br/>[[File:Vol5 page 0822 eq 003.png|RTENOTITLE]]<br/><br/>Calculate the partial penetration skin, s<sub>p</sub> , using the appropriate equation from among Eqs. A-25 through A-35 in '''Table 8.A-2'''. First, calculate<br/><br/>[[File:Vol5 page 0822 eq 004.png|RTENOTITLE]]<br/><br/>to determine whether "Case 1" or "Case 2" applies:<br/><br/>[[File:Vol5 page 0822 eq 005.png|RTENOTITLE]]<br/><br/>Because<br/><br/>[[File:Vol5 page 0822 eq 006.png|RTENOTITLE]]<br/><br/>this is Case 1 (Eq. A-26). Use Eqs. A-25 through A-31 from the table. From Eq. A-27, ''s''<sub>''p''</sub> = ''p''<sub>''xyz''</sub> + ''p''′<sub>''xy''</sub>. From Eq. A-25,<br/><br/>[[File:Vol5 page 0823 eq 001.png|RTENOTITLE]]<br/><br/>From Eq. A-28,<br/><br/>[[File:Vol5 page 0823 eq 002.png|RTENOTITLE]]<br/><br/>Here, from Eq. A-29,<br/><br/>[[File:Vol5 page 0823 eq 003.png|RTENOTITLE]]<br/><br/>The well is centered, so ''d''<sub>''y''</sub> = ''D''<sub>''y''</sub> = 500 ft.<br/><br/>[[File:Vol5 page 0823 eq 004.png|RTENOTITLE]]<br/><br/>From Eq. A-30,<br/><br/>[[File:Vol5 page 0823 eq 005.png|RTENOTITLE]]<br/><br/>Also,<br/><br/>[[File:Vol5 page 0823 eq 006.png|RTENOTITLE]]<br/><br/>From Eq. A-31,<br/><br/>[[File:Vol5 page 0823 eq 007.png|RTENOTITLE]]<br/><br/>and<br/><br/>[[File:Vol5 page 0824 eq 001.png|RTENOTITLE]]<br/><br/>Then,<br/><br/>[[File:Vol5 page 0824 eq 002.png|RTENOTITLE]]<br/><br/>Then,<br/><br/>[[File:Vol5 page 0824 eq 003.png|RTENOTITLE]]<br/><br/>Now check the expected duration of the late-linear flow regime. The start is the larger of values from '''Eqs. 8.212''' and '''8.213'''. From '''Eq. 8.212''',<br/><br/>[[File:Vol5 page 0824 eq 004.png|RTENOTITLE]]<br/><br/>From '''Eq. 8.213''',<br/><br/>[[File:Vol5 page 0824 eq 005.png|RTENOTITLE]]<br/><br/>Thus, the start is expected to be at approximately 162 hours. '''Eq. 8.214''' gives the end of the flow regime.<br/><br/>[[File:Vol5 page 0824 eq 006.png|RTENOTITLE]]<br/><br/>The data in this example spanned the time range from 60 to 240 hours. Some of the data that fall on the straight line are, in theory, from times before the start of the late-linear flow regime, but they appear to cause no problem in determining the slope of the line.
<br>
+
 
<br>
 
[[File:Vol5 page 0822 eq 001.png]]
 
<br>
 
<br>
 
Then, ''k''<sub>''x''</sub> = 100 md. Because ''k''<sub>''k''</sub>''k''<sub>''z''</sub> = 100 md<sup>2</sup>, ''k''<sub>''z''</sub> = 10 md. Also, because ''k''<sub>''x''</sub>''k''<sub>''y''</sub> = 5,000 md<sup>2</sup>, ''k''<sub>''y''</sub> =50 md. From '''Eq. 8.217''',
 
<br>
 
<br>
 
[[File:Vol5 page 0822 eq 002.png]]
 
<br>
 
<br>
 
From '''Eq. 8.203''',
 
<br>
 
<br>
 
[[File:Vol5 page 0822 eq 003.png]]
 
<br>
 
<br>
 
Calculate the partial penetration skin, s<sub>p</sub> , using the appropriate equation from among Eqs. A-25 through A-35 in '''Table 8.A-2'''. First, calculate
 
<br>
 
<br>
 
[[File:Vol5 page 0822 eq 004.png]]
 
<br>
 
<br>
 
to determine whether "Case 1" or "Case 2" applies:
 
<br>
 
<br>
 
[[File:Vol5 page 0822 eq 005.png]]
 
<br>
 
<br>
 
Because
 
<br>
 
<br>
 
[[File:Vol5 page 0822 eq 006.png]]
 
<br>
 
<br>
 
this is Case 1 (Eq. A-26). Use Eqs. A-25 through A-31 from the table. From Eq. A-27, ''s''<sub>''p''</sub> = ''p''<sub>''xyz''</sub> + ''p''′<sub>''xy''</sub>. From Eq. A-25,
 
<br>
 
<br>
 
[[File:Vol5 page 0823 eq 001.png]]
 
<br>
 
<br>
 
From Eq. A-28,
 
<br>
 
<br>
 
[[File:Vol5 page 0823 eq 002.png]]
 
<br>
 
<br>
 
Here, from Eq. A-29,
 
<br>
 
<br>
 
[[File:Vol5 page 0823 eq 003.png]]
 
<br>
 
<br>
 
The well is centered, so ''d''<sub>''y''</sub> = ''D''<sub>''y''</sub> = 500 ft.
 
<br>
 
<br>
 
[[File:Vol5 page 0823 eq 004.png]]
 
<br>
 
<br>
 
From Eq. A-30,
 
<br>
 
<br>
 
[[File:Vol5 page 0823 eq 005.png]]
 
<br>
 
<br>
 
Also,
 
<br>
 
<br>
 
[[File:Vol5 page 0823 eq 006.png]]
 
<br>
 
<br>
 
From Eq. A-31,
 
<br>
 
<br>
 
[[File:Vol5 page 0823 eq 007.png]]
 
<br>
 
<br>
 
and
 
<br>
 
<br>
 
[[File:Vol5 page 0824 eq 001.png]]
 
<br>
 
<br>
 
Then,
 
<br>
 
<br>
 
[[File:Vol5 page 0824 eq 002.png]]
 
<br>
 
<br>
 
Then,
 
<br>
 
<br>
 
[[File:Vol5 page 0824 eq 003.png]]
 
<br>
 
<br>
 
Now check the expected duration of the late-linear flow regime. The start is the larger of values from '''Eqs. 8.212''' and '''8.213'''. From '''Eq. 8.212''',
 
<br>
 
<br>
 
[[File:Vol5 page 0824 eq 004.png]]
 
<br>
 
<br>
 
From '''Eq. 8.213''',
 
<br>
 
<br>
 
[[File:Vol5 page 0824 eq 005.png]]
 
<br>
 
<br>
 
Thus, the start is expected to be at approximately 162 hours. '''Eq. 8.214''' gives the end of the flow regime.
 
<br>
 
<br>
 
[[File:Vol5 page 0824 eq 006.png]]
 
<br>
 
<br>
 
The data in this example spanned the time range from 60 to 240 hours. Some of the data that fall on the straight line are, in theory, from times before the start of the late-linear flow regime, but they appear to cause no problem in determining the slope of the line.  
 
<br>
 
 
----
 
----
<br>
 
  
=== Field Examples<ref name="r41" /> ===
 
  
The following field examples illustrate the procedures used in analyzing horizontal well-test data.  
+
 
<br>
+
=== Field Examples<ref name="r41">_</ref> ===
<br>
+
 
'''''Field Example Well A.''''' '''Table 8.7''' summarizes the reservoir and completion properties for Well A. The target for Well A, a horizontal exploration well, was vertical tectonic fracture development in a low-permeability shale. Because of the fractures, the permeability is assumed to be isotropic (''k''<sub>''h''</sub> = ''k''<sub>''z''</sub>) and a result of the fractures. '''Fig. 8.107''' is a diagnostic plot for Well A and includes a history match using an analytical model.
+
The following field examples illustrate the procedures used in analyzing horizontal well-test data.<br/><br/>'''''Field Example Well A.''''' '''Table 8.7''' summarizes the reservoir and completion properties for Well A. The target for Well A, a horizontal exploration well, was vertical tectonic fracture development in a low-permeability shale. Because of the fractures, the permeability is assumed to be isotropic (''k''<sub>''h''</sub> = ''k''<sub>''z''</sub>) and a result of the fractures. '''Fig. 8.107''' is a diagnostic plot for Well A and includes a history match using an analytical model.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
<gallery widths=300px heights=200px>
 
 
File:Vol5 Page 0826 Image 0001.png|'''Table 8.7'''
 
File:Vol5 Page 0826 Image 0001.png|'''Table 8.7'''
  
 
File:vol5 Page 0826 Image 0002.png|'''Fig. 8.107 – Diagnostic plot for Well A.'''
 
File:vol5 Page 0826 Image 0002.png|'''Fig. 8.107 – Diagnostic plot for Well A.'''
 
</gallery>
 
</gallery>
<br/>
 
  
During the early part of the test there is a unit-slope line representing wellbore storage. Following that wellbore storage, there is a transition to radial flow. The final few data points may be in radial flow. On the Horner plot ('''Fig. 8.108'''), the last few data points fall on a straight line. From the slope of the straight line, the apparent permeability is 0.011 md and the altered zone skin is 2.9. There is no evidence of boundary effects on this Horner plot. The existence of the semilog straight line is not assured, but the data are at least on the verge of reaching it.  
+
During the early part of the test there is a unit-slope line representing wellbore storage. Following that wellbore storage, there is a transition to radial flow. The final few data points may be in radial flow. On the Horner plot ('''Fig. 8.108'''), the last few data points fall on a straight line. From the slope of the straight line, the apparent permeability is 0.011 md and the altered zone skin is 2.9. There is no evidence of boundary effects on this Horner plot. The existence of the semilog straight line is not assured, but the data are at least on the verge of reaching it.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0827 Image 0001.png|'''Fig. 8.108 – Horner plot suggests beginning of middle-time region appears in test data.'''
 
File:vol5 Page 0827 Image 0001.png|'''Fig. 8.108 – Horner plot suggests beginning of middle-time region appears in test data.'''
 
</gallery>
 
</gallery>
<br/>
 
  
Guided by the Horner analysis results, engineers simulated these data with an analytical horizontal well model. The initial match was poor. The match was improved considerably by introducing a no-flow boundary approximately 16 ft above the wellbore, which led to a permeability estimate of 0.027 md and an apparent skin of 11.5. It was concluded that the flow regime observed in the test was hemiradial flow.  
+
Guided by the Horner analysis results, engineers simulated these data with an analytical horizontal well model. The initial match was poor. The match was improved considerably by introducing a no-flow boundary approximately 16 ft above the wellbore, which led to a permeability estimate of 0.027 md and an apparent skin of 11.5. It was concluded that the flow regime observed in the test was hemiradial flow.<br/><br/>The final match, shown on the type-curve plot in '''Fig. 8.107''', is still not a good match at all times, but the author stated that the poor match in the transition region could be the result of phase-redistribution effects in the wellbore. The distance to the no-flow boundary that led to the best match compared favorably with well survey data, which indicated that the well was drilled approximately 20 ft below the upper limit of the productive horizon.<br/><br/>Well B is in a west Texas carbonate formation. It was expected to have isotropic permeability caused by fracturing and dissolution. '''Table 8.8''' gives the field data for this well.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>The final match, shown on the type-curve plot in '''Fig. 8.107''', is still not a good match at all times, but the author stated that the poor match in the transition region could be the result of phase-redistribution effects in the wellbore. The distance to the no-flow boundary that led to the best match compared favorably with well survey data, which indicated that the well was drilled approximately 20 ft below the upper limit of the productive horizon.  
 
<br>
 
<br>
 
Well B is in a west Texas carbonate formation. It was expected to have isotropic permeability caused by fracturing and dissolution. '''Table 8.8''' gives the field data for this well.
 
<br>
 
<br>
 
<gallery widths=300px heights=200px>
 
 
File:Vol5 Page 0827 Image 0002.png|'''Table 8.8'''
 
File:Vol5 Page 0827 Image 0002.png|'''Table 8.8'''
 
</gallery>
 
</gallery>
<br>
 
  
'''Fig. 8.109''' is the diagnostic plot for this well. After wellbore storage, a short period of radial flow appears to be followed by the onset of linear flow, because p′ approaches a slope of 0.5. In the time region where the derivative is horizontal, a straight line on the Horner plot ('''Fig. 8.110''') yields ''k'' = 0.14 md. Using these results in the analysis shows that the end of radial flow occurs at ''t''<sub>Erf</sub> = 165 hours.  
+
'''Fig. 8.109''' is the diagnostic plot for this well. After wellbore storage, a short period of radial flow appears to be followed by the onset of linear flow, because p′ approaches a slope of 0.5. In the time region where the derivative is horizontal, a straight line on the Horner plot ('''Fig. 8.110''') yields ''k'' = 0.14 md. Using these results in the analysis shows that the end of radial flow occurs at ''t''<sub>Erf</sub> = 165 hours.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0828 Image 0001.png|'''Fig. 8.109 – Diagnostic plot for Well B indicates radial and linear flow regimes.'''
 
File:vol5 Page 0828 Image 0001.png|'''Fig. 8.109 – Diagnostic plot for Well B indicates radial and linear flow regimes.'''
  
 
File:vol5 Page 0828 Image 0002.png|'''Fig. 8.110 – Horner plot confirms early-radial flow.'''
 
File:vol5 Page 0828 Image 0002.png|'''Fig. 8.110 – Horner plot confirms early-radial flow.'''
 
</gallery>
 
</gallery>
<br/>
 
  
A tandem-root plot ('''Fig. 8.111''') indicates linear flow and also suggests a distance to the nearest boundary of 29 ft. This is in good agreement with geological observations and helps to verify the assumption of isotropic permeability. The history match with an analytical horizontal well model, shown in '''Fig. 8.109''', confirms the results of the Horner and tandem-root plots.  
+
A tandem-root plot ('''Fig. 8.111''') indicates linear flow and also suggests a distance to the nearest boundary of 29 ft. This is in good agreement with geological observations and helps to verify the assumption of isotropic permeability. The history match with an analytical horizontal well model, shown in '''Fig. 8.109''', confirms the results of the Horner and tandem-root plots.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0829 Image 0001.png|'''Fig. 8.111 – Tandem-root plot confirms linear flow.'''
 
File:vol5 Page 0829 Image 0001.png|'''Fig. 8.111 – Tandem-root plot confirms linear flow.'''
 
</gallery>
 
</gallery>
<br/>
 
  
'''''Field Example Well C.''''' Well C data are from a buildup test of a horizontal well in a high-permeability sandstone where a 54-ft oil column overlies an extensive aquifer estimated to be approximately 180 ft thick. '''Table 8.9''' shows the available data.
+
'''''Field Example Well C.''''' Well C data are from a buildup test of a horizontal well in a high-permeability sandstone where a 54-ft oil column overlies an extensive aquifer estimated to be approximately 180 ft thick. '''Table 8.9''' shows the available data.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
<gallery widths=300px heights=200px>
 
 
File:Vol5 Page 0829 Image 0002.png|'''Table 8.9'''
 
File:Vol5 Page 0829 Image 0002.png|'''Table 8.9'''
 
</gallery>
 
</gallery>
<br>
 
  
The diagnostic plot ('''Fig. 8.112''') shows essentially no wellbore storage and a constant derivative, indicating radial, hemiradial, or elliptical flow at early times. The rapid decline of the derivative at the end of the test is caused by the aquifer underlying the oil column, which is acting like a constant-pressure boundary. A history match with an analytical horizontal well model with one no-flow and one constant-pressure boundary (the lower boundary), yielded ''k''<sub>''h''</sub> = 313 md, ''k''<sub>''z''</sub> = 7.5 md, ''s''<sub>''a''</sub> = 1.5, and ''L''<sub>''w''</sub> = 356 ft. The no-flow boundary was estimated to be approximately 112 ft below the wellbore.  
+
The diagnostic plot ('''Fig. 8.112''') shows essentially no wellbore storage and a constant derivative, indicating radial, hemiradial, or elliptical flow at early times. The rapid decline of the derivative at the end of the test is caused by the aquifer underlying the oil column, which is acting like a constant-pressure boundary. A history match with an analytical horizontal well model with one no-flow and one constant-pressure boundary (the lower boundary), yielded ''k''<sub>''h''</sub> = 313 md, ''k''<sub>''z''</sub> = 7.5 md, ''s''<sub>''a''</sub> = 1.5, and ''L''<sub>''w''</sub> = 356 ft. The no-flow boundary was estimated to be approximately 112 ft below the wellbore.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0830 Image 0001.png|'''Fig. 8.112 – Diagnostic plot for Well C shows no wellbore storage and constant derivative, with aquifer underlying oil column.'''
 
File:vol5 Page 0830 Image 0001.png|'''Fig. 8.112 – Diagnostic plot for Well C shows no wellbore storage and constant derivative, with aquifer underlying oil column.'''
 
</gallery>
 
</gallery>
<br/>
 
  
The time at which the early radial flow regime ends—the time where the derivative ceases to be flat on the diagnostic plot—is approximately 1.5 hours. For a wellbore with a volume of 130 bbl filled with a fluid of compressibility of 3.5 × 10<sup>–6</sup> psi, the duration of wellbore storage (the unit-slope line) is estimated to be 0.0005 hours. With the gauge sampling rate set at 0.017 hours, the wellbore-storage unit slope simply could not be detected and does not appear at all on this plot. '''Fig. 8.113''' is the Horner plot for this test. A straight line appears in the same range as the flat derivative on a diagnostic plot. From the slope of the line, the permeability is estimated to be 53 md, close to the regression analysis match value of 48 md.  
+
The time at which the early radial flow regime ends—the time where the derivative ceases to be flat on the diagnostic plot—is approximately 1.5 hours. For a wellbore with a volume of 130 bbl filled with a fluid of compressibility of 3.5 × 10<sup>–6</sup> psi, the duration of wellbore storage (the unit-slope line) is estimated to be 0.0005 hours. With the gauge sampling rate set at 0.017 hours, the wellbore-storage unit slope simply could not be detected and does not appear at all on this plot. '''Fig. 8.113''' is the Horner plot for this test. A straight line appears in the same range as the flat derivative on a diagnostic plot. From the slope of the line, the permeability is estimated to be 53 md, close to the regression analysis match value of 48 md.<br/><br/><gallery widths="300px" heights="200px">
<br/>
 
<br/>
 
<gallery widths="300px" heights="200px">
 
 
File:vol5 Page 0830 Image 0002.png|'''Fig. 8.113 – Horner plot gives permeability of 5.3 md, close to that from regression analysis.'''
 
File:vol5 Page 0830 Image 0002.png|'''Fig. 8.113 – Horner plot gives permeability of 5.3 md, close to that from regression analysis.'''
 
</gallery>
 
</gallery>
<br/>
+
 
  
  
 
=== Running Horizontal Well Tests ===
 
=== Running Horizontal Well Tests ===
  
The measurements in horizontal wells are usually made above the wellbore with the pressure gauge still in the vertical section.  
+
The measurements in horizontal wells are usually made above the wellbore with the pressure gauge still in the vertical section.<br/><br/>The test string may often be too rigid to pass through the wellbore. However, in most cases, conventional hardware can be used for horizontal well tests. With longer horizontal wellbores, wellbore storage is an inherent problem for testing, even for buildup tests with downhole shut-in. As mentioned previously, problems arise in conducting buildup tests with short-duration production periods because superposition is inappropriate; therefore, Horner plots and tandem-root plots, which depend on superposition being applicable, are often inappropriate.<br/><br/>Another problem in conducting buildup test following a short production period is that significant pressure gradients along the length of the wellbore may cause crossflow within the wellbore during shut-in, so fluid may flow from one region to another in the wellbore. These gradients can be removed and this crossflow eliminated by a longer-duration flow period preceding a buildup test.<br/><br/>'''''Factors That Affect Transient Responses.''''' A number of factors may affect the transient response of a horizontal well test: horizontal permeability (normal and parallel to well trajectory), vertical permeability, drilling damage, completion damage, producing interval that may be effectively much less than drilled length, and variations in standoff along length of well.<br/><br/>In summary, seven or more factors may affect interpretation for horizontal wells in homogeneous reservoirs before the effects from boundaries. The problem is complex, so test results are frequently inconclusive. Furthermore, wellbore storage inhibits determination of properties associated with early-time transient behavior such as vertical permeability and damage from drilling and completion. Middle- and late-time behavior may require several hours, days, or months to appear in transient data.<br/><br/>Some practical steps will help ensure interpretable test data. First, it is helpful to run tests in the pilot hole before kicking off to drill the horizontal borehole section. From a test in the vertical section, it is possible to get usable estimates of horizontal and vertical permeabilities using modern wireline test tools. Second, a good directional drilling survey can frequently provide an adequate estimate of standoff. A production-log flow survey conducted with coiled tubing can determine what part of the wellbore is actually producing and, therefore, help provide an estimate of effective productive length. Wells in developed reservoirs should be flowed long enough to bring pressures along the wellbore to equilibrium and thus minimize crossflow. For high-rate wells, continuous borehole pressure and flow-rate measurements acquired during production can be used to interpret the pressure-drawdown transient response. If the downhole rates are not measured, the buildup test should be conducted with downhole shut-in to minimize wellbore storage distortion of test data.
<br>
 
<br>
 
The test string may often be too rigid to pass through the wellbore. However, in most cases, conventional hardware can be used for horizontal well tests. With longer horizontal wellbores, wellbore storage is an inherent problem for testing, even for buildup tests with downhole shut-in. As mentioned previously, problems arise in conducting buildup tests with short-duration production periods because superposition is inappropriate; therefore, Horner plots and tandem-root plots, which depend on superposition being applicable, are often inappropriate.  
 
<br>
 
<br>
 
Another problem in conducting buildup test following a short production period is that significant pressure gradients along the length of the wellbore may cause crossflow within the wellbore during shut-in, so fluid may flow from one region to another in the wellbore. These gradients can be removed and this crossflow eliminated by a longer-duration flow period preceding a buildup test.  
 
<br>
 
<br>
 
'''''Factors That Affect Transient Responses.''''' A number of factors may affect the transient response of a horizontal well test: horizontal permeability (normal and parallel to well trajectory), vertical permeability, drilling damage, completion damage, producing interval that may be effectively much less than drilled length, and variations in standoff along length of well.  
 
<br>
 
<br>
 
In summary, seven or more factors may affect interpretation for horizontal wells in homogeneous reservoirs before the effects from boundaries. The problem is complex, so test results are frequently inconclusive. Furthermore, wellbore storage inhibits determination of properties associated with early-time transient behavior such as vertical permeability and damage from drilling and completion. Middle- and late-time behavior may require several hours, days, or months to appear in transient data.  
 
<br>
 
<br>
 
Some practical steps will help ensure interpretable test data. First, it is helpful to run tests in the pilot hole before kicking off to drill the horizontal borehole section. From a test in the vertical section, it is possible to get usable estimates of horizontal and vertical permeabilities using modern wireline test tools. Second, a good directional drilling survey can frequently provide an adequate estimate of standoff. A production-log flow survey conducted with coiled tubing can determine what part of the wellbore is actually producing and, therefore, help provide an estimate of effective productive length. Wells in developed reservoirs should be flowed long enough to bring pressures along the wellbore to equilibrium and thus minimize crossflow. For high-rate wells, continuous borehole pressure and flow-rate measurements acquired during production can be used to interpret the pressure-drawdown transient response. If the downhole rates are not measured, the buildup test should be conducted with downhole shut-in to minimize wellbore storage distortion of test data.  
 
  
 
=== Estimating Horizontal Well Productivity ===
 
=== Estimating Horizontal Well Productivity ===
  
Because of two fundamental problems, estimating the productivity of a horizontal well accurately is even more difficult than estimating the productivity of a vertical well. The theoretical models available have a number of simplifying assumptions and the data required for even these simplified models are not likely to be available. Still, we must make estimates and decisions based on those estimates. In this section, two productivity models that have proved useful in practice are discussed. The first, published by Babu and Odeh<ref name="r42" /> in 1989, is limited to single-horizontal wells. The second, published by Economides, Brand, and Frick<ref name="r43" /> in 1996, is more general and is useful for multilateral wells.  
+
Because of two fundamental problems, estimating the productivity of a horizontal well accurately is even more difficult than estimating the productivity of a vertical well. The theoretical models available have a number of simplifying assumptions and the data required for even these simplified models are not likely to be available. Still, we must make estimates and decisions based on those estimates. In this section, two productivity models that have proved useful in practice are discussed. The first, published by Babu and Odeh<ref name="r42">_</ref> in 1989, is limited to single-horizontal wells. The second, published by Economides, Brand, and Frick<ref name="r43">_</ref> in 1996, is more general and is useful for multilateral wells.<br/><br/>'''''Babu-Odeh Method.''''' Babu and Odeh<ref name="r42">_</ref> obtained a rigorous solution to the diffusivity equation for a well in a box-shaped reservoir, subject to certain limiting assumptions. The assumptions include the following:
<br>
+
 
<br>
+
*Fluid flows to the well uniformly at all points along the wellbore (uniform flux) and the well is completed uniformly.
'''''Babu-Odeh Method.''''' Babu and Odeh<ref name="r42" /> obtained a rigorous solution to the diffusivity equation for a well in a box-shaped reservoir, subject to certain limiting assumptions. The assumptions include the following:
+
*The sides of the drainage volume are aligned with the principal permeability direction.
<br>
+
*The wellbore is parallel to the sides of the drainage area and is oriented parallel to one direction of principal permeability and perpendicular to the other two.
* Fluid flows to the well uniformly at all points along the wellbore (uniform flux) and the well is completed uniformly.
+
*The boundaries of the reservoir are all no-flow boundaries and the well reaches stabilized, pseudosteady-state flow.
* The sides of the drainage volume are aligned with the principal permeability direction.
+
*The formation damage around the wellbore is uniform at all points along the wellbore.
* The wellbore is parallel to the sides of the drainage area and is oriented parallel to one direction of principal permeability and perpendicular to the other two.
+
 
* The boundaries of the reservoir are all no-flow boundaries and the well reaches stabilized, pseudosteady-state flow.
+
 
* The formation damage around the wellbore is uniform at all points along the wellbore.
+
 
<br>
+
'''Fig. 8.95''' introduces the nomenclature in the Babu and Odeh solution. The solution is quite complex but is approximated accurately with an equation written in the same form as the pseudosteady-state flow equation for a vertical oil well producing a single-phase, slightly incompressible liquid.<br/><br/>[[File:Vol5 page 0831 eq 001.png|RTENOTITLE]]....................(8.218)<br/><br/>[[File:Vol5 page 0831 eq 002.png|RTENOTITLE]]....................(8.219)<br/><br/>'''Table 8.A-2''' gives equations to estimate ''C''<sub>''H''</sub> and ''s''<sub>''p''</sub>. Two examples adapted from Babu and Odeh<ref name="r42">_</ref> illustrate the application of these equations.
  
'''Fig. 8.95''' introduces the nomenclature in the Babu and Odeh solution. The solution is quite complex but is approximated accurately with an equation written in the same form as the pseudosteady-state flow equation for a vertical oil well producing a single-phase, slightly incompressible liquid.
 
<br>
 
<br>
 
[[File:Vol5 page 0831 eq 001.png]]....................(8.218)
 
<br>
 
<br>
 
[[File:Vol5 page 0831 eq 002.png]]....................(8.219)
 
<br>
 
<br>
 
'''Table 8.A-2''' gives equations to estimate ''C''<sub>''H''</sub> and ''s''<sub>''p''</sub>. Two examples adapted from Babu and Odeh<ref name="r42" /> illustrate the application of these equations.
 
<br>
 
 
----
 
----
<br>
+
 
'''''Example 8.6'''''
+
<br/>'''''Example 8.6''''' A horizontal well 1,000 ft long (''L''<sub>''w''</sub>) is drilled in a box-shaped drainage volume 4,000 ft long (''a''<sub>''H''</sub>), 2,000 ft wide (''b''<sub>''H''</sub>), and 100 ft thick (''h''). The well is off-center in the y -direction (parallel to the well), so ''d''<sub>''y''</sub> = 250 ft and ''D''<sub>''y''</sub> = 750 ft. The well is also off-center in the ''x''-direction so that ''d''<sub>''x''</sub> = 1,000 ft and ''D''<sub>''x''</sub> = 3,000 ft. Finally, the well is centered in the ''z''-direction so that ''d''<sub>''z''</sub> = ''D''<sub>''z''</sub> = 50 ft. The wellbore radius is 0.25 ft; ''k''<sub>''x''</sub> = ''k''<sub>''y''</sub> = 200 md and ''k''<sub>''z''</sub> = 50 md. Fluid properties are ''B''<sub>''o''</sub> = 1.25 RB/STB and ''μ'' = 1 cp. Calculate the productivity index.<br/><br/>''Solution.''<br/><br/>From Eq. A-38,<br/><br/>[[File:Vol5 page 0831 eq 003.png|RTENOTITLE]]<br/><br/>and [[File:Vol5 page 0831 eq 004.png|RTENOTITLE]]<br/><br/>Thus, use Case 1 equations (Eqs. A-26 through A-31) to calculate ''s''<sub>''p''</sub>.<br/><br/>[[File:Vol5 page 0832 eq 001.png|RTENOTITLE]]<br/><br/>To calculate ''p''′<sub>''xy''</sub>, determine ''y''<sub>''m''</sub>, ''L''<sub>''w''</sub>/2''b''<sub>''H''</sub>, (4''y''<sub>''m''</sub> − ''L''<sub>''w''</sub>)/2''b''<sub>''H''</sub>, and (4''y''<sub>''m''</sub> + ''L''<sub>''w''</sub>)/2''b''<sub>''H''</sub>.<br/><br/>[[File:Vol5 page 0832 eq 002.png|RTENOTITLE]]<br/><br/>Thus,<br/><br/>[[File:Vol5 page 0832 eq 003.png|RTENOTITLE]]<br/><br/>Then,<br/><br/>[[File:Vol5 page 0832 eq 004.png|RTENOTITLE]]<br/><br/>Then, ''s''<sub>''p''</sub> = ''p''<sub>''xyz''</sub> + ''p''′<sub>''xy''</sub> =4.50+6.54=11.0, and<br/><br/>[[File:Vol5 page 0832 eq 005.png|RTENOTITLE]]
A horizontal well 1,000 ft long (''L''<sub>''w''</sub>) is drilled in a box-shaped drainage volume 4,000 ft long (''a''<sub>''H''</sub>), 2,000 ft wide (''b''<sub>''H''</sub>), and 100 ft thick (''h''). The well is off-center in the y -direction (parallel to the well), so ''d''<sub>''y''</sub> = 250 ft and ''D''<sub>''y''</sub> = 750 ft. The well is also off-center in the ''x''-direction so that ''d''<sub>''x''</sub> = 1,000 ft and ''D''<sub>''x''</sub> = 3,000 ft. Finally, the well is centered in the ''z''-direction so that ''d''<sub>''z''</sub> = ''D''<sub>''z''</sub> = 50 ft. The wellbore radius is 0.25 ft; ''k''<sub>''x''</sub> = ''k''<sub>''y''</sub> = 200 md and ''k''<sub>''z''</sub> = 50 md. Fluid properties are ''B''<sub>''o''</sub> = 1.25 RB/STB and ''μ'' = 1 cp. Calculate the productivity index.  
+
 
<br>
 
<br>
 
''Solution.''  
 
<br>
 
<br>
 
From Eq. A-38,
 
<br>
 
<br>
 
[[File:Vol5 page 0831 eq 003.png]]
 
<br>
 
<br>
 
and [[File:Vol5 page 0831 eq 004.png]]
 
<br>
 
<br>
 
Thus, use Case 1 equations (Eqs. A-26 through A-31) to calculate ''s''<sub>''p''</sub>.
 
<br>
 
<br>
 
[[File:Vol5 page 0832 eq 001.png]]
 
<br>
 
<br>
 
To calculate ''p''′<sub>''xy''</sub>, determine ''y''<sub>''m''</sub>, ''L''<sub>''w''</sub>/2''b''<sub>''H''</sub>, (4''y''<sub>''m''</sub> − ''L''<sub>''w''</sub>)/2''b''<sub>''H''</sub>, and (4''y''<sub>''m''</sub> + ''L''<sub>''w''</sub>)/2''b''<sub>''H''</sub>.
 
<br>
 
<br>
 
[[File:Vol5 page 0832 eq 002.png]]
 
<br>
 
<br>
 
Thus,
 
<br>
 
<br>
 
[[File:Vol5 page 0832 eq 003.png]]
 
<br>
 
<br>
 
Then,
 
<br>
 
<br>
 
[[File:Vol5 page 0832 eq 004.png]]
 
<br>
 
<br>
 
Then, ''s''<sub>''p''</sub> = ''p''<sub>''xyz''</sub> + ''p''′<sub>''xy''</sub> =4.50+6.54=11.0, and
 
<br>
 
<br>
 
[[File:Vol5 page 0832 eq 005.png]]
 
<br>
 
 
----
 
----
<br>
+
 
'''''Example 8.7'''''
+
<br/>'''''Example 8.7''''' A horizontal well is drilled in a box-shaped reservoir with the following characteristics: ''L''<sub>''w''</sub> = 1,000 ft, ''a''<sub>''H''</sub> = 2,000 ft long, ''b''<sub>''H''</sub> = 4,000 ft wide, and ''h''<sub>''w''</sub> = 2,000 ft thick. The well is off-center in the ''y''-direction (''d''<sub>''y''</sub> = 1,000 ft; ''D''<sub>''y''</sub> = 2,000 ft), centered in the ''x''-direction (''d''<sub>''x''</sub> = ''D''<sub>''x''</sub> = 1,000 ft), and off-center in the ''z''-direction (''d''<sub>''z''</sub> = 50 ft; ''D''<sub>''z''</sub> = 150 ft). Permeabilities are ''k''<sub>''x''</sub> = ''k''<sub>''y''</sub> =100 md and ''k''<sub>''z''</sub> = 20 md. Wellbore radius is 0.25 ft, ''B''<sub>''o''</sub> = 1.25 RB/STB, ''μ'' = 1 cp, and ''s''<sub>''d''</sub> = 0. Find the productivity index, ''J''.<br/><br/>''Solution''.<br/><br/>From Eq. A-38 ('''Table 8.A-2'''),<br/><br/>[[File:Vol5 page 0833 eq 001.png|RTENOTITLE]]<br/><br/>Note that<br/><br/>[[File:Vol5 page 0833 eq 002.png|RTENOTITLE]]<br/><br/>Thus, use Case 2 equations to calculate ''s''<sub>''p''</sub>.<br/><br/>[[File:Vol5 page 0833 eq 003.png|RTENOTITLE]]<br/><br/>To calculate ''p''<sub>''y''</sub>, determine ''y''<sub>''m''</sub>. From Eq. A-29 ('''Table 8.A-2'''),<br/><br/>[[File:Vol5 page 0833 eq 004.png|RTENOTITLE]]<br/><br/>From Eq. A-35 ('''Table 8.A-2'''),<br/><br/>[[File:Vol5 page 0833 eq 005.png|RTENOTITLE]]<br/><br/>Thus, ''s''<sub>''p''</sub> = 16.79 + 7.90 + 7.02 = 31.7. Then, from Eq. A-37 ('''Table 8.A-2'''),<br/><br/>[[File:Vol5 page 0834 eq 001.png|RTENOTITLE]]
A horizontal well is drilled in a box-shaped reservoir with the following characteristics: ''L''<sub>''w''</sub> = 1,000 ft, ''a''<sub>''H''</sub> = 2,000 ft long, ''b''<sub>''H''</sub> = 4,000 ft wide, and ''h''<sub>''w''</sub> = 2,000 ft thick. The well is off-center in the ''y''-direction (''d''<sub>''y''</sub> = 1,000 ft; ''D''<sub>''y''</sub> = 2,000 ft), centered in the ''x''-direction (''d''<sub>''x''</sub> = ''D''<sub>''x''</sub> = 1,000 ft), and off-center in the ''z''-direction (''d''<sub>''z''</sub> = 50 ft; ''D''<sub>''z''</sub> = 150 ft). Permeabilities are ''k''<sub>''x''</sub> = ''k''<sub>''y''</sub> =100 md and ''k''<sub>''z''</sub> = 20 md. Wellbore radius is 0.25 ft, ''B''<sub>''o''</sub> = 1.25 RB/STB, ''μ'' = 1 cp, and ''s''<sub>''d''</sub> = 0. Find the productivity index, ''J''.  
+
 
<br>
 
<br>
 
''Solution''.  
 
<br>
 
<br>
 
From Eq. A-38 ('''Table 8.A-2'''),
 
<br>
 
<br>
 
[[File:Vol5 page 0833 eq 001.png]]
 
<br>
 
<br>
 
Note that
 
<br>
 
<br>
 
[[File:Vol5 page 0833 eq 002.png]]
 
<br>
 
<br>
 
Thus, use Case 2 equations to calculate ''s''<sub>''p''</sub>.
 
<br>
 
<br>
 
[[File:Vol5 page 0833 eq 003.png]]
 
<br>
 
<br>
 
To calculate ''p''<sub>''y''</sub>, determine ''y''<sub>''m''</sub>. From Eq. A-29 ('''Table 8.A-2'''),
 
<br>
 
<br>
 
[[File:Vol5 page 0833 eq 004.png]]
 
<br>
 
<br>
 
From Eq. A-35 ('''Table 8.A-2'''),
 
<br>
 
<br>
 
[[File:Vol5 page 0833 eq 005.png]]
 
<br>
 
<br>
 
Thus, ''s''<sub>''p''</sub> = 16.79 + 7.90 + 7.02 = 31.7. Then, from Eq. A-37 ('''Table 8.A-2'''),
 
<br>
 
<br>
 
[[File:Vol5 page 0834 eq 001.png]]
 
<br>
 
 
----
 
----
<br>
+
 
'''''Economides et al. Method.''''' Economides ''et al.''<ref name="r45" /> presented a more general method to estimate productivity index for a horizontal well. The method has the advantage that it is applicable to multilateral wells in the same plane and is not limited to wells aligned with principal permeabilities. It includes solutions for wells with no pressure drop in the wellbore (infinite conductivity, as opposed to wells with uniform flux). It has the disadvantage that it requires interpolation in a table in which only certain drainage area shapes are given.  
+
<br/>'''''Economides et al. Method.''''' Economides ''et al.''<ref name="r45">_</ref> presented a more general method to estimate productivity index for a horizontal well. The method has the advantage that it is applicable to multilateral wells in the same plane and is not limited to wells aligned with principal permeabilities. It includes solutions for wells with no pressure drop in the wellbore (infinite conductivity, as opposed to wells with uniform flux). It has the disadvantage that it requires interpolation in a table in which only certain drainage area shapes are given.<br/><br/>The basic working equation for the productivity index in this method is<br/><br/>[[File:Vol5 page 0834 eq 002.png|RTENOTITLE]] ....................(8.220)<br/><br/>where Σ''s'' refers to damage skin, turbulence, and other pseudoskin factors. In '''Eq. 8.220''',<br/><br/>[[File:Vol5 page 0834 eq 003.png|RTENOTITLE]] ....................(8.221)<br/><br/>where<br/><br/>[[File:Vol5 page 0834 eq 004.png|RTENOTITLE]] ....................(8.222)<br/><br/>and ''s''<sub>''e''</sub>, describing eccentricity effects in the vertical direction, is<br/><br/>[[File:Vol5 page 0834 eq 005.png|RTENOTITLE]] ....................(8.223)<br/><br/>''s''<sub>''e''</sub> = 0 when a well is centered in the vertical plane. This convergence skin differs only slightly from that used by Babu and Odeh. The difference is 0.25 ln (''k''<sub>''x''</sub> / ''k''<sub>''z''</sub>) + ''h'' / ''L''<sub>''w''</sub>[2''d''<sub>''z'' </sub>/ ''h'' - 1/2(2''d''<sub>''z''</sub> / ''h'')<sup>2</sup> - 2/3], which is usually small (< 0.5). '''Table 8.10''' gives values of ''C''<sub>''H''</sub> for several drainage areas and multilateral configurations. The equations as written are for isotropic reservoirs. Certain variable transformations are required before substituting into the working equation:<br/><br/>[[File:Vol5 page 0834 eq 006.png|RTENOTITLE]]....................(8.224)<br/><br/>[[File:Vol5 page 0835 eq 001.png|RTENOTITLE]]....................(8.225)<br/><br/>where<br/><br/>[[File:Vol5 page 0835 eq 002.png|RTENOTITLE]]....................(8.226)<br/><br/>and [[File:Vol5 page 0835 eq 003.png|RTENOTITLE]]....................(8.227)<br/><br/>''ϕ'' is the azimuth of the well trajectory (relative to the y-axis). Reservoir dimensions:<br/><br/>[[File:Vol5 page 0835 eq 004.png|RTENOTITLE]]....................(8.228)<br/><br/>[[File:Vol5 page 0835 eq 005.png|RTENOTITLE]]....................(8.229)<br/><br/>[[File:Vol5 page 0835 eq 006.png|RTENOTITLE]]....................(8.230)<br/><br/>and [[File:Vol5 page 0835 eq 007.png|RTENOTITLE]]....................(8.231)<br/><br/>Two examples, one from an isotropic reservoir and one from an anisotropic reservoir, illustrate this method.<br/><br/><gallery widths="300px" heights="200px">
<br>
 
<br>
 
The basic working equation for the productivity index in this method is
 
<br>
 
<br>
 
[[File:Vol5 page 0834 eq 002.png]] ....................(8.220)
 
<br>
 
<br>
 
where Σ''s'' refers to damage skin, turbulence, and other pseudoskin factors. In '''Eq. 8.220''',
 
<br>
 
<br>
 
[[File:Vol5 page 0834 eq 003.png]] ....................(8.221)
 
<br>
 
<br>
 
where
 
<br>
 
<br>
 
[[File:Vol5 page 0834 eq 004.png]] ....................(8.222)
 
<br>
 
<br>
 
and ''s''<sub>''e''</sub>, describing eccentricity effects in the vertical direction, is
 
<br>
 
<br>
 
[[File:Vol5 page 0834 eq 005.png]] ....................(8.223)
 
<br>
 
<br>
 
''s''<sub>''e''</sub> = 0 when a well is centered in the vertical plane. This convergence skin differs only slightly from that used by Babu and Odeh. The difference is 0.25 ln (''k''<sub>''x''</sub> / ''k''<sub>''z''</sub>) + ''h'' / ''L''<sub>''w''</sub>[2''d''<sub>''z'' </sub>/ ''h'' - 1/2(2''d''<sub>''z''</sub> / ''h'')<sup>2</sup> - 2/3], which is usually small (< 0.5). '''Table 8.10''' gives values of ''C''<sub>''H''</sub> for several drainage areas and multilateral configurations. The equations as written are for isotropic reservoirs. Certain variable transformations are required before substituting into the working equation:
 
<br>
 
<br>
 
[[File:Vol5 page 0834 eq 006.png]]....................(8.224)
 
<br>
 
<br>
 
[[File:Vol5 page 0835 eq 001.png]]....................(8.225)
 
<br>
 
<br>
 
where
 
<br>
 
<br>
 
[[File:Vol5 page 0835 eq 002.png]]....................(8.226)
 
<br>
 
<br>
 
and [[File:Vol5 page 0835 eq 003.png]]....................(8.227)
 
<br>
 
<br>
 
''ϕ'' is the azimuth of the well trajectory (relative to the y-axis). Reservoir dimensions:
 
<br>
 
<br>
 
[[File:Vol5 page 0835 eq 004.png]]....................(8.228)
 
<br>
 
<br>
 
[[File:Vol5 page 0835 eq 005.png]]....................(8.229)
 
<br>
 
<br>
 
[[File:Vol5 page 0835 eq 006.png]]....................(8.230)
 
<br>
 
<br>
 
and [[File:Vol5 page 0835 eq 007.png]]....................(8.231)
 
<br>
 
<br>
 
Two examples, one from an isotropic reservoir and one from an anisotropic reservoir, illustrate this method.  
 
<br>
 
<br>
 
<gallery widths=300px heights=200px>
 
 
File:Vol5 Page 0836 Image 0001.png|'''Table 8.10'''
 
File:Vol5 Page 0836 Image 0001.png|'''Table 8.10'''
</gallery>
+
</gallery><br/>
<br>
+
 
<br>
 
 
----
 
----
<br>
+
 
'''''Example 8.8'''''
+
<br/>'''''Example 8.8''''' Economides ''et al.''<ref name="r43">_</ref> provide this example. Consider a horizontal well 1,500 ft long in a reservoir with ''b''<sub>''H''</sub> = 2,000 ft, ''a''<sub>''H''</sub> = 4,000 ft, ''h'' = 20 ft, ''r''<sub>''w''</sub> = 0.4, ''k''<sub>''x''</sub> = ''k''<sub>''y''</sub> = ''k''<sub>''z''</sub> = 10 md, ''B''<sub>''o''</sub> = 1.25 RB/STB, and ''μ'' = 1 cp. Assume that the well is centered vertically so that ''s''<sub>''e''</sub> = 0. Also, assume Σ''s'' = 0.<br/><br/>''Solution.''<br/><br/>From '''Eq. 8.223''',<br/><br/>[[File:Vol5 page 0835 eq 008.png|RTENOTITLE]]<br/><br/>(As a matter of interest, the Babu and Odeh ''s''<sub>''c''</sub> for this case is also 2.07.) From '''Table 8.10''', for 2''b''<sub>''H''</sub> = ''a''<sub>''H''</sub> and ''L''<sub>''w''</sub>/''b''<sub>''H''</sub> = 1,500/2,000 = 0.75, ''C''<sub>''H''</sub> = 2.53. From '''Eq. 8.221''',<br/><br/>[[File:Vol5 page 0836 eq 001.png|RTENOTITLE]]<br/><br/>Then, from '''Eq. 8.220''',<br/><br/>[[File:Vol5 page 0837 eq 001.png|RTENOTITLE]]
Economides ''et al.''<ref name="r43" /> provide this example. Consider a horizontal well 1,500 ft long in a reservoir with ''b''<sub>''H''</sub> = 2,000 ft, ''a''<sub>''H''</sub> = 4,000 ft, ''h'' = 20 ft, ''r''<sub>''w''</sub> = 0.4, ''k''<sub>''x''</sub> = ''k''<sub>''y''</sub> = ''k''<sub>''z''</sub> = 10 md, ''B''<sub>''o''</sub> = 1.25 RB/STB, and ''μ'' = 1 cp. Assume that the well is centered vertically so that ''s''<sub>''e''</sub> = 0. Also, assume Σ''s'' = 0.  
+
 
<br>
 
<br>
 
''Solution.''
 
<br>
 
<br>
 
From '''Eq. 8.223''',
 
<br>
 
<br>
 
[[File:Vol5 page 0835 eq 008.png]]
 
<br>
 
<br>
 
(As a matter of interest, the Babu and Odeh ''s''<sub>''c''</sub> for this case is also 2.07.) From '''Table 8.10''', for 2''b''<sub>''H''</sub> = ''a''<sub>''H''</sub> and ''L''<sub>''w''</sub>/''b''<sub>''H''</sub> = 1,500/2,000 = 0.75, ''C''<sub>''H''</sub> = 2.53. From '''Eq. 8.221''',
 
<br>
 
<br>
 
[[File:Vol5 page 0836 eq 001.png]]
 
<br>
 
<br>
 
Then, from '''Eq. 8.220''',
 
<br>
 
<br>
 
[[File:Vol5 page 0837 eq 001.png]]
 
<br>
 
 
----
 
----
<br>
+
 
'''''Example 8.9'''''
+
<br/>'''''Example 8.9'''''<br/><br/>Rework the Babu-Odeh '''Example 8.7''' using the Economides ''et al.'' method.<br/><br/>''Solution.''<br/><br/>First transform the variables. From '''Eqs. 8.226''' and '''8.227''',<br/><br/>[[File:Vol5 page 0837 eq 002.png|RTENOTITLE]]<br/><br/>Because the well is parallel to the ''x''-axis, ''ϕ'' = 0, and<br/><br/>[[File:Vol5 page 0837 eq 003.png|RTENOTITLE]]<br/><br/>From '''Eq. 8.231''',<br/><br/>[[File:Vol5 page 0837 eq 004.png|RTENOTITLE]]<br/><br/>From '''Eq. 8.224''',<br/><br/>[[File:Vol5 page 0837 eq 005.png|RTENOTITLE]]<br/><br/>From '''Eq. 8.225''',<br/><br/>[[File:Vol5 page 0837 eq 006.png|RTENOTITLE]]<br/><br/>From '''Eqs. 8.228''' through '''8.230''',<br/><br/>[[File:Vol5 page 0838 eq 001.png|RTENOTITLE]]<br/><br/>Thus, the equivalent system is a rectangular-shaped drainage area twice as long parallel to the wellbore (3,050 ft) as perpendicular, with ''L''′/''b''<sub>''H''</sub> ′ = 765/3,058 = 0.25. In the original example, the well was off-center in the horizontal plane; here, assume that a centered well is an adequate approximation. From '''Table 8.10''', ''C''<sub>''H''</sub> = 3.19.<br/><br/>From '''Eq. 8.223''',<br/><br/>[[File:Vol5 page 0838 eq 002.png|RTENOTITLE]]<br/><br/>Then,<br/><br/>[[File:Vol5 page 0838 eq 003.png|RTENOTITLE]]<br/><br/>(The Babu-Odeh ''s''<sub>''c''</sub> is 5.60 for this case.) Then, from '''Eq. 8.221''',<br/><br/>[[File:Vol5 page 0838 eq 004.png|RTENOTITLE]]<br/><br/>Finally, from '''Eq. 8.220''',<br/><br/>[[File:Vol5 page 0838 eq 005.png|RTENOTITLE]]<br/><br/>The result is slightly larger than the result using the Babu-Odeh method (''J'' = 25.6 STB/D/psi). At least part of the reason for the difference is that, in this example, it was necessary to assume that the well was centered in its drainage volume, which was not true in the original example. The optimal location of a horizontal well to maximize productivity is to center it within its drainage volume.
<br>
+
 
<br>
 
Rework the Babu-Odeh '''Example 8.7''' using the Economides ''et al.'' method.  
 
<br>
 
<br>
 
''Solution.''  
 
<br>
 
<br>
 
First transform the variables. From '''Eqs. 8.226''' and '''8.227''',
 
<br>
 
<br>
 
[[File:Vol5 page 0837 eq 002.png]]
 
<br>
 
<br>
 
Because the well is parallel to the ''x''-axis, ''ϕ'' = 0, and
 
<br>
 
<br>
 
[[File:Vol5 page 0837 eq 003.png]]
 
<br>
 
<br>
 
From '''Eq. 8.231''',
 
<br>
 
<br>
 
[[File:Vol5 page 0837 eq 004.png]]
 
<br>
 
<br>
 
From '''Eq. 8.224''',
 
<br>
 
<br>
 
[[File:Vol5 page 0837 eq 005.png]]
 
<br>
 
<br>
 
From '''Eq. 8.225''',
 
<br>
 
<br>
 
[[File:Vol5 page 0837 eq 006.png]]
 
<br>
 
<br>
 
From '''Eqs. 8.228''' through '''8.230''',
 
<br>
 
<br>
 
[[File:Vol5 page 0838 eq 001.png]]
 
<br>
 
<br>
 
Thus, the equivalent system is a rectangular-shaped drainage area twice as long parallel to the wellbore (3,050 ft) as perpendicular, with ''L''′/''b''<sub>''H''</sub> ′ = 765/3,058 = 0.25. In the original example, the well was off-center in the horizontal plane; here, assume that a centered well is an adequate approximation. From '''Table 8.10''', ''C''<sub>''H''</sub> = 3.19.  
 
<br>
 
<br>
 
From '''Eq. 8.223''',
 
<br>
 
<br>
 
[[File:Vol5 page 0838 eq 002.png]]
 
<br>
 
<br>
 
Then,
 
<br>
 
<br>
 
[[File:Vol5 page 0838 eq 003.png]]
 
<br>
 
<br>
 
(The Babu-Odeh ''s''<sub>''c''</sub> is 5.60 for this case.) Then, from '''Eq. 8.221''',
 
<br>
 
<br>
 
[[File:Vol5 page 0838 eq 004.png]]
 
<br>
 
<br>
 
Finally, from '''Eq. 8.220''',
 
<br>
 
<br>
 
[[File:Vol5 page 0838 eq 005.png]]
 
<br>
 
<br>
 
The result is slightly larger than the result using the Babu-Odeh method (''J'' = 25.6 STB/D/psi). At least part of the reason for the difference is that, in this example, it was necessary to assume that the well was centered in its drainage volume, which was not true in the original example. The optimal location of a horizontal well to maximize productivity is to center it within its drainage volume.  
 
<br>
 
 
----
 
----
<br>
 
'''''Comparison of Recent and Older Horizontal Well Models.''''' Ozkan<ref name="r44" /> compared "contemporary" (generally 1990s) and "conventional" horizontal well models in a paper published in 2001. He pointed out that the older models are used for both pressure-transient test analysis and for estimating well productivity. Ozkan stressed three limitations of the conventional models, which include the Babu-Odeh model and other pioneering work.
 
<br>
 
<br>
 
Conventional models usually assume that the horizontal well is parallel to one of the principal permeability directions (preferably the minimum permeability direction in the horizontal plane). In many cases, this is not true. In fact, in many cases the principal permeability directions are unknown. When the principal permeability directions are known, corrections to length are possible (as in the Economides ''et al.'' model); if they are not known, there is no way to correct the analysis. Contemporary models show that the error in permeability estimates approaches 50% when the deviation angle exceeds 50°. Unfortunately, the models also indicate that there is nothing in a well’s response that provides any indication that the assumption that the well is parallel to a principal permeability direction is incorrect.
 </