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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 I – General Engineering
|volume = Volume I – General Engineering
|vol editor = John R. Fanchi, Editor
|vol editor = John R. Fanchi, Editor
|date         = 2007
|date = 2007
|publisher   = Society of Petroleum Engineers
|publisher = Society of Petroleum Engineers
|image       = [[File:Vol1GECover.png|center|120px]]
|image = [[File:Vol1GECover.png|center|120px]]
|imagestyle   =  
|imagestyle =  
|chapter = Chapter 3 – Mathematics of Transient Analysis
|chapter = Chapter 3 – Mathematics of Transient Analysis
|ch author = Erdal Ozkan, Colorado School of Mines
|ch author = Erdal Ozkan, Colorado School of Mines
|ch author info =  
|ch author info =  
|page numbers   = 77-172
|page numbers = 77-172
|ISBN   = 978-1-55563-108-6
|ISBN = 978-1-55563-108-6
}}
}}<br/><br/>This chapter explains how fluid flow in porous media can be translated into a mathematical statement and how mathematical analysis can be used to answer transient-flow problems. This broad area is common to many other disciplines, such as heat conduction in solids and groundwater hydrology. The objective of this chapter is to introduce the fundamentals of transient analysis, present examples, and guide the interested reader to relevant references.<br/><br/>__TOC__
<br>
<div class="toccolours mw-collapsible mw-collapsed">
<br>
This chapter explains how fluid flow in porous media can be translated into a mathematical statement and how mathematical analysis can be used to answer transient-flow problems. This broad area is common to many other disciplines, such as heat conduction in solids and groundwater hydrology. The objective of this chapter is to introduce the fundamentals of transient analysis, present examples, and guide the interested reader to relevant references.
<br>
<br>
__TOC__
<br>
<div class="toccolours mw-collapsible mw-collapsed" >
== Introduction ==
== Introduction ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/>Most physical phenomena in the domain of transient fluid flow in porous media can be described generally by partial differential equations (PDEs). With appropriate boundary conditions and sometimes with simplifying assumptions, the PDE leads to an initial boundary value problem (IBVP) that is solved to find a mathematical statement of the resulting flow in the porous medium. This section briefly discusses the statement of the IBVP for transient fluid flow in porous media.
Most physical phenomena in the domain of transient fluid flow in porous media can be described generally by partial differential equations (PDEs). With appropriate boundary conditions and sometimes with simplifying assumptions, the PDE leads to an initial boundary value problem (IBVP) that is solved to find a mathematical statement of the resulting flow in the porous medium. This section briefly discusses the statement of the IBVP for transient fluid flow in porous media.


=== Equations of Transient Fluid Flow in Porous Media ===
=== Equations of Transient Fluid Flow in Porous Media ===


In essence, fluid motion in porous media can be specified by the knowledge of the velocity vector, [[File:Vol1 page 0077 inline 001.png]], and the density of the fluid, ''ρ'', as a function of the position (''x'', ''y'', ''z'') and time, ''t''; that is, [[File:Vol1 page 0077 inline 001.png]] = [[File:Vol1 page 0077 inline 001.png]] (''x'', ''y'', ''z'', ''t'') and ''ρ''= ''ρ'' (''x'', ''y'', ''z'', ''t''). Relative to the fixed Cartesian axes, the velocity vector can be written as
In essence, fluid motion in porous media can be specified by the knowledge of the velocity vector, [[File:Vol1 page 0077 inline 001.png|RTENOTITLE]], and the density of the fluid, ''ρ'', as a function of the position (''x'', ''y'', ''z'') and time, ''t''; that is, [[File:Vol1 page 0077 inline 001.png|RTENOTITLE]] = [[File:Vol1 page 0077 inline 001.png|RTENOTITLE]] (''x'', ''y'', ''z'', ''t'') and ''ρ''= ''ρ'' (''x'', ''y'', ''z'', ''t''). Relative to the fixed Cartesian axes, the velocity vector can be written as<br/><br/>[[File:Vol1 page 0077 eq 001.png|RTENOTITLE]]....................(3.1)<br/><br/>where ''v''<sub>''x''</sub>, ''v''<sub>''y''</sub>, and ''v''<sub>''z''</sub> are the velocity components, and [[File:Vol1 page 0077 inline 002.png|RTENOTITLE]], [[File:Vol1 page 0077 inline 003.png|RTENOTITLE]], and [[File:Vol1 page 0077 inline 004.png|RTENOTITLE]] are the unit vectors in the ''x'', ''y'', and ''z'' directions, respectively.<br/><br/>The physical law governing the macroscopic fluid-flow phenomena in porous media is the conservation of mass, which states that mass is neither created nor destroyed. The mathematical formula of this rule is developed by considering the flow through a fixed arbitrary closed surface, Γ, lying entirely within a porous medium of porosity ''Φ'', which is filled with a fluid of viscosity ''μ''. '''Fig. 3.1''' illustrates an arbitrary closed surface in porous medium.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0077 eq 001.png]]....................(3.1)
<br>
<br>
where ''v''<sub>''x''</sub>, ''v''<sub>''y''</sub>, and ''v''<sub>''z''</sub> are the velocity components, and [[File:Vol1 page 0077 inline 002.png]], [[File:Vol1 page 0077 inline 003.png]], and [[File:Vol1 page 0077 inline 004.png]] are the unit vectors in the ''x'', ''y'', and ''z'' directions, respectively.
<br>
<br>
The physical law governing the macroscopic fluid-flow phenomena in porous media is the conservation of mass, which states that mass is neither created nor destroyed. The mathematical formula of this rule is developed by considering the flow through a fixed arbitrary closed surface, Γ, lying entirely within a porous medium of porosity ''Φ'', which is filled with a fluid of viscosity ''μ''. '''Fig. 3.1''' illustrates an arbitrary closed surface in porous medium.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 078 Image 0001.png|'''Fig. 3.1 – Arbitrary closed surface Γ in porous medium.'''
File:vol1 Page 078 Image 0001.png|'''Fig. 3.1 – Arbitrary closed surface Γ in porous medium.'''
</gallery>
</gallery><br/>The conservation of mass principle requires that the difference between the rates at which fluid enters and leaves the volume through its surface must equal the rate at which mass accumulates within the volume. The total mass within the volume at any time is given by<br/><br/>[[File:Vol1 page 0078 eq 001.png|RTENOTITLE]]....................(3.2)<br/><br/>Then, the time rate of change of mass within Γ is<br/><br/>[[File:Vol1 page 0078 eq 002.png|RTENOTITLE]]....................(3.3)<br/><br/>which, by the conservation of mass law, is equal to the rate at which mass enters ''V'' through the surface.<br/><br/>Consider the differential surface element, dΓ, shown in '''Fig. 3.1'''. The mass entering the volume through dΓ at the normal velocity, [[File:Vol1 page 0078 inline 001.png|RTENOTITLE]], in a time increment, Δ''t'', is [[File:Vol1 page 0078 inline 002.png|RTENOTITLE]], and the total mass of the fluid passing through Γ during Δ''t'' is<br/><br/>[[File:Vol1 page 0078 eq 003.png|RTENOTITLE]]....................(3.4)<br/><br/>The surface integral in '''Eq. 3.4''' accounts for both influx and outflux through the surface of the volume; that is, Δ''M''<sub>''g''</sub> is the difference between the masses entering and leaving the control volume during the time increment, Δ''t''. Then, the mass rate entering the volume, ''V'', through its surface, Γ, can be written as<br/><br/>[[File:Vol1 page 0078 eq 004.png|RTENOTITLE]]....................(3.5)<br/><br/>By the principle of conservation of mass, equating the right sides of '''Eqs. 3.3''' and '''3.5''' yields<br/><br/>[[File:Vol1 page 0078 eq 005.png|RTENOTITLE]]....................(3.6)<br/><br/>A more useful relation is found with the divergence theorem, which states that the flux of [[File:Vol1 page 0079 inline 001.png|RTENOTITLE]] through the closed surface, Γ, is identical to the volume integral of [[File:Vol1 page 0079 inline 002.png|RTENOTITLE]] (the divergence of [[File:Vol1 page 0079 inline 001.png|RTENOTITLE]]) taken throughout ''V''; that is,<br/><br/>[[File:Vol1 page 0079 eq 001.png|RTENOTITLE]]....................(3.7)<br/><br/>Here, ∇ is the gradient operator, which in 3D Cartesian and cylindrical coordinates is given, respectively, by<br/><br/>[[File:Vol1 page 0079 eq 002.png|RTENOTITLE]]....................(3.8)<br/><br/>and<br/><br/>[[File:Vol1 page 0079 eq 003.png|RTENOTITLE]]....................(3.9)<br/><br/>With the relation in '''Eq. 3.7''', '''Eq. 3.6''' can be recast into<br/><br/>[[File:Vol1 page 0079 eq 004.png|RTENOTITLE]]....................(3.10)<br/><br/>If the functions involved in the argument of the integral in '''Eq. 3.10''' are continuous, then the integral is identically zero if and only if its argument is zero (because the volume integral in '''Eq. 3.10''' is identically zero for any arbitrarily chosen volume). Then, the following continuity equation can be obtained.<br/><br/>[[File:Vol1 page 0079 eq 005.png|RTENOTITLE]]....................(3.11)<br/><br/>'''Eq. 3.11''' is a PDE that is equivalent to the statement of the conservation of mass for fluid flow in porous media. For practical purposes, however, '''Eq. 3.11''' is expressed in terms of pressure because density and velocity cannot be measured directly. To express density, ''ρ'', and velocity, [[File:Vol1 page 0077 inline 001.png|RTENOTITLE]], in terms of pressure, we use an equation of state and a flux law, known as Darcy’s law, respectively.<br/><br/>The following definition of isothermal fluid compressibility, ''c'', is a useful equation of state that relates density to pressure.<br/><br/>[[File:Vol1 page 0079 eq 006.png|RTENOTITLE]]....................(3.12)<br/><br/>If ''c'' is a constant (the compressibility of many reservoir liquids may be considered as constant), then '''Eq. 3.12''' can be integrated to yield<br/><br/>[[File:Vol1 page 0079 eq 007.png|RTENOTITLE]]....................(3.13)<br/><br/>where subscript 0 indicates the conditions at the datum. Similarly, the compressibility of the porous rock, ''c''<sub>''f''</sub>, is defined by<br/><br/>[[File:Vol1 page 0080 eq 001.png|RTENOTITLE]]....................(3.14)<br/><br/>and the total system compressibility, ''c''<sub>''t''</sub>, is given by<br/><br/>[[File:Vol1 page 0080 eq 002.png|RTENOTITLE]]....................(3.15)<br/><br/>These definitions of compressibility help recast '''Eq. 3.11''' in terms of pressure.<br/><br/>Darcy’s law for fluid flow in porous media is a flux law. Neglecting the gravity effect, it is expressed by<br/><br/>[[File:Vol1 page 0080 eq 003.png|RTENOTITLE]]....................(3.16)<br/><br/>In '''Eq. 3.16''', ''μ'' is the viscosity of the fluid, and ''k'' is the permeability tensor of the formation given by<br/><br/>[[File:Vol1 page 0080 eq 004.png|RTENOTITLE]]....................(3.17)<br/><br/>where ''α'', ''β'', and ''γ'' are the directions, and ''k''<sub>''ij''</sub> is the permeability in the ''i'' direction as a result of the pressure gradient in the ''j'' direction.<br/><br/>If '''Eqs. 3.13''' through '''3.16''' are used in '''Eq. 3.11''', an alternative statement of the conservation of mass principle for fluid flow in porous media is obtained:<br/><br/>[[File:Vol1 page 0080 eq 005.png|RTENOTITLE]]....................(3.18)<br/><br/>'''Eq. 3.18''' is the PDE that governs transient fluid flow in porous media. In the present form, '''Eq. 3.18''' is not very useful in obtaining practical solutions because of the nonlinearity displayed in the second term of the left side. For liquid flow, the viscosity, ''μ'', is constant and '''Eq. 3.18''' can be linearized by assuming that the pressure gradients, ∇''p'', are small in the reservoir and the compressibility of the reservoir liquids, ''c'', is on the order of 10<sup>−5</sup> or smaller. Then, the second term of the left side of '''Eq. 3.18''' may be neglected compared with the remaining terms and the following linear expression is obtained:<br/><br/>[[File:Vol1 page 0080 eq 006.png|RTENOTITLE]]....................(3.19)<br/><br/>'''Eq. 3.19''' (or '''Eq. 3.18''') is known as the diffusivity equation. As an example in Cartesian coordinates, assuming that the coordinate axes can be chosen in the directions of the principal permeabilities, ''k'', in '''Eq. 3.19''', may be represented by the following diagonal tensor:<br/><br/>[[File:Vol1 page 0081 eq 001.png|RTENOTITLE]]....................(3.20)<br/><br/>Then, '''Eq. 3.19''' may be written<br/><br/>[[File:Vol1 page 0081 eq 002.png|RTENOTITLE]]....................(3.21)<br/><br/>If each coordinate, ''j'' = ''x'', ''y'', or ''z'', is multiplied by [[File:Vol1 page 0081 inline 001.png|RTENOTITLE]], where ''k'' may be chosen arbitrarily (to preserve the material balance, ''k'' is usually chosen to be [[File:Vol1 page 0081 inline 002.png|RTENOTITLE]]), '''Eq. 3.21''' may be transformed into the diffusion equation for an isotropic domain:<br/><br/>[[File:Vol1 page 0081 eq 003.png|RTENOTITLE]]....................(3.22)<br/><br/>where ''η'' is the diffusivity constant defined by<br/><br/>[[File:Vol1 page 0081 eq 004.png|RTENOTITLE]]....................(3.23)<br/><br/>If the same transformation is also applied to the boundary conditions (see '''Sec. 3.1.2'''), the problems in anisotropic reservoirs may be transformed into those in isotropic reservoirs provided that the system is infinite or bounded by planes perpendicular to the principal axes of permeability. In all other cases, this transformation distorts the bounding surfaces.<br/><br/>For the flow of gases, the assumptions of small fluid compressibility and pressure gradient may not be appropriate and the ''c''(∇''p'')<sup>2</sup> term in '''Eq. 3.18''' may not be negligible. In these cases, an expression similar to '''Eq. 3.21''' may be obtained from '''Eq. 3.18''' in terms of pseudopressure, ''m'', as<br/><br/>[[File:Vol1 page 0081 eq 005.png|RTENOTITLE]]....................(3.24)<br/><br/>Here, the pseudopressure is defined by<ref name="r1">Al-Hussainy, R., Ramey Jr., H.J., and Crawford, P.B. 1966. The Flow of Real Gases Through Porous Media. J Pet Technol 18 (5): 624–636. SPE-1243-A-PA. http://dx.doi.org/10.2118/1243-A-PA</ref><br/><br/>[[File:Vol1 page 0081 eq 006.png|RTENOTITLE]]....................(3.25)<br/><br/>where ''Z'' is the compressibility factor. To define a complete physical problem, '''Eq. 3.21''' (or '''3.24''') should be subject to the initial and boundary conditions discussed in '''Sec. 3.1.2'''.
<br/>
The conservation of mass principle requires that the difference between the rates at which fluid enters and leaves the volume through its surface must equal the rate at which mass accumulates within the volume. The total mass within the volume at any time is given by
<br>
<br>
[[File:Vol1 page 0078 eq 001.png]]....................(3.2)
<br>
<br>
Then, the time rate of change of mass within Γ is
<br>
<br>
[[File:Vol1 page 0078 eq 002.png]]....................(3.3)
<br>
<br>
which, by the conservation of mass law, is equal to the rate at which mass enters ''V'' through the surface.
<br>
<br>
Consider the differential surface element, dΓ, shown in '''Fig. 3.1'''. The mass entering the volume through dΓ at the normal velocity, [[File:Vol1 page 0078 inline 001.png]], in a time increment, Δ''t'', is [[File:Vol1 page 0078 inline 002.png]], and the total mass of the fluid passing through Γ during Δ''t'' is
<br>
<br>
[[File:Vol1 page 0078 eq 003.png]]....................(3.4)
<br>
<br>
The surface integral in '''Eq. 3.4''' accounts for both influx and outflux through the surface of the volume; that is, Δ''M''<sub>''g''</sub> is the difference between the masses entering and leaving the control volume during the time increment, Δ''t''. Then, the mass rate entering the volume, ''V'', through its surface, Γ, can be written as
<br>
<br>
[[File:Vol1 page 0078 eq 004.png]]....................(3.5)
<br>
<br>
By the principle of conservation of mass, equating the right sides of '''Eqs. 3.3''' and '''3.5''' yields
<br>
<br>
[[File:Vol1 page 0078 eq 005.png]]....................(3.6)
<br>
<br>
A more useful relation is found with the divergence theorem, which states that the flux of [[File:Vol1 page 0079 inline 001.png]] through the closed surface, Γ, is identical to the volume integral of [[File:Vol1 page 0079 inline 002.png]] (the divergence of [[File:Vol1 page 0079 inline 001.png]]) taken throughout ''V''; that is,
<br>
<br>
[[File:Vol1 page 0079 eq 001.png]]....................(3.7)
<br>
<br>
Here, ∇ is the gradient operator, which in 3D Cartesian and cylindrical coordinates is given, respectively, by
<br>
<br>
[[File:Vol1 page 0079 eq 002.png]]....................(3.8)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0079 eq 003.png]]....................(3.9)
<br>
<br>
With the relation in '''Eq. 3.7''', '''Eq. 3.6''' can be recast into
<br>
<br>
[[File:Vol1 page 0079 eq 004.png]]....................(3.10)
<br>
<br>
If the functions involved in the argument of the integral in '''Eq. 3.10''' are continuous, then the integral is identically zero if and only if its argument is zero (because the volume integral in '''Eq. 3.10''' is identically zero for any arbitrarily chosen volume). Then, the following continuity equation can be obtained.
<br>
<br>
[[File:Vol1 page 0079 eq 005.png]]....................(3.11)
<br>
<br>
'''Eq. 3.11''' is a PDE that is equivalent to the statement of the conservation of mass for fluid flow in porous media. For practical purposes, however, '''Eq. 3.11''' is expressed in terms of pressure because density and velocity cannot be measured directly. To express density, ''ρ'', and velocity, [[File:Vol1 page 0077 inline 001.png]], in terms of pressure, we use an equation of state and a flux law, known as Darcy’s law, respectively.
<br>
<br>
The following definition of isothermal fluid compressibility, ''c'', is a useful equation of state that relates density to pressure.
<br>
<br>
[[File:Vol1 page 0079 eq 006.png]]....................(3.12)
<br>
<br>
If ''c'' is a constant (the compressibility of many reservoir liquids may be considered as constant), then '''Eq. 3.12''' can be integrated to yield
<br>
<br>
[[File:Vol1 page 0079 eq 007.png]]....................(3.13)
<br>
<br>
where subscript 0 indicates the conditions at the datum. Similarly, the compressibility of the porous rock, ''c''<sub>''f''</sub>, is defined by
<br>
<br>
[[File:Vol1 page 0080 eq 001.png]]....................(3.14)
<br>
<br>
and the total system compressibility, ''c''<sub>''t''</sub>, is given by
<br>
<br>
[[File:Vol1 page 0080 eq 002.png]]....................(3.15)
<br>
<br>
These definitions of compressibility help recast '''Eq. 3.11''' in terms of pressure.
<br>
<br>
Darcy’s law for fluid flow in porous media is a flux law. Neglecting the gravity effect, it is expressed by
<br>
<br>
[[File:Vol1 page 0080 eq 003.png]]....................(3.16)
<br>
<br>
In '''Eq. 3.16''', ''μ'' is the viscosity of the fluid, and ''k'' is the permeability tensor of the formation given by
<br>
<br>
[[File:Vol1 page 0080 eq 004.png]]....................(3.17)
<br>
<br>
where ''α'', ''β'', and ''γ'' are the directions, and ''k''<sub>''ij''</sub> is the permeability in the ''i'' direction as a result of the pressure gradient in the ''j'' direction.
<br>
<br>
If '''Eqs. 3.13''' through '''3.16''' are used in '''Eq. 3.11''', an alternative statement of the conservation of mass principle for fluid flow in porous media is obtained:
<br>
<br>
[[File:Vol1 page 0080 eq 005.png]]....................(3.18)
<br>
<br>
'''Eq. 3.18''' is the PDE that governs transient fluid flow in porous media. In the present form, '''Eq. 3.18''' is not very useful in obtaining practical solutions because of the nonlinearity displayed in the second term of the left side. For liquid flow, the viscosity, ''μ'', is constant and '''Eq. 3.18''' can be linearized by assuming that the pressure gradients, ∇''p'', are small in the reservoir and the compressibility of the reservoir liquids, ''c'', is on the order of 10<sup>−5</sup> or smaller. Then, the second term of the left side of '''Eq. 3.18''' may be neglected compared with the remaining terms and the following linear expression is obtained:
<br>
<br>
[[File:Vol1 page 0080 eq 006.png]]....................(3.19)
<br>
<br>
'''Eq. 3.19''' (or '''Eq. 3.18''') is known as the diffusivity equation. As an example in Cartesian coordinates, assuming that the coordinate axes can be chosen in the directions of the principal permeabilities, ''k'', in '''Eq. 3.19''', may be represented by the following diagonal tensor:
<br>
<br>
[[File:Vol1 page 0081 eq 001.png]]....................(3.20)
<br>
<br>
Then, '''Eq. 3.19''' may be written
<br>
<br>
[[File:Vol1 page 0081 eq 002.png]]....................(3.21)
<br>
<br>
If each coordinate, ''j'' = ''x'', ''y'', or ''z'', is multiplied by [[File:Vol1 page 0081 inline 001.png]], where ''k'' may be chosen arbitrarily (to preserve the material balance, ''k'' is usually chosen to be [[File:Vol1 page 0081 inline 002.png]]), '''Eq. 3.21''' may be transformed into the diffusion equation for an isotropic domain:
<br>
<br>
[[File:Vol1 page 0081 eq 003.png]]....................(3.22)
<br>
<br>
where ''η'' is the diffusivity constant defined by
<br>
<br>
[[File:Vol1 page 0081 eq 004.png]]....................(3.23)
<br>
<br>
If the same transformation is also applied to the boundary conditions (see '''Sec. 3.1.2'''), the problems in anisotropic reservoirs may be transformed into those in isotropic reservoirs provided that the system is infinite or bounded by planes perpendicular to the principal axes of permeability. In all other cases, this transformation distorts the bounding surfaces.
<br>
<br>
For the flow of gases, the assumptions of small fluid compressibility and pressure gradient may not be appropriate and the ''c''(∇''p'')<sup>2</sup> term in '''Eq. 3.18''' may not be negligible. In these cases, an expression similar to '''Eq. 3.21''' may be obtained from '''Eq. 3.18''' in terms of pseudopressure, ''m'', as
<br>
<br>
[[File:Vol1 page 0081 eq 005.png]]....................(3.24)
<br>
<br>
Here, the pseudopressure is defined by<ref name="r1" />
<br>
<br>
[[File:Vol1 page 0081 eq 006.png]]....................(3.25)
<br>
<br>
where ''Z'' is the compressibility factor. To define a complete physical problem, '''Eq. 3.21''' (or '''3.24''') should be subject to the initial and boundary conditions discussed in '''Sec. 3.1.2'''.


=== Initial and Boundary Conditions ===
=== Initial and Boundary Conditions ===


The solution of the diffusivity equation ('''Eq. 3.19''') should satisfy the initial condition in the porous medium. The initial condition is normally expressed in terms of a known pressure distribution at time zero; that is,
The solution of the diffusivity equation ('''Eq. 3.19''') should satisfy the initial condition in the porous medium. The initial condition is normally expressed in terms of a known pressure distribution at time zero; that is,<br/><br/>[[File:Vol1 page 0082 eq 001.png|RTENOTITLE]]....................(3.26)<br/><br/>The most common initial condition is the uniform pressure distribution in the entire porous medium; that is, ''f'' (''x'', ''y'', ''z'') = ''p''<sub>''i''</sub>.<br/><br/>The boundary conditions are specified at the inner (wellbore) and outer boundaries of the reservoir. These are usually in the form of prescribed flux or pressure at the boundaries. The condition of prescribed flux can be formulated as<br/><br/>[[File:Vol1 page 0082 eq 002.png|RTENOTITLE]]....................(3.27)<br/><br/>where Γ is the surface of the boundary, and ''n'' indicates the outward normal direction of the boundary surface. The prescribed flux condition may be used at the inner and outer boundaries of the reservoir. The most common use of the prescribed flux condition at the inner boundary is for the production at a constant rate. In this case, the function, ''g''(''t''), is related to a constant production rate, ''q''. At the outer boundary, the prescribed flux condition is usually used to indicate impermeable boundaries [''g''(''t'')=0] and leads to a pseudosteady state under the influence of boundaries.
<br>
<br>
[[File:Vol1 page 0082 eq 001.png]]....................(3.26)
<br>
<br>
The most common initial condition is the uniform pressure distribution in the entire porous medium; that is, ''f'' (''x'', ''y'', ''z'') = ''p''<sub>''i''</sub>.
<br>
<br>
The boundary conditions are specified at the inner (wellbore) and outer boundaries of the reservoir. These are usually in the form of prescribed flux or pressure at the boundaries. The condition of prescribed flux can be formulated as
<br>
<br>
[[File:Vol1 page 0082 eq 002.png]]....................(3.27)
<br>
<br>
where Γ is the surface of the boundary, and ''n'' indicates the outward normal direction of the boundary surface. The prescribed flux condition may be used at the inner and outer boundaries of the reservoir. The most common use of the prescribed flux condition at the inner boundary is for the production at a constant rate. In this case, the function, ''g''(''t''), is related to a constant production rate, ''q''. At the outer boundary, the prescribed flux condition is usually used to indicate impermeable boundaries [''g''(''t'')=0] and leads to a pseudosteady state under the influence of boundaries.


For some applications, pressure may be specified at the inner and outer boundaries. In this case,
For some applications, pressure may be specified at the inner and outer boundaries. In this case,<br/><br/>[[File:Vol1 page 0082 eq 003.png|RTENOTITLE]]....................(3.28)<br/><br/>When used at the inner boundary, this condition represents production at a constant pressure, ''p''<sub>''wf''</sub>; that is, ''h''(''t'') = ''p''<sub>''wf''</sub>. At the outer boundary, specified pressure, ''p''<sub>''e''</sub>, is usually a result of injection or influx from an adjacent aquifer, which usually leads to steady state in the reservoir.<br/><br/>It is also possible to have boundary conditions of mixed type. These usually correspond to interface conditions in porous media. Raghavan<ref name="r2">Raghavan, R. 1993. Well Test Analysis, 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.</ref> contains more details about the common boundary conditions for the diffusion equation.
<br>
<br>
[[File:Vol1 page 0082 eq 003.png]]....................(3.28)
<br>
<br>
When used at the inner boundary, this condition represents production at a constant pressure, ''p''<sub>''wf''</sub>; that is, ''h''(''t'') = ''p''<sub>''wf''</sub>. At the outer boundary, specified pressure, ''p''<sub>''e''</sub>, is usually a result of injection or influx from an adjacent aquifer, which usually leads to steady state in the reservoir.
<br>
<br>
It is also possible to have boundary conditions of mixed type. These usually correspond to interface conditions in porous media. Raghavan<ref name="r2" /> contains more details about the common boundary conditions for the diffusion equation.


=== Assumptions and Limits ===
=== Assumptions and Limits ===


Some assumptions have been made in the derivation of the diffusivity equation given by '''Eq. 3.19'''. These assumptions determine the limits of applicability of the solutions obtained from '''Eq. 3.19'''. One of the most important assumptions involved is the continuity of the properties involved in '''Eq. 3.19'''. (This was required to obtain '''Eq. 3.19''' from the more general integral form in '''Eq. 3.10'''.) Therefore, sharp changes in the properties of the reservoir rock and fluid (such as faults and fluid banks) should be incorporated in the form of boundary or interface conditions in the solution of '''Eq. 3.19'''.
Some assumptions have been made in the derivation of the diffusivity equation given by '''Eq. 3.19'''. These assumptions determine the limits of applicability of the solutions obtained from '''Eq. 3.19'''. One of the most important assumptions involved is the continuity of the properties involved in '''Eq. 3.19'''. (This was required to obtain '''Eq. 3.19''' from the more general integral form in '''Eq. 3.10'''.) Therefore, sharp changes in the properties of the reservoir rock and fluid (such as faults and fluid banks) should be incorporated in the form of boundary or interface conditions in the solution of '''Eq. 3.19'''.<br/><br/>The second important assumption is that Darcy’s law describes the flux in porous media. This assumption is valid at relatively low fluid velocities that may be appropriate to describe liquid flow. At high velocities (when Reynolds number based on average sand grain diameter approaches unity) such as those observed in gas reservoirs, Darcy’s law is not valid.<ref name="r3">Fancher, G.H., Lewis, J.A., and Barnes, K.B. 1933. Some Physical Characteristics of Oil Sands. Bulletin 12, Mineral Industries Experimental Station, Pennsylvania State University, University Park, Pennsylvania, 65–167.</ref> In this case, Forchheimer’s equation,<ref name="r4">Forchheimer, P.F. 1901. Wasserbewegung durch Boden. Zeitschrift des Vereines deutscher Ingenieure 45 (5): 1781–1788.</ref> which accounts for the inertial effects, should be used. In petroleum engineering, it is a common practice to consider the additional pressure drop as a result of non-Darcy flow in the form of a pseudoskin because it is usually effective in a small vicinity of the wellbore. Therefore, in this chapter, we do not consider non-Darcy flow in the reservoir.
<br>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<br>
The second important assumption is that Darcy’s law describes the flux in porous media. This assumption is valid at relatively low fluid velocities that may be appropriate to describe liquid flow. At high velocities (when Reynolds number based on average sand grain diameter approaches unity) such as those observed in gas reservoirs, Darcy’s law is not valid.<ref name="r3" /> In this case, Forchheimer’s equation,<ref name="r4" /> which accounts for the inertial effects, should be used. In petroleum engineering, it is a common practice to consider the additional pressure drop as a result of non-Darcy flow in the form of a pseudoskin because it is usually effective in a small vicinity of the wellbore. Therefore, in this chapter, we do not consider non-Darcy flow in the reservoir.
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
== Bessel Functions ==
== Bessel Functions ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/>As '''Sec. 3.3''' illustrates, the Laplace transform of the diffusion equation in radial coordinates yields a modified Bessel’s equation, and its solutions are obtained in terms of modified Bessel functions. This section introduces Bessel functions and discusses some of their properties to the extent that they are encountered in the solutions of more common petroleum engineering problems.
As '''Sec. 3.3''' illustrates, the Laplace transform of the diffusion equation in radial coordinates yields a modified Bessel’s equation, and its solutions are obtained in terms of modified Bessel functions. This section introduces Bessel functions and discusses some of their properties to the extent that they are encountered in the solutions of more common petroleum engineering problems.


=== Preliminary Definitions ===
=== Preliminary Definitions ===


A differential equation of the type
A differential equation of the type<br/><br/>[[File:Vol1 page 0083 eq 001.png|RTENOTITLE]]....................(3.29)<br/><br/>is called a Bessel’s equation of order ''v''. A solution of Bessel’s equation of order ''v'' is called a Bessel function of order ''v''. A differential equation of the type<br/><br/>[[File:Vol1 page 0083 eq 002.png|RTENOTITLE]]....................(3.30)<br/><br/>is called a modified Bessel’s equation of order ''v''. '''Eq. 3.30''' is obtained by substituting ''λz'' for ''z'' in '''Eq. 3.29'''. Of particular interest is the case in which ''λ''=''ki'' so that '''Eq. 3.30''' becomes<br/><br/>[[File:Vol1 page 0083 eq 003.png|RTENOTITLE]]....................(3.31)<br/><br/>'''Eq. 3.31''' is called the modified Bessel’s equation of order ''v''. A solution of the modified Bessel’s equation of order ''v'' is called a modified Bessel function of order ''v''.
<br>
<br>
[[File:Vol1 page 0083 eq 001.png]]....................(3.29)
<br>
<br>
is called a Bessel’s equation of order ''v''. A solution of Bessel’s equation of order ''v'' is called a Bessel function of order ''v''. A differential equation of the type
<br>
<br>
[[File:Vol1 page 0083 eq 002.png]]....................(3.30)
<br>
<br>
is called a modified Bessel’s equation of order ''v''. '''Eq. 3.30''' is obtained by substituting ''λz'' for ''z'' in '''Eq. 3.29'''. Of particular interest is the case in which ''λ''=''ki'' so that '''Eq. 3.30''' becomes
<br>
<br>
[[File:Vol1 page 0083 eq 003.png]]....................(3.31)
<br>
<br>
'''Eq. 3.31''' is called the modified Bessel’s equation of order ''v''. A solution of the modified Bessel’s equation of order ''v'' is called a modified Bessel function of order ''v''.


=== Solutions of Bessel’s Equations and Bessel Functions ===
=== Solutions of Bessel’s Equations and Bessel Functions ===


There are many methods of obtaining or constructing Bessel functions.<ref name="r5" /> Only the final form of the Bessel functions that are of interest are presented here.
There are many methods of obtaining or constructing Bessel functions.<ref name="r5">Watson, G. N. 1944. A Treatise on the Theory of Bessel Functions. London: Cambridge University Press.</ref> Only the final form of the Bessel functions that are of interest are presented here.<br/><br/>If ''v'' is not a positive integer, then the general solution of Bessel’s equation of order ''v'' ('''Eq. 3.29''') is given by<br/><br/>[[File:Vol1 page 0083 eq 004.png|RTENOTITLE]]....................(3.32)<br/><br/>where ''A'' and ''B'' are arbitrary constants, and ''J''<sub>''v''</sub>(''z'') is the Bessel function of order ''v'' of the first kind given by<br/><br/>[[File:Vol1 page 0083 eq 005.png|RTENOTITLE]]....................(3.33)<br/><br/>In '''Eq. 3.33''', Γ(''x'') is the gamma function defined by<br/><br/>[[File:Vol1 page 0083 eq 006.png|RTENOTITLE]]....................(3.34)<br/><br/>If ''v'' is a positive integer, ''n'', then ''J''<sub>''v''</sub> and ''J−''<sub>''v''</sub> are linearly dependent, and the solution of '''Eq. 3.29''' is written as<br/><br/>[[File:Vol1 page 0083 eq 007.png|RTENOTITLE]]....................(3.35)<br/><br/>In '''Eq. 3.35''', ''Y''<sub>''n''</sub>(''z'') is the Bessel function of order ''n'' of the second kind and is defined by<br/><br/>[[File:Vol1 page 0084 eq 001.png|RTENOTITLE]]....................(3.36)<br/><br/>Similarly, if ''v'' is not a positive integer, the general solution of the modified Bessel’s equation of order ''v'' ('''Eq. 3.31''') is given by<br/><br/>[[File:Vol1 page 0084 eq 002.png|RTENOTITLE]]....................(3.37)<br/><br/>where ''I''<sub>''v''</sub>(''z'') is the modified Bessel function of order ''v'' of the first kind defined by<br/><br/>[[File:Vol1 page 0084 eq 003.png|RTENOTITLE]]....................(3.38)<br/><br/>If ''v'' is a positive integer, ''n'', ''I''<sub>''v''</sub>, and ''I''<sub>''−v''</sub> are linearly dependent. The solution for this case is<br/><br/>[[File:Vol1 page 0084 eq 004.png|RTENOTITLE]]....................(3.39)<br/><br/>where ''K''<sub>''n''</sub>(''z'') is the modified Bessel function of order ''n'' of the second kind and is defined by<br/><br/>[[File:Vol1 page 0084 eq 005.png|RTENOTITLE]]....................(3.40)<br/><br/>The modified Bessel functions of order zero and one are of special interest, and '''Sec. 3.2.3''' discusses some of their special features.
<br>
<br>
If ''v'' is not a positive integer, then the general solution of Bessel’s equation of order ''v'' ('''Eq. 3.29''') is given by
<br>
<br>
[[File:Vol1 page 0083 eq 004.png]]....................(3.32)
<br>
<br>
where ''A'' and ''B'' are arbitrary constants, and ''J''<sub>''v''</sub>(''z'') is the Bessel function of order ''v'' of the first kind given by
<br>
<br>
[[File:Vol1 page 0083 eq 005.png]]....................(3.33)
<br>
<br>
In '''Eq. 3.33''', Γ(''x'') is the gamma function defined by
<br>
<br>
[[File:Vol1 page 0083 eq 006.png]]....................(3.34)
<br>
<br>
If ''v'' is a positive integer, ''n'', then ''J''<sub>''v''</sub> and ''J−''<sub>''v''</sub> are linearly dependent, and the solution of '''Eq. 3.29''' is written as
<br>
<br>
[[File:Vol1 page 0083 eq 007.png]]....................(3.35)
<br>
<br>
In '''Eq. 3.35''', ''Y''<sub>''n''</sub>(''z'') is the Bessel function of order ''n'' of the second kind and is defined by
<br>
<br>
[[File:Vol1 page 0084 eq 001.png]]....................(3.36)
<br>
<br>
Similarly, if ''v'' is not a positive integer, the general solution of the modified Bessel’s equation of order ''v'' ('''Eq. 3.31''') is given by
<br>
<br>
[[File:Vol1 page 0084 eq 002.png]]....................(3.37)
<br>
<br>
where ''I''<sub>''v''</sub>(''z'') is the modified Bessel function of order ''v'' of the first kind defined by
<br>
<br>
[[File:Vol1 page 0084 eq 003.png]]....................(3.38)
<br>
<br>
If ''v'' is a positive integer, ''n'', ''I''<sub>''v''</sub>, and ''I''<sub>''−v''</sub> are linearly dependent. The solution for this case is
<br>
<br>
[[File:Vol1 page 0084 eq 004.png]]....................(3.39)
<br>
<br>
where ''K''<sub>''n''</sub>(''z'') is the modified Bessel function of order ''n'' of the second kind and is defined by
<br>
<br>
[[File:Vol1 page 0084 eq 005.png]]....................(3.40)
<br>
<br>
The modified Bessel functions of order zero and one are of special interest, and '''Sec. 3.2.3''' discusses some of their special features.


=== Modified Bessel Functions of Order Zero and One ===
=== Modified Bessel Functions of Order Zero and One ===


Modified Bessel functions of order zero and one are related to each other by the following relations:
Modified Bessel functions of order zero and one are related to each other by the following relations:<br/><br/>[[File:Vol1 page 0084 eq 006.png|RTENOTITLE]]....................(3.41)<br/><br/>and<br/><br/>[[File:Vol1 page 0084 eq 007.png|RTENOTITLE]]....................(3.42)<br/><br/>'''Fig. 3.2''' shows these functions graphically.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0084 eq 006.png]]....................(3.41)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0084 eq 007.png]]....................(3.42)
<br>
<br>
'''Fig. 3.2''' shows these functions graphically.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 085 Image 0001.png|'''Fig. 3.2 – Bessel functions of order zero and one.'''
File:vol1 Page 085 Image 0001.png|'''Fig. 3.2 – Bessel functions of order zero and one.'''
</gallery>
</gallery><br/>For small arguments, the following asymptotic expansions may be used for the modified Bessel functions of order zero and one:<ref name="r5">Watson, G. N. 1944. A Treatise on the Theory of Bessel Functions. London: Cambridge University Press.</ref><br/><br/>[[File:Vol1 page 0084 eq 008.png|RTENOTITLE]]....................(3.43)<br/><br/>[[File:Vol1 page 0085 eq 001.png|RTENOTITLE]]....................(3.44)<br/><br/>[[File:Vol1 page 0085 eq 002.png|RTENOTITLE]]....................(3.45)<br/><br/>where ''γ'' = 0.5772…, and<br/><br/>[[File:Vol1 page 0085 eq 003.png|RTENOTITLE]]....................(3.46)<br/><br/>Also, for large arguments, the following relations may be useful:<br/><br/>[[File:Vol1 page 0085 eq 004.png|RTENOTITLE]]....................(3.47)<br/><br/>for |''arg'' ''z''| < ''π'' / 2, and<br/><br/>[[File:Vol1 page 0085 eq 005.png|RTENOTITLE]]....................(3.48)<br/><br/>for |''arg'' ''z''| < 3''π'' / 2. On the basis of the relations given by '''Eqs. 3.43''' through '''3.48''', the following limiting forms may be written:<br/><br/>[[File:Vol1 page 0086 eq 001.png|RTENOTITLE]]....................(3.49)<br/><br/>[[File:Vol1 page 0086 eq 002.png|RTENOTITLE]]....................(3.50)<br/><br/>[[File:Vol1 page 0086 eq 003.png|RTENOTITLE]]....................(3.51)<br/><br/>[[File:Vol1 page 0086 eq 004.png|RTENOTITLE]]....................(3.52)<br/><br/>[[File:Vol1 page 0086 eq 005.png|RTENOTITLE]]....................(3.53)<br/><br/>[[File:Vol1 page 0086 eq 006.png|RTENOTITLE]]....................(3.54)<br/><br/>[[File:Vol1 page 0086 eq 007.png|RTENOTITLE]]....................(3.55)<br/><br/>[[File:Vol1 page 0086 eq 008.png|RTENOTITLE]]....................(3.56)<br/><br/>and<br/><br/>[[File:Vol1 page 0086 eq 009.png|RTENOTITLE]]....................(3.57)<br/><br/>These relations are useful in the evaluation of the asymptotic behavior of transient-pressure solutions.
<br/>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
For small arguments, the following asymptotic expansions may be used for the modified Bessel functions of order zero and one:<ref name="r5" />
<br>
<br>
[[File:Vol1 page 0084 eq 008.png]]....................(3.43)
<br>
<br>
[[File:Vol1 page 0085 eq 001.png]]....................(3.44)
<br>
<br>
[[File:Vol1 page 0085 eq 002.png]]....................(3.45)
<br>
<br>
where ''γ'' = 0.5772…, and
<br>
<br>
[[File:Vol1 page 0085 eq 003.png]]....................(3.46)
<br>
<br>
Also, for large arguments, the following relations may be useful:
<br>
<br>
[[File:Vol1 page 0085 eq 004.png]]....................(3.47)
<br>
<br>
for |''arg'' ''z''| < ''π'' / 2, and
<br>
<br>
[[File:Vol1 page 0085 eq 005.png]]....................(3.48)
<br>
<br>
for |''arg'' ''z''| < 3''π'' / 2. On the basis of the relations given by '''Eqs. 3.43''' through '''3.48''', the following limiting forms may be written:
<br>
<br>
[[File:Vol1 page 0086 eq 001.png]]....................(3.49)
<br>
<br>
[[File:Vol1 page 0086 eq 002.png]]....................(3.50)
<br>
<br>
[[File:Vol1 page 0086 eq 003.png]]....................(3.51)
<br>
<br>
[[File:Vol1 page 0086 eq 004.png]]....................(3.52)
<br>
<br>
[[File:Vol1 page 0086 eq 005.png]]....................(3.53)
<br>
<br>
[[File:Vol1 page 0086 eq 006.png]]....................(3.54)
<br>
<br>
[[File:Vol1 page 0086 eq 007.png]]....................(3.55)
<br>
<br>
[[File:Vol1 page 0086 eq 008.png]]....................(3.56)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0086 eq 009.png]]....................(3.57)
<br>
<br>
These relations are useful in the evaluation of the asymptotic behavior of transient-pressure solutions.
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
 
== Laplace Transformation ==
== Laplace Transformation ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/>Integral transforms are useful in solving differential equations. A special form of the linear integral transforms, known as the Laplace transformation, is particularly useful in the solution of the diffusion equation. The Laplace transformation of a function, ''F''(''t''), denoted by ''L''{''F''(''t'')}, is defined by<br/><br/>[[File:Vol1 page 0086 eq 010.png|RTENOTITLE]]....................(3.58)<br/><br/>where ''s'' is a number whose real part is positive and large enough for the integral in '''Eq. 3.58''' to exist. In this chapter, a bar over the function indicates the image or the Laplace transform of the function; that is,<br/><br/>[[File:Vol1 page 0086 eq 011.png|RTENOTITLE]]....................(3.59)<br/>
Integral transforms are useful in solving differential equations. A special form of the linear integral transforms, known as the Laplace transformation, is particularly useful in the solution of the diffusion equation. The Laplace transformation of a function, ''F''(''t''), denoted by ''L''{''F''(''t'')}, is defined by
<br>
<br>
[[File:Vol1 page 0086 eq 010.png]]....................(3.58)
<br>
<br>
where ''s'' is a number whose real part is positive and large enough for the integral in '''Eq. 3.58''' to exist. In this chapter, a bar over the function indicates the image or the Laplace transform of the function; that is,
<br>
<br>
[[File:Vol1 page 0086 eq 011.png]]....................(3.59)
<br>
<br>


=== Fundamental Properties of the Laplace Transformation ===
=== Fundamental Properties of the Laplace Transformation ===


The following fundamental properties of the Laplace transformation are useful in the solution of common transient-flow problems.
The following fundamental properties of the Laplace transformation are useful in the solution of common transient-flow problems.<br/><br/>'''''Transforms of Derivatives.'''''<br/><br/>[[File:Vol1 page 0087 eq 001.png|RTENOTITLE]]....................(3.60)<br/><br/>[[File:Vol1 page 0087 eq 002.png|RTENOTITLE]]....................(3.61)<br/><br/>[[File:Vol1 page 0087 eq 003.png|RTENOTITLE]]....................(3.62)<br/><br/>'''''Transforms of Integrals.'''''<br/><br/>[[File:Vol1 page 0087 eq 004.png|RTENOTITLE]]....................(3.63)<br/><br/>'''''Substitution.'''''<br/><br/>[[File:Vol1 page 0087 eq 005.png|RTENOTITLE]]....................(3.64)<br/><br/>[[File:Vol1 page 0087 eq 006.png|RTENOTITLE]]....................(3.65)<br/><br/>where [[File:Vol1 page 0087 inline 001.png|RTENOTITLE]].<br/><br/>'''''Translation.'''''<br/><br/>[[File:Vol1 page 0087 eq 007.png|RTENOTITLE]]....................(3.66)<br/><br/>where ''H''(''t'' - ''t''<sub>0</sub>) is Heaviside’s unit step function defined by<br/><br/>[[File:Vol1 page 0087 eq 008.png|RTENOTITLE]]....................(3.67)<br/><br/>'''''Convolution.'''''<br/><br/>[[File:Vol1 page 0087 eq 009.png|RTENOTITLE]]....................(3.68)<br/>
<br>
<br>
'''''Transforms of Derivatives.'''''
<br>
<br>
[[File:Vol1 page 0087 eq 001.png]]....................(3.60)
<br>
<br>
[[File:Vol1 page 0087 eq 002.png]]....................(3.61)
<br>
<br>
[[File:Vol1 page 0087 eq 003.png]]....................(3.62)
<br>
<br>
'''''Transforms of Integrals.'''''
<br>
<br>
[[File:Vol1 page 0087 eq 004.png]]....................(3.63)
<br>
<br>
'''''Substitution.'''''
<br>
<br>
[[File:Vol1 page 0087 eq 005.png]]....................(3.64)
<br>
<br>
[[File:Vol1 page 0087 eq 006.png]]....................(3.65)
<br>
<br>
where [[File:Vol1 page 0087 inline 001.png]].
<br>
<br>
'''''Translation.'''''
<br>
<br>
[[File:Vol1 page 0087 eq 007.png]]....................(3.66)
<br>
<br>
where ''H''(''t'' - ''t''<sub>0</sub>) is Heaviside’s unit step function defined by
<br>
<br>
[[File:Vol1 page 0087 eq 008.png]]....................(3.67)
<br>
<br>
'''''Convolution.'''''
<br>
<br>
[[File:Vol1 page 0087 eq 009.png]]....................(3.68)
<br>
<br>


=== Inverse Laplace Transformation and Asymptotic Forms ===
=== Inverse Laplace Transformation and Asymptotic Forms ===


For the Laplace transform to be useful, the inverse Laplace transformation must be uniquely defined. L<sup>−1</sup> denotes the inverse Laplace transform operator; that is,
For the Laplace transform to be useful, the inverse Laplace transformation must be uniquely defined. L<sup>−1</sup> denotes the inverse Laplace transform operator; that is,<br/><br/>[[File:Vol1 page 0088 eq 001.png|RTENOTITLE]]....................(3.69)<br/><br/>In this operation, ''p''(''t'') represents the inverse (transform) of the Laplace domain function, [[File:Vol1 page 0088 inline 001.png|RTENOTITLE]]. A uniqueness theorem of the inversion guarantees that no two functions of the class ''ε'' have the same Laplace transform.<ref name="r6">Churchill, R.V. 1972. Operational Mathematics, Vol. 2. New York: McGraw-Hill Book Co.</ref> The class ''ε'' is defined as the set of sectionally continuous functions ''F''(''t'') that are continuous on each bounded interval over the half line ''t'' > 0 except at a finite number of points, ''t''<sub>''i''</sub>, where they are defined by<br/><br/>[[File:Vol1 page 0088 eq 002.png|RTENOTITLE]]....................(3.70)<br/><br/>and | ''F''(''t'') | < ''Me''<sup>''αt''</sup> for any constants ''M'' and ''α''.<br/><br/>The most rigorous technique to find the inverse Laplace transform of a Laplace domain function is the use of the inversion integral,<ref name="r6">Churchill, R.V. 1972. Operational Mathematics, Vol. 2. New York: McGraw-Hill Book Co.</ref> but its discussion is outside the scope of this chapter. For petroleum engineering applications, a simple table look-up procedure is usually the first resort. '''Table 3.1''' shows an example table of Laplace transform pairs that may be used to find the Laplace transforms of real-space functions or the inverse Laplace transforms of the Laplace domain functions. Fairly large tables of Laplace transform pairs can be found in a couple of sources.<ref name="r6">Churchill, R.V. 1972. Operational Mathematics, Vol. 2. New York: McGraw-Hill Book Co.</ref><ref name="r7">Abramowitz, M. and Stegun, I.A. ed. 1972. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, ninth edition, 1020–1029. New York: Dover Publications.</ref> The relations given in the Laplace transform tables may be extended to more complex functions with the fundamental properties of the Laplace transforms noted in '''Sec. 3.3.1'''.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0088 eq 001.png]]....................(3.69)
<br>
<br>
In this operation, ''p''(''t'') represents the inverse (transform) of the Laplace domain function, [[File:Vol1 page 0088 inline 001.png]]. A uniqueness theorem of the inversion guarantees that no two functions of the class ''ε'' have the same Laplace transform.<ref name="r6" /> The class ''ε'' is defined as the set of sectionally continuous functions ''F''(''t'') that are continuous on each bounded interval over the half line ''t'' > 0 except at a finite number of points, ''t''<sub>''i''</sub>, where they are defined by
<br>
<br>
[[File:Vol1 page 0088 eq 002.png]]....................(3.70)
<br>
<br>
and | ''F''(''t'') | < ''Me''<sup>''αt''</sup> for any constants ''M'' and ''α''.
<br>
<br>
The most rigorous technique to find the inverse Laplace transform of a Laplace domain function is the use of the inversion integral,<ref name="r6" /> but its discussion is outside the scope of this chapter. For petroleum engineering applications, a simple table look-up procedure is usually the first resort. '''Table 3.1''' shows an example table of Laplace transform pairs that may be used to find the Laplace transforms of real-space functions or the inverse Laplace transforms of the Laplace domain functions. Fairly large tables of Laplace transform pairs can be found in a couple of sources.<ref name="r6" /><ref name="r7" /> The relations given in the Laplace transform tables may be extended to more complex functions with the fundamental properties of the Laplace transforms noted in '''Sec. 3.3.1'''.
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 089 Image 0001.png|'''Table 3.1'''
File:Vol1 Page 089 Image 0001.png|'''Table 3.1'''
</gallery>
</gallery><br/>When a simple analytical inversion is not possible, numerical inversion of a Laplace domain function is an alternate procedure. Many numerical inversion algorithms have been proposed in the literature. For the inversion of the transient-flow solutions in Laplace domain, the numerical inversion algorithm suggested by Stehfest<ref name="r8">Stehfest, H. 1970. Algorithm 368: Numerical inversion of Laplace transforms. Commun. ACM 13 (1): 47–49. http://dx.doi.org/10.1145/361953.361969</ref> is the most popular algorithm.<br/><br/>The Stehfest algorithm is based on a stochastic process and suggests that an approximate value, ''p''<sub>''a''</sub>(''T''), of the inverse of the Laplace domain function, [[File:Vol1 page 0088 inline 002.png|RTENOTITLE]], may be obtained at time ''t'' = ''T'' by<br/><br/>[[File:Vol1 page 0088 eq 003.png|RTENOTITLE]]....................(3.71)<br/><br/>where<br/><br/>[[File:Vol1 page 0088 eq 004.png|RTENOTITLE]]....................(3.72)<br/><br/>In '''Eqs. 3.71''' and '''3.72''', ''N'' is an even integer. Although, theoretically, the accuracy of the inversion should increase as ''N'' tends to infinity [''p''<sub>''a''</sub> (''T'') should tend to ''p''(''T'')], the accuracy may be lost because of round-off errors when ''N'' becomes large. Normally, the optimum value of ''N'' is determined as a result of a numerical experiment. As a reference, however, the range of 6 ≤ ''N'' ≤ 18 covers the most common values of ''N'' for transient-flow problems. The Stehfest algorithm is not appropriate for the numerical inversion of oscillatory and discontinuous functions. In these cases, a more complex algorithm proposed by Crump<ref name="r9">Crump, K.S. 1976. Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation. J. ACM 23 (1): 89-96. http://dx.doi.org/10.1145/321921.321931</ref> may be used.<br/><br/>In some cases, obtaining asymptotic solutions for small and large values of time may be of interest. These asymptotic results may be obtained without inverting the full solution into the real-time domain. The limiting forms of the full solution as ''s'' → ∞ and ''s'' → 0 correspond to the limiting forms in the time domain for short and long time, respectively. The inversion of the limiting forms may be easier than the inversion of the full solution. '''Examples 3.1''' through '''3.4''' demonstrate the use of Laplace transformation in the solution of transient-flow problems.
<br>
When a simple analytical inversion is not possible, numerical inversion of a Laplace domain function is an alternate procedure. Many numerical inversion algorithms have been proposed in the literature. For the inversion of the transient-flow solutions in Laplace domain, the numerical inversion algorithm suggested by Stehfest<ref name="r8" /> is the most popular algorithm.
<br>
<br>
The Stehfest algorithm is based on a stochastic process and suggests that an approximate value, ''p''<sub>''a''</sub>(''T''), of the inverse of the Laplace domain function, [[File:Vol1 page 0088 inline 002.png]], may be obtained at time ''t'' = ''T'' by
<br>
<br>
[[File:Vol1 page 0088 eq 003.png]]....................(3.71)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0088 eq 004.png]]....................(3.72)
<br>
<br>
In '''Eqs. 3.71''' and '''3.72''', ''N'' is an even integer. Although, theoretically, the accuracy of the inversion should increase as ''N'' tends to infinity [''p''<sub>''a''</sub> (''T'') should tend to ''p''(''T'')], the accuracy may be lost because of round-off errors when ''N'' becomes large. Normally, the optimum value of ''N'' is determined as a result of a numerical experiment. As a reference, however, the range of 6 ≤ ''N'' ≤ 18 covers the most common values of ''N'' for transient-flow problems. The Stehfest algorithm is not appropriate for the numerical inversion of oscillatory and discontinuous functions. In these cases, a more complex algorithm proposed by Crump<ref name="r9" /> may be used.
<br>
<br>
In some cases, obtaining asymptotic solutions for small and large values of time may be of interest. These asymptotic results may be obtained without inverting the full solution into the real-time domain. The limiting forms of the full solution as ''s'' → ∞ and ''s'' → 0 correspond to the limiting forms in the time domain for short and long time, respectively. The inversion of the limiting forms may be easier than the inversion of the full solution. '''Examples 3.1''' through '''3.4''' demonstrate the use of Laplace transformation in the solution of transient-flow problems.


----
----


'''Example 3.1'''
'''Example 3.1'''<br/><br/>Consider transient flow toward a fully penetrating vertical well in an infinite homogeneous reservoir of uniform thickness, ''h'', and initial pressure, ''p''<sub>''i''</sub>.<br/><br/>''Solution''. This problem may be formulated most conveniently in the radial coordinates. The diffusivity equation governing fluid flow in porous media is given, in radial coordinates, by<br/><br/>[[File:Vol1 page 0089 eq 001.png|RTENOTITLE]]....................(3.73)<br/><br/>where ∆''p'' = ''p''<sub>''i''</sub> – ''p''. '''Eq. 3.73''' is the same in absolute (cgs or SI) or Darcy units. (In field units, some conversion coefficients are involved in '''Eq. 3.73'''.) The initial condition is<br/><br/>[[File:Vol1 page 0090 eq 001.png|RTENOTITLE]]....................(3.74)<br/><br/>which means that the pressure is uniform and equal to ''p''<sub>''i''</sub> initially throughout the reservoir. The outer boundary condition for an infinite reservoir is<br/><br/>[[File:Vol1 page 0090 eq 002.png|RTENOTITLE]]....................(3.75)<br/><br/>which physically means that for any given time, ''t'', there is a large enough distance, ''r'', in the reservoir at which the initial pressure, ''p''<sub>''i''</sub>, has been preserved.<br/><br/>The inner boundary condition depends on the production conditions at the surface of the wellbore (''r'' = ''r''<sub>''w''</sub>). Assuming that the well is produced at a constant rate, ''q'', for all times,<br/><br/>[[File:Vol1 page 0090 eq 003.png|RTENOTITLE]]....................(3.76)<br/><br/>The inner boundary condition given in '''Eq. 3.76''' is simply a restatement of the flux law (Darcy’s law given by '''Eq. 3.16''') at the surface of the wellbore.<br/><br/>'''Eqs. 3.73''' through '''3.76''' define the IBVP to be solved to obtain the transient-pressure distribution for the given system. Application of the Laplace transforms to '''Eq. 3.73''' yields<br/><br/>[[File:Vol1 page 0090 eq 004.png|RTENOTITLE]]....................(3.77)<br/><br/>or, rearranging, we obtain<br/><br/>[[File:Vol1 page 0090 eq 005.png|RTENOTITLE]]....................(3.78)<br/><br/>In obtaining the right side of '''Eq. 3.77''', the initial condition ('''Eq. 3.74''') has been used. Similarly, '''Eqs. 3.75''' and '''3.76''' are transformed into the following forms, respectively.<br/><br/>[[File:Vol1 page 0090 eq 006.png|RTENOTITLE]]....................(3.79)<br/><br/>and<br/><br/>[[File:Vol1 page 0090 eq 007.png|RTENOTITLE]]....................(3.80)<br/><br/>Comparing '''Eq. 3.78''' with '''Eq. 3.31''', we recognize '''Eq. 3.78''' as the modified Bessel’s equation of order zero. The solution of '''Eq. 3.78''' may be written directly from '''Eq. 3.39''' as<br/><br/>[[File:Vol1 page 0090 eq 008.png|RTENOTITLE]]....................(3.81)<br/><br/>The constants ''C''<sub>1</sub> and ''C''<sub>2</sub> in '''Eq. 3.81''' are obtained from the boundary conditions. The outer boundary condition ('''Eq. 3.79''') indicates that ''C''<sub>1</sub> = 0 [because [[File:Vol1 page 0091 inline 001.png|RTENOTITLE]], '''Eq. 3.79''' is satisfied only if ''C''<sub>1</sub> = 0]; therefore,<br/><br/>[[File:Vol1 page 0091 eq 001.png|RTENOTITLE]]....................(3.82)<br/><br/>From '''Eqs. 3.80''' and '''3.82''', we obtain<br/><br/>[[File:Vol1 page 0091 eq 002.png|RTENOTITLE]]....................(3.83)<br/><br/>which yields<br/><br/>[[File:Vol1 page 0091 eq 003.png|RTENOTITLE]]....................(3.84)<br/><br/>Then, the solution for the transient-pressure distribution is given, in the Laplace transform domain, by<br/><br/>[[File:Vol1 page 0091 eq 004.png|RTENOTITLE]]....................(3.85)<br/><br/>To complete the solution of the problem, '''Eq. 3.85''' should be inverted into the real-time domain. The real inversion of '''Eq. 3.85''', however, is not available in terms of standard functions. One option is to use Stehfest’s numerical inversion algorithm<ref name="r8">Stehfest, H. 1970. Algorithm 368: Numerical inversion of Laplace transforms. Commun. ACM 13 (1): 47–49. http://dx.doi.org/10.1145/361953.361969</ref> as discussed in '''Sec. 3.3.2'''. The dashed line in '''Fig. 3.3''' represents the numerical inversion of the solution in '''Eq. 3.85'''. Another option is to find an approximate inversion. One of these asymptotic forms is known as the line-source solution and commonly used in transient-pressure analysis.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
Consider transient flow toward a fully penetrating vertical well in an infinite homogeneous reservoir of uniform thickness, ''h'', and initial pressure, ''p''<sub>''i''</sub>.
<br>
<br>
''Solution''. This problem may be formulated most conveniently in the radial coordinates. The diffusivity equation governing fluid flow in porous media is given, in radial coordinates, by
<br>
<br>
[[File:Vol1 page 0089 eq 001.png]]....................(3.73)
<br>
<br>
where ∆''p'' = ''p''<sub>''i''</sub> – ''p''. '''Eq. 3.73''' is the same in absolute (cgs or SI) or Darcy units. (In field units, some conversion coefficients are involved in '''Eq. 3.73'''.) The initial condition is
<br>
<br>
[[File:Vol1 page 0090 eq 001.png]]....................(3.74)
<br>
<br>
which means that the pressure is uniform and equal to ''p''<sub>''i''</sub> initially throughout the reservoir. The outer boundary condition for an infinite reservoir is
<br>
<br>
[[File:Vol1 page 0090 eq 002.png]]....................(3.75)
<br>
<br>
which physically means that for any given time, ''t'', there is a large enough distance, ''r'', in the reservoir at which the initial pressure, ''p''<sub>''i''</sub>, has been preserved.
<br>
<br>
The inner boundary condition depends on the production conditions at the surface of the wellbore (''r'' = ''r''<sub>''w''</sub>). Assuming that the well is produced at a constant rate, ''q'', for all times,
<br>
<br>
[[File:Vol1 page 0090 eq 003.png]]....................(3.76)
<br>
<br>
The inner boundary condition given in '''Eq. 3.76''' is simply a restatement of the flux law (Darcy’s law given by '''Eq. 3.16''') at the surface of the wellbore.
<br>
<br>
'''Eqs. 3.73''' through '''3.76''' define the IBVP to be solved to obtain the transient-pressure distribution for the given system. Application of the Laplace transforms to '''Eq. 3.73''' yields
<br>
<br>
[[File:Vol1 page 0090 eq 004.png]]....................(3.77)
<br>
<br>
or, rearranging, we obtain
<br>
<br>
[[File:Vol1 page 0090 eq 005.png]]....................(3.78)
<br>
<br>
In obtaining the right side of '''Eq. 3.77''', the initial condition ('''Eq. 3.74''') has been used. Similarly, '''Eqs. 3.75''' and '''3.76''' are transformed into the following forms, respectively.
<br>
<br>
[[File:Vol1 page 0090 eq 006.png]]....................(3.79)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0090 eq 007.png]]....................(3.80)
<br>
<br>
Comparing '''Eq. 3.78''' with '''Eq. 3.31''', we recognize '''Eq. 3.78''' as the modified Bessel’s equation of order zero. The solution of '''Eq. 3.78''' may be written directly from '''Eq. 3.39''' as
<br>
<br>
[[File:Vol1 page 0090 eq 008.png]]....................(3.81)
<br>
<br>
The constants ''C''<sub>1</sub> and ''C''<sub>2</sub> in '''Eq. 3.81''' are obtained from the boundary conditions. The outer boundary condition ('''Eq. 3.79''') indicates that ''C''<sub>1</sub> = 0 [because [[File:Vol1 page 0091 inline 001.png]], '''Eq. 3.79''' is satisfied only if ''C''<sub>1</sub> = 0]; therefore,
<br>
<br>
[[File:Vol1 page 0091 eq 001.png]]....................(3.82)
<br>
<br>
From '''Eqs. 3.80''' and '''3.82''', we obtain
<br>
<br>
[[File:Vol1 page 0091 eq 002.png]]....................(3.83)
<br>
<br>
which yields
<br>
<br>
[[File:Vol1 page 0091 eq 003.png]]....................(3.84)
<br>
<br>
Then, the solution for the transient-pressure distribution is given, in the Laplace transform domain, by
<br>
<br>
[[File:Vol1 page 0091 eq 004.png]]....................(3.85)
<br>
<br>
To complete the solution of the problem, '''Eq. 3.85''' should be inverted into the real-time domain. The real inversion of '''Eq. 3.85''', however, is not available in terms of standard functions. One option is to use Stehfest’s numerical inversion algorithm<ref name="r8" /> as discussed in '''Sec. 3.3.2'''. The dashed line in '''Fig. 3.3''' represents the numerical inversion of the solution in '''Eq. 3.85'''. Another option is to find an approximate inversion. One of these asymptotic forms is known as the line-source solution and commonly used in transient-pressure analysis.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 091 Image 0001.png|'''Fig. 3.3 – Finite wellbore radius (Eq. 3.85) and line-source (Eq. 3.87 or 3.91) solutions for Example 3.1.'''
File:vol1 Page 091 Image 0001.png|'''Fig. 3.3 – Finite wellbore radius (Eq. 3.85) and line-source (Eq. 3.87 or 3.91) solutions for Example 3.1.'''
</gallery>
</gallery><br/>To obtain the line-source approximation of the solution given in '''Eq. 3.85''', we assume that the radius of the wellbore is small compared with the other dimensions of the reservoir. Thus, if we assume ''r''<sub>''w''</sub>→0 and use the relation given in '''Eq. 3.56''', we obtain<br/><br/>[[File:Vol1 page 0092 eq 001.png|RTENOTITLE]]....................(3.86)<br/><br/>Using this relation in '''Eq. 3.85''', we obtain the line-source solution in Laplace domain as<br/><br/>[[File:Vol1 page 0092 eq 002.png|RTENOTITLE]]....................(3.87)<br/><br/>The inversion of '''Eq. 3.87''' can be accomplished by using a Laplace transform table. From '''Table 3.1''' (or from the tables in two sources<ref name="r6">Churchill, R.V. 1972. Operational Mathematics, Vol. 2. New York: McGraw-Hill Book Co.</ref><ref name="r7">Abramowitz, M. and Stegun, I.A. ed. 1972. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, ninth edition, 1020–1029. New York: Dover Publications.</ref>), we have<br/><br/>[[File:Vol1 page 0092 eq 003.png|RTENOTITLE]]....................(3.88)<br/><br/>With '''Eq. 3.88''' and the Laplace transform property noted in '''Eq. 3.63''', we obtain the following inversion of '''Eq. 3.87''' in the real-time domain:<br/><br/>[[File:Vol1 page 0092 eq 004.png|RTENOTITLE]]....................(3.89)<br/><br/>Making the substitution ''u'' = r<sup>2</sup> / (4''ηt′'') and noting the definition of the exponential integral function, ''Ei''(''x''), given by<br/><br/>[[File:Vol1 page 0092 eq 005.png|RTENOTITLE]]....................(3.90)<br/><br/>we obtain the line-source solution as<br/><br/>[[File:Vol1 page 0092 eq 006.png|RTENOTITLE]]....................(3.91)<br/><br/>'''Fig. 3.3''' shows a comparison of the results computed from '''Eq. 3.85''' (finite-wellbore radius) and '''Eq. 3.91''' (line source) for the data noted in the figure. The two solutions yield different results at early times but become the same at later times. In fact, it can be shown analytically that the long-time approximation of the finite-wellbore radius solution ('''Eq. 3.85''') is the same as the line-source well solution. To show this, we note that the long-time approximation of the solution in the time domain corresponds to the limiting form of the solution in the Laplace domain as s → 0. Then, with the property of the Bessel function given in '''Eq. 3.56''', we can show that<br/><br/>[[File:Vol1 page 0092 eq 007.png|RTENOTITLE]]....................(3.92)<br/>
<br/>
To obtain the line-source approximation of the solution given in '''Eq. 3.85''', we assume that the radius of the wellbore is small compared with the other dimensions of the reservoir. Thus, if we assume ''r''<sub>''w''</sub>→0 and use the relation given in '''Eq. 3.56''', we obtain
<br>
<br>
[[File:Vol1 page 0092 eq 001.png]]....................(3.86)
<br>
<br>
Using this relation in '''Eq. 3.85''', we obtain the line-source solution in Laplace domain as
<br>
<br>
[[File:Vol1 page 0092 eq 002.png]]....................(3.87)
<br>
<br>
The inversion of '''Eq. 3.87''' can be accomplished by using a Laplace transform table. From '''Table 3.1''' (or from the tables in two sources<ref name="r6" /><ref name="r7" />), we have
<br>
<br>
[[File:Vol1 page 0092 eq 003.png]]....................(3.88)
<br>
<br>
With '''Eq. 3.88''' and the Laplace transform property noted in '''Eq. 3.63''', we obtain the following inversion of '''Eq. 3.87''' in the real-time domain:
<br>
<br>
[[File:Vol1 page 0092 eq 004.png]]....................(3.89)
<br>
<br>
Making the substitution ''u'' = r<sup>2</sup> / (4''ηt′'') and noting the definition of the exponential integral function, ''Ei''(''x''), given by
<br>
<br>
[[File:Vol1 page 0092 eq 005.png]]....................(3.90)
<br>
<br>
we obtain the line-source solution as
<br>
<br>
[[File:Vol1 page 0092 eq 006.png]]....................(3.91)
<br>
<br>
'''Fig. 3.3''' shows a comparison of the results computed from '''Eq. 3.85''' (finite-wellbore radius) and '''Eq. 3.91''' (line source) for the data noted in the figure. The two solutions yield different results at early times but become the same at later times. In fact, it can be shown analytically that the long-time approximation of the finite-wellbore radius solution ('''Eq. 3.85''') is the same as the line-source well solution. To show this, we note that the long-time approximation of the solution in the time domain corresponds to the limiting form of the solution in the Laplace domain as s → 0. Then, with the property of the Bessel function given in '''Eq. 3.56''', we can show that
<br>
<br>
[[File:Vol1 page 0092 eq 007.png]]....................(3.92)
<br>
<br>


----
----


'''Example 3.2'''
'''Example 3.2'''<br/><br/>Consider transient flow as a result of constant-rate production from a fully penetrating vertical well in a closed cylindrical reservoir initially at uniform initial pressure, ''p''<sub>''i''</sub>.<br/><br/>''Solution''. Fluid flow in cylindrical porous media is described by the diffusion equation in radial coordinates given by<br/><br/>[[File:Vol1 page 0093 eq 001.png|RTENOTITLE]]....................(3.93)<br/><br/>The initial condition corresponding to the uniform pressure distribution equal to ''p''<sub>''i''</sub> is<br/><br/>[[File:Vol1 page 0093 eq 002.png|RTENOTITLE]]....................(3.94)<br/><br/>and the inner boundary condition for a constant production rate, ''q'', for all times is<br/><br/>[[File:Vol1 page 0093 eq 003.png|RTENOTITLE]]....................(3.95)<br/><br/>The closed outer boundary condition is represented mathematically by zero flux at the outer boundary (''r'' = ''r''<sub>''e''</sub>) that corresponds to<br/><br/>[[File:Vol1 page 0093 eq 004.png|RTENOTITLE]]....................(3.96)<br/><br/>The Laplace transforms of '''Eqs. 3.93''' through '''3.96''' yield, respectively,<br/><br/>[[File:Vol1 page 0093 eq 005.png|RTENOTITLE]]....................(3.97)<br/><br/>[[File:Vol1 page 0093 eq 006.png|RTENOTITLE]]....................(3.98)<br/><br/>and<br/><br/>[[File:Vol1 page 0093 eq 007.png|RTENOTITLE]]....................(3.99)<br/><br/>(The initial condition given by '''Eq. 3.94''' has been used to obtain '''Eq. 3.97'''.) Because '''Eq. 3.97''' is the modified Bessel’s equation of order zero, its general solution is given by<br/><br/>[[File:Vol1 page 0093 eq 008.png|RTENOTITLE]]....................(3.100)<br/><br/>With the outer boundary condition given by '''Eq. 3.99''', we obtain<br/><br/>[[File:Vol1 page 0093 eq 009.png|RTENOTITLE]]....................(3.101)<br/><br/>which yields<br/><br/>[[File:Vol1 page 0094 eq 001.png|RTENOTITLE]]....................(3.102)<br/><br/>and thus<br/><br/>[[File:Vol1 page 0094 eq 002.png|RTENOTITLE]]....................(3.103)<br/><br/>Using the inner boundary condition given by '''Eq. 3.98''' yields<br/><br/>[[File:Vol1 page 0094 eq 003.png|RTENOTITLE]]....................(3.104)<br/><br/>From '''Eqs. 3.102''' and '''3.104''', we obtain the coefficients ''C''<sub>1</sub> and ''C''<sub>2</sub> as follows:<br/><br/>[[File:Vol1 page 0094 eq 004.png|RTENOTITLE]]....................(3.105)<br/><br/>and<br/><br/>[[File:Vol1 page 0094 eq 005.png|RTENOTITLE]]....................(3.106)<br/><br/>Substituting ''C''<sub>1</sub> and ''C''<sub>2</sub> into '''Eq. 3.100''' yields<br/><br/>[[File:Vol1 page 0094 eq 006.png|RTENOTITLE]]....................(3.107)<br/><br/>The inverse of the solution given by '''Eq. 3.107''' may not be found in the Laplace transform tables. van Everdingen and Hurst<ref name="r10">van Everdingen, A.F. and Hurst, W. 1953. The Application of the Laplace Transformation to Flow Problems in Reservoirs. In Transactions of the American Institute of Mining and Metallurgical Engineers, Vol. 198, 171. </ref> provided the following analytical inversion of '''Eq. 3.107''' with the inversion integral.<br/><br/>[[File:Vol1 page 0094 eq 007.png|RTENOTITLE]] [[File:Vol1 page 0095 eq 001.png|RTENOTITLE]]....................(3.108)<br/><br/>In '''Eq. 3.108''', ''β''<sub>1</sub>, ''β''<sub>2</sub>, etc. are the roots of<br/><br/>[[File:Vol1 page 0095 eq 002.png|RTENOTITLE]]....................(3.109)<br/><br/>The solution given in '''Eq. 3.107''' may also be inverted numerically with the Stehfest algorithm.<ref name="r8">Stehfest, H. 1970. Algorithm 368: Numerical inversion of Laplace transforms. Commun. ACM 13 (1): 47–49. http://dx.doi.org/10.1145/361953.361969</ref> '''Fig. 3.4''' shows the results of the numerical inversion of '''Eq. 3.107'''.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
Consider transient flow as a result of constant-rate production from a fully penetrating vertical well in a closed cylindrical reservoir initially at uniform initial pressure, ''p''<sub>''i''</sub>.
<br>
<br>
''Solution''. Fluid flow in cylindrical porous media is described by the diffusion equation in radial coordinates given by
<br>
<br>
[[File:Vol1 page 0093 eq 001.png]]....................(3.93)
<br>
<br>
The initial condition corresponding to the uniform pressure distribution equal to ''p''<sub>''i''</sub> is
<br>
<br>
[[File:Vol1 page 0093 eq 002.png]]....................(3.94)
<br>
<br>
and the inner boundary condition for a constant production rate, ''q'', for all times is
<br>
<br>
[[File:Vol1 page 0093 eq 003.png]]....................(3.95)
<br>
<br>
The closed outer boundary condition is represented mathematically by zero flux at the outer boundary (''r'' = ''r''<sub>''e''</sub>) that corresponds to
<br>
<br>
[[File:Vol1 page 0093 eq 004.png]]....................(3.96)
<br>
<br>
The Laplace transforms of '''Eqs. 3.93''' through '''3.96''' yield, respectively,
<br>
<br>
[[File:Vol1 page 0093 eq 005.png]]....................(3.97)
<br>
<br>
[[File:Vol1 page 0093 eq 006.png]]....................(3.98)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0093 eq 007.png]]....................(3.99)
<br>
<br>
(The initial condition given by '''Eq. 3.94''' has been used to obtain '''Eq. 3.97'''.) Because '''Eq. 3.97''' is the modified Bessel’s equation of order zero, its general solution is given by
<br>
<br>
[[File:Vol1 page 0093 eq 008.png]]....................(3.100)
<br>
<br>
With the outer boundary condition given by '''Eq. 3.99''', we obtain
<br>
<br>
[[File:Vol1 page 0093 eq 009.png]]....................(3.101)
<br>
<br>
which yields
<br>
<br>
[[File:Vol1 page 0094 eq 001.png]]....................(3.102)
<br>
<br>
and thus
<br>
<br>
[[File:Vol1 page 0094 eq 002.png]]....................(3.103)
<br>
<br>
Using the inner boundary condition given by '''Eq. 3.98''' yields
<br>
<br>
[[File:Vol1 page 0094 eq 003.png]]....................(3.104)
<br>
<br>
From '''Eqs. 3.102''' and '''3.104''', we obtain the coefficients ''C''<sub>1</sub> and ''C''<sub>2</sub> as follows:
<br>
<br>
[[File:Vol1 page 0094 eq 004.png]]....................(3.105)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0094 eq 005.png]]....................(3.106)
<br>
<br>
Substituting ''C''<sub>1</sub> and ''C''<sub>2</sub> into '''Eq. 3.100''' yields
<br>
<br>
[[File:Vol1 page 0094 eq 006.png]]....................(3.107)
<br>
<br>
The inverse of the solution given by '''Eq. 3.107''' may not be found in the Laplace transform tables. van Everdingen and Hurst<ref name="r10" /> provided the following analytical inversion of '''Eq. 3.107''' with the inversion integral.
<br>
<br>
[[File:Vol1 page 0094 eq 007.png]]
[[File:Vol1 page 0095 eq 001.png]]....................(3.108)
<br>
<br>
In '''Eq. 3.108''', ''β''<sub>1</sub>, ''β''<sub>2</sub>, etc. are the roots of
<br>
<br>
[[File:Vol1 page 0095 eq 002.png]]....................(3.109)
<br>
<br>
The solution given in '''Eq. 3.107''' may also be inverted numerically with the Stehfest algorithm.<ref name="r8" /> '''Fig. 3.4''' shows the results of the numerical inversion of '''Eq. 3.107'''.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 095 Image 0001.png|'''Fig. 3.4 – Bounded reservoir solution (Eq. 3.107) for Example 3.2.'''
File:vol1 Page 095 Image 0001.png|'''Fig. 3.4 – Bounded reservoir solution (Eq. 3.107) for Example 3.2.'''
</gallery>
</gallery>
<br/>


----
----


'''Example 3.3'''
'''Example 3.3'''<br/><br/>Consider the flowing wellbore pressure of a fully penetrating vertical well with wellbore storage and skin in an infinite reservoir.<br/><br/>''Solution''. Revisit the case in '''Example 3.1''' and add the effect of a skin zone around the wellbore. Assume that the constant production rate is specified at the surface so that the storage capacity of the wellbore needs to be taken into account. Before presenting the initial-boundary value problem, skin factor and surface production rate should be defined.<br/><br/>Using van Everdingen and Hurst’s thin-skin concept<ref name="r10">van Everdingen, A.F. and Hurst, W. 1953. The Application of the Laplace Transformation to Flow Problems in Reservoirs. In Transactions of the American Institute of Mining and Metallurgical Engineers, Vol. 198, 171. </ref> (vanishingly small skin-zone radius), the skin factor is defined by<br/><br/>[[File:Vol1 page 0095 eq 003.png|RTENOTITLE]]....................(3.110)<br/><br/>where ''q''<sub>''sf''</sub> is the sandface production rate, ''p''(''r''<sub>''w''</sub> +) denotes the reservoir pressure immediately outside the skin-zone boundary, and ''p''<sub>''wf''</sub> is the flowing wellbore pressure measured inside the wellbore. Rearranging '''Eq. 3.110''', we obtain the following relation for the flowing wellbore pressure.<br/><br/>[[File:Vol1 page 0096 eq 001.png|RTENOTITLE]]....................(3.111)<br/><br/>When the production rate is specified at the surface, it is necessary to account for the fact that the wellbore can store and unload fluids. The surface production rate, ''q'', is equal to the sum of the wellbore unloading rate, ''q''<sub>''wb''</sub>, and the sandface production rate, ''q''<sub>''sf''</sub>; that is,<br/><br/>[[File:Vol1 page 0096 eq 002.png|RTENOTITLE]]....................(3.112)<br/><br/>where<br/><br/>[[File:Vol1 page 0096 eq 003.png|RTENOTITLE]]....................(3.113)<br/><br/>and<br/><br/>[[File:Vol1 page 0096 eq 004.png|RTENOTITLE]]....................(3.114)<br/><br/>In '''Eq. 3.113''', ''C'' is the wellbore-storage coefficient. Substituting '''Eqs. 3.113''' and '''3.114''' into '''Eq. 3.112''', we obtain the following expression for the surface production rate.<br/><br/>[[File:Vol1 page 0096 eq 005.png|RTENOTITLE]]....................(3.115)<br/><br/>The mathematical statement of the problem under consideration is similar to that in '''Example 3.1''', except that the inner-boundary condition should be replaced by '''Eq. 3.115''', and '''Eq. 3.111''' should be incorporated to account for the skin effect. The IBVP is defined by the following set of equations in the Laplace domain:<br/><br/>[[File:Vol1 page 0096 eq 006.png|RTENOTITLE]]....................(3.116)<br/><br/>[[File:Vol1 page 0096 eq 007.png|RTENOTITLE]]....................(3.117)<br/><br/>[[File:Vol1 page 0096 eq 008.png|RTENOTITLE]]....................(3.118)<br/><br/>and<br/><br/>[[File:Vol1 page 0096 eq 009.png|RTENOTITLE]]....................(3.119)<br/><br/>The general solution of '''Eq. 3.116''' is<br/><br/>[[File:Vol1 page 0096 eq 010.png|RTENOTITLE]]....................(3.120)<br/><br/>The condition in '''Eq. 3.117''' requires that ''C''<sub>1</sub> = 0; therefore,<br/><br/>[[File:Vol1 page 0097 eq 001.png|RTENOTITLE]]....................(3.121)<br/><br/>The use of '''Eq. 3.121''' in '''Eq. 3.119''' yields<br/><br/>[[File:Vol1 page 0097 eq 002.png|RTENOTITLE]]....................(3.122)<br/><br/>From '''Eqs. 3.118''', '''3.121''', and '''3.122''', we obtain<br/><br/>[[File:Vol1 page 0097 eq 003.png|RTENOTITLE]]....................(3.123)<br/><br/>which yields<br/><br/>[[File:Vol1 page 0097 eq 004.png|RTENOTITLE]]....................(3.124)<br/><br/>Substituting '''Eq. 3.124''' for ''C''<sub>2</sub> in '''Eq. 3.122''', we obtain the solution for the transient-pressure distribution in the Laplace transform domain as<br/><br/>[[File:Vol1 page 0097 eq 005.png|RTENOTITLE]]....................(3.125)<br/><br/>The real inversion of the solution in '''Eq. 3.125''' has been obtained by Agarwal ''et al''.<ref name="r11">Agarwal, R.G., Al-Hussainy, R., and Ramey Jr., H.J. 1970. An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment. SPE J. 10 (3): 279-290. SPE-2466-PA. http://dx.doi.org/10.2118/2466-PA</ref> with the inversion integral. It is also possible to invert '''Eq. 3.125''' numerically. '''Fig. 3.5''' shows the results of the numerical inversion of '''Eq. 3.125''' with the Stehfest’s algorithm.<ref name="r8">Stehfest, H. 1970. Algorithm 368: Numerical inversion of Laplace transforms. Commun. ACM 13 (1): 47–49. http://dx.doi.org/10.1145/361953.361969</ref> Also shown in '''Fig. 3.5''' are the logarithmic derivatives of Δ''p''<sub>''wf''</sub>. These derivatives are computed by applying the Laplace transformation property given in '''Eq. 3.60''' to '''Eq. 3.125''' as follows:<br/><br/>[[File:Vol1 page 0097 eq 006.png|RTENOTITLE]]....................(3.126)<br/><br/>Here, we have used Δ''p''<sub>''w f''</sub>(''t'' = 0) = 0. To obtain the logarithmic derivatives, we simply note that<br/><br/>[[File:Vol1 page 0097 eq 007.png|RTENOTITLE]]....................(3.127)<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
Consider the flowing wellbore pressure of a fully penetrating vertical well with wellbore storage and skin in an infinite reservoir.
<br>
<br>
''Solution''. Revisit the case in '''Example 3.1''' and add the effect of a skin zone around the wellbore. Assume that the constant production rate is specified at the surface so that the storage capacity of the wellbore needs to be taken into account. Before presenting the initial-boundary value problem, skin factor and surface production rate should be defined.
<br>
<br>
Using van Everdingen and Hurst’s thin-skin concept<ref name="r10" /> (vanishingly small skin-zone radius), the skin factor is defined by
<br>
<br>
[[File:Vol1 page 0095 eq 003.png]]....................(3.110)
<br>
<br>
where ''q''<sub>''sf''</sub> is the sandface production rate, ''p''(''r''<sub>''w''</sub> +) denotes the reservoir pressure immediately outside the skin-zone boundary, and ''p''<sub>''wf''</sub> is the flowing wellbore pressure measured inside the wellbore. Rearranging '''Eq. 3.110''', we obtain the following relation for the flowing wellbore pressure.
<br>
<br>
[[File:Vol1 page 0096 eq 001.png]]....................(3.111)
<br>
<br>
When the production rate is specified at the surface, it is necessary to account for the fact that the wellbore can store and unload fluids. The surface production rate, ''q'', is equal to the sum of the wellbore unloading rate, ''q''<sub>''wb''</sub>, and the sandface production rate, ''q''<sub>''sf''</sub>; that is,
<br>
<br>
[[File:Vol1 page 0096 eq 002.png]]....................(3.112)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0096 eq 003.png]]....................(3.113)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0096 eq 004.png]]....................(3.114)
<br>
<br>
In '''Eq. 3.113''', ''C'' is the wellbore-storage coefficient. Substituting '''Eqs. 3.113''' and '''3.114''' into '''Eq. 3.112''', we obtain the following expression for the surface production rate.
<br>
<br>
[[File:Vol1 page 0096 eq 005.png]]....................(3.115)
<br>
<br>
The mathematical statement of the problem under consideration is similar to that in '''Example 3.1''', except that the inner-boundary condition should be replaced by '''Eq. 3.115''', and '''Eq. 3.111''' should be incorporated to account for the skin effect. The IBVP is defined by the following set of equations in the Laplace domain:
<br>
<br>
[[File:Vol1 page 0096 eq 006.png]]....................(3.116)
<br>
<br>
[[File:Vol1 page 0096 eq 007.png]]....................(3.117)
<br>
<br>
[[File:Vol1 page 0096 eq 008.png]]....................(3.118)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0096 eq 009.png]]....................(3.119)
<br>
<br>
The general solution of '''Eq. 3.116''' is
<br>
<br>
[[File:Vol1 page 0096 eq 010.png]]....................(3.120)
<br>
<br>
The condition in '''Eq. 3.117''' requires that ''C''<sub>1</sub> = 0; therefore,
<br>
<br>
[[File:Vol1 page 0097 eq 001.png]]....................(3.121)
<br>
<br>
The use of '''Eq. 3.121''' in '''Eq. 3.119''' yields
<br>
<br>
[[File:Vol1 page 0097 eq 002.png]]....................(3.122)
<br>
<br>
From '''Eqs. 3.118''', '''3.121''', and '''3.122''', we obtain
<br>
<br>
[[File:Vol1 page 0097 eq 003.png]]....................(3.123)
<br>
<br>
which yields
<br>
<br>
[[File:Vol1 page 0097 eq 004.png]]....................(3.124)
<br>
<br>
Substituting '''Eq. 3.124''' for ''C''<sub>2</sub> in '''Eq. 3.122''', we obtain the solution for the transient-pressure distribution in the Laplace transform domain as
<br>
<br>
[[File:Vol1 page 0097 eq 005.png]]....................(3.125)
<br>
<br>
The real inversion of the solution in '''Eq. 3.125''' has been obtained by Agarwal ''et al''.<ref name="r11" /> with the inversion integral. It is also possible to invert '''Eq. 3.125''' numerically. '''Fig. 3.5''' shows the results of the numerical inversion of '''Eq. 3.125''' with the Stehfest’s algorithm.<ref name="r8" /> Also shown in '''Fig. 3.5''' are the logarithmic derivatives of Δ''p''<sub>''wf''</sub>. These derivatives are computed by applying the Laplace transformation property given in '''Eq. 3.60''' to '''Eq. 3.125''' as follows:
<br>
<br>
[[File:Vol1 page 0097 eq 006.png]]....................(3.126)
<br>
<br>
Here, we have used Δ''p''<sub>''w f''</sub>(''t'' = 0) = 0. To obtain the logarithmic derivatives, we simply note that
<br>
<br>
[[File:Vol1 page 0097 eq 007.png]]....................(3.127)
<br>
<br>
<gallery widths="300px" heights="200px">
File:vol1 Page 098 Image 0001.png|'''Fig. 3.5 – Wellbore-storage and skin solution (Eq. 3.125) for Example 3.3.'''
File:vol1 Page 098 Image 0001.png|'''Fig. 3.5 – Wellbore-storage and skin solution (Eq. 3.125) for Example 3.3.'''
</gallery>
</gallery>
<br/>


----
----


'''Example 3.4'''
'''Example 3.4'''<br/><br/>Consider pressure buildup with wellbore storage and skin following a drawdown period at a constant rate in an infinite reservoir.<br/><br/>''Solution''. This example is similar to '''Example 3.3''' except, at time ''t''<sub>''p''</sub>, the well is shut in and pressure buildup begins. The system of equations to define this problem is<br/><br/>[[File:Vol1 page 0098 eq 001.png|RTENOTITLE]]....................(3.128)<br/><br/>[[File:Vol1 page 0098 eq 002.png|RTENOTITLE]]....................(3.129)<br/><br/>[[File:Vol1 page 0098 eq 003.png|RTENOTITLE]]....................(3.130)<br/><br/>[[File:Vol1 page 0098 eq 004.png|RTENOTITLE]]....................(3.131)<br/><br/>where ''H''(''t'' - ''t''<sub>''p''</sub>) is Heaviside’s unit function ('''Eq. 3.67'''), and<br/><br/>[[File:Vol1 page 0098 eq 005.png|RTENOTITLE]]....................(3.132)<br/><br/>The right side of the boundary condition in '''Eq. 3.131''' accounts for a constant surface production rate, ''q'', for 0 < ''t'' < ''t''<sub>''p''</sub> and for shut in (''q'' = 0) for ''t'' > ''t''<sub>''p''</sub>. Taking the Laplace transforms of '''Eqs. 3.128''' through '''3.132''', we obtain<br/><br/>[[File:Vol1 page 0098 eq 006.png|RTENOTITLE]]....................(3.133)<br/><br/>[[File:Vol1 page 0098 eq 007.png|RTENOTITLE]]....................(3.134)<br/><br/>[[File:Vol1 page 0099 eq 001.png|RTENOTITLE]]....................(3.135)<br/><br/>and<br/><br/>[[File:Vol1 page 0099 eq 002.png|RTENOTITLE]]....................(3.136)<br/><br/>The general solution of '''Eq. 3.133''' is<br/><br/>[[File:Vol1 page 0099 eq 003.png|RTENOTITLE]]....................(3.137)<br/><br/>The condition in '''Eq. 3.134''' requires that ''C''<sub>1</sub>= 0; therefore,<br/><br/>[[File:Vol1 page 0099 eq 004.png|RTENOTITLE]]....................(3.138)<br/><br/>From '''Eqs. 3.138''' and '''3.136''', we obtain<br/><br/>[[File:Vol1 page 0099 eq 005.png|RTENOTITLE]]....................(3.139)<br/><br/>Substituting the results of '''Eqs. 3.138''' and '''3.139''' into '''Eq. 3.135''', we have<br/><br/>[[File:Vol1 page 0099 eq 006.png|RTENOTITLE]]....................(3.140)<br/><br/>which yields<br/><br/>[[File:Vol1 page 0099 eq 007.png|RTENOTITLE]]....................(3.141)<br/><br/>Substituting '''Eq. 3.141''' into '''Eq. 3.139''', we obtain the following solution in the Laplace transform domain, which covers both the drawdown and buildup periods.<br/><br/>[[File:Vol1 page 0099 eq 008.png|RTENOTITLE]]....................(3.142)<br/><br/>The [[File:Vol1 page 0100 inline 001.png|RTENOTITLE]] term contributed by the discontinuity at time ''t'' = ''t''<sub>''p''</sub> causes difficulties in the numerical inversion of the right side of '''Eq. 3.142''' with the use of the Stehfest algorithm.<ref name="r8">Stehfest, H. 1970. Algorithm 368: Numerical inversion of Laplace transforms. Commun. ACM 13 (1): 47–49. http://dx.doi.org/10.1145/361953.361969</ref> As suggested by Chen and Raghavan,<ref name="r12">Chen, C.-C. and Raghavan, R. 1996. An Approach To Handle Discontinuities by the Stehfest Algorithm. SPE J. 1 (4): 363-368. SPE-28419-PA. http://dx.doi.org/10.2118/28419-PA</ref> this problem may be solved by noting that<br/><br/>[[File:Vol1 page 0100 eq 001.png|RTENOTITLE]]....................(3.143)<br/><br/>and applying the Stehfest algorithm term by term to the right side of '''Eq. 3.143'''. '''Fig. 3.6''' shows sample results obtained by the numerical inversion of '''Eq. 3.142'''.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
Consider pressure buildup with wellbore storage and skin following a drawdown period at a constant rate in an infinite reservoir.
<br>
<br>
''Solution''. This example is similar to '''Example 3.3''' except, at time ''t''<sub>''p''</sub>, the well is shut in and pressure buildup begins. The system of equations to define this problem is
<br>
<br>
[[File:Vol1 page 0098 eq 001.png]]....................(3.128)
<br>
<br>
[[File:Vol1 page 0098 eq 002.png]]....................(3.129)
<br>
<br>
[[File:Vol1 page 0098 eq 003.png]]....................(3.130)
<br>
<br>
[[File:Vol1 page 0098 eq 004.png]]....................(3.131)
<br>
<br>
where ''H''(''t'' - ''t''<sub>''p''</sub>) is Heaviside’s unit function ('''Eq. 3.67'''), and
<br>
<br>
[[File:Vol1 page 0098 eq 005.png]]....................(3.132)
<br>
<br>
The right side of the boundary condition in '''Eq. 3.131''' accounts for a constant surface production rate, ''q'', for 0 < ''t'' < ''t''<sub>''p''</sub> and for shut in (''q'' = 0) for ''t'' > ''t''<sub>''p''</sub>. Taking the Laplace transforms of '''Eqs. 3.128''' through '''3.132''', we obtain
<br>
<br>
[[File:Vol1 page 0098 eq 006.png]]....................(3.133)
<br>
<br>
[[File:Vol1 page 0098 eq 007.png]]....................(3.134)
<br>
<br>
[[File:Vol1 page 0099 eq 001.png]]....................(3.135)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0099 eq 002.png]]....................(3.136)
<br>
<br>
The general solution of '''Eq. 3.133''' is
<br>
<br>
[[File:Vol1 page 0099 eq 003.png]]....................(3.137)
<br>
<br>
The condition in '''Eq. 3.134''' requires that ''C''<sub>1</sub>= 0; therefore,
<br>
<br>
[[File:Vol1 page 0099 eq 004.png]]....................(3.138)
<br>
<br>
From '''Eqs. 3.138''' and '''3.136''', we obtain
<br>
<br>
[[File:Vol1 page 0099 eq 005.png]]....................(3.139)
<br>
<br>
Substituting the results of '''Eqs. 3.138''' and '''3.139''' into '''Eq. 3.135''', we have
<br>
<br>
[[File:Vol1 page 0099 eq 006.png]]....................(3.140)
<br>
<br>
which yields
<br>
<br>
[[File:Vol1 page 0099 eq 007.png]]....................(3.141)
<br>
<br>
Substituting '''Eq. 3.141''' into '''Eq. 3.139''', we obtain the following solution in the Laplace transform domain, which covers both the drawdown and buildup periods.
<br>
<br>
[[File:Vol1 page 0099 eq 008.png]]....................(3.142)
<br>
<br>
The [[File:Vol1 page 0100 inline 001.png]] term contributed by the discontinuity at time ''t'' = ''t''<sub>''p''</sub> causes difficulties in the numerical inversion of the right side of '''Eq. 3.142''' with the use of the Stehfest algorithm.<ref name="r8" /> As suggested by Chen and Raghavan,<ref name="r12" /> this problem may be solved by noting that
<br>
<br>
[[File:Vol1 page 0100 eq 001.png]]....................(3.143)
<br>
<br>
and applying the Stehfest algorithm term by term to the right side of '''Eq. 3.143'''. '''Fig. 3.6''' shows sample results obtained by the numerical inversion of '''Eq. 3.142'''.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 100 Image 0001.png|'''Fig. 3.6 – Drawdown and buildup results with wellbore-storage and skin solution (Eq. 3.142) for Example 3.4.'''
File:vol1 Page 100 Image 0001.png|'''Fig. 3.6 – Drawdown and buildup results with wellbore-storage and skin solution (Eq. 3.142) for Example 3.4.'''
</gallery>
</gallery>
<br/>


----
----


<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >


</div></div><div class="toccolours mw-collapsible mw-collapsed">
== Green’s Functions and Source Functions ==
== Green’s Functions and Source Functions ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/>Green’s function and source functions are used to solve 2D and 3D transient-flow problems that may result from complex well geometries, such as partially penetrating vertical and inclined wells, hydraulically fractured wells, and horizontal wells. Before introducing these techniques, it is useful to clarify the terminology.<br/><br/>In our terminology, a source is a point, line, surface, or volume at which fluids are withdrawn from the reservoir. Strictly speaking, fluid withdrawal should be associated with a sink, and the injection of fluids should be related to a source. Here, however, the term source is used for both production and injection with the convention that a negative withdrawal rate indicates injection.<br/><br/>Green’s functions and source functions are closely related. A Green’s function is defined for a differential equation with specified boundary conditions (prescribed flux or pressure) and corresponds to an instantaneous point-source solution. A source function, on the other hand, is the solution of the given differential equation with specified boundary conditions and source geometry.<br/><br/>The details of the theory and application of Green’s function and source functions for the solution of transient-flow problems in porous media can be found in many sources.<ref name="r2">Raghavan, R. 1993. Well Test Analysis, 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.</ref><ref name="r13">Gringarten, A.C. and Ramey Jr., H.J. 1973. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. SPE J. 13 (5): 285-296. SPE-3818-PA. http://dx.doi.org/10.2118/3818-PA</ref><ref name="r14">Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, second edition, 353–386. Oxford, UK: Oxford University Press.</ref><ref name="r15">Stakgold, I. 1979. Green’s Functions and Boundary Value Problems, 86–104. New York: John Wiley & Sons.</ref><ref name="r16">Kelvin, W.T. 1884. Mathematical and Physical Papers, Vol. 2, 41. Cambridge, UK: Cambridge University Press.</ref><ref name="r17">Ozkan, E. and Raghavan, R. 1991a. New Solutions for Well-Test-Analysis Problems: Part 1—Analytical Considerations. SPE Form Eval 6 (3): 359–368. SPE-18615-PA. http://dx.doi.org/10.2118/18615-PA</ref><ref name="r18">Ozkan, E. and Raghavan, R. 1991b. New Solutions for Well-Test-Analysis Problems: Part 2—Computational Considerations and Applications. SPE Form Eval 6 (3): 369–378. SPE-18616-PA. http://dx.doi.org/10.2118/18616-PA</ref><ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref><ref name="r20">Raghavan, R. 1993. The Method of Sources and Sinks. In Well Test Analysis, Chap. 10, 336-435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.</ref> A brief account of the use of these techniques is presented here, as well as an introduction of the fundamental solution and point-source concepts.
Green’s function and source functions are used to solve 2D and 3D transient-flow problems that may result from complex well geometries, such as partially penetrating vertical and inclined wells, hydraulically fractured wells, and horizontal wells. Before introducing these techniques, it is useful to clarify the terminology.
<br>
<br>
In our terminology, a source is a point, line, surface, or volume at which fluids are withdrawn from the reservoir. Strictly speaking, fluid withdrawal should be associated with a sink, and the injection of fluids should be related to a source. Here, however, the term source is used for both production and injection with the convention that a negative withdrawal rate indicates injection.
<br>
<br>
Green’s functions and source functions are closely related. A Green’s function is defined for a differential equation with specified boundary conditions (prescribed flux or pressure) and corresponds to an instantaneous point-source solution. A source function, on the other hand, is the solution of the given differential equation with specified boundary conditions and source geometry.
<br>
<br>
The details of the theory and application of Green’s function and source functions for the solution of transient-flow problems in porous media can be found in many sources.<ref name="r2" /><ref name="r13" /><ref name="r14" /><ref name="r15" /><ref name="r16" /><ref name="r17" /><ref name="r18" /><ref name="r19" /><ref name="r20" /> A brief account of the use of these techniques is presented here, as well as an introduction of the fundamental solution and point-source concepts.


=== Fundamental Solution of the Diffusion Equation ===
=== Fundamental Solution of the Diffusion Equation ===


The fundamental solution, ''γ''<sub>''f''</sub>(''M'', ''M′'', ''t'', ''τ''), of the diffusion equation for fluid flow in porous media satisfies the following differential equation:
The fundamental solution, ''γ''<sub>''f''</sub>(''M'', ''M′'', ''t'', ''τ''), of the diffusion equation for fluid flow in porous media satisfies the following differential equation:<br/><br/>[[File:Vol1 page 0101 eq 001.png|RTENOTITLE]]....................(3.144)<br/><br/>where ''δ''(''M'', ''M′'', ''t'', ''τ'') is a generalized (symbolic) function<ref name="r15">Stakgold, I. 1979. Green’s Functions and Boundary Value Problems, 86–104. New York: John Wiley & Sons.</ref> called the Dirac delta function and is defined on the basis of its following properties:<br/><br/>[[File:Vol1 page 0101 eq 002.png|RTENOTITLE]]....................(3.145)<br/><br/>and<br/><br/>[[File:Vol1 page 0101 eq 003.png|RTENOTITLE]]....................(3.146)<br/><br/>The delta function is symmetric in ''M'' and ''M′'' and also in ''t'' and ''τ''. In this formulation, the delta function represents the symbolic density of a unit-strength, concentrated source located at ''M′'' and acting at time ''τ''. In physical terms, this source corresponds to an infinitesimally small well (located at point ''M′'') at which a finite amount of fluid is withdrawn (or injected) instantaneously (at time ''τ''). Therefore, the solution of '''Eq. 3.144''' (the fundamental solution) is also known as the instantaneous point-source solution. Formally, the point-source solution corresponds to the pressure drop, Δ''p'' = ''p''<sub>''i''</sub> − ''p'', at a point ''M'' and time ''t'' in an infinite porous medium (reservoir) because of a point source of unit strength located at point ''M′'' and acting at ''τ'' <''t''.
<br>
<br>
[[File:Vol1 page 0101 eq 001.png]]....................(3.144)
<br>
<br>
where ''δ''(''M'', ''M′'', ''t'', ''τ'') is a generalized (symbolic) function<ref name="r15" /> called the Dirac delta function and is defined on the basis of its following properties:
<br>
<br>
[[File:Vol1 page 0101 eq 002.png]]....................(3.145)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0101 eq 003.png]]....................(3.146)
<br>
<br>
The delta function is symmetric in ''M'' and ''M′'' and also in ''t'' and ''τ''. In this formulation, the delta function represents the symbolic density of a unit-strength, concentrated source located at ''M′'' and acting at time ''τ''. In physical terms, this source corresponds to an infinitesimally small well (located at point ''M′'') at which a finite amount of fluid is withdrawn (or injected) instantaneously (at time ''τ''). Therefore, the solution of '''Eq. 3.144''' (the fundamental solution) is also known as the instantaneous point-source solution. Formally, the point-source solution corresponds to the pressure drop, Δ''p'' = ''p''<sub>''i''</sub> − ''p'', at a point ''M'' and time ''t'' in an infinite porous medium (reservoir) because of a point source of unit strength located at point ''M′'' and acting at ''τ'' <''t''.


=== The Source-Function Solutions of the Diffusion Equation ===
=== The Source-Function Solutions of the Diffusion Equation ===


The point-source solution was first introduced by Lord Kelvin<ref name="r16" /> for the solution of heat conduction problems and was extensively discussed by Carslaw and Jaeger.<ref name="r14" /> The point-source solution is usually obtained by finding the limiting form of the pressure drop resulting from a spherical source as the source volume vanishes. To demonstrate the derivation of the instantaneous point-source solution, consider the transient flow of a slightly compressible fluid of constant compressibility and viscosity toward a spherical source of radius ''r'' = ''a'' in an infinite, homogeneous, and isotropic porous medium. Because of the spherical symmetry of the physical problem, we can conveniently express the governing equation of fluid flow in porous media in spherical coordinates as
The point-source solution was first introduced by Lord Kelvin<ref name="r16">Kelvin, W.T. 1884. Mathematical and Physical Papers, Vol. 2, 41. Cambridge, UK: Cambridge University Press.</ref> for the solution of heat conduction problems and was extensively discussed by Carslaw and Jaeger.<ref name="r14">Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, second edition, 353–386. Oxford, UK: Oxford University Press.</ref> The point-source solution is usually obtained by finding the limiting form of the pressure drop resulting from a spherical source as the source volume vanishes. To demonstrate the derivation of the instantaneous point-source solution, consider the transient flow of a slightly compressible fluid of constant compressibility and viscosity toward a spherical source of radius ''r'' = ''a'' in an infinite, homogeneous, and isotropic porous medium. Because of the spherical symmetry of the physical problem, we can conveniently express the governing equation of fluid flow in porous media in spherical coordinates as<br/><br/>[[File:Vol1 page 0101 eq 004.png|RTENOTITLE]]....................(3.147)<br/><br/>Assume that the initial pressure drop satisfies<br/><br/>[[File:Vol1 page 0102 eq 001.png|RTENOTITLE]]....................(3.148)<br/><br/>and we have the condition that<br/><br/>[[File:Vol1 page 0102 eq 002.png|RTENOTITLE]]....................(3.149)<br/><br/>On substitution of ''u'' = ''r''Δ''p'', '''Eqs. 3.147''' through '''3.149''' become, respectively,<br/><br/>[[File:Vol1 page 0102 eq 003.png|RTENOTITLE]]....................(3.150)<br/><br/>[[File:Vol1 page 0102 eq 004.png|RTENOTITLE]]....................(3.151)<br/><br/>and<br/><br/>[[File:Vol1 page 0102 eq 005.png|RTENOTITLE]]....................(3.152)<br/><br/>The solution of the problem described by '''Eqs. 3.150''' through '''3.152''' is given by<ref name="r14">Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, second edition, 353–386. Oxford, UK: Oxford University Press.</ref><br/><br/>[[File:Vol1 page 0102 eq 006.png|RTENOTITLE]]....................(3.153)<br/><br/>If we expand the exponential terms in the integrand in '''Eq. 3.153''' in powers of ''r′'' and neglect the terms with powers higher than four, we obtain<br/><br/>[[File:Vol1 page 0102 eq 007.png|RTENOTITLE]]....................(3.154)<br/><br/>In '''Eq. 3.154''', 4''πα''<sup>3</sup>/3=''V'' where ''V'' is the volume of the spherical source. If [[File:Vol1 page 0102 inline 001.png|RTENOTITLE]] denotes the volume of the liquid released as a result of the change in the volume of the source, Δ''V'', which is caused by the change in pressure, Δ''p''<sub>''i''</sub>, then [[File:Vol1 page 0102 inline 002.png|RTENOTITLE]]. With the definition of compressibility, ''c'' = -(1 / ''V'')(Δ''V'' / Δ''p''<sub>''i''</sub>), we obtain [[File:Vol1 page 0102 inline 003.png|RTENOTITLE]]. Then, we can show that<br/><br/>[[File:Vol1 page 0102 eq 008.png|RTENOTITLE]]....................(3.155)<br/><br/>Substituting '''Eq. 3.155''' into '''Eq. 3.154''', we obtain<br/><br/>[[File:Vol1 page 0102 eq 009.png|RTENOTITLE]]....................(3.156)<br/><br/>If we let the radius of the spherical source, ''a'', tend to zero while [[File:Vol1 page 0102 inline 001.png|RTENOTITLE]] remains constant, '''Eq. 3.156''' yields the point-source solution in spherical coordinates given by<br/><br/>[[File:Vol1 page 0103 eq 001.png|RTENOTITLE]]....................(3.157)<br/><br/>This solution may be interpreted as the pressure drop at a distance ''r'' because of a volume of fluid, [[File:Vol1 page 0102 inline 001.png|RTENOTITLE]], instantaneously withdrawn at ''r'' = 0 and ''t'' = 0. Consistent with this interpretation, [[File:Vol1 page 0103 inline 001.png|RTENOTITLE]] is the strength of the source, which is the pressure drop in a unit volume of the porous medium caused by the instantaneous withdrawal of a volume of fluid, [[File:Vol1 page 0102 inline 001.png|RTENOTITLE]] (see '''Eq. 3.155''').<br/><br/>'''''Instantaneous Point Source in an Infinite Reservoir'''''. Nisle<ref name="r21">Nisle, R.G. 1958. The Effect of Partial Penetration on Pressure Build-Up in Oil Wells. In Transactions of the American Institute of Mining, Metallurgical, and Petroleum Engineers, Vol. 213, Paper 971-G, 85-90. Dallas, Texas: Society of Petroleum Engineers.</ref> presented a more general solution for an instantaneous point source of strength [[File:Vol1 page 0103 inline 001.png|RTENOTITLE]] acting at ''t'' = ''τ'' in an infinite, homogeneous, but anisotropic reservoir as<br/><br/>[[File:Vol1 page 0103 eq 002.png|RTENOTITLE]]....................(3.158)<br/><br/>In '''Eq. 3.158''', ''M'' and ''M′'' indicate the locations of the observation point and the source, respectively. For a 3D Cartesian coordinate system, [[File:Vol1 page 0103 inline 002.png|RTENOTITLE]] with ''η''<sub>''x''</sub>, ''η''<sub>''y''</sub>, and ''η''<sub>''z''</sub> representing the diffusivity constants (defined in '''Eq. 3.23''') in the ''x'', ''y'', and ''z'' directions, respectively.<br/><br/>'''''Continuous Point Source in an Infinite Reservoir'''''. If the fluid withdrawal is at a continuous rate, [[File:Vol1 page 0103 inline 003.png|RTENOTITLE]], from time 0 to ''t'', then the pressure drop as a result of a continuous point source in an infinite reservoir is obtained by distributing the point sources of strength [[File:Vol1 page 0103 inline 001.png|RTENOTITLE]] over a time interval 0 ≤ ''τ'' ≤ ''t''. This is given by<br/><br/>[[File:Vol1 page 0103 eq 003.png|RTENOTITLE]]....................(3.159)<br/><br/>where ''S''(''M'', ''M′'', ''t''−''τ'') corresponds to a unit-strength [[File:Vol1 page 0103 inline 004.png|RTENOTITLE]], instantaneous point source in an infinite reservoir; that is,<br/><br/>[[File:Vol1 page 0103 eq 004.png|RTENOTITLE]]....................(3.160)<br/><br/>'''''Instantaneous and Continuous Line, Surface, and Volumetric Sources in an Infinite Reservoir'''''. Similarly, the distribution of instantaneous point sources of strength [[File:Vol1 page 0103 inline 005.png|RTENOTITLE]] over a line, surface, or volume, Γ<sub>''w''</sub>, in an infinite reservoir leads to the following solution corresponding to the pressure drop because of production from a line, surface, or volumetric source, respectively.<br/><br/>[[File:Vol1 page 0104 eq 005.png|RTENOTITLE]]....................(3.161)<br/><br/>In '''Eq. 3.161''', ''M''<sub>''w''</sub> indicates a point on the source (Γ<sub>''w''</sub>) and [[File:Vol1 page 0104 inline 001.png|RTENOTITLE]] is the instantaneous withdrawal volume of fluids per unit length, area, or volume of the source, depending on the source geometry. For example, the pressure drop as a result of an infinite line source at ''x′'', ''y′'' and -∞≤ ''z′'' ≤ ∞ may be obtained as follows:<br/><br/>[[File:Vol1 page 0104 eq 001.png|RTENOTITLE]]....................(3.162)<br/><br/>If we assume that the flux is uniform along the line source and the source strength is unity [[File:Vol1 page 0104 inline 002.png|RTENOTITLE]], then we can write the instantaneous, infinite line-source solution in an infinite reservoir as<br/><br/>[[File:Vol1 page 0104 eq 002.png|RTENOTITLE]]....................(3.163)<br/><br/>As another example, if we consider an instantaneous, infinite plane source at ''x ''= ''x′'', -∞ ≤ ''y′'' ≤ ∞, and -∞ ≤ ''z′'' ≤ ∞ in an infinite reservoir, we can write<br/><br/>[[File:Vol1 page 0104 eq 003.png|RTENOTITLE]]....................(3.164)<br/><br/>which also leads to the following uniform-flux, unit-strength, instantaneous, infinite plane-source solution in an infinite reservoir:<br/><br/>[[File:Vol1 page 0104 eq 004.png|RTENOTITLE]]....................(3.165)<br/><br/>If the fluid withdrawal is at a continuous rate from time 0 to ''t'', then the continuous line-, surface-, or volumetric-source solution for an infinite reservoir is given by<br/><br/>[[File:Vol1 page 0104 eq 005.png|RTENOTITLE]]....................(3.166)<br/><br/>'''''Source Functions for Bounded Reservoirs'''''. The source solutions discussed previously can be extended to bounded reservoirs. The method of images provides a convenient means of generating the bounded-reservoir solutions with the use of the infinite reservoir solutions, especially when the reservoir boundaries consist of impermeable and constant-pressure planes. The method of images is an application of the principle of superposition, which states that if ''f''<sub>1</sub> and ''f''<sub>2</sub> are two linearly independent solutions of a linear PDE and ''c''<sub>1</sub> and ''c''<sub>2</sub> are two arbitrary constants, then ''f''<sub>3</sub> = ''c''<sub>1</sub>''f''<sub>1</sub> + ''c''<sub>2</sub>''f''<sub>2</sub> is also a solution of the PDE. Examples of source functions in bounded reservoirs are presented here.<br/><br/>'''''Instantaneous Point Source Near a Single Linear Boundary'''''. To generate the effect of an impermeable planar boundary at a distance d from a unit-strength, instantaneous point source in an infinite reservoir (see '''Fig. 3.7'''), we can apply the method of images to the instantaneous point-source solution given in '''Eq. 3.157''' as<br/><br/>[[File:Vol1 page 0105 eq 001.png|RTENOTITLE]]....................(3.167)<br/><br/>It can be shown from '''Eq. 3.167''' that (''∂S''/''∂x'')<sub>x=d</sub> = 0; that is, the bisector of the distance between the two sources is a no-flow boundary. Similarly, to generate the effect of a constant-pressure boundary, we use the method of images and the unit-strength, instantaneous point-source solution ('''Eq. 3.160''') as follows:<br/><br/>[[File:Vol1 page 0105 eq 002.png|RTENOTITLE]]....................(3.168)<br/><br/>'''''Instantaneous Point Source in an Infinite-Slab Reservoir'''''. Using the method of images and considering the geometry shown in Col. A of '''Fig. 3.8''', we can generate the solution for a unit-strength, instantaneous point source in an infinite-slab reservoir with impermeable boundaries at ''z'' = 0 and ''h''. The result is given by<br/><br/>[[File:Vol1 page 0105 eq 003.png|RTENOTITLE]]....................(3.169)<br/><br/>which, with Poisson’s summation formula given by<ref name="r14">Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, second edition, 353–386. Oxford, UK: Oxford University Press.</ref><br/><br/>[[File:Vol1 page 0106 eq 001.png|RTENOTITLE]]....................(3.170)<br/><br/>may be transformed into<br/><br/>[[File:Vol1 page 0106 eq 002.png|RTENOTITLE]]....................(3.171)<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0101 eq 004.png]]....................(3.147)
<br>
<br>
Assume that the initial pressure drop satisfies
<br>
<br>
[[File:Vol1 page 0102 eq 001.png]]....................(3.148)
<br>
<br>
and we have the condition that
<br>
<br>
[[File:Vol1 page 0102 eq 002.png]]....................(3.149)
<br>
<br>
On substitution of ''u'' = ''r''Δ''p'', '''Eqs. 3.147''' through '''3.149''' become, respectively,
<br>
<br>
[[File:Vol1 page 0102 eq 003.png]]....................(3.150)
<br>
<br>
[[File:Vol1 page 0102 eq 004.png]]....................(3.151)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0102 eq 005.png]]....................(3.152)
<br>
<br>
The solution of the problem described by '''Eqs. 3.150''' through '''3.152''' is given by<ref name="r14" />
<br>
<br>
[[File:Vol1 page 0102 eq 006.png]]....................(3.153)
<br>
<br>
If we expand the exponential terms in the integrand in '''Eq. 3.153''' in powers of ''r′'' and neglect the terms with powers higher than four, we obtain
<br>
<br>
[[File:Vol1 page 0102 eq 007.png]]....................(3.154)
<br>
<br>
In '''Eq. 3.154''', 4''πα''<sup>3</sup>/3=''V'' where ''V'' is the volume of the spherical source. If [[File:Vol1 page 0102 inline 001.png]] denotes the volume of the liquid released as a result of the change in the volume of the source, Δ''V'', which is caused by the change in pressure, Δ''p''<sub>''i''</sub>, then [[File:Vol1 page 0102 inline 002.png]]. With the definition of compressibility, ''c'' = -(1 / ''V'')(Δ''V'' / Δ''p''<sub>''i''</sub>), we obtain [[File:Vol1 page 0102 inline 003.png]]. Then, we can show that
<br>
<br>
[[File:Vol1 page 0102 eq 008.png]]....................(3.155)
<br>
<br>
Substituting '''Eq. 3.155''' into '''Eq. 3.154''', we obtain
<br>
<br>
[[File:Vol1 page 0102 eq 009.png]]....................(3.156)
<br>
<br>
If we let the radius of the spherical source, ''a'', tend to zero while [[File:Vol1 page 0102 inline 001.png]] remains constant, '''Eq. 3.156''' yields the point-source solution in spherical coordinates given by
<br>
<br>
[[File:Vol1 page 0103 eq 001.png]]....................(3.157)
<br>
<br>
This solution may be interpreted as the pressure drop at a distance ''r'' because of a volume of fluid, [[File:Vol1 page 0102 inline 001.png]], instantaneously withdrawn at ''r'' = 0 and ''t'' = 0. Consistent with this interpretation, [[File:Vol1 page 0103 inline 001.png]] is the strength of the source, which is the pressure drop in a unit volume of the porous medium caused by the instantaneous withdrawal of a volume of fluid, [[File:Vol1 page 0102 inline 001.png]] (see '''Eq. 3.155''').
<br>
<br>
'''''Instantaneous Point Source in an Infinite Reservoir'''''. Nisle<ref name="r21" /> presented a more general solution for an instantaneous point source of strength [[File:Vol1 page 0103 inline 001.png]] acting at ''t'' = ''τ'' in an infinite, homogeneous, but anisotropic reservoir as
<br>
<br>
[[File:Vol1 page 0103 eq 002.png]]....................(3.158)
<br>
<br>
In '''Eq. 3.158''', ''M'' and ''M′'' indicate the locations of the observation point and the source, respectively. For a 3D Cartesian coordinate system, [[File:Vol1 page 0103 inline 002.png]] with ''η''<sub>''x''</sub>, ''η''<sub>''y''</sub>, and ''η''<sub>''z''</sub> representing the diffusivity constants (defined in '''Eq. 3.23''') in the ''x'', ''y'', and ''z'' directions, respectively.
<br>
<br>
'''''Continuous Point Source in an Infinite Reservoir'''''. If the fluid withdrawal is at a continuous rate, [[File:Vol1 page 0103 inline 003.png]], from time 0 to ''t'', then the pressure drop as a result of a continuous point source in an infinite reservoir is obtained by distributing the point sources of strength [[File:Vol1 page 0103 inline 001.png]] over a time interval 0 ≤ ''τ'' ≤ ''t''. This is given by
<br>
<br>
[[File:Vol1 page 0103 eq 003.png]]....................(3.159)
<br>
<br>
where ''S''(''M'', ''M′'', ''t''−''τ'') corresponds to a unit-strength [[File:Vol1 page 0103 inline 004.png]], instantaneous point source in an infinite reservoir; that is,
<br>
<br>
[[File:Vol1 page 0103 eq 004.png]]....................(3.160)
<br>
<br>
'''''Instantaneous and Continuous Line, Surface, and Volumetric Sources in an Infinite Reservoir'''''. Similarly, the distribution of instantaneous point sources of strength [[File:Vol1 page 0103 inline 005.png]] over a line, surface, or volume, Γ<sub>''w''</sub>, in an infinite reservoir leads to the following solution corresponding to the pressure drop because of production from a line, surface, or volumetric source, respectively.
<br>
<br>
[[File:Vol1 page 0104 eq 005.png]]....................(3.161)
<br>
<br>
In '''Eq. 3.161''', ''M''<sub>''w''</sub> indicates a point on the source (Γ<sub>''w''</sub>) and [[File:Vol1 page 0104 inline 001.png]] is the instantaneous withdrawal volume of fluids per unit length, area, or volume of the source, depending on the source geometry. For example, the pressure drop as a result of an infinite line source at ''x′'', ''y′'' and -∞≤ ''z′'' ≤ ∞ may be obtained as follows:
<br>
<br>
[[File:Vol1 page 0104 eq 001.png]]....................(3.162)
<br>
<br>
If we assume that the flux is uniform along the line source and the source strength is unity [[File:Vol1 page 0104 inline 002.png]], then we can write the instantaneous, infinite line-source solution in an infinite reservoir as
<br>
<br>
[[File:Vol1 page 0104 eq 002.png]]....................(3.163)
<br>
<br>
As another example, if we consider an instantaneous, infinite plane source at ''x ''= ''x′'', -∞ ≤ ''y′'' ≤ ∞, and -∞ ≤ ''z′'' ≤ ∞ in an infinite reservoir, we can write
<br>
<br>
[[File:Vol1 page 0104 eq 003.png]]....................(3.164)
<br>
<br>
which also leads to the following uniform-flux, unit-strength, instantaneous, infinite plane-source solution in an infinite reservoir:
<br>
<br>
[[File:Vol1 page 0104 eq 004.png]]....................(3.165)
<br>
<br>
If the fluid withdrawal is at a continuous rate from time 0 to ''t'', then the continuous line-, surface-, or volumetric-source solution for an infinite reservoir is given by
<br>
<br>
[[File:Vol1 page 0104 eq 005.png]]....................(3.166)
<br>
<br>
'''''Source Functions for Bounded Reservoirs'''''. The source solutions discussed previously can be extended to bounded reservoirs. The method of images provides a convenient means of generating the bounded-reservoir solutions with the use of the infinite reservoir solutions, especially when the reservoir boundaries consist of impermeable and constant-pressure planes. The method of images is an application of the principle of superposition, which states that if ''f''<sub>1</sub> and ''f''<sub>2</sub> are two linearly independent solutions of a linear PDE and ''c''<sub>1</sub> and ''c''<sub>2</sub> are two arbitrary constants, then ''f''<sub>3</sub> = ''c''<sub>1</sub>''f''<sub>1</sub> + ''c''<sub>2</sub>''f''<sub>2</sub> is also a solution of the PDE. Examples of source functions in bounded reservoirs are presented here.
<br>
<br>
'''''Instantaneous Point Source Near a Single Linear Boundary'''''. To generate the effect of an impermeable planar boundary at a distance d from a unit-strength, instantaneous point source in an infinite reservoir (see '''Fig. 3.7'''), we can apply the method of images to the instantaneous point-source solution given in '''Eq. 3.157''' as
<br>
<br>
[[File:Vol1 page 0105 eq 001.png]]....................(3.167)
<br>
<br>
It can be shown from '''Eq. 3.167''' that (''∂S''/''∂x'')<sub>x=d</sub> = 0; that is, the bisector of the distance between the two sources is a no-flow boundary. Similarly, to generate the effect of a constant-pressure boundary, we use the method of images and the unit-strength, instantaneous point-source solution ('''Eq. 3.160''') as follows:
<br>
<br>
[[File:Vol1 page 0105 eq 002.png]]....................(3.168)
<br>
<br>
'''''Instantaneous Point Source in an Infinite-Slab Reservoir'''''. Using the method of images and considering the geometry shown in Col. A of '''Fig. 3.8''', we can generate the solution for a unit-strength, instantaneous point source in an infinite-slab reservoir with impermeable boundaries at ''z'' = 0 and ''h''. The result is given by
<br>
<br>
[[File:Vol1 page 0105 eq 003.png]]....................(3.169)
<br>
<br>
which, with Poisson’s summation formula given by<ref name="r14" />
<br>
<br>
[[File:Vol1 page 0106 eq 001.png]]....................(3.170)
<br>
<br>
may be transformed into
<br>
<br>
[[File:Vol1 page 0106 eq 002.png]]....................(3.171)
<br>
<br>
<gallery widths="300px" heights="200px">
File:vol1 Page 105 Image 0001.png|'''Fig. 3.7 – Application of the method of images to generate the effect of a linear boundary.'''
File:vol1 Page 105 Image 0001.png|'''Fig. 3.7 – Application of the method of images to generate the effect of a linear boundary.'''


File:vol1 Page 107 Image 0001.png|'''Fig. 3.8 – Application of the method of images to generate the solutions for infinite-slab reservoirs.'''
File:vol1 Page 107 Image 0001.png|'''Fig. 3.8 – Application of the method of images to generate the solutions for infinite-slab reservoirs.'''
</gallery>
</gallery><br/>Following similar lines, if the slab boundaries at ''z'' = 0 and h are at a constant pressure equal to ''p''<sub>''i''</sub>, we obtain<br/><br/>[[File:Vol1 page 0106 eq 003.png|RTENOTITLE]]....................(3.172)<br/><br/>Similarly, for the case in which the slab boundary at ''z'' = 0 is impermeable while the boundary at ''z'' = ''h'' is at a constant pressure equal to ''p''<sub>''i''</sub>, the following solution may be derived:<br/><br/>[[File:Vol1 page 0106 eq 004.png|RTENOTITLE]]....................(3.173)<br/><br/>'''''Instantaneous Point Source in a Closed Parallelepiped.''''' The ideas used previously for slab reservoirs may be used to develop solutions for reservoirs bounded by planes in all three directions. For example, if the reservoir is bounded in all three directions (i.e., 0 ≤ ''x'' ≤ ''x''<sub>''e''</sub>, 0 ≤ ''y'' ≤ ''y''<sub>''e''</sub>, and 0 ≤ ''z'' ≤ ''h'') and the bounding planes are impermeable, then we can use '''Eq. 3.157''' and the method of images to write<br/><br/>[[File:Vol1 page 0106 eq 001.png|RTENOTITLE]] [[File:Vol1 page 0107 eq 001.png|RTENOTITLE]]....................(3.174)<br/><br/>which, with Poisson’s summation formula ('''Eq. 3.170'''), may be recast into the following form:<br/><br/>[[File:Vol1 page 0107 eq 002.png|RTENOTITLE]]....................(3.175)<br/><br/>'''''Instantaneous Infinite-Plane Source in an Infinite-Slab Reservoir With Impermeable Boundaries.''''' The instantaneous point-source solutions of '''Eqs. 3.171''' through '''3.173''' may be extended to different source geometries with '''Eq. 3.161'''. For example, the solution for an instantaneous infinite-plane source at ''z'' = ''z′'' in an infinite-slab reservoir with impermeable boundaries is obtained by substituting '''Eq. 3.171''' for ''S'' in '''Eq. 3.161'''. This yields<br/><br/>[[File:Vol1 page 0108 eq 001.png|RTENOTITLE]]....................(3.176)<br/><br/>Assuming a unit-strength, uniform-flux source [[File:Vol1 page 0108 inline 001.png|RTENOTITLE]], we obtain the following instantaneous infinite-plane source solution in an infinite-slab reservoir with impermeable boundaries:<br/><br/>[[File:Vol1 page 0108 eq 002.png|RTENOTITLE]]....................(3.177)<br/><br/>'''''Instantaneous Infinite-Slab Source in an Infinite-Slab Reservoir With Impermeable Boundaries.''''' Following similar lines, we can obtain the solution for an instantaneous, infinite-slab source of thickness, ''h''<sub>''p''</sub>, located at ''z'' = ''z''<sub>''w''</sub> (''z''<sub>''w''</sub> is the ''z''-coordinate of the midpoint of the slab source) in an infinite-slab reservoir with impermeable boundaries.<br/><br/>[[File:Vol1 page 0108 eq 003.png|RTENOTITLE]]....................(3.178)<br/><br/>If we assume a uniform-flux slab source [[File:Vol1 page 0108 inline 002.png|RTENOTITLE]], then '''Eq. 3.178''' yields<br/><br/>[[File:Vol1 page 0108 eq 004.png|RTENOTITLE]]....................(3.179)<br/><br/>'''''Uniform-Flux, Continuous, Infinite-Slab Source in an Infinite-Slab Reservoir With Impermeable Boundaries.''''' Solutions for continuous plane and slab sources can be obtained as indicated by '''Eq. 3.159''' (or '''Eq. 3.166'''). For example, the solution for a uniform-flux, continuous, infinite-slab source in an infinite-slab reservoir with impermeable top and bottom boundaries may be obtained by substituting the right side of '''Eq. 3.179''' for ''S'' in '''Eq. 3.159''' and is given by<br/><br/>[[File:Vol1 page 0108 eq 005.png|RTENOTITLE]]....................(3.180)<br/><br/>The same solution could have been obtained by substituting the unit-strength instantaneous point-source solution given by '''Eq. 3.171''' for ''S'' in '''Eq. 3.166'''.
<br/>
Following similar lines, if the slab boundaries at ''z'' = 0 and h are at a constant pressure equal to ''p''<sub>''i''</sub>, we obtain
<br>
<br>
[[File:Vol1 page 0106 eq 003.png]]....................(3.172)
<br>
<br>
Similarly, for the case in which the slab boundary at ''z'' = 0 is impermeable while the boundary at ''z'' = ''h'' is at a constant pressure equal to ''p''<sub>''i''</sub>, the following solution may be derived:
<br>
<br>
[[File:Vol1 page 0106 eq 004.png]]....................(3.173)
<br>
<br>
'''''Instantaneous Point Source in a Closed Parallelepiped.''''' The ideas used previously for slab reservoirs may be used to develop solutions for reservoirs bounded by planes in all three directions. For example, if the reservoir is bounded in all three directions (i.e., 0 ≤ ''x'' ≤ ''x''<sub>''e''</sub>, 0 ≤ ''y'' ≤ ''y''<sub>''e''</sub>, and 0 ≤ ''z'' ≤ ''h'') and the bounding planes are impermeable, then we can use '''Eq. 3.157''' and the method of images to write
<br>
<br>
[[File:Vol1 page 0106 eq 001.png]]
[[File:Vol1 page 0107 eq 001.png]]....................(3.174)
<br>
<br>
which, with Poisson’s summation formula ('''Eq. 3.170'''), may be recast into the following form:
<br>
<br>
[[File:Vol1 page 0107 eq 002.png]]....................(3.175)
<br>
<br>
'''''Instantaneous Infinite-Plane Source in an Infinite-Slab Reservoir With Impermeable Boundaries.''''' The instantaneous point-source solutions of '''Eqs. 3.171''' through '''3.173''' may be extended to different source geometries with '''Eq. 3.161'''. For example, the solution for an instantaneous infinite-plane source at ''z'' = ''z′'' in an infinite-slab reservoir with impermeable boundaries is obtained by substituting '''Eq. 3.171''' for ''S'' in '''Eq. 3.161'''. This yields
<br>
<br>
[[File:Vol1 page 0108 eq 001.png]]....................(3.176)
<br>
<br>
Assuming a unit-strength, uniform-flux source [[File:Vol1 page 0108 inline 001.png]], we obtain the following instantaneous infinite-plane source solution in an infinite-slab reservoir with impermeable boundaries:
<br>
<br>
[[File:Vol1 page 0108 eq 002.png]]....................(3.177)
<br>
<br>
'''''Instantaneous Infinite-Slab Source in an Infinite-Slab Reservoir With Impermeable Boundaries.''''' Following similar lines, we can obtain the solution for an instantaneous, infinite-slab source of thickness, ''h''<sub>''p''</sub>, located at ''z'' = ''z''<sub>''w''</sub> (''z''<sub>''w''</sub> is the ''z''-coordinate of the midpoint of the slab source) in an infinite-slab reservoir with impermeable boundaries.
<br>
<br>
[[File:Vol1 page 0108 eq 003.png]]....................(3.178)
<br>
<br>
If we assume a uniform-flux slab source [[File:Vol1 page 0108 inline 002.png]], then '''Eq. 3.178''' yields
<br>
<br>
[[File:Vol1 page 0108 eq 004.png]]....................(3.179)
<br>
<br>
'''''Uniform-Flux, Continuous, Infinite-Slab Source in an Infinite-Slab Reservoir With Impermeable Boundaries.''''' Solutions for continuous plane and slab sources can be obtained as indicated by '''Eq. 3.159''' (or '''Eq. 3.166'''). For example, the solution for a uniform-flux, continuous, infinite-slab source in an infinite-slab reservoir with impermeable top and bottom boundaries may be obtained by substituting the right side of '''Eq. 3.179''' for ''S'' in '''Eq. 3.159''' and is given by
<br>
<br>
[[File:Vol1 page 0108 eq 005.png]]....................(3.180)
<br>
<br>
The same solution could have been obtained by substituting the unit-strength instantaneous point-source solution given by '''Eq. 3.171''' for ''S'' in '''Eq. 3.166'''.


----
----


'''Example 3.5'''
'''Example 3.5'''<br/><br/>Consider transient flow toward a partially penetrating vertical well of penetration length, ''h''<sub>''w''</sub>, in an infinite, homogeneous, slab reservoir of uniform thickness, ''h'', and initial pressure, ''p''<sub>''i''</sub>, with impermeable top and bottom boundaries.<br/><br/>''Solution''. '''Fig. 3.9''' shows the geometry of the well and reservoir system of interest. The solution for this problem can be obtained by assuming that the well may be represented by a vertical line source. Then, starting with '''Eq. 3.166''' and substituting the unit-strength, instantaneous point-source solution in an infinite-slab reservoir with impermeable boundaries ['''Eq. 3.171''' with [[File:Vol1 page 0109 inline 001.png|RTENOTITLE]]] for ''S'', we obtain<br/><br/>[[File:Vol1 page 0109 eq 001.png|RTENOTITLE]]....................(3.181)<br/><br/>If we assume that the strength of the source is uniformly distributed along its length (this physically corresponds to a uniform-flux distribution) and the production rate is constant over time [i.e., [[File:Vol1 page 0109 inline 002.png|RTENOTITLE]], where ''q'' is the constant production rate of the well], then '''Eq. 3.181''' yields<br/><br/>[[File:Vol1 page 0109 eq 002.png|RTENOTITLE]] [[File:Vol1 page 0110 eq 001.png|RTENOTITLE]]....................(3.182)<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
Consider transient flow toward a partially penetrating vertical well of penetration length, ''h''<sub>''w''</sub>, in an infinite, homogeneous, slab reservoir of uniform thickness, ''h'', and initial pressure, ''p''<sub>''i''</sub>, with impermeable top and bottom boundaries.
<br>
<br>
''Solution''. '''Fig. 3.9''' shows the geometry of the well and reservoir system of interest. The solution for this problem can be obtained by assuming that the well may be represented by a vertical line source. Then, starting with '''Eq. 3.166''' and substituting the unit-strength, instantaneous point-source solution in an infinite-slab reservoir with impermeable boundaries ['''Eq. 3.171''' with [[File:Vol1 page 0109 inline 001.png]]] for ''S'', we obtain
<br>
<br>
[[File:Vol1 page 0109 eq 001.png]]....................(3.181)
<br>
<br>
If we assume that the strength of the source is uniformly distributed along its length (this physically corresponds to a uniform-flux distribution) and the production rate is constant over time [i.e., [[File:Vol1 page 0109 inline 002.png]], where ''q'' is the constant production rate of the well], then '''Eq. 3.181''' yields
<br>
<br>
[[File:Vol1 page 0109 eq 002.png]]
[[File:Vol1 page 0110 eq 001.png]]....................(3.182)
<br>
<br>
<gallery widths="300px" heights="200px">
File:vol1 Page 109 Image 0001.png|'''Fig. 3.9 – Geometry of the well/reservoir system for a partially penetrating vertical well in an infinite-slab reservoir with impermeable boundaries for Example 3.5.'''
File:vol1 Page 109 Image 0001.png|'''Fig. 3.9 – Geometry of the well/reservoir system for a partially penetrating vertical well in an infinite-slab reservoir with impermeable boundaries for Example 3.5.'''
</gallery>
</gallery>
<br/>


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Line 1,302: Line 121:
=== The Use of Green’s Functions and Source Functions in Solving Unsteady-Flow Problems ===
=== The Use of Green’s Functions and Source Functions in Solving Unsteady-Flow Problems ===


As discussed in '''Sec. 3.4.2''', the conventional development of the source-function solutions uses the instantaneous point-source solution as the building block with the appropriate integration (superposition) in space and time. In 1973, Gringarten and Ramey<ref name="r13" /> introduced the use of the source and Green’s function method to the petroleum engineering literature with a more efficient method of developing the source solutions. Specifically, they suggested the use of infinite-plane sources as the building block with Newman’s product method.<ref name="r22" /> In this section, we discuss the use of Green’s functions and source functions in solving unsteady-flow problems in reservoirs.
As discussed in '''Sec. 3.4.2''', the conventional development of the source-function solutions uses the instantaneous point-source solution as the building block with the appropriate integration (superposition) in space and time. In 1973, Gringarten and Ramey<ref name="r13">Gringarten, A.C. and Ramey Jr., H.J. 1973. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. SPE J. 13 (5): 285-296. SPE-3818-PA. http://dx.doi.org/10.2118/3818-PA</ref> introduced the use of the source and Green’s function method to the petroleum engineering literature with a more efficient method of developing the source solutions. Specifically, they suggested the use of infinite-plane sources as the building block with Newman’s product method.<ref name="r22">Newman, A.B. 1936. Heating and Cooling Rectangular and Cylindrical Solids. Ind. Eng. Chem. 28 (5): 545–548. http://dx.doi.org/10.1021/ie50317a010</ref> In this section, we discuss the use of Green’s functions and source functions in solving unsteady-flow problems in reservoirs.<br/><br/>Green’s function for transient flow in a porous medium is defined as the pressure at ''M'' (''x'', ''y'', ''z'') at time ''t'' because of an instantaneous point source of unit strength generated at point ''M′''(''x′'', ''y′'', ''z′'') at time ''τ'' < ''t'' with the porous medium initially at zero pressure and the boundary of the medium kept at zero pressure or impermeable to flow.<ref name="r13">Gringarten, A.C. and Ramey Jr., H.J. 1973. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. SPE J. 13 (5): 285-296. SPE-3818-PA. http://dx.doi.org/10.2118/3818-PA</ref><ref name="r14">Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, second edition, 353–386. Oxford, UK: Oxford University Press.</ref> If we let ''G''(''M'', ''M′'', ''t'' − ''τ'') denote the Green’s function, then it should satisfy the diffusion equation; that is,<br/><br/>[[File:Vol1 page 0110 eq 002.png|RTENOTITLE]]....................(3.183)<br/><br/>Because ''G'' is a function of ''t'' − ''τ'', it should also satisfy the adjoint diffusion equation,<br/><br/>[[File:Vol1 page 0110 eq 003.png|RTENOTITLE]]....................(3.184)<br/><br/>Green’s function also has the following properties: <ref name="r13">Gringarten, A.C. and Ramey Jr., H.J. 1973. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. SPE J. 13 (5): 285-296. SPE-3818-PA. http://dx.doi.org/10.2118/3818-PA</ref><ref name="r14">Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, second edition, 353–386. Oxford, UK: Oxford University Press.</ref>
<br>
 
<br>
Green’s function for transient flow in a porous medium is defined as the pressure at ''M'' (''x'', ''y'', ''z'') at time ''t'' because of an instantaneous point source of unit strength generated at point ''M′''(''x′'', ''y′'', ''z′'') at time ''τ'' < ''t'' with the porous medium initially at zero pressure and the boundary of the medium kept at zero pressure or impermeable to flow.<ref name="r13" /><ref name="r14" /> If we let ''G''(''M'', ''M′'', ''t'' − ''τ'') denote the Green’s function, then it should satisfy the diffusion equation; that is,
<br>
<br>
[[File:Vol1 page 0110 eq 002.png]]....................(3.183)
<br>
<br>
Because ''G'' is a function of ''t'' − ''τ'', it should also satisfy the adjoint diffusion equation,
<br>
<br>
[[File:Vol1 page 0110 eq 003.png]]....................(3.184)
<br>
<br>
Green’s function also has the following properties: <ref name="r13" /><ref name="r14" />
<br>
#''G'' is symmetrical in the two points ''M'' and ''M′''; that is, Green’s function is invariant as the source and the observation points are interchanged.
#''G'' is symmetrical in the two points ''M'' and ''M′''; that is, Green’s function is invariant as the source and the observation points are interchanged.
#As ''t'' → ''τ'', ''G'' vanishes at all points in the porous medium; that is, [[File:Vol1 page 0110 inline 001.png]], except at the source location, ''M'' = ''M′'', where it becomes infinite, so that ''G'' satisfies the delta function property,
#As ''t'' → ''τ'', ''G'' vanishes at all points in the porous medium; that is, [[File:Vol1 page 0110 inline 001.png|RTENOTITLE]], except at the source location, ''M'' = ''M′'', where it becomes infinite, so that ''G'' satisfies the delta function property,
<br>
 
<br>
<br/><br/>[[File:Vol1 page 0110 eq 004.png|RTENOTITLE]]....................(3.185)<br/><br/>where ''D'' indicates the domain of the porous medium, and ''φ''(''M'') is any continuous function.
[[File:Vol1 page 0110 eq 004.png]]....................(3.185)
 
<br>
<br>
where ''D'' indicates the domain of the porous medium, and ''φ''(''M'') is any continuous function.
#Because ''G'' corresponds to the pressure because of an instantaneous point source of unit strength, it satisfies
#Because ''G'' corresponds to the pressure because of an instantaneous point source of unit strength, it satisfies
<br>
 
<br>
<br/><br/>[[File:Vol1 page 0110 eq 005.png|RTENOTITLE]]....................(3.186)<br/>
[[File:Vol1 page 0110 eq 005.png]]....................(3.186)
 
<br>
<br>
#''G'' or its normal derivative, ''∂G''/''∂n'', vanishes at the boundary, Γ, of the porous medium. If the porous medium is infinite, then ''G'' vanishes as ''M'' or ''M′''→∞.
#''G'' or its normal derivative, ''∂G''/''∂n'', vanishes at the boundary, Γ, of the porous medium. If the porous medium is infinite, then ''G'' vanishes as ''M'' or ''M′''→∞.
<br>
 
<br>
<br/><br/>Let ''p''(''M′'' , ''τ'') be the pressure in the porous medium and ''G''(''M'', ''M′'', ''t'' - ''τ'') be the Green’s function. Let ''D'' denote the domain of the porous medium. Then, ''p'' and ''G'' satisfy the following differential equations:<br/><br/>[[File:Vol1 page 0111 eq 001.png|RTENOTITLE]]....................(3.187)<br/><br/>and<br/><br/>[[File:Vol1 page 0111 eq 002.png|RTENOTITLE]]....................(3.188)<br/><br/>Then, we can write<br/><br/>[[File:Vol1 page 0111 eq 003.png|RTENOTITLE]]....................(3.189)<br/><br/>or<br/><br/>[[File:Vol1 page 0111 eq 004.png|RTENOTITLE]]....................(3.190)<br/><br/>where ''ε'' is a small positive number. Changing the order of integration and applying the Green’s theorem,<br/><br/>[[File:Vol1 page 0111 eq 005.png|RTENOTITLE]]....................(3.191)<br/><br/>where ''D'' and Γ indicate the volume and boundary surface of the domain, respectively; ''S'' denotes the points on the boundary; and ''∂''/''∂n'' indicates differentiation in the normal direction of the surface Γ; we obtain<br/><br/>[[File:Vol1 page 0111 eq 006.png|RTENOTITLE]]....................(3.192)<br/><br/>Taking the limit as ''ε''→0 and noting the delta-function property of the Green’s function ('''Eq. 3.185'''), '''Eq. 3.192''' yields<br/><br/>[[File:Vol1 page 0111 eq 007.png|RTENOTITLE]]....................(3.193)<br/><br/>where ''p''<sub>''i''</sub>(''M'') = ''p''(''M'', ''t'' = 0) is the initial pressure at point ''M''.<br/><br/>In '''Eq. 3.193''', the boundary of the porous medium consists of two surfaces: the inner boundary that corresponds to the surface of the wellbore, Γ<sub>''w''</sub>, and the outer boundary of the reservoir, Γ<sub>''e''</sub>. '''Eq. 3.193''' may be written as<br/><br/>[[File:Vol1 page 0112 eq 001.png|RTENOTITLE]]....................(3.194)<br/><br/>As the fourth property of Green’s function noted previously requires, if the outer boundary of the reservoir is impermeable, [[File:Vol1 page 0112 inline 001.png|RTENOTITLE]] or at infinity, then ''G'' vanishes at the outer boundary; that is, ''G''(Γ<sub>''e''</sub>) = 0. Thus, '''Eq. 3.194''' becomes<br/><br/>[[File:Vol1 page 0112 eq 002.png|RTENOTITLE]]....................(3.195)<br/><br/>Similarly, if the flux, [[File:Vol1 page 0112 inline 002.png|RTENOTITLE]], is specified at the inner boundary, then the normal derivative of Green’s function, [[File:Vol1 page 0112 inline 003.png|RTENOTITLE]], vanishes at that boundary. This yields<br/><br/>[[File:Vol1 page 0112 eq 003.png|RTENOTITLE]]....................(3.196)<br/><br/>If the initial pressure, ''p''<sub>''i''</sub>, is uniform over the entire domain (porous medium), ''D'', then, by the third property of Green’s function ('''Eq. 3.186'''), we should have<br/><br/>[[File:Vol1 page 0112 eq 004.png|RTENOTITLE]]....................(3.197)<br/><br/>Also, the flux law for porous medium (Darcy’s law) requires that the volume of fluid passing through the point, ''M′''<sub>''w''</sub>, on the inner-boundary surface, Γ<sub>''w''</sub>, at time ''t'' be equal to<br/><br/>[[File:Vol1 page 0112 eq 005.png|RTENOTITLE]]....................(3.198)<br/><br/>The substitution of '''Eqs. 3.197''' and '''3.198''' into '''Eq. 3.196''' yields<br/><br/>[[File:Vol1 page 0112 eq 006.png|RTENOTITLE]]....................(3.199)<br/><br/>Not surprisingly, '''Eq. 3.199''' is the same as '''Eq. 3.166''' because ''G'' in '''Eq. 3.199''' is the instantaneous point-source solution of unit strength denoted by ''S'' in '''Eq. 3.166'''.<br/><br/>The expression given in '''Eq. 3.199''' may be simplified further by assuming that the flux, [[File:Vol1 page 0112 inline 004.png|RTENOTITLE]], is uniformly distributed on the inner-boundary surface (wellbore), Γ<sub>''w''</sub>. This yields<br/><br/>[[File:Vol1 page 0113 eq 001.png|RTENOTITLE]]....................(3.200)<br/><br/>where<br/><br/>[[File:Vol1 page 0113 eq 002.png|RTENOTITLE]]....................(3.201)<br/><br/>The integration in the right side of '''Eq. 3.201''' represents the distribution of instantaneous point sources over the length, area, or volume of the source (well), and ''S'' denotes the corresponding instantaneous source function. In '''Sec. 3.4.2''', we discussed the conventional derivation of the source functions starting from the basic instantaneous point-source solution. Here, we discuss the use of infinite-plane sources as the building block with Newman’s product method.<ref name="r22">Newman, A.B. 1936. Heating and Cooling Rectangular and Cylindrical Solids. Ind. Eng. Chem. 28 (5): 545–548. http://dx.doi.org/10.1021/ie50317a010</ref><br/><br/>Newman’s product method<ref name="r22">Newman, A.B. 1936. Heating and Cooling Rectangular and Cylindrical Solids. Ind. Eng. Chem. 28 (5): 545–548. http://dx.doi.org/10.1021/ie50317a010</ref> may be stated for transient-flow problems in porous media as follows: <ref name="r13">Gringarten, A.C. and Ramey Jr., H.J. 1973. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. SPE J. 13 (5): 285-296. SPE-3818-PA. http://dx.doi.org/10.2118/3818-PA</ref> if a well/reservoir system can be visualized as the intersection of 1D or 2D systems, then the instantaneous source or Green’s function for this well/reservoir system can be constructed by the product of the source or Green’s functions for the 1D and/or 2D systems. For example, an infinite line-source well at ''x'' = ''x′'', ''y'' = ''y′'', and −∞ ≤ ''z′'' ≤ +∞ in an infinite reservoir may be visualized as the intersection of two infinite, 1D plane sources; one at ''x'' = ''x′'', −∞ ≤ ''y′'' ≤ +∞, and −∞ ≤ ''z′'' ≤ +∞, and the other at −∞ ≤ ''x′'' ≤ +∞, ''y'' = ''y′'', and −∞ ≤ ''z′'' ≤ +∞. Then, the instantaneous source function for this infinite line-source well, ''S''(''x'', ''x′'', ''y'', ''y′'', ''t'' − ''τ''), may be obtained as the product of two infinite, 1D plane sources, given by<br/><br/>[[File:Vol1 page 0113 eq 003.png|RTENOTITLE]]....................(3.202)<br/><br/>as follows<br/><br/>[[File:Vol1 page 0113 eq 004.png|RTENOTITLE]]....................(3.203)<br/><br/>As expected, this solution is the same as '''Eq. 3.163''', which was obtained by integration of an instantaneous point source in an infinite reservoir. For a radially isotropic reservoir (''η''<sub>''x''</sub> = ''η''<sub>''y''</sub> = ''η''<sub>''z''</sub>), '''Eq. 3.203''' yields<br/><br/>[[File:Vol1 page 0113 eq 005.png|RTENOTITLE]]....................(3.204)<br/><br/>where ''d'' is the distance between the line source and the observation line in the ''x''-''y'' plane (see '''Fig. 3.10''') and is given by<br/><br/>[[File:Vol1 page 0113 eq 006.png|RTENOTITLE]]....................(3.205)<br/><br/>Similarly, intersecting three infinite instantaneous plane sources (or a line source and a plane source), we can obtain the instantaneous point-source solution in an infinite reservoir as<br/><br/>[[File:Vol1 page 0114 eq 001.png|RTENOTITLE]]....................(3.206)<br/><br/>Instantaneous plane sources in slab reservoirs can be generated with the plane sources in infinite reservoirs and the method of images as discussed in '''Sec. 3.4.2'''. Similarly, the instantaneous slab sources can be obtained by integrating plane sources over the thickness of the slab source (see '''Sec. 3.4.2'''). '''Table 3.2''', compiled from the work of Gringarten and Ramey,<ref name="r13">Gringarten, A.C. and Ramey Jr., H.J. 1973. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. SPE J. 13 (5): 285-296. SPE-3818-PA. http://dx.doi.org/10.2118/3818-PA</ref> presents the basic instantaneous source functions in infinite reservoirs, and '''Table 3.3''' shows the corresponding geometries of the source-reservoir systems. The basic instantaneous source functions given in '''Table 3.3''' may be used to construct the source functions that represent the appropriate well geometry by Newman’s product method.<br/><br/><gallery widths="300px" heights="200px">
Let ''p''(''M′'' , ''τ'') be the pressure in the porous medium and ''G''(''M'', ''M′'', ''t'' - ''τ'') be the Green’s function. Let ''D'' denote the domain of the porous medium. Then, ''p'' and ''G'' satisfy the following differential equations:
<br>
<br>
[[File:Vol1 page 0111 eq 001.png]]....................(3.187)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0111 eq 002.png]]....................(3.188)
<br>
<br>
Then, we can write
<br>
<br>
[[File:Vol1 page 0111 eq 003.png]]....................(3.189)
<br>
<br>
or
<br>
<br>
[[File:Vol1 page 0111 eq 004.png]]....................(3.190)
<br>
<br>
where ''ε'' is a small positive number. Changing the order of integration and applying the Green’s theorem,
<br>
<br>
[[File:Vol1 page 0111 eq 005.png]]....................(3.191)
<br>
<br>
where ''D'' and Γ indicate the volume and boundary surface of the domain, respectively; ''S'' denotes the points on the boundary; and ''∂''/''∂n'' indicates differentiation in the normal direction of the surface Γ; we obtain
<br>
<br>
[[File:Vol1 page 0111 eq 006.png]]....................(3.192)
<br>
<br>
Taking the limit as ''ε''→0 and noting the delta-function property of the Green’s function ('''Eq. 3.185'''), '''Eq. 3.192''' yields
<br>
<br>
[[File:Vol1 page 0111 eq 007.png]]....................(3.193)
<br>
<br>
where ''p''<sub>''i''</sub>(''M'') = ''p''(''M'', ''t'' = 0) is the initial pressure at point ''M''.
<br>
<br>
In '''Eq. 3.193''', the boundary of the porous medium consists of two surfaces: the inner boundary that corresponds to the surface of the wellbore, Γ<sub>''w''</sub>, and the outer boundary of the reservoir, Γ<sub>''e''</sub>. '''Eq. 3.193''' may be written as
<br>
<br>
[[File:Vol1 page 0112 eq 001.png]]....................(3.194)
<br>
<br>
As the fourth property of Green’s function noted previously requires, if the outer boundary of the reservoir is impermeable, [[File:Vol1 page 0112 inline 001.png]] or at infinity, then ''G'' vanishes at the outer boundary; that is, ''G''(Γ<sub>''e''</sub>) = 0. Thus, '''Eq. 3.194''' becomes
<br>
<br>
[[File:Vol1 page 0112 eq 002.png]]....................(3.195)
<br>
<br>
Similarly, if the flux, [[File:Vol1 page 0112 inline 002.png]], is specified at the inner boundary, then the normal derivative of Green’s function, [[File:Vol1 page 0112 inline 003.png]], vanishes at that boundary. This yields
<br>
<br>
[[File:Vol1 page 0112 eq 003.png]]....................(3.196)
<br>
<br>
If the initial pressure, ''p''<sub>''i''</sub>, is uniform over the entire domain (porous medium), ''D'', then, by the third property of Green’s function ('''Eq. 3.186'''), we should have
<br>
<br>
[[File:Vol1 page 0112 eq 004.png]]....................(3.197)
<br>
<br>
Also, the flux law for porous medium (Darcy’s law) requires that the volume of fluid passing through the point, ''M′''<sub>''w''</sub>, on the inner-boundary surface, Γ<sub>''w''</sub>, at time ''t'' be equal to
<br>
<br>
[[File:Vol1 page 0112 eq 005.png]]....................(3.198)
<br>
<br>
The substitution of '''Eqs. 3.197''' and '''3.198''' into '''Eq. 3.196''' yields
<br>
<br>
[[File:Vol1 page 0112 eq 006.png]]....................(3.199)
<br>
<br>
Not surprisingly, '''Eq. 3.199''' is the same as '''Eq. 3.166''' because ''G'' in '''Eq. 3.199''' is the instantaneous point-source solution of unit strength denoted by ''S'' in '''Eq. 3.166'''.
<br>
<br>
The expression given in '''Eq. 3.199''' may be simplified further by assuming that the flux, [[File:Vol1 page 0112 inline 004.png]], is uniformly distributed on the inner-boundary surface (wellbore), Γ<sub>''w''</sub>. This yields
<br>
<br>
[[File:Vol1 page 0113 eq 001.png]]....................(3.200)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0113 eq 002.png]]....................(3.201)
<br>
<br>
The integration in the right side of '''Eq. 3.201''' represents the distribution of instantaneous point sources over the length, area, or volume of the source (well), and ''S'' denotes the corresponding instantaneous source function. In '''Sec. 3.4.2''', we discussed the conventional derivation of the source functions starting from the basic instantaneous point-source solution. Here, we discuss the use of infinite-plane sources as the building block with Newman’s product method.<ref name="r22" />
<br>
<br>
Newman’s product method<ref name="r22" /> may be stated for transient-flow problems in porous media as follows: <ref name="r13" /> if a well/reservoir system can be visualized as the intersection of 1D or 2D systems, then the instantaneous source or Green’s function for this well/reservoir system can be constructed by the product of the source or Green’s functions for the 1D and/or 2D systems. For example, an infinite line-source well at ''x'' = ''x′'', ''y'' = ''y′'', and −∞ ≤ ''z′'' ≤ +∞ in an infinite reservoir may be visualized as the intersection of two infinite, 1D plane sources; one at ''x'' = ''x′'', −∞ ≤ ''y′'' ≤ +∞, and −∞ ≤ ''z′'' ≤ +∞, and the other at −∞ ≤ ''x′'' ≤ +∞, ''y'' = ''y′'', and −∞ ≤ ''z′'' ≤ +∞. Then, the instantaneous source function for this infinite line-source well, ''S''(''x'', ''x′'', ''y'', ''y′'', ''t'' − ''τ''), may be obtained as the product of two infinite, 1D plane sources, given by
<br>
<br>
[[File:Vol1 page 0113 eq 003.png]]....................(3.202)
<br>
<br>
as follows
<br>
<br>
[[File:Vol1 page 0113 eq 004.png]]....................(3.203)
<br>
<br>
As expected, this solution is the same as '''Eq. 3.163''', which was obtained by integration of an instantaneous point source in an infinite reservoir. For a radially isotropic reservoir (''η''<sub>''x''</sub> = ''η''<sub>''y''</sub> = ''η''<sub>''z''</sub>), '''Eq. 3.203''' yields
<br>
<br>
[[File:Vol1 page 0113 eq 005.png]]....................(3.204)
<br>
<br>
where ''d'' is the distance between the line source and the observation line in the ''x''-''y'' plane (see '''Fig. 3.10''') and is given by
<br>
<br>
[[File:Vol1 page 0113 eq 006.png]]....................(3.205)
<br>
<br>
Similarly, intersecting three infinite instantaneous plane sources (or a line source and a plane source), we can obtain the instantaneous point-source solution in an infinite reservoir as
<br>
<br>
[[File:Vol1 page 0114 eq 001.png]]....................(3.206)
<br>
<br>
Instantaneous plane sources in slab reservoirs can be generated with the plane sources in infinite reservoirs and the method of images as discussed in '''Sec. 3.4.2'''. Similarly, the instantaneous slab sources can be obtained by integrating plane sources over the thickness of the slab source (see '''Sec. 3.4.2'''). '''Table 3.2''', compiled from the work of Gringarten and Ramey,<ref name="r13" /> presents the basic instantaneous source functions in infinite reservoirs, and '''Table 3.3''' shows the corresponding geometries of the source-reservoir systems. The basic instantaneous source functions given in '''Table 3.3''' may be used to construct the source functions that represent the appropriate well geometry by Newman’s product method.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 114 Image 0001.png|'''Fig. 3.10 – Geometry of a line source in 3D Cartesian and radial coordinates.'''
File:vol1 Page 114 Image 0001.png|'''Fig. 3.10 – Geometry of a line source in 3D Cartesian and radial coordinates.'''


Line 1,474: Line 140:


File:Vol1 Page 116 Image 0001.png|'''Table 3.3'''
File:Vol1 Page 116 Image 0001.png|'''Table 3.3'''
</gallery>
</gallery><br/>Gringarten and Ramey<ref name="r13">Gringarten, A.C. and Ramey Jr., H.J. 1973. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. SPE J. 13 (5): 285-296. SPE-3818-PA. http://dx.doi.org/10.2118/3818-PA</ref> have also presented the approximating forms of the instantaneous linear sources and the time ranges for these approximations to be valid. The approximate solutions are very useful in obtaining expressions for pressure distributions at early and late times and identifying the flow regimes during the corresponding time periods. '''Table 3.4''' presents the short- and long-time approximating forms for instantaneous linear sources and their time ranges. Examples 3.6 and 3.7 present some applications of the product-solution method and the derivation of the approximate solutions for pressure distributions.<br/><br/><gallery widths="300px" heights="200px">
<br>
Gringarten and Ramey<ref name="r13" /> have also presented the approximating forms of the instantaneous linear sources and the time ranges for these approximations to be valid. The approximate solutions are very useful in obtaining expressions for pressure distributions at early and late times and identifying the flow regimes during the corresponding time periods. '''Table 3.4''' presents the short- and long-time approximating forms for instantaneous linear sources and their time ranges. Examples 3.6 and 3.7 present some applications of the product-solution method and the derivation of the approximate solutions for pressure distributions.
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 117 Image 0001.png|'''Table 3.4'''
File:Vol1 Page 117 Image 0001.png|'''Table 3.4'''
</gallery>
</gallery>
<br>


----
----


'''Example 3.6'''
'''Example 3.6'''<br/><br/>Consider transient flow toward a partially penetrating vertical fracture of vertical penetration ''h''<sub>''f''</sub> and horizontal penetration 2''x''<sub>''f''</sub> in an infinite, homogeneous, slab reservoir of uniform thickness, ''h'', and initial pressure, ''p''<sub>''i''</sub>, with impermeable top and bottom boundaries.<br/><br/>''Solution''. '''Fig. 3.11''' shows the geometry of the well reservoir system of interest. Approximate the fracture by a vertical plane of height ''h''<sub>''f''</sub> and length 2''x''<sub>''f''</sub>. The corresponding source geometry may be visualized as the intersection of an infinite plane source at ''y'' = ''y′'' in an infinite reservoir (Source I in '''Tables 3.2''' and '''3.3'''), an infinite-slab source of thickness 2''x''<sub>''f''</sub> at ''x'' = ''x′'' in an infinite reservoir (Source IV), and an infinite-slab source of thickness ''h''<sub>''p''</sub> = ''h''<sub>''f''</sub> at ''z'' = ''z''<sub>''w''</sub> in an infinite-slab reservoir of thickness ''h'' (Source VIII). Then, by Newman’s product method, the appropriate source function is given by<br/><br/>[[File:Vol1 page 0115 eq 001.png|RTENOTITLE]] [[File:Vol1 page 0116 eq 001.png|RTENOTITLE]]....................(3.207)<br/><br/>Assuming that the production is at a constant rate, [[File:Vol1 page 0116 inline 001.png|RTENOTITLE]] and using '''Eq. 3.207''' for the source function, ''S'', in '''Eq. 3.200''', we obtain<br/><br/>[[File:Vol1 page 0116 eq 002.png|RTENOTITLE]] [[File:Vol1 page 0117 eq 001.png|RTENOTITLE]]....................(3.208)<br/><br/>If the fracture penetrates the entire thickness of the reservoir (i.e., ''h''<sub>''f''</sub> = ''h'') as shown in '''Fig. 3.12''', then '''Eq. 3.208''' yields<br/><br/>[[File:Vol1 page 0117 eq 002.png|RTENOTITLE]]....................(3.209)<br/><br/>The fully penetrating fracture solution given in '''Eq. 3.209''' also could be obtained by constructing the source function as the product of an infinite plane source at ''y'' = ''y′'' in an infinite reservoir (Source I in '''Tables 3.2''' and '''3.3''') and an infinite-slab source of thickness 2''x''<sub>''f''</sub> at ''x'' = ''x′'' in an infinite reservoir (Source IV). This source function then could be used in '''Eq. 3.200'''.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
Consider transient flow toward a partially penetrating vertical fracture of vertical penetration ''h''<sub>''f''</sub> and horizontal penetration 2''x''<sub>''f''</sub> in an infinite, homogeneous, slab reservoir of uniform thickness, ''h'', and initial pressure, ''p''<sub>''i''</sub>, with impermeable top and bottom boundaries.
<br>
<br>
''Solution''. '''Fig. 3.11''' shows the geometry of the well reservoir system of interest. Approximate the fracture by a vertical plane of height ''h''<sub>''f''</sub> and length 2''x''<sub>''f''</sub>. The corresponding source geometry may be visualized as the intersection of an infinite plane source at ''y'' = ''y′'' in an infinite reservoir (Source I in '''Tables 3.2''' and '''3.3'''), an infinite-slab source of thickness 2''x''<sub>''f''</sub> at ''x'' = ''x′'' in an infinite reservoir (Source IV), and an infinite-slab source of thickness ''h''<sub>''p''</sub> = ''h''<sub>''f''</sub> at ''z'' = ''z''<sub>''w''</sub> in an infinite-slab reservoir of thickness ''h'' (Source VIII). Then, by Newman’s product method, the appropriate source function is given by
<br>
<br>
[[File:Vol1 page 0115 eq 001.png]]
[[File:Vol1 page 0116 eq 001.png]]....................(3.207)
<br>
<br>
Assuming that the production is at a constant rate, [[File:Vol1 page 0116 inline 001.png]] and using '''Eq. 3.207''' for the source function, ''S'', in '''Eq. 3.200''', we obtain
<br>
<br>
[[File:Vol1 page 0116 eq 002.png]]
[[File:Vol1 page 0117 eq 001.png]]....................(3.208)
<br>
<br>
If the fracture penetrates the entire thickness of the reservoir (i.e., ''h''<sub>''f''</sub> = ''h'') as shown in '''Fig. 3.12''', then '''Eq. 3.208''' yields
<br>
<br>
[[File:Vol1 page 0117 eq 002.png]]....................(3.209)
<br>
<br>
The fully penetrating fracture solution given in '''Eq. 3.209''' also could be obtained by constructing the source function as the product of an infinite plane source at ''y'' = ''y′'' in an infinite reservoir (Source I in '''Tables 3.2''' and '''3.3''') and an infinite-slab source of thickness 2''x''<sub>''f''</sub> at ''x'' = ''x′'' in an infinite reservoir (Source IV). This source function then could be used in '''Eq. 3.200'''.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 118 Image 0001.png|'''Fig. 3.11 – Geometry of the well/reservoir system for a partially penetrating vertical fracture in an infinite-slab reservoir with impermeable boundaries for Example 3.6.'''
File:vol1 Page 118 Image 0001.png|'''Fig. 3.11 – Geometry of the well/reservoir system for a partially penetrating vertical fracture in an infinite-slab reservoir with impermeable boundaries for Example 3.6.'''


File:vol1 Page 118 Image 0002.png|'''Fig. 3.12 – A fully penetrating vertical fracture in an infinite-slab reservoir with impermeable boundaries.'''
File:vol1 Page 118 Image 0002.png|'''Fig. 3.12 – A fully penetrating vertical fracture in an infinite-slab reservoir with impermeable boundaries.'''
</gallery>
</gallery><br/>'''Fig. 3.13''' presents an example of transient-pressure responses computed from '''Eq. 3.209'''. To obtain the results shown in '''Fig. 3.13''', numerical integration has been used to evaluate the right side of '''Eq. 3.209'''. It is also of interest to obtain an early-time approximation for the solution given in '''Eq. 3.209'''. Substituting the early-time approximating forms for the slab sources in an infinite reservoir (approximations given in '''Table 3.4''' for Source Functions IV and VIII), we obtain<br/><br/>[[File:Vol1 page 0118 eq 001.png|RTENOTITLE]]....................(3.210)<br/><br/>where<br/><br/>[[File:Vol1 page 0119 eq 001.png|RTENOTITLE]]....................(3.211)<br/><br/>and<br/><br/>[[File:Vol1 page 0119 eq 002.png|RTENOTITLE]]....................(3.212)<br/><br/><gallery widths="300px" heights="200px">
<br/>
'''Fig. 3.13''' presents an example of transient-pressure responses computed from '''Eq. 3.209'''. To obtain the results shown in '''Fig. 3.13''', numerical integration has been used to evaluate the right side of '''Eq. 3.209'''. It is also of interest to obtain an early-time approximation for the solution given in '''Eq. 3.209'''. Substituting the early-time approximating forms for the slab sources in an infinite reservoir (approximations given in '''Table 3.4''' for Source Functions IV and VIII), we obtain
<br>
<br>
[[File:Vol1 page 0118 eq 001.png]]....................(3.210)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0119 eq 001.png]]....................(3.211)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0119 eq 002.png]]....................(3.212)
<br>
<br>
<gallery widths="300px" heights="200px">
File:vol1 Page 119 Image 0001.png|'''Fig. 3.13 – Transient pressure responses of a uniform-flux fracture computed from Eq. 3.209.'''
File:vol1 Page 119 Image 0001.png|'''Fig. 3.13 – Transient pressure responses of a uniform-flux fracture computed from Eq. 3.209.'''
</gallery>
</gallery><br/>Assuming a constant production rate, [[File:Vol1 page 0116 inline 001.png|RTENOTITLE]], and substituting the source function given by '''Eq. 3.210''' in '''Eq. 3.200''', we obtain<br/><br/>[[File:Vol1 page 0119 eq 003.png|RTENOTITLE]]....................(3.213)<br/><br/>where erfc (''z'') is the complementary error function defined by<br/><br/>[[File:Vol1 page 0119 eq 004.png|RTENOTITLE]]....................(3.214)<br/>
<br/>
Assuming a constant production rate, [[File:Vol1 page 0116 inline 001.png]], and substituting the source function given by '''Eq. 3.210''' in '''Eq. 3.200''', we obtain
<br>
<br>
[[File:Vol1 page 0119 eq 003.png]]....................(3.213)
<br>
<br>
where erfc (''z'') is the complementary error function defined by
<br>
<br>
[[File:Vol1 page 0119 eq 004.png]]....................(3.214)
<br>
<br>


----
----


'''Example 3.7'''
'''Example 3.7'''<br/><br/>Consider transient flow toward a uniform-flux horizontal well of length ''L''<sub>''h''</sub> located at (''x′'', ''y′'', ''z''<sub>''w''</sub>) in a closed, homogeneous, rectangular parallelepiped of dimensions 0 ≤ ''x'' ≤ ''x''<sub>''e''</sub>, 0 ≤ ''y'' ≤ ''y''<sub>''e''</sub>, 0 ≤ ''z'' ≤ ''h'' and of initial pressure, ''p''<sub>''i''</sub>.<br/><br/>''Solution''. '''Fig. 3.14''' shows the sketch of the horizontal-well/reservoir system considered in this example. If we approximate the horizontal well by a horizontal line source of length ''L''<sub>''h''</sub>, then the resulting source/reservoir system may be visualized as the intersection of three sources: an infinite plane source at ''y'' = ''y′'' in an infinite-slab reservoir of thickness ye with impermeable boundaries (Source V in '''Tables 3.2''' and '''3.3'''), an infinite-slab source of thickness ''L''<sub>''h''</sub> at ''x'' = ''x′'' in an infinite-slab reservoir of thickness ''x''<sub>''e''</sub> with impermeable boundaries (Source VIII), and an infinite-plane source at ''z'' = ''z''<sub>''w''</sub> in an infinite-slab reservoir of thickness ''h'' with impermeable boundaries (Source V). Then, by Newman’s product method, the appropriate source function can be obtained as<br/><br/>[[File:Vol1 page 0120 eq 001.png|RTENOTITLE]]....................(3.215)<br/><br/>Assuming that the production is at a constant rate, [[File:Vol1 page 0120 inline 001.png|RTENOTITLE]], and using '''Eq. 3.215''' for the source function, ''S'', in '''Eq. 3.200''', we obtain<br/><br/>[[File:Vol1 page 0120 eq 002.png|RTENOTITLE]]....................(3.216)<br/><br/>'''Table 3.5''' presents the pressure responses for a uniform-flux horizontal well in a closed square computed from '''Eq. 3.216'''. We may obtain a short-time approximation for '''Eq. 3.216''' with the early-time approximations given in '''Table 3.4''' for Source Functions V and VIII. This yields<br/><br/>[[File:Vol1 page 0120 eq 003.png|RTENOTITLE]]....................(3.217)<br/><br/>where Ei(−''x'') is the exponential integral function defined by '''Eq. 3.90'''. '''Eq. 3.217''' indicates that the early-time flow is radial in the ''y''-''z'' plane around the axis of the horizontal well. This solution corresponds to the time period during which none of the reservoir boundaries influence the pressure response.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
Consider transient flow toward a uniform-flux horizontal well of length ''L''<sub>''h''</sub> located at (''x′'', ''y′'', ''z''<sub>''w''</sub>) in a closed, homogeneous, rectangular parallelepiped of dimensions 0 ≤ ''x'' ≤ ''x''<sub>''e''</sub>, 0 ≤ ''y'' ≤ ''y''<sub>''e''</sub>, 0 ≤ ''z'' ≤ ''h'' and of initial pressure, ''p''<sub>''i''</sub>.
<br>
<br>
''Solution''. '''Fig. 3.14''' shows the sketch of the horizontal-well/reservoir system considered in this example. If we approximate the horizontal well by a horizontal line source of length ''L''<sub>''h''</sub>, then the resulting source/reservoir system may be visualized as the intersection of three sources: an infinite plane source at ''y'' = ''y′'' in an infinite-slab reservoir of thickness ye with impermeable boundaries (Source V in '''Tables 3.2''' and '''3.3'''), an infinite-slab source of thickness ''L''<sub>''h''</sub> at ''x'' = ''x′'' in an infinite-slab reservoir of thickness ''x''<sub>''e''</sub> with impermeable boundaries (Source VIII), and an infinite-plane source at ''z'' = ''z''<sub>''w''</sub> in an infinite-slab reservoir of thickness ''h'' with impermeable boundaries (Source V). Then, by Newman’s product method, the appropriate source function can be obtained as
<br>
<br>
[[File:Vol1 page 0120 eq 001.png]]....................(3.215)
<br>
<br>
Assuming that the production is at a constant rate, [[File:Vol1 page 0120 inline 001.png]], and using '''Eq. 3.215''' for the source function, ''S'', in '''Eq. 3.200''', we obtain
<br>
<br>
[[File:Vol1 page 0120 eq 002.png]]....................(3.216)
<br>
<br>
'''Table 3.5''' presents the pressure responses for a uniform-flux horizontal well in a closed square computed from '''Eq. 3.216'''. We may obtain a short-time approximation for '''Eq. 3.216''' with the early-time approximations given in '''Table 3.4''' for Source Functions V and VIII. This yields
<br>
<br>
[[File:Vol1 page 0120 eq 003.png]]....................(3.217)
<br>
<br>
where Ei(−''x'') is the exponential integral function defined by '''Eq. 3.90'''. '''Eq. 3.217''' indicates that the early-time flow is radial in the ''y''-''z'' plane around the axis of the horizontal well. This solution corresponds to the time period during which none of the reservoir boundaries influence the pressure response.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 121 Image 0001.png|'''Fig. 3.14 – A horizontal well in a closed rectangular parallelepiped (Example 3.7).'''
File:vol1 Page 121 Image 0001.png|'''Fig. 3.14 – A horizontal well in a closed rectangular parallelepiped (Example 3.7).'''


File:Vol1 Page 122 Image 0001.png|'''Table 3.5'''
File:Vol1 Page 122 Image 0001.png|'''Table 3.5'''
</gallery>
</gallery><br/>It is also possible to obtain another approximation for '''Eq. 3.216''' that covers the intermediate time-flow behavior. If we approximate the source function in the ''x'' direction (Source Function VIII) by its early and intermediate time approximation and the source function in the ''y'' direction (Source Function V) by its early time approximation given in '''Table 3.4''', we obtain<br/><br/>[[File:Vol1 page 0121 eq 001.png|RTENOTITLE]]....................(3.218)<br/><br/>This approximation should correspond to the time period during which the influence of the top and/or bottom boundaries may be evident but the lateral boundaries in the ''x'' and ''y'' directions do not have an influence on the pressure response. This solution also could have been obtained by assuming a laterally infinite reservoir. In this case, the source function would have been constructed as the product of three source functions: an infinite-plane source at ''y'' = ''y′'' in an infinite reservoir (Source I in '''Tables 3.2''' and '''3.3'''), an infinite-slab source of thickness ''L''<sub>''h''</sub> at ''x'' = ''x′'' in an infinite reservoir (Source IV), and an infinite-plane source at ''z'' = ''z''<sub>''w''</sub> in an infinite-slab reservoir of thickness ''h'' with impermeable boundaries (Source V).
<br>
It is also possible to obtain another approximation for '''Eq. 3.216''' that covers the intermediate time-flow behavior. If we approximate the source function in the ''x'' direction (Source Function VIII) by its early and intermediate time approximation and the source function in the ''y'' direction (Source Function V) by its early time approximation given in '''Table 3.4''', we obtain
<br>
<br>
[[File:Vol1 page 0121 eq 001.png]]....................(3.218)
<br>
<br>
This approximation should correspond to the time period during which the influence of the top and/or bottom boundaries may be evident but the lateral boundaries in the ''x'' and ''y'' directions do not have an influence on the pressure response. This solution also could have been obtained by assuming a laterally infinite reservoir. In this case, the source function would have been constructed as the product of three source functions: an infinite-plane source at ''y'' = ''y′'' in an infinite reservoir (Source I in '''Tables 3.2''' and '''3.3'''), an infinite-slab source of thickness ''L''<sub>''h''</sub> at ''x'' = ''x′'' in an infinite reservoir (Source IV), and an infinite-plane source at ''z'' = ''z''<sub>''w''</sub> in an infinite-slab reservoir of thickness ''h'' with impermeable boundaries (Source V).


----
----
Line 1,603: Line 166:
=== The Use of Source Functions in the Laplace Domain To Solve Unsteady-Flow Problems ===
=== The Use of Source Functions in the Laplace Domain To Solve Unsteady-Flow Problems ===


There are many advantages of developing transient-flow solutions in the Laplace transform domain. For example, in the Laplace transform domain, Duhamel’s theorem<ref name="r23" /> provides a convenient means of developing transient-flow solutions for variable-rate production problems using the solutions for the corresponding constant-rate production problem. Duhamel’s theorem states that if Δ''p'' and Δ''p''<sub>''c''</sub> denote the pressure drawdown corresponding to the variable production rate, ''q''(''t''), and the constant production rate, ''q''<sub>''c''</sub>, respectively, then
There are many advantages of developing transient-flow solutions in the Laplace transform domain. For example, in the Laplace transform domain, Duhamel’s theorem<ref name="r23">Duhamel, J.M.C. 1833. Mémoire sur la méthode générale relative au mouvement de la chaleur dans les corps solides polongé dans les milieux dont la température varie avec le temps. Journal de l’École Polytechnique 14 (22): 20-66.</ref> provides a convenient means of developing transient-flow solutions for variable-rate production problems using the solutions for the corresponding constant-rate production problem. Duhamel’s theorem states that if Δ''p'' and Δ''p''<sub>''c''</sub> denote the pressure drawdown corresponding to the variable production rate, ''q''(''t''), and the constant production rate, ''q''<sub>''c''</sub>, respectively, then<br/><br/>[[File:Vol1 page 0121 eq 002.png|RTENOTITLE]]....................(3.219)<br/><br/>Applying the Laplace transform converts the convolution integral in '''Eq. 3.219''' to an algebraic expression, and Duhamel’s theorem is given in the Laplace transform domain as<br/><br/>[[File:Vol1 page 0121 eq 003.png|RTENOTITLE]]....................(3.220)<br/><br/>The simplicity of the expression given in '''Eq. 3.220''' explains our interest in obtaining transient-flow solutions in the Laplace transform domain.<br/><br/>Another example to explain the convenience of the Laplace domain solutions is for the naturally fractured reservoirs. Common transient-flow models of naturally fractured reservoirs lead to the following differential equation in radial coordinates in the Laplace transform domain: <ref name="r2">Raghavan, R. 1993. Well Test Analysis, 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.</ref><br/><br/>[[File:Vol1 page 0122 eq 001.png|RTENOTITLE]]....................(3.221)<br/><br/>where the subscript ''f'' indicates the fracture property, and ''t''<sub>''D''</sub> and ''r''<sub>''D''</sub> are the dimensionless time and distance (as defined in '''Eqs. 3.230''' and '''3.234''').<br/><br/>The naturally fractured reservoir function, ''f'' (''s''), is a function of matrix and fracture properties and depends on the model chosen to represent the naturally fractured reservoir.<ref name="r2">Raghavan, R. 1993. Well Test Analysis, 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.</ref> The corresponding differential equation for a homogeneous reservoir is obtained by setting ''f'' (''s'') = 1 and is given by<br/><br/>[[File:Vol1 page 0123 eq 001.png|RTENOTITLE]]....................(3.222)<br/><br/>The general solutions for '''Eqs. 3.221''' and '''3.222''' are given, respectively, by<br/><br/>[[File:Vol1 page 0123 eq 002.png|RTENOTITLE]]....................(3.223)<br/><br/>and<br/><br/>[[File:Vol1 page 0123 eq 003.png|RTENOTITLE]]....................(3.224)<br/><br/>To obtain a solution for constant-rate production from an infinite reservoir, for example, the following boundary conditions are imposed:<br/><br/>[[File:Vol1 page 0123 eq 004.png|RTENOTITLE]]....................(3.225)<br/><br/>and<br/><br/>[[File:Vol1 page 0123 eq 005.png|RTENOTITLE]]....................(3.226)<br/><br/>Then, it may be shown that<br/><br/>[[File:Vol1 page 0123 eq 006.png|RTENOTITLE]]....................(3.227)<br/><br/>where the right side of '''Eq. 3.227''' indicates the substitution of ''sf'' (''s'') for ''s'' in ''s''Δ''p''(''s''). This discussion demonstrates that it is possible to derive transient-flow solutions for naturally fractured reservoirs by following the same lines as those for the homogeneous reservoirs. Furthermore, if the solution for the corresponding homogeneous reservoir system is known in the Laplace transform domain, then the solution for the naturally fractured reservoir problem may be directly obtained from '''Eq. 3.227'''.<br/><br/>Obtaining the Laplace transforms of the Green’s and source function solutions developed in the time domain with the methods explained in '''Secs. 3.4.2''' and '''3.4.3''' usually poses a difficult problem. The problems arise mainly because of the use of the product method solution. For a specific class of functions, Chen ''et al''.<ref name="r24">Chen, H.Y., Poston, S.W., and Raghavan, R. 1991. An Application of the Product Solution Principle for Instantaneous Source and Green's Functions. SPE Form Eval 6 (2): 161-167. SPE-20801-PA. http://dx.doi.org/10.2118/20801-PA</ref> presented a technique that may be used to apply the Laplace transform to the product-solution technique. For a more general procedure to develop source function solutions in the Laplace transform domain, however, the product solution technique should be avoided.<ref name="r20">Raghavan, R. 1993. The Method of Sources and Sinks. In Well Test Analysis, Chap. 10, 336-435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.</ref><br/><br/>Ozkan and Raghavan<ref name="r18">Ozkan, E. and Raghavan, R. 1991b. New Solutions for Well-Test-Analysis Problems: Part 2—Computational Considerations and Applications. SPE Form Eval 6 (3): 369–378. SPE-18616-PA. http://dx.doi.org/10.2118/18616-PA</ref><ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref> have shown that it is more convenient to develop source-function solutions in the Laplace transform domain if the point-source solution is used as a building block. Then, other source geometries may be obtained by the superposition (integration) of the point sources along the length, surface, or volume of the source.<br/><br/>'''''Point-Source Solution in the Laplace Domain.''''' Consider the flow of a slightly compressible fluid in an infinite, naturally fractured reservoir. We can use the double-porosity model suggested by Barenblatt ''et al''.<ref name="r25">Barenblatt, G.I., Zheltov, I.P., and Kochina, I.N. 1960. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24 (5): 1286–1303. http://dx.doi.org/10.1016/0021-8928(60)90107-6</ref> and Warren and Root<ref name="r26">Warren, J.E. and Root, P.J. 1963. The Behavior of Naturally Fractured Reservoirs. SPE J. 3 (3): 245–255. SPE-426-PA. http://dx.doi.org/10.2118/426-PA</ref> to develop the governing flow equations for naturally fractured reservoirs. The results, however, will be applicable to the model suggested by Kazemi<ref name="r27">Kazemi, H. 1969. Pressure Transient Analysis of Naturally Fractured Reservoir with Uniform Fracture Distribution. SPE J. 9 (4): 451–462. SPE-2156-PA. http://dx.doi.org/10.2118/2156-PA</ref> and de Swaan-O<ref name="r28">de Swaan O., A. 1976. Analytical Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing. SPE J. 16 (3): 117–122. SPE-5346-PA. http://dx.doi.org/10.2118/5346-PA</ref> with a simple modification.<br/><br/>Flow around a point source in an infinite porous medium may be expressed conveniently in spherical coordinates. The differential equations governing flow in a naturally fractured reservoir are given in spherical coordinates by<br/><br/>[[File:Vol1 page 0124 eq 001.png|RTENOTITLE]]....................(3.228)<br/><br/>and<br/><br/>[[File:Vol1 page 0124 eq 002.png|RTENOTITLE]]....................(3.229)<br/><br/>In '''Eqs. 3.228''' and '''3.229''', subscripts f and m indicate the property of the fracture and matrix systems, respectively. Initial pressure, ''p''<sub>''i''</sub>, is assumed to be uniform in the entire system; that is, ''p''<sub>''fi''</sub> = ''p''<sub>''mi''</sub> = ''p''<sub>''i''</sub>. The dimensionless time, ''t''<sub>''D''</sub>, is defined by<br/><br/>[[File:Vol1 page 0124 eq 003.png|RTENOTITLE]]....................(3.230)<br/><br/>where ''ℓ'' is a characteristic length in the system, and<br/><br/>[[File:Vol1 page 0124 eq 004.png|RTENOTITLE]]....................(3.231)<br/><br/>The definitions of the other variables used in '''Eqs. 3.228''' and '''3.229''' are<br/><br/>[[File:Vol1 page 0124 eq 005.png|RTENOTITLE]]....................(3.232)<br/><br/>[[File:Vol1 page 0124 eq 006.png|RTENOTITLE]]....................(3.233)<br/><br/>and<br/><br/>[[File:Vol1 page 0124 eq 007.png|RTENOTITLE]]....................(3.234)<br/><br/>where<br/><br/>[[File:Vol1 page 0125 eq 001.png|RTENOTITLE]]....................(3.235)<br/><br/>The initial and outer-boundary conditions are given, respectively, by<br/><br/>[[File:Vol1 page 0125 eq 002.png|RTENOTITLE]]....................(3.236)<br/><br/>and<br/><br/>[[File:Vol1 page 0125 eq 003.png|RTENOTITLE]]....................(3.237)<br/><br/>The inner-boundary condition corresponding to the instantaneous withdrawal of an amount of fluid, [[File:Vol1 page 0102 inline 001.png|RTENOTITLE]], at ''t'' = 0 from a point source is obtained by considering the mass balance on a small sphere. If we require that at any time ''t'' = ''T'' > 0, the sum of the flux through the surface of a small sphere around the source location must equal the volume of the fluid, [[File:Vol1 page 0102 inline 001.png|RTENOTITLE]], instantaneously withdrawn from the sphere at ''t'' = 0, we can write<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0125 eq 004.png|RTENOTITLE]]....................(3.238)<br/><br/>Although the withdrawal of fluids from the sphere is instantaneous, the resulting flow in the porous medium, and the flux across the surface of the sphere, is continuous. Therefore, if ''q'' represents the total flux across the surface of the small sphere during the time interval 0 ≤ ''t'' ≤ ''T'', then the mass balance requires that the cumulative production (flux across the surface of the small sphere) at time ''T'' be equal to the instantaneous withdrawal volume of fluid from the sphere. That is,<br/><br/>[[File:Vol1 page 0125 eq 005.png|RTENOTITLE]]....................(3.239)<br/><br/>For the condition expressed in '''Eq. 3.239''' to hold for every ''T'' ≥ 0, we must have<br/><br/>[[File:Vol1 page 0125 eq 006.png|RTENOTITLE]]....................(3.240)<br/><br/>where ''δ''(''t'') is the Dirac delta function satisfying the properties expressed by '''Eqs. 3.185''' and '''3.186'''.<br/><br/>Using the results given by '''Eqs. 3.239''' and '''3.240''' in '''Eq. 3.238''', we obtain<br/><br/>[[File:Vol1 page 0125 eq 007.png|RTENOTITLE]]....................(3.241)<br/><br/>The Laplace transform of '''Eqs. 3.228''', '''3.229''', '''3.237''', and '''3.241''' yields<br/><br/>[[File:Vol1 page 0125 eq 008.png|RTENOTITLE]]....................(3.242)<br/><br/>where<br/><br/>[[File:Vol1 page 0126 eq 001.png|RTENOTITLE]]....................(3.243)<br/><br/>[[File:Vol1 page 0126 eq 002.png|RTENOTITLE]]....................(3.244)<br/><br/>and<br/><br/>[[File:Vol1 page 0126 eq 003.png|RTENOTITLE]]....................(3.245)<br/><br/>In deriving these results, we have used the initial condition given by '''Eq. 3.236''' and noted that<br/><br/>[[File:Vol1 page 0126 eq 004.png|RTENOTITLE]]....................(3.246)<br/><br/>In '''Eq. 3.245''', the term [[File:Vol1 page 0126 inline 001.png|RTENOTITLE]] represents the strength of the source for the naturally fractured porous medium.<br/><br/>The solution of '''Eqs. 3.242''', '''3.244''', and '''3.245''' yields the following solution for the pressure distribution in the reservoir, except at the source location (the origin), because of an instantaneous point source of strength [[File:Vol1 page 0126 inline 001.png|RTENOTITLE]] acting at ''t'' = 0:<br/><br/>[[File:Vol1 page 0126 eq 005.png|RTENOTITLE]]....................(3.247)<br/><br/>If the source is located at ''x′''<sub>''D''</sub>, ''y′''<sub>''D''</sub>, ''z′''<sub>''D''</sub>, then, by translation, we can write<br/><br/>[[File:Vol1 page 0126 eq 006.png|RTENOTITLE]]....................(3.248)<br/><br/>where<br/><br/>[[File:Vol1 page 0126 eq 007.png|RTENOTITLE]]....................(3.249)<br/><br/>and<br/><br/>[[File:Vol1 page 0126 eq 008.png|RTENOTITLE]]....................(3.250)<br/><br/>The instantaneous point-source solution for the model suggested by Barenblatt ''et al''.<ref name="r25">Barenblatt, G.I., Zheltov, I.P., and Kochina, I.N. 1960. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24 (5): 1286–1303. http://dx.doi.org/10.1016/0021-8928(60)90107-6</ref> and Warren and Root<ref name="r26">Warren, J.E. and Root, P.J. 1963. The Behavior of Naturally Fractured Reservoirs. SPE J. 3 (3): 245–255. SPE-426-PA. http://dx.doi.org/10.2118/426-PA</ref> can also be used for the model suggested by Kazemi<ref name="r27">Kazemi, H. 1969. Pressure Transient Analysis of Naturally Fractured Reservoir with Uniform Fracture Distribution. SPE J. 9 (4): 451–462. SPE-2156-PA. http://dx.doi.org/10.2118/2156-PA</ref> and de Swaan-O,<ref name="r28">de Swaan O., A. 1976. Analytical Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing. SPE J. 16 (3): 117–122. SPE-5346-PA. http://dx.doi.org/10.2118/5346-PA</ref> provided that the appropriate ''f''(''s'') function is invoked. To obtain the solution for a homogeneous reservoir, ''f''(''s'') should be set to unity, ''V''<sub>''f''</sub> = 1, and ''V''<sub>''m''</sub> = 0.<br/><br/>If we consider continuous withdrawal of fluids from the point source, then, by the principle of superposition, we should have<br/><br/>[[File:Vol1 page 0128 eq 001.png|RTENOTITLE]]....................(3.251)<br/><br/>The Laplace transform of '''Eq. 3.251''' yields the following continuous point-source solution in an infinite reservoir:<br/><br/>[[File:Vol1 page 0128 eq 002.png|RTENOTITLE]]....................(3.252)<br/><br/>where we have substituted '''Eq. 3.249''' for ''S'', dropped the subscript ''f'', and defined<br/><br/>[[File:Vol1 page 0128 eq 003.png|RTENOTITLE]]....................(3.253)<br/><br/>'''''Line-, Surface-, and Volumetric-Source Solution in the Laplace Domain.''''' The point-source solution in the Laplace domain may be used to obtain the source solutions for different source geometries. If we define<br/><br/>[[File:Vol1 page 0129 eq 001.png|RTENOTITLE]]....................(3.254)<br/><br/>where Δ''p''<sub>''p''</sub> represents the appropriate point-source solution, then, by the application of the superposition principle, the solution for the withdrawal of fluids from a line, surface, or volume, Γ<sub>''w''</sub>, is given by<br/><br/>[[File:Vol1 page 0129 eq 002.png|RTENOTITLE]]....................(3.255)<br/><br/>If we assume a uniform-flux distribution in time and over the length, surface, or volume of the source, then<br/><br/>[[File:Vol1 page 0129 eq 003.png|RTENOTITLE]]....................(3.256)<br/><br/>The following presentation of the source function approach in the Laplace domain assumes that the flux distribution is uniform, and [[File:Vol1 page 0129 inline 001.png|RTENOTITLE]]. Also, the constant production rate from the length, area, or the volume of the source, Γ<sub>''w''</sub>, is denoted by q so that [[File:Vol1 page 0129 inline 002.png|RTENOTITLE]].<br/><br/>Only sources in infinite reservoirs have been considered so far. These solutions may be easily extended to bounded reservoirs. The following sections present some useful solutions for transient-flow problems in bounded porous media. The first group of solutions is for laterally infinite reservoirs bounded by parallel planes in the vertical direction (infinite-slab reservoirs). The second and third groups comprise the solutions for cylindrical and rectangular reservoirs, respectively.<br/><br/>'''''Solutions for Infinite-Slab Reservoirs.''''' In this section, we consider one of the most common reservoir geometries used in pressure-transient analysis of wells in porous media. It is assumed that the lateral boundaries of the reservoir are far enough not to influence the pressure response during the time period of interest. The top and bottom boundaries of the reservoir at ''z'' = 0 and ''z'' = ''h'' are parallel planes and may be of impermeable, constant pressure, or mixed type. '''Table 3.6''' presents the solutions for the most common well geometries (point-source, vertical, fractured, and horizontal wells) in infinite-slab reservoirs. Next, we briefly discuss the derivation of these solutions.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0121 eq 002.png]]....................(3.219)
<br>
<br>
Applying the Laplace transform converts the convolution integral in '''Eq. 3.219''' to an algebraic expression, and Duhamel’s theorem is given in the Laplace transform domain as
<br>
<br>
[[File:Vol1 page 0121 eq 003.png]]....................(3.220)
<br>
<br>
The simplicity of the expression given in '''Eq. 3.220''' explains our interest in obtaining transient-flow solutions in the Laplace transform domain.
<br>
<br>
Another example to explain the convenience of the Laplace domain solutions is for the naturally fractured reservoirs. Common transient-flow models of naturally fractured reservoirs lead to the following differential equation in radial coordinates in the Laplace transform domain: <ref name="r2" />
<br>
<br>
[[File:Vol1 page 0122 eq 001.png]]....................(3.221)
<br>
<br>
where the subscript ''f'' indicates the fracture property, and ''t''<sub>''D''</sub> and ''r''<sub>''D''</sub> are the dimensionless time and distance (as defined in '''Eqs. 3.230''' and '''3.234''').
<br>
<br>
The naturally fractured reservoir function, ''f'' (''s''), is a function of matrix and fracture properties and depends on the model chosen to represent the naturally fractured reservoir.<ref name="r2" /> The corresponding differential equation for a homogeneous reservoir is obtained by setting ''f'' (''s'') = 1 and is given by
<br>
<br>
[[File:Vol1 page 0123 eq 001.png]]....................(3.222)
<br>
<br>
The general solutions for '''Eqs. 3.221''' and '''3.222''' are given, respectively, by
<br>
<br>
[[File:Vol1 page 0123 eq 002.png]]....................(3.223)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0123 eq 003.png]]....................(3.224)
<br>
<br>
To obtain a solution for constant-rate production from an infinite reservoir, for example, the following boundary conditions are imposed:
<br>
<br>
[[File:Vol1 page 0123 eq 004.png]]....................(3.225)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0123 eq 005.png]]....................(3.226)
<br>
<br>
Then, it may be shown that
<br>
<br>
[[File:Vol1 page 0123 eq 006.png]]....................(3.227)
<br>
<br>
where the right side of '''Eq. 3.227''' indicates the substitution of ''sf'' (''s'') for ''s'' in ''s''Δ''p''(''s''). This discussion demonstrates that it is possible to derive transient-flow solutions for naturally fractured reservoirs by following the same lines as those for the homogeneous reservoirs. Furthermore, if the solution for the corresponding homogeneous reservoir system is known in the Laplace transform domain, then the solution for the naturally fractured reservoir problem may be directly obtained from '''Eq. 3.227'''.
<br>
<br>
Obtaining the Laplace transforms of the Green’s and source function solutions developed in the time domain with the methods explained in '''Secs. 3.4.2''' and '''3.4.3''' usually poses a difficult problem. The problems arise mainly because of the use of the product method solution. For a specific class of functions, Chen ''et al''.<ref name="r24" /> presented a technique that may be used to apply the Laplace transform to the product-solution technique. For a more general procedure to develop source function solutions in the Laplace transform domain, however, the product solution technique should be avoided.<ref name="r20" />
<br>
<br>
Ozkan and Raghavan<ref name="r18" /><ref name="r19" /> have shown that it is more convenient to develop source-function solutions in the Laplace transform domain if the point-source solution is used as a building block. Then, other source geometries may be obtained by the superposition (integration) of the point sources along the length, surface, or volume of the source.
<br>
<br>
'''''Point-Source Solution in the Laplace Domain.''''' Consider the flow of a slightly compressible fluid in an infinite, naturally fractured reservoir. We can use the double-porosity model suggested by Barenblatt ''et al''.<ref name="r25" /> and Warren and Root<ref name="r26" /> to develop the governing flow equations for naturally fractured reservoirs. The results, however, will be applicable to the model suggested by Kazemi<ref name="r27" /> and de Swaan-O<ref name="r28" /> with a simple modification.
<br>
<br>
Flow around a point source in an infinite porous medium may be expressed conveniently in spherical coordinates. The differential equations governing flow in a naturally fractured reservoir are given in spherical coordinates by
<br>
<br>
[[File:Vol1 page 0124 eq 001.png]]....................(3.228)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0124 eq 002.png]]....................(3.229)
<br>
<br>
In '''Eqs. 3.228''' and '''3.229''', subscripts f and m indicate the property of the fracture and matrix systems, respectively. Initial pressure, ''p''<sub>''i''</sub>, is assumed to be uniform in the entire system; that is, ''p''<sub>''fi''</sub> = ''p''<sub>''mi''</sub> = ''p''<sub>''i''</sub>. The dimensionless time, ''t''<sub>''D''</sub>, is defined by
<br>
<br>
[[File:Vol1 page 0124 eq 003.png]]....................(3.230)
<br>
<br>
where '''' is a characteristic length in the system, and
<br>
<br>
[[File:Vol1 page 0124 eq 004.png]]....................(3.231)
<br>
<br>
The definitions of the other variables used in '''Eqs. 3.228''' and '''3.229''' are
<br>
<br>
[[File:Vol1 page 0124 eq 005.png]]....................(3.232)
<br>
<br>
[[File:Vol1 page 0124 eq 006.png]]....................(3.233)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0124 eq 007.png]]....................(3.234)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0125 eq 001.png]]....................(3.235)
<br>
<br>
The initial and outer-boundary conditions are given, respectively, by
<br>
<br>
[[File:Vol1 page 0125 eq 002.png]]....................(3.236)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0125 eq 003.png]]....................(3.237)
<br>
<br>
The inner-boundary condition corresponding to the instantaneous withdrawal of an amount of fluid, [[File:Vol1 page 0102 inline 001.png]], at ''t'' = 0 from a point source is obtained by considering the mass balance on a small sphere. If we require that at any time ''t'' = ''T'' > 0, the sum of the flux through the surface of a small sphere around the source location must equal the volume of the fluid, [[File:Vol1 page 0102 inline 001.png]], instantaneously withdrawn from the sphere at ''t'' = 0, we can write<ref name="r29" />
<br>
<br>
[[File:Vol1 page 0125 eq 004.png]]....................(3.238)
<br>
<br>
Although the withdrawal of fluids from the sphere is instantaneous, the resulting flow in the porous medium, and the flux across the surface of the sphere, is continuous. Therefore, if ''q'' represents the total flux across the surface of the small sphere during the time interval 0 ≤ ''t'' ≤ ''T'', then the mass balance requires that the cumulative production (flux across the surface of the small sphere) at time ''T'' be equal to the instantaneous withdrawal volume of fluid from the sphere. That is,
<br>
<br>
[[File:Vol1 page 0125 eq 005.png]]....................(3.239)
<br>
<br>
For the condition expressed in '''Eq. 3.239''' to hold for every ''T'' ≥ 0, we must have
<br>
<br>
[[File:Vol1 page 0125 eq 006.png]]....................(3.240)
<br>
<br>
where ''δ''(''t'') is the Dirac delta function satisfying the properties expressed by '''Eqs. 3.185''' and '''3.186'''.
<br>
<br>
Using the results given by '''Eqs. 3.239''' and '''3.240''' in '''Eq. 3.238''', we obtain
<br>
<br>
[[File:Vol1 page 0125 eq 007.png]]....................(3.241)
<br>
<br>
The Laplace transform of '''Eqs. 3.228''', '''3.229''', '''3.237''', and '''3.241''' yields
<br>
<br>
[[File:Vol1 page 0125 eq 008.png]]....................(3.242)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0126 eq 001.png]]....................(3.243)
<br>
<br>
[[File:Vol1 page 0126 eq 002.png]]....................(3.244)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0126 eq 003.png]]....................(3.245)
<br>
<br>
In deriving these results, we have used the initial condition given by '''Eq. 3.236''' and noted that
<br>
<br>
[[File:Vol1 page 0126 eq 004.png]]....................(3.246)
<br>
<br>
In '''Eq. 3.245''', the term [[File:Vol1 page 0126 inline 001.png]] represents the strength of the source for the naturally fractured porous medium.
<br>
<br>
The solution of '''Eqs. 3.242''', '''3.244''', and '''3.245''' yields the following solution for the pressure distribution in the reservoir, except at the source location (the origin), because of an instantaneous point source of strength [[File:Vol1 page 0126 inline 001.png]] acting at ''t'' = 0:
<br>
<br>
[[File:Vol1 page 0126 eq 005.png]]....................(3.247)
<br>
<br>
If the source is located at ''x′''<sub>''D''</sub>, ''y′''<sub>''D''</sub>, ''z′''<sub>''D''</sub>, then, by translation, we can write
<br>
<br>
[[File:Vol1 page 0126 eq 006.png]]....................(3.248)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0126 eq 007.png]]....................(3.249)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0126 eq 008.png]]....................(3.250)
<br>
<br>
The instantaneous point-source solution for the model suggested by Barenblatt ''et al''.<ref name="r25" /> and Warren and Root<ref name="r26" /> can also be used for the model suggested by Kazemi<ref name="r27" /> and de Swaan-O,<ref name="r28" /> provided that the appropriate ''f''(''s'') function is invoked. To obtain the solution for a homogeneous reservoir, ''f''(''s'') should be set to unity, ''V''<sub>''f''</sub> = 1, and ''V''<sub>''m''</sub> = 0.
<br>
<br>
If we consider continuous withdrawal of fluids from the point source, then, by the principle of superposition, we should have
<br>
<br>
[[File:Vol1 page 0128 eq 001.png]]....................(3.251)
<br>
<br>
The Laplace transform of '''Eq. 3.251''' yields the following continuous point-source solution in an infinite reservoir:
<br>
<br>
[[File:Vol1 page 0128 eq 002.png]]....................(3.252)
<br>
<br>
where we have substituted '''Eq. 3.249''' for ''S'', dropped the subscript ''f'', and defined
<br>
<br>
[[File:Vol1 page 0128 eq 003.png]]....................(3.253)
<br>
<br>
'''''Line-, Surface-, and Volumetric-Source Solution in the Laplace Domain.''''' The point-source solution in the Laplace domain may be used to obtain the source solutions for different source geometries. If we define
<br>
<br>
[[File:Vol1 page 0129 eq 001.png]]....................(3.254)
<br>
<br>
where Δ''p''<sub>''p''</sub> represents the appropriate point-source solution, then, by the application of the superposition principle, the solution for the withdrawal of fluids from a line, surface, or volume, Γ<sub>''w''</sub>, is given by
<br>
<br>
[[File:Vol1 page 0129 eq 002.png]]....................(3.255)
<br>
<br>
If we assume a uniform-flux distribution in time and over the length, surface, or volume of the source, then
<br>
<br>
[[File:Vol1 page 0129 eq 003.png]]....................(3.256)
<br>
<br>
The following presentation of the source function approach in the Laplace domain assumes that the flux distribution is uniform, and [[File:Vol1 page 0129 inline 001.png]]. Also, the constant production rate from the length, area, or the volume of the source, Γ<sub>''w''</sub>, is denoted by q so that [[File:Vol1 page 0129 inline 002.png]].
<br>
<br>
Only sources in infinite reservoirs have been considered so far. These solutions may be easily extended to bounded reservoirs. The following sections present some useful solutions for transient-flow problems in bounded porous media. The first group of solutions is for laterally infinite reservoirs bounded by parallel planes in the vertical direction (infinite-slab reservoirs). The second and third groups comprise the solutions for cylindrical and rectangular reservoirs, respectively.
<br>
<br>
'''''Solutions for Infinite-Slab Reservoirs.''''' In this section, we consider one of the most common reservoir geometries used in pressure-transient analysis of wells in porous media. It is assumed that the lateral boundaries of the reservoir are far enough not to influence the pressure response during the time period of interest. The top and bottom boundaries of the reservoir at ''z'' = 0 and ''z'' = ''h'' are parallel planes and may be of impermeable, constant pressure, or mixed type. '''Table 3.6''' presents the solutions for the most common well geometries (point-source, vertical, fractured, and horizontal wells) in infinite-slab reservoirs. Next, we briefly discuss the derivation of these solutions.
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 127 Image 0001.png|'''Table 3.6'''
File:Vol1 Page 127 Image 0001.png|'''Table 3.6'''


File:Vol1 Page 128 Image 0001.png|'''Table 3.6 (continued)'''
File:Vol1 Page 128 Image 0001.png|'''Table 3.6 (continued)'''
</gallery>
</gallery><br/>Consider a point source in an infinite-slab reservoir with impermeable boundaries at the bottom, ''z'' = 0, and the top, ''z'' = ''h''. To obtain the point-source solution for this case, we use the point-source solution in an infinite reservoir given by '''Eq. 3.252''' with the method of images. The result is given by<br/><br/>[[File:Vol1 page 0129 eq 004.png|RTENOTITLE]]....................(3.257)<br/><br/>where<br/><br/>[[File:Vol1 page 0129 eq 005.png|RTENOTITLE]]....................(3.258)<br/><br/>[[File:Vol1 page 0130 eq 001.png|RTENOTITLE]]....................(3.259)<br/><br/>[[File:Vol1 page 0130 eq 002.png|RTENOTITLE]]....................(3.260)<br/><br/>and<br/><br/>[[File:Vol1 page 0130 eq 003.png|RTENOTITLE]]....................(3.261)<br/><br/>The solution given in '''Eq. 3.257''' is not very convenient for computational purposes. To obtain a computationally convenient form of the solution, we use the summation formula given by<ref name="r17">Ozkan, E. and Raghavan, R. 1991a. New Solutions for Well-Test-Analysis Problems: Part 1—Analytical Considerations. SPE Form Eval 6 (3): 359–368. SPE-18615-PA. http://dx.doi.org/10.2118/18615-PA</ref><ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0130 eq 004.png|RTENOTITLE]]....................(3.262)<br/><br/>and recast '''Eq. 3.257''' as<br/><br/>[[File:Vol1 page 0130 eq 005.png|RTENOTITLE]]....................(3.263)<br/><br/>The point-source solutions for infinite-slab reservoirs with constant pressure and mixed boundaries at the top and bottom are obtained in a similar manner<ref name="r17">Ozkan, E. and Raghavan, R. 1991a. New Solutions for Well-Test-Analysis Problems: Part 1—Analytical Considerations. SPE Form Eval 6 (3): 359–368. SPE-18615-PA. http://dx.doi.org/10.2118/18615-PA</ref> and are given in '''Table 3.6'''. The point-source solutions can be used with '''Eqs. 3.254''' and '''3.256''' to generate the solutions for the other well geometries given in '''Table 3.6'''. For example, to generate the solution for a partially penetrating vertical line-source well of length hw in an infinite-slab reservoir with impermeable slab boundaries, we integrate the right side of '''Eq. 3.263''' from ''z''<sub>''w''</sub> ''h''<sub>''w''</sub> / 2 to ''z''<sub>''w''</sub> + ''h''<sub>''w''</sub> / 2 with respect to ''z′'', where ''z''<sub>''w''</sub> is the vertical coordinate of the midpoint of the open interval. If ''h''<sub>''w''</sub> = ''h'' (i.e., the well penetrates the entire thickness of the slab reservoir), then this procedure yields the solution for a fully penetrating vertical line-source well. The solution for a partially penetrating fracture of height ''h''<sub>''f''</sub> and half-length ''x''<sub>''f''</sub> is obtained if the point-source solution is integrated once with respect to ''z′'' from ''z''<sub>''w''</sub> ''h''<sub>''f''</sub> / 2 to ''z''<sub>''w''</sub> + ''h''<sub>''f''</sub> / 2 and then with respect to ''x′'' from ''x''<sub>''w''</sub> ''x''<sub>''f''</sub> to ''x''<sub>''w''</sub> + ''x''<sub>''f''</sub>, where ''x''<sub>''w''</sub> and ''z''<sub>''w''</sub> are the coordinates of the midpoint of the fracture. Similarly, the solution for a horizontal-line source well of length ''L''<sub>''h''</sub> is obtained by integrating the point-source solution with respect to ''x′'' from ''x''<sub>''w''</sub> ''L''<sub>''h''</sub> / 2 to ''x''<sub>''w''</sub> + ''L''<sub>''h''</sub> / 2, where ''x''<sub>''w''</sub> is the ''x''-coordinate of the midpoint of the horizontal well.<br/><br/>'''Solutions for Cylindrical Reservoir.''' Solutions for cylindrical reservoirs may also be obtained by starting from the point-source solution in the Laplace transform domain. The Laplace domain solution for a point source located at ''r′''<sub>''D''</sub>, ''θ′'', ''z′''<sub>''D''</sub> should satisfy the following diffusion equation in cylindrical coordinates.<ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref><br/><br/>[[File:Vol1 page 0130 eq 006.png|RTENOTITLE]]....................(3.264)<br/><br/>where<br/><br/>[[File:Vol1 page 0130 eq 007.png|RTENOTITLE]]....................(3.265)<br/><br/>The point-source solution is also required to satisfy the following flux condition at the source location (''r''<sub>''D''</sub> →0+, ''θ'' = ''θ''′, ''z''<sub>''D''</sub> = ''z′''<sub>''D''</sub>):<br/><br/>[[File:Vol1 page 0131 eq 001.png|RTENOTITLE]]....................(3.266)<br/><br/>Assuming that the reservoir is bounded by a cylindrical surface at ''r''<sub>''D''</sub> = ''r''<sub>''eD''</sub> and by the parallel planes at ''z''<sub>''D''</sub> = 0 and ''h''<sub>''D''</sub>, we should also impose the appropriate physical conditions at these boundaries.<br/><br/>We seek a point-source solution for a cylindrical reservoir in the following form:<br/><br/>[[File:Vol1 page 0131 eq 002.png|RTENOTITLE]]....................(3.267)<br/><br/>In '''Eq. 3.267''', [[File:Vol1 page 0131 inline 001.png|RTENOTITLE]] is a solution of '''Eq. 3.264''' that satisfies '''Eq. 3.266''' and the boundary conditions at ''z''<sub>''D''</sub> = 0 and ''h''<sub>''D''</sub>. [[File:Vol1 page 0131 inline 002.png|RTENOTITLE]] may be chosen as one of the point-source solutions in an infinite-slab reservoir given in '''Table 3.6''', depending on the conditions imposed at the boundaries at ''z''<sub>''D''</sub> = 0 and ''h''<sub>''D''</sub>. If [[File:Vol1 page 0131 inline 003.png|RTENOTITLE]] is chosen such that it satisfies the boundary conditions at ''z''<sub>''D''</sub> = 0 and ''h''<sub>''D''</sub>, its contribution to the flux vanishes at the source location, and [[File:Vol1 page 0131 inline 003.png|RTENOTITLE]] + [[File:Vol1 page 0131 inline 002.png|RTENOTITLE]] satisfies the appropriate boundary condition at ''r''<sub>''D''</sub> = ''r''<sub>''eD''</sub>, then '''Eq. 3.267''' should yield the point-source solution for a cylindrical reservoir with appropriate boundary conditions.<br/><br/>Consider the example of a closed cylindrical reservoir in which the boundary conditions are given by<br/><br/>[[File:Vol1 page 0131 eq 003.png|RTENOTITLE]]....................(3.268)<br/><br/>and<br/><br/>[[File:Vol1 page 0131 eq 004.png|RTENOTITLE]]....................(3.269)<br/><br/>According to the boundary condition given by '''Eq. 3.268''', we should choose [[File:Vol1 page 0131 inline 002.png|RTENOTITLE]] as the point-source solution given in '''Table 3.6''' (or by '''Eq. 3.263'''). Then, with the addition theorem for the Bessel function ''K''<sub>0</sub>(''aR''<sub>D</sub>) given by<ref name="r14">Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, second edition, 353–386. Oxford, UK: Oxford University Press.</ref><br/><br/>[[File:Vol1 page 0131 eq 005.png|RTENOTITLE]]....................(3.270)<br/><br/>where<br/><br/>[[File:Vol1 page 0131 eq 006.png|RTENOTITLE]]....................(3.271)<br/><br/>we can write<br/><br/>[[File:Vol1 page 0132 eq 001.png|RTENOTITLE]]....................(3.272)<br/><br/>for ''r''<sub>''D''</sub> < ''r′''<sub>''D''</sub>. If ''r''<sub>''D''</sub> > ''r′''<sub>''D''</sub>, we interchange ''r''<sub>''D''</sub> and ''r′''<sub>''D''</sub> in '''Eq. 3.272'''. If we choose [[File:Vol1 page 0131 inline 003.png|RTENOTITLE]] in '''Eq. 3.267''' as<br/><br/>[[File:Vol1 page 0132 eq 002.png|RTENOTITLE]]....................(3.273)<br/><br/>where ''a''<sub>''k''</sub> and ''b''<sub>''k''</sub> are constants, then [[File:Vol1 page 0132 inline 001.png|RTENOTITLE]] satisfies the boundary condition given by '''Eq. 3.268''', and the contribution of [[File:Vol1 page 0131 inline 003.png|RTENOTITLE]] to the flux at the source location vanishes. If we also choose the constants ''a''<sub>''k''</sub> and ''b''<sub>''k''</sub> in '''Eq. 3.273''' as<br/><br/>[[File:Vol1 page 0132 eq 003.png|RTENOTITLE]]....................(3.274)<br/><br/>and<br/><br/>[[File:Vol1 page 0132 eq 004.png|RTENOTITLE]]....................(3.275)<br/><br/>then [[File:Vol1 page 0132 inline 002.png|RTENOTITLE]] satisfies the impermeable boundary condition at ''r''<sub>''D''</sub> = ''r''<sub>''eD''</sub> given by '''Eq. 3.269'''. Thus, the point-source solution for a closed cylindrical reservoir is given by<br/><br/>[[File:Vol1 page 0132 eq 005.png|RTENOTITLE]]....................(3.276)<br/><br/>This solution procedure may be extended to the cases in which the boundaries are at constant pressure or of mixed type.<ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref> '''Table 3.7''' presents the point-source solutions for cylindrical reservoirs for all possible combinations of boundary conditions. Solutions for other source geometries in cylindrical reservoirs may be obtained by using the point-source solutions in '''Table 3.7''' in '''Eq. 3.255''' (or '''Eq. 3.256''').<br/><br/><gallery widths="300px" heights="200px">
<br>
Consider a point source in an infinite-slab reservoir with impermeable boundaries at the bottom, ''z'' = 0, and the top, ''z'' = ''h''. To obtain the point-source solution for this case, we use the point-source solution in an infinite reservoir given by '''Eq. 3.252''' with the method of images. The result is given by
<br>
<br>
[[File:Vol1 page 0129 eq 004.png]]....................(3.257)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0129 eq 005.png]]....................(3.258)
<br>
<br>
[[File:Vol1 page 0130 eq 001.png]]....................(3.259)
<br>
<br>
[[File:Vol1 page 0130 eq 002.png]]....................(3.260)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0130 eq 003.png]]....................(3.261)
<br>
<br>
The solution given in '''Eq. 3.257''' is not very convenient for computational purposes. To obtain a computationally convenient form of the solution, we use the summation formula given by<ref name="r17" /><ref name="r29" />
<br>
<br>
[[File:Vol1 page 0130 eq 004.png]]....................(3.262)
<br>
<br>
and recast '''Eq. 3.257''' as
<br>
<br>
[[File:Vol1 page 0130 eq 005.png]]....................(3.263)
<br>
<br>
The point-source solutions for infinite-slab reservoirs with constant pressure and mixed boundaries at the top and bottom are obtained in a similar manner<ref name="r17" /> and are given in '''Table 3.6'''. The point-source solutions can be used with '''Eqs. 3.254''' and '''3.256''' to generate the solutions for the other well geometries given in '''Table 3.6'''. For example, to generate the solution for a partially penetrating vertical line-source well of length hw in an infinite-slab reservoir with impermeable slab boundaries, we integrate the right side of '''Eq. 3.263''' from ''z''<sub>''w''</sub> − ''h''<sub>''w''</sub> / 2 to ''z''<sub>''w''</sub> + ''h''<sub>''w''</sub> / 2 with respect to ''z′'', where ''z''<sub>''w''</sub> is the vertical coordinate of the midpoint of the open interval. If ''h''<sub>''w''</sub> = ''h'' (i.e., the well penetrates the entire thickness of the slab reservoir), then this procedure yields the solution for a fully penetrating vertical line-source well. The solution for a partially penetrating fracture of height ''h''<sub>''f''</sub> and half-length ''x''<sub>''f''</sub> is obtained if the point-source solution is integrated once with respect to ''z′'' from ''z''<sub>''w''</sub> − ''h''<sub>''f''</sub> / 2 to ''z''<sub>''w''</sub> + ''h''<sub>''f''</sub> / 2 and then with respect to ''x′'' from ''x''<sub>''w''</sub> ''x''<sub>''f''</sub> to ''x''<sub>''w''</sub> + ''x''<sub>''f''</sub>, where ''x''<sub>''w''</sub> and ''z''<sub>''w''</sub> are the coordinates of the midpoint of the fracture. Similarly, the solution for a horizontal-line source well of length ''L''<sub>''h''</sub> is obtained by integrating the point-source solution with respect to ''x′'' from ''x''<sub>''w''</sub> ''L''<sub>''h''</sub> / 2 to ''x''<sub>''w''</sub> + ''L''<sub>''h''</sub> / 2, where ''x''<sub>''w''</sub> is the ''x''-coordinate of the midpoint of the horizontal well.
<br>
<br>
'''Solutions for Cylindrical Reservoir.''' Solutions for cylindrical reservoirs may also be obtained by starting from the point-source solution in the Laplace transform domain. The Laplace domain solution for a point source located at ''r′''<sub>''D''</sub>, ''θ′'', ''z′''<sub>''D''</sub> should satisfy the following diffusion equation in cylindrical coordinates.<ref name="r19" />
<br>
<br>
[[File:Vol1 page 0130 eq 006.png]]....................(3.264)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0130 eq 007.png]]....................(3.265)
<br>
<br>
The point-source solution is also required to satisfy the following flux condition at the source location (''r''<sub>''D''</sub> →0+, ''θ'' = ''θ''′, ''z''<sub>''D''</sub> = ''z′''<sub>''D''</sub>):
<br>
<br>
[[File:Vol1 page 0131 eq 001.png]]....................(3.266)
<br>
<br>
Assuming that the reservoir is bounded by a cylindrical surface at ''r''<sub>''D''</sub> = ''r''<sub>''eD''</sub> and by the parallel planes at ''z''<sub>''D''</sub> = 0 and ''h''<sub>''D''</sub>, we should also impose the appropriate physical conditions at these boundaries.
<br>
<br>
We seek a point-source solution for a cylindrical reservoir in the following form:
<br>
<br>
[[File:Vol1 page 0131 eq 002.png]]....................(3.267)
<br>
<br>
In '''Eq. 3.267''', [[File:Vol1 page 0131 inline 001.png]] is a solution of '''Eq. 3.264''' that satisfies '''Eq. 3.266''' and the boundary conditions at ''z''<sub>''D''</sub> = 0 and ''h''<sub>''D''</sub>. [[File:Vol1 page 0131 inline 002.png]] may be chosen as one of the point-source solutions in an infinite-slab reservoir given in '''Table 3.6''', depending on the conditions imposed at the boundaries at ''z''<sub>''D''</sub> = 0 and ''h''<sub>''D''</sub>. If [[File:Vol1 page 0131 inline 003.png]] is chosen such that it satisfies the boundary conditions at ''z''<sub>''D''</sub> = 0 and ''h''<sub>''D''</sub>, its contribution to the flux vanishes at the source location, and [[File:Vol1 page 0131 inline 003.png]] + [[File:Vol1 page 0131 inline 002.png]] satisfies the appropriate boundary condition at ''r''<sub>''D''</sub> = ''r''<sub>''eD''</sub>, then '''Eq. 3.267''' should yield the point-source solution for a cylindrical reservoir with appropriate boundary conditions.
<br>
<br>
Consider the example of a closed cylindrical reservoir in which the boundary conditions are given by
<br>
<br>
[[File:Vol1 page 0131 eq 003.png]]....................(3.268)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0131 eq 004.png]]....................(3.269)
<br>
<br>
According to the boundary condition given by '''Eq. 3.268''', we should choose [[File:Vol1 page 0131 inline 002.png]] as the point-source solution given in '''Table 3.6''' (or by '''Eq. 3.263'''). Then, with the addition theorem for the Bessel function ''K''<sub>0</sub>(''aR''<sub>D</sub>) given by<ref name="r14" />
<br>
<br>
[[File:Vol1 page 0131 eq 005.png]]....................(3.270)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0131 eq 006.png]]....................(3.271)
<br>
<br>
we can write
<br>
<br>
[[File:Vol1 page 0132 eq 001.png]]....................(3.272)
<br>
<br>
for ''r''<sub>''D''</sub> < ''r′''<sub>''D''</sub>. If ''r''<sub>''D''</sub> > ''r′''<sub>''D''</sub>, we interchange ''r''<sub>''D''</sub> and ''r′''<sub>''D''</sub> in '''Eq. 3.272'''. If we choose [[File:Vol1 page 0131 inline 003.png]] in '''Eq. 3.267''' as
<br>
<br>
[[File:Vol1 page 0132 eq 002.png]]....................(3.273)
<br>
<br>
where ''a''<sub>''k''</sub> and ''b''<sub>''k''</sub> are constants, then [[File:Vol1 page 0132 inline 001.png]] satisfies the boundary condition given by '''Eq. 3.268''', and the contribution of [[File:Vol1 page 0131 inline 003.png]] to the flux at the source location vanishes. If we also choose the constants ''a''<sub>''k''</sub> and ''b''<sub>''k''</sub> in '''Eq. 3.273''' as
<br>
<br>
[[File:Vol1 page 0132 eq 003.png]]....................(3.274)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0132 eq 004.png]]....................(3.275)
<br>
<br>
then [[File:Vol1 page 0132 inline 002.png]] satisfies the impermeable boundary condition at ''r''<sub>''D''</sub> = ''r''<sub>''eD''</sub> given by '''Eq. 3.269'''. Thus, the point-source solution for a closed cylindrical reservoir is given by
<br>
<br>
[[File:Vol1 page 0132 eq 005.png]]....................(3.276)
<br>
<br>
This solution procedure may be extended to the cases in which the boundaries are at constant pressure or of mixed type.<ref name="r19" /> '''Table 3.7''' presents the point-source solutions for cylindrical reservoirs for all possible combinations of boundary conditions. Solutions for other source geometries in cylindrical reservoirs may be obtained by using the point-source solutions in '''Table 3.7''' in '''Eq. 3.255''' (or '''Eq. 3.256''').
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 134 Image 0001.png|'''Table 3.7'''
File:Vol1 Page 134 Image 0001.png|'''Table 3.7'''
</gallery>
</gallery>
<br>


----
----


'''Example 3.8'''
'''Example 3.8'''<br/><br/>Consider a partially penetrating, uniform-flux fracture of height ''h''<sub>''f''</sub> and half-length ''x''<sub>''f''</sub> in an isotropic and closed cylindrical reservoir. The center of the fracture is at ''r′'' = 0, ''θ''′ =0, ''z′'' = ''z''<sub>''w''</sub>, and the fracture tips extend from (''r′'' = ''x''<sub>''f''</sub>, ''θ'' = α + ''π'') to (''r′'' = ''x''<sub>''f''</sub>, ''θ'' = ''α'').<br/><br/>''Solution''. '''Fig. 3.15''' shows the geometry of the fracture/reservoir system considered in this example. The solution for this problem is obtained by first generating a partially penetrating line source and then using this line-source solution to generate the plane source. The solution for a partially penetrating line source at ''r′''<sub>''D''</sub>, ''θ′'', ''z''<sub>''w''</sub> is obtained by integrating the corresponding point-source solution given in '''Table 3.7''' with respect to ''z′'' from ''z''<sub>''w''</sub> – ''h''<sub>''f''</sub> / 2 to ''z''<sub>''w''</sub> + ''h''<sub>''f''</sub> / 2 and is given by<br/><br/>[[File:Vol1 page 0133 eq 001.png|RTENOTITLE]]....................(3.277)<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
Consider a partially penetrating, uniform-flux fracture of height ''h''<sub>''f''</sub> and half-length ''x''<sub>''f''</sub> in an isotropic and closed cylindrical reservoir. The center of the fracture is at ''r′'' = 0, ''θ''′ =0, ''z′'' = ''z''<sub>''w''</sub>, and the fracture tips extend from (''r′'' = ''x''<sub>''f''</sub>, ''θ'' = α + ''π'') to (''r′'' = ''x''<sub>''f''</sub>, ''θ'' = ''α'').
<br>
<br>
''Solution''. '''Fig. 3.15''' shows the geometry of the fracture/reservoir system considered in this example. The solution for this problem is obtained by first generating a partially penetrating line source and then using this line-source solution to generate the plane source. The solution for a partially penetrating line source at ''r′''<sub>''D''</sub>, ''θ′'', ''z''<sub>''w''</sub> is obtained by integrating the corresponding point-source solution given in '''Table 3.7''' with respect to ''z′'' from ''z''<sub>''w''</sub> – ''h''<sub>''f''</sub> / 2 to ''z''<sub>''w''</sub> + ''h''<sub>''f''</sub> / 2 and is given by
<br>
<br>
[[File:Vol1 page 0133 eq 001.png]]....................(3.277)
<br>
<br>
<gallery widths="300px" heights="200px">
File:vol1 Page 135 Image 0001.png|'''Fig. 3.15 – Geometry of a partially penetrating fracture in a closed cylindrical reservoir (Example 3.8).'''
File:vol1 Page 135 Image 0001.png|'''Fig. 3.15 – Geometry of a partially penetrating fracture in a closed cylindrical reservoir (Example 3.8).'''
</gallery>
</gallery><br/>To generate the solution for a partially penetrating plane source that represents the fracture, the partially penetrating line-source solution given in '''Eq. 3.277''' is integrated with respect to ''r′'' from 0 to ''x''<sub>''f''</sub> with ''θ''′ = ''α'' + ''π'' in the third quadrant and with ''θ''′ = ''α'' in the first quadrant. This procedure yields<br/><br/>[[File:Vol1 page 0133 eq 002.png|RTENOTITLE]] [[File:Vol1 page 0135 eq 001.png|RTENOTITLE]]....................(3.278)<br/><br/>It is possible to obtain an alternate representation of the solution given in '''Eq. 3.278'''. With the addition theorem of the Bessel function ''K''<sub>0</sub>(''x'') given by '''Eq. 3.270''', the solution in '''Eq. 3.277''' may be written as<br/><br/>[[File:Vol1 page 0135 eq 002.png|RTENOTITLE]]....................(3.279)<br/><br/>where<br/><br/>[[File:Vol1 page 0135 eq 003.png|RTENOTITLE]]....................(3.280)<br/><br/>and<br/><br/>[[File:Vol1 page 0135 eq 004.png|RTENOTITLE]]....................(3.281)<br/><br/>The integration of the partially penetrating vertical well solution given in '''Eq. 3.279''' with respect to ''r′'' from 0 to ''x''<sub>''f''</sub> (with ''θ''′ = ''α'' + ''π'' in the third quadrant and with ''θ''′ = ''α'' in the first quadrant) yields the following alternative form of the partially penetrating fracture solution:<br/><br/>[[File:Vol1 page 0136 eq 001.png|RTENOTITLE]]....................(3.282)<br/><br/>where<br/><br/>[[File:Vol1 page 0136 eq 002.png|RTENOTITLE]]....................(3.283)<br/>
<br/>
To generate the solution for a partially penetrating plane source that represents the fracture, the partially penetrating line-source solution given in '''Eq. 3.277''' is integrated with respect to ''r′'' from 0 to ''x''<sub>''f''</sub> with ''θ''′ = ''α'' + ''π'' in the third quadrant and with ''θ''′ = ''α'' in the first quadrant. This procedure yields
<br>
<br>
[[File:Vol1 page 0133 eq 002.png]]
[[File:Vol1 page 0135 eq 001.png]]....................(3.278)
<br>
<br>
It is possible to obtain an alternate representation of the solution given in '''Eq. 3.278'''. With the addition theorem of the Bessel function ''K''<sub>0</sub>(''x'') given by '''Eq. 3.270''', the solution in '''Eq. 3.277''' may be written as
<br>
<br>
[[File:Vol1 page 0135 eq 002.png]]....................(3.279)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0135 eq 003.png]]....................(3.280)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0135 eq 004.png]]....................(3.281)
<br>
<br>
The integration of the partially penetrating vertical well solution given in '''Eq. 3.279''' with respect to ''r′'' from 0 to ''x''<sub>''f''</sub> (with ''θ''′ = ''α'' + ''π'' in the third quadrant and with ''θ''′ = ''α'' in the first quadrant) yields the following alternative form of the partially penetrating fracture solution:
<br>
<br>
[[File:Vol1 page 0136 eq 001.png]]....................(3.282)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0136 eq 002.png]]....................(3.283)
<br>
<br>


----
----


'''Example 3.9'''
'''Example 3.9''' Consider a uniform-flux, horizontal line-source well of length ''L''<sub>''h''</sub> in an isotropic and closed cylindrical reservoir. The well extends from (''r′'' = ''L''<sub>''h''</sub>/2, ''θ'' = ''α'' + ''π'') to (''r′'' = ''L''<sub>''h''</sub>/2, ''θ'' = ''α''), and the center of the well is at ''r′'' = 0, ''θ''′ = 0, ''z′'' = ''z''<sub>''w''</sub>.<br/><br/>''Solution''. The solution for a horizontal line-source well in a closed cylindrical reservoir is obtained by integrating the corresponding point-source solution in '''Table 3.7''' with respect to ''r′'' from 0 to ''L''<sub>''h''</sub> / 2 with ''θ''′ = ''α'' + ''π'' in the third quadrant and with ''θ''′ = ''α'' in the first quadrant. The final form of the solution is given by<br/><br/>[[File:Vol1 page 0136 eq 001.png|RTENOTITLE]] [[File:Vol1 page 0137 eq 001.png|RTENOTITLE]]....................(3.284)<br/>
Consider a uniform-flux, horizontal line-source well of length ''L''<sub>''h''</sub> in an isotropic and closed cylindrical reservoir. The well extends from (''r′'' = ''L''<sub>''h''</sub>/2, ''θ'' = ''α'' + ''π'') to (''r′'' = ''L''<sub>''h''</sub>/2, ''θ'' = ''α''), and the center of the well is at ''r′'' = 0, ''θ''′ = 0, ''z′'' = ''z''<sub>''w''</sub>.
<br>
<br>
''Solution''. The solution for a horizontal line-source well in a closed cylindrical reservoir is obtained by integrating the corresponding point-source solution in '''Table 3.7''' with respect to ''r′'' from 0 to ''L''<sub>''h''</sub> / 2 with ''θ''′ = ''α'' + ''π'' in the third quadrant and with ''θ''′ = ''α'' in the first quadrant. The final form of the solution is given by
<br>
<br>
[[File:Vol1 page 0136 eq 001.png]]
[[File:Vol1 page 0137 eq 001.png]]....................(3.284)
<br>
<br>


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'''''Solutions for Rectangular Parallelepiped Reservoir.''''' Solutions for rectangular parallelepiped reservoirs may also be obtained by starting from the point-source solution in the Laplace transform domain in an infinite reservoir and using the method of images to generate the effects of the planar boundaries. Although the formal procedure to obtain the solution is fairly easy, the use of the method of images in three directions (''x'', ''y'', ''z'') yields triple infinite Fourier series, which may pose computational inconveniences. As an example, the solution for a continuous point source located at ''x′'', ''y′'', ''z′'' in a rectangular porous medium occupying the region 0 < ''x'' < ''x''<sub>''e''</sub>, 0 < ''y'' < ''y''<sub>''e''</sub>, and 0 < ''z'' < ''h'' is obtained by applying the method of images to the point-source solution given by '''Eq. 3.252''': <ref name="r19" /><ref name="r29" />
'''''Solutions for Rectangular Parallelepiped Reservoir.''''' Solutions for rectangular parallelepiped reservoirs may also be obtained by starting from the point-source solution in the Laplace transform domain in an infinite reservoir and using the method of images to generate the effects of the planar boundaries. Although the formal procedure to obtain the solution is fairly easy, the use of the method of images in three directions (''x'', ''y'', ''z'') yields triple infinite Fourier series, which may pose computational inconveniences. As an example, the solution for a continuous point source located at ''x′'', ''y′'', ''z′'' in a rectangular porous medium occupying the region 0 < ''x'' < ''x''<sub>''e''</sub>, 0 < ''y'' < ''y''<sub>''e''</sub>, and 0 < ''z'' < ''h'' is obtained by applying the method of images to the point-source solution given by '''Eq. 3.252''': <ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref><ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0137 eq 002.png|RTENOTITLE]]....................(3.285)<br/><br/>where<br/><br/>[[File:Vol1 page 0137 eq 003.png|RTENOTITLE]]....................(3.286)<br/><br/>and<br/><br/>[[File:Vol1 page 0137 eq 004.png|RTENOTITLE]]....................(3.287)<br/><br/>[[File:Vol1 page 0137 eq 005.png|RTENOTITLE]]....................(3.288)<br/><br/>[[File:Vol1 page 0137 eq 006.png|RTENOTITLE]]....................(3.289)<br/><br/>Ozkan<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref> shows that the triple infinite sums in '''Eq. 3.285''' may be reduced to double infinite sums with<br/><br/>[[File:Vol1 page 0138 eq 001.png|RTENOTITLE]]....................(3.290)<br/><br/>where<br/><br/>[[File:Vol1 page 0138 eq 002.png|RTENOTITLE]]....................(3.291)<br/><br/>The resulting continuous point-source solution for a closed rectangular reservoir is given by<br/><br/>[[File:Vol1 page 0138 eq 003.png|RTENOTITLE]]....................(3.292)<br/><br/>where<br/><br/>[[File:Vol1 page 0138 eq 004.png|RTENOTITLE]]....................(3.293)<br/><br/>[[File:Vol1 page 0138 eq 005.png|RTENOTITLE]]....................(3.294)<br/><br/>[[File:Vol1 page 0138 eq 006.png|RTENOTITLE]]....................(3.295)<br/><br/>[[File:Vol1 page 0139 eq 001.png|RTENOTITLE]]....................(3.296)<br/><br/>and<br/><br/>[[File:Vol1 page 0139 eq 002.png|RTENOTITLE]]....................(3.297)<br/><br/>Following a procedure similar to the one explained here, it is possible to derive the point-source solutions in rectangular parallelepiped reservoirs for different combinations of boundary conditions.<ref name="r17">Ozkan, E. and Raghavan, R. 1991a. New Solutions for Well-Test-Analysis Problems: Part 1—Analytical Considerations. SPE Form Eval 6 (3): 359–368. SPE-18615-PA. http://dx.doi.org/10.2118/18615-PA</ref><ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref> '''Table 3.8''' gives these solutions, which may be used to derive the solutions for the other source geometries with '''Eq. 3.255''' (or '''Eq. 3.256'''). '''Examples 3.10''' and '''3.11''' demonstrate the derivation of the solutions for the other source geometries in rectangular reservoirs.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0137 eq 002.png]]....................(3.285)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0137 eq 003.png]]....................(3.286)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0137 eq 004.png]]....................(3.287)
<br>
<br>
[[File:Vol1 page 0137 eq 005.png]]....................(3.288)
<br>
<br>
[[File:Vol1 page 0137 eq 006.png]]....................(3.289)
<br>
<br>
Ozkan<ref name="r29" /> shows that the triple infinite sums in '''Eq. 3.285''' may be reduced to double infinite sums with
<br>
<br>
[[File:Vol1 page 0138 eq 001.png]]....................(3.290)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0138 eq 002.png]]....................(3.291)
<br>
<br>
The resulting continuous point-source solution for a closed rectangular reservoir is given by
<br>
<br>
[[File:Vol1 page 0138 eq 003.png]]....................(3.292)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0138 eq 004.png]]....................(3.293)
<br>
<br>
[[File:Vol1 page 0138 eq 005.png]]....................(3.294)
<br>
<br>
[[File:Vol1 page 0138 eq 006.png]]....................(3.295)
<br>
<br>
[[File:Vol1 page 0139 eq 001.png]]....................(3.296)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0139 eq 002.png]]....................(3.297)
<br>
<br>
Following a procedure similar to the one explained here, it is possible to derive the point-source solutions in rectangular parallelepiped reservoirs for different combinations of boundary conditions.<ref name="r17" /><ref name="r29" /> '''Table 3.8''' gives these solutions, which may be used to derive the solutions for the other source geometries with '''Eq. 3.255''' (or '''Eq. 3.256'''). '''Examples 3.10''' and '''3.11''' demonstrate the derivation of the solutions for the other source geometries in rectangular reservoirs.
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 140 Image 0001.png|'''Table 3.8'''
File:Vol1 Page 140 Image 0001.png|'''Table 3.8'''


Line 2,145: Line 195:
File:Vol1 Page 143 Image 0001.png|'''Table 3.8'''
File:Vol1 Page 143 Image 0001.png|'''Table 3.8'''
</gallery>
</gallery>
<br>


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'''Example 3.10'''
'''Example 3.10'''<br/><br/>Consider a fully penetrating vertical fracture of half-length ''x''<sub>''f''</sub> located at ''x′'' = ''x''<sub>''w''</sub> and ''y′'' = ''y''<sub>''w''</sub> in a closed rectangular reservoir.<br/><br/>''Solution''. Assuming uniform-flux distribution along the fracture surface, the solution for this problem is obtained by integrating the corresponding point-source solution in '''Table 3.8''', first with respect to ''z′'' from 0 to ''h'' and then with respect to ''x′'' from ''x''<sub>''w''</sub> – ''x''<sub>''f''</sub> to ''x''<sub>''w''</sub> + ''x''<sub>''f''</sub>. The result is<br/><br/>[[File:Vol1 page 0139 eq 003.png|RTENOTITLE]]....................(3.298)<br/><br/>where [[File:Vol1 page 0139 inline 001.png|RTENOTITLE]], [[File:Vol1 page 0139 inline 002.png|RTENOTITLE]], and ''ε''<sub>''k''</sub> are given respectively by '''Eqs. 3.293''', '''3.294''', and '''3.296'''.
<br>
<br>
Consider a fully penetrating vertical fracture of half-length ''x''<sub>''f''</sub> located at ''x′'' = ''x''<sub>''w''</sub> and ''y′'' = ''y''<sub>''w''</sub> in a closed rectangular reservoir.
<br>
<br>
''Solution''. Assuming uniform-flux distribution along the fracture surface, the solution for this problem is obtained by integrating the corresponding point-source solution in '''Table 3.8''', first with respect to ''z′'' from 0 to ''h'' and then with respect to ''x′'' from ''x''<sub>''w''</sub> – ''x''<sub>''f''</sub> to ''x''<sub>''w''</sub> + ''x''<sub>''f''</sub>. The result is
<br>
<br>
[[File:Vol1 page 0139 eq 003.png]]....................(3.298)
<br>
<br>
where [[File:Vol1 page 0139 inline 001.png]], [[File:Vol1 page 0139 inline 002.png]], and ''ε''<sub>''k''</sub> are given respectively by '''Eqs. 3.293''', '''3.294''', and '''3.296'''.


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'''Example 3.11'''
'''Example 3.11'''<br/><br/>Consider a horizontal well of length ''L''<sub>''h''</sub> in the ''x''-direction located at ''x′'' = ''x''<sub>''w''</sub>, ''y′'' = ''y''<sub>''w''</sub>, and ''z′'' = ''z''<sub>''w''</sub> in a closed rectangular reservoir.<br/><br/>''Solution''. The solution for a horizontal line-source well is obtained by integrating the corresponding point-source solution in '''Table 3.8''', with respect to ''x′'' from ''x''<sub>''w''</sub>–''L''<sub>''h''</sub> /2 to ''x''<sub>''w''</sub>+''L''<sub>''h''</sub> /2, and is given by<br/><br/>[[File:Vol1 page 0139 eq 004.png|RTENOTITLE]]....................(3.299)<br/><br/>where<br/><br/>[[File:Vol1 page 0139 eq 005.png|RTENOTITLE]]....................(3.300)<br/><br/>and<br/><br/>[[File:Vol1 page 0139 eq 006.png|RTENOTITLE]] [[File:Vol1 page 0143 eq 001.png|RTENOTITLE]]....................(3.301)<br/><br/>In '''Eq. 3.301''', [[File:Vol1 page 0139 inline 001.png|RTENOTITLE]], [[File:Vol1 page 0139 inline 002.png|RTENOTITLE]], ''ε''<sub>''n''</sub>, ''ε''<sub>''k''</sub>, and ''ε''<sub>''k, n''</sub> are given by '''Eqs. 3.293''' through '''3.297'''.
<br>
<br>
Consider a horizontal well of length ''L''<sub>''h''</sub> in the ''x''-direction located at ''x′'' = ''x''<sub>''w''</sub>, ''y′'' = ''y''<sub>''w''</sub>, and ''z′'' = ''z''<sub>''w''</sub> in a closed rectangular reservoir.
<br>
<br>
''Solution''. The solution for a horizontal line-source well is obtained by integrating the corresponding point-source solution in '''Table 3.8''', with respect to ''x′'' from ''x''<sub>''w''</sub>–''L''<sub>''h''</sub> /2 to ''x''<sub>''w''</sub>+''L''<sub>''h''</sub> /2, and is given by
<br>
<br>
[[File:Vol1 page 0139 eq 004.png]]....................(3.299)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0139 eq 005.png]]....................(3.300)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0139 eq 006.png]]
[[File:Vol1 page 0143 eq 001.png]]....................(3.301)
<br>
<br>
In '''Eq. 3.301''', [[File:Vol1 page 0139 inline 001.png]], [[File:Vol1 page 0139 inline 002.png]], ''ε''<sub>''n''</sub>, ''ε''<sub>''k''</sub>, and ''ε''<sub>''k, n''</sub> are given by '''Eqs. 3.293''' through '''3.297'''.


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'''''Conversion From 3D to 2D Anisotropy.''''' The solutions previously presented assume that the reservoir is anisotropic in all three principal directions, ''x'', ''y'', and ''z'' with ''k''<sub>''x''</sub>, ''k''<sub>''y''</sub>, and ''k''<sub>''z''</sub> denoting the corresponding permeabilities. In these solutions, an equivalent isotropic permeability, ''k'', has been defined by
'''''Conversion From 3D to 2D Anisotropy.''''' The solutions previously presented assume that the reservoir is anisotropic in all three principal directions, ''x'', ''y'', and ''z'' with ''k''<sub>''x''</sub>, ''k''<sub>''y''</sub>, and ''k''<sub>''z''</sub> denoting the corresponding permeabilities. In these solutions, an equivalent isotropic permeability, ''k'', has been defined by<br/><br/>[[File:Vol1 page 0144 eq 001.png|RTENOTITLE]]....................(3.302)<br/><br/>For some applications, it may be more convenient to define an equivalent horizontal permeability by<br/><br/>[[File:Vol1 page 0144 eq 002.png|RTENOTITLE]]....................(3.303)<br/><br/>and replace ''k'' in the solutions given in this section ('''Sec. 3.4.4''') by ''k''<sub>''h''</sub>. Note that ''k'' takes place in the definition of the dimensionless time ''t''<sub>''D''</sub> ('''Eq. 3.230'''). Then, if we define a dimensionless time [[File:Vol1 page 0144 inline 001.png|RTENOTITLE]] based on ''k''<sub>''h''</sub>, the relation between [[File:Vol1 page 0144 inline 001.png|RTENOTITLE]] and ''t''<sub>''D''</sub> is given by<br/><br/>[[File:Vol1 page 0144 eq 003.png|RTENOTITLE]]....................(3.304)<br/><br/>Because in the solutions given in this section the Laplace transformation is with respect to ''t''<sub>''D''</sub>, conversion from 3D to 2D anisotropy requires the use of the following property of the Laplace transforms:<br/><br/>[[File:Vol1 page 0144 eq 004.png|RTENOTITLE]]....................(3.305)<br/><br/>As an example, consider the solution for a horizontal well in an infinite-slab reservoir. Assuming that the midpoint of the well is the origin (''x''<sub>''wD''</sub> = 0, ''y''<sub>''wD''</sub> = 0) and choosing the half-length of the horizontal well as the characteristic length (i.e., ℓ = ''L''<sub>''h''</sub> / 2), the horizontal-well solution given in '''Table 3.6''' may be written as<br/><br/>[[File:Vol1 page 0144 eq 005.png|RTENOTITLE]]....................(3.306)<br/><br/>In '''Eq. 3.306''', ''s'' is the Laplace transform variable with respect to dimensionless time, ''t''<sub>''D''</sub>, based on ''k'' and<br/><br/>[[File:Vol1 page 0144 eq 006.png|RTENOTITLE]]....................(3.307)<br/><br/>[[File:Vol1 page 0144 eq 007.png|RTENOTITLE]]....................(3.308)<br/><br/>[[File:Vol1 page 0144 eq 008.png|RTENOTITLE]]....................(3.309)<br/><br/>and<br/><br/>[[File:Vol1 page 0145 eq 001.png|RTENOTITLE]]....................(3.310)<br/><br/>If we define the following variables based on ''k''<sub>''h''</sub>,<br/><br/>[[File:Vol1 page 0145 eq 002.png|RTENOTITLE]]....................(3.311)<br/><br/>[[File:Vol1 page 0145 eq 003.png|RTENOTITLE]]....................(3.312)<br/><br/>[[File:Vol1 page 0145 eq 004.png|RTENOTITLE]]....................(3.313)<br/><br/>and also note that<br/><br/>[[File:Vol1 page 0145 eq 005.png|RTENOTITLE]]....................(3.314)<br/><br/>then, we may rearrange '''Eq. 3.306''' in terms of the dimensionless variables based on ''k''<sub>''h''</sub> as<br/><br/>[[File:Vol1 page 0145 eq 006.png|RTENOTITLE]]....................(3.315)<br/><br/>where<br/><br/>[[File:Vol1 page 0145 eq 007.png|RTENOTITLE]]....................(3.316)<br/><br/>and<br/><br/>[[File:Vol1 page 0145 eq 008.png|RTENOTITLE]]....................(3.317)<br/><br/>If we compare '''Eqs. 3.306''' and '''3.315''', we can show that<br/><br/>[[File:Vol1 page 0145 eq 009.png|RTENOTITLE]]....................(3.318)<br/><br/>where we have used the relation given by '''Eq. 3.305'''. If we now define [[File:Vol1 page 0145 inline 001.png|RTENOTITLE]] as the Laplace transform variable with respect to [[File:Vol1 page 0144 inline 001.png|RTENOTITLE]], we may write<br/><br/>[[File:Vol1 page 0145 eq 010.png|RTENOTITLE]]....................(3.319)<br/><br/>With the relation given by '''Eq. 3.319''' and '''Eq. 3.306''', we obtain the following horizontal-well solution in terms of dimensionless variables based on ''k''<sub>''h''</sub>:<br/><br/>[[File:Vol1 page 0146 eq 001.png|RTENOTITLE]]....................(3.320)<br/><br/>'''''Computational Considerations and Applications.''''' The numerical evaluation of the solutions given previously may be sometimes difficult, inefficient, or even impossible. Alternative computational forms of some of these solutions have been presented in a few sources.<ref name="r18">Ozkan, E. and Raghavan, R. 1991b. New Solutions for Well-Test-Analysis Problems: Part 2—Computational Considerations and Applications. SPE Form Eval 6 (3): 369–378. SPE-18616-PA. http://dx.doi.org/10.2118/18616-PA</ref><ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref><ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref> Here, we present a summary of the alternative formulas to be used in the computation of the source functions in the Laplace transform domain. Some of these formulas are for computations at early or late times and may be useful to derive asymptotic approximations of the solutions during the corresponding time periods.<br/><br/>As '''Sec. 3.3.1''' notes, the short- and long-time approximations of the solutions correspond to the limiting forms of the solution in the Laplace transform domain as s→∞ and s→0, respectively. In the solutions given in this section, we have defined ''u'' = ''sf''(''s''). From elementary considerations, it is possible to show that the definition of ''f''(''s'') given in '''Eq. 3.243''' yields the following limiting forms:<br/><br/>[[File:Vol1 page 0146 eq 002.png|RTENOTITLE]]....................(3.321)<br/><br/>and<br/><br/>[[File:Vol1 page 0146 eq 003.png|RTENOTITLE]]....................(3.322)<br/><br/>These limiting forms are used in the derivation of the short- and long-time asymptotic approximations. In the following expressions, homogeneous reservoir solutions are obtained by substituting ''ω'' = 1.<br/><br/>''The Integral.''<br/><br/>[[File:Vol1 page 0146 eq 004.png|RTENOTITLE]]....................(3.323)<br/><br/>This integral arises in the computation of many practical transient-pressure solutions and may not be numerically evaluated, especially as ''y''<sub>''D''</sub>→0; however, the following alternate forms of the integral are numerically computable.<ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref><br/><br/>[[File:Vol1 page 0146 eq 005.png|RTENOTITLE]]....................(3.324)<br/><br/>[[File:Vol1 page 0147 eq 001.png|RTENOTITLE]]....................(3.325)<br/><br/>and<br/><br/>[[File:Vol1 page 0147 eq 002.png|RTENOTITLE]]....................(3.326)<br/><br/>The integrals in '''Eqs. 3.324''' through '''3.326''' may be evaluated with the standard numerical integration algorithms for ''y''<sub>''D''</sub> ≠ 0. For ''y''<sub>''D''</sub> = 0, the polynomial approximations given by Luke or the following power series expansion given by Abramowitz and Stegun<ref name="r7">Abramowitz, M. and Stegun, I.A. ed. 1972. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, ninth edition, 1020–1029. New York: Dover Publications.</ref> may be used in the computation of the integrals in '''Eqs. 3.324''' through '''3.326''':<br/><br/>[[File:Vol1 page 0147 eq 003.png|RTENOTITLE]]....................(3.327)<br/><br/>For numerical computations and asymptotic evaluations, it may also be useful to note the following relations: <ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref><br/><br/>[[File:Vol1 page 0147 eq 004.png|RTENOTITLE]]....................(3.328)<br/><br/>and<br/><br/>[[File:Vol1 page 0147 eq 005.png|RTENOTITLE]]....................(3.329)<br/><br/>It can be shown from '''Eqs. 3.328''' and '''3.329''' that, for practical purposes, when ''z'' ≥ 20, the right sides of '''Eqs. 3.327''' and '''3.328''' may be approximated by ''π''/2 and ''π'' exp (−|''c''|)/2, respectively.<ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref><ref name="r27">Kazemi, H. 1969. Pressure Transient Analysis of Naturally Fractured Reservoir with Uniform Fracture Distribution. SPE J. 9 (4): 451–462. SPE-2156-PA. http://dx.doi.org/10.2118/2156-PA</ref><br/><br/>As a few sources<ref name="r18">Ozkan, E. and Raghavan, R. 1991b. New Solutions for Well-Test-Analysis Problems: Part 2—Computational Considerations and Applications. SPE Form Eval 6 (3): 369–378. SPE-18616-PA. http://dx.doi.org/10.2118/18616-PA</ref><ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref><ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref> show, it is possible to derive the following short- and long-time approximations (i.e., the limiting forms as ''s''→∞ and ''s''→0, respectively) for the integral [[File:Vol1 page 0147 inline 001.png|RTENOTITLE]] given, respectively, by<br/><br/>[[File:Vol1 page 0147 eq 006.png|RTENOTITLE]]....................(3.330)<br/><br/>where<br/><br/>[[File:Vol1 page 0148 eq 001.png|RTENOTITLE]]....................(3.331)<br/><br/>and<br/><br/>[[File:Vol1 page 0148 eq 002.png|RTENOTITLE]]....................(3.332)<br/><br/>where ''γ''=0.5772… and<br/><br/>[[File:Vol1 page 0148 eq 003.png|RTENOTITLE]]....................(3.333)<br/><br/>It is also useful to note the real inversions of '''Eqs. 3.330''' and '''3.332''' given, respectively, by<br/><br/>[[File:Vol1 page 0148 eq 004.png|RTENOTITLE]]....................(3.334)<br/><br/>and<br/><br/>[[File:Vol1 page 0148 eq 005.png|RTENOTITLE]]....................(3.335)<br/><br/>''The Series.'' [[File:Vol1 page 0148 inline 001.png|RTENOTITLE]]<br/><br/>[[File:Vol1 page 0148 eq 006.png|RTENOTITLE]]....................(3.336)<br/><br/>Two alternative expressions for the series ''S''<sub>1</sub> may be convenient for the large and small values of u (i.e., for short and long times).<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref> When ''u'' is large,<br/><br/>[[File:Vol1 page 0148 eq 007.png|RTENOTITLE]]....................(3.337)<br/><br/>and when ''u'' + ''a''<sup>2</sup> << ''n''<sup>2</sup>''π''<sup>2</sup>/''h''<sup>2</sup><sub>''D''</sub>,<br/><br/>[[File:Vol1 page 0149 eq 001.png|RTENOTITLE]]....................(3.338)<br/><br/>''The Series'' [[File:Vol1 page 0149 inline 001.png|RTENOTITLE]].<br/><br/>[[File:Vol1 page 0149 eq 002.png|RTENOTITLE]]....................(3.339)<br/><br/>Alternative computational forms for the series ''S''<sub>2</sub> are given next.<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref> When ''u'' is large,<br/><br/>[[File:Vol1 page 0149 eq 003.png|RTENOTITLE]]....................(3.340)<br/><br/>and when ''u'' + ''a''<sup>2</sup> << ''n''<sup>2</sup>''π''<sup>2</sup>/''h''<sup>2</sup><sub>''D''</sub>,<br/><br/>[[File:Vol1 page 0149 eq 004.png|RTENOTITLE]]....................(3.341)<br/><br/>''The Series'' [[File:Vol1 page 0149 inline 003.png|RTENOTITLE]].<br/><br/>[[File:Vol1 page 0149 eq 005.png|RTENOTITLE]]....................(3.342)<br/><br/>The following alternative forms for the series [[File:Vol1 page 0149 inline 002.png|RTENOTITLE]] may be convenient for the large and small values of ''u'' (i.e., for short and long times).<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref> When ''u'' is large,<br/><br/>[[File:Vol1 page 0149 eq 006.png|RTENOTITLE]]....................(3.343)<br/><br/>and when ''u'' + ''a''<sup>2</sup> << (2''n'' − 1)<sup>2</sup> ''π''<sup>2</sup>/(4''h''<sup>2</sup><sub>''D''</sub>),<br/><br/>[[File:Vol1 page 0150 eq 001.png|RTENOTITLE]]....................(3.344)<br/><br/>''The Series [[File:Vol1 page 0150 inline 001.png|RTENOTITLE]]''.<br/><br/>[[File:Vol1 page 0150 eq 002.png|RTENOTITLE]]....................(3.345)<br/><br/>where<br/><br/>[[File:Vol1 page 0150 eq 003.png|RTENOTITLE]]....................(3.346)<br/><br/>The series [[File:Vol1 page 0150 inline 002.png|RTENOTITLE]] may be written in the following forms with the use of '''Eqs. 3.324''' through '''3.326'''.<br/><br/>[[File:Vol1 page 0150 eq 004.png|RTENOTITLE]]....................(3.347)<br/><br/>[[File:Vol1 page 0150 eq 005.png|RTENOTITLE]]....................(3.348)<br/><br/>and<br/><br/>[[File:Vol1 page 0150 eq 006.png|RTENOTITLE]] [[File:Vol1 page 0151 eq 001.png|RTENOTITLE]]....................(3.349)<br/><br/>The computation of the series in '''Eqs. 3.347''' and '''3.348''' should not pose numerical difficulties; however, the series in '''Eq. 3.349''' converges slowly. With the relation given in '''Eq. 3.328''', we may write '''Eq. 3.349''' as<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0151 eq 002.png|RTENOTITLE]]....................(3.350)<br/><br/>where<br/><br/>[[File:Vol1 page 0151 eq 003.png|RTENOTITLE]]....................(3.351)<br/><br/>Before discussing the computation of the series given in '''Eq. 3.351''', we first discuss the derivation of the asymptotic approximations for the series [[File:Vol1 page 0150 inline 002.png|RTENOTITLE]]. When ''s'' is large (small times), [[File:Vol1 page 0150 inline 002.png|RTENOTITLE]] may be approximated by<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0151 eq 004.png|RTENOTITLE]]....................(3.352)<br/><br/>where ''β'' is given by '''Eq. 3.331'''. If s is sufficiently large, then '''Eq. 3.352''' may be further approximated by<br/><br/>[[File:Vol1 page 0151 eq 005.png|RTENOTITLE]]....................(3.353)<br/><br/>The inverse Laplace transform of '''Eq. 3.353''' yields<br/><br/>[[File:Vol1 page 0151 eq 006.png|RTENOTITLE]]....................(3.354)<br/><br/>For small ''s'' (large times), depending on the value of ''x''<sub>''D''</sub>, [[File:Vol1 page 0150 inline 002.png|RTENOTITLE]] may be approximated by one of the following equations: <ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0152 eq 001.png|RTENOTITLE]]....................(3.355)<br/><br/>[[File:Vol1 page 0152 eq 002.png|RTENOTITLE]]....................(3.356)<br/><br/>[[File:Vol1 page 0152 eq 003.png|RTENOTITLE]]....................(3.357)<br/><br/>where [[File:Vol1 page 0152 inline 001.png|RTENOTITLE]] is given by '''Eq. 3.364'''.
<br>
<br>
[[File:Vol1 page 0144 eq 001.png]]....................(3.302)
<br>
<br>
For some applications, it may be more convenient to define an equivalent horizontal permeability by
<br>
<br>
[[File:Vol1 page 0144 eq 002.png]]....................(3.303)
<br>
<br>
and replace ''k'' in the solutions given in this section ('''Sec. 3.4.4''') by ''k''<sub>''h''</sub>. Note that ''k'' takes place in the definition of the dimensionless time ''t''<sub>''D''</sub> ('''Eq. 3.230'''). Then, if we define a dimensionless time [[File:Vol1 page 0144 inline 001.png]] based on ''k''<sub>''h''</sub>, the relation between [[File:Vol1 page 0144 inline 001.png]] and ''t''<sub>''D''</sub> is given by
<br>
<br>
[[File:Vol1 page 0144 eq 003.png]]....................(3.304)
<br>
<br>
Because in the solutions given in this section the Laplace transformation is with respect to ''t''<sub>''D''</sub>, conversion from 3D to 2D anisotropy requires the use of the following property of the Laplace transforms:
<br>
<br>
[[File:Vol1 page 0144 eq 004.png]]....................(3.305)
<br>
<br>
As an example, consider the solution for a horizontal well in an infinite-slab reservoir. Assuming that the midpoint of the well is the origin (''x''<sub>''wD''</sub> = 0, ''y''<sub>''wD''</sub> = 0) and choosing the half-length of the horizontal well as the characteristic length (i.e., ℓ = ''L''<sub>''h''</sub> / 2), the horizontal-well solution given in '''Table 3.6''' may be written as
<br>
<br>
[[File:Vol1 page 0144 eq 005.png]]....................(3.306)
<br>
<br>
In '''Eq. 3.306''', ''s'' is the Laplace transform variable with respect to dimensionless time, ''t''<sub>''D''</sub>, based on ''k'' and
<br>
<br>
[[File:Vol1 page 0144 eq 006.png]]....................(3.307)
<br>
<br>
[[File:Vol1 page 0144 eq 007.png]]....................(3.308)
<br>
<br>
[[File:Vol1 page 0144 eq 008.png]]....................(3.309)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0145 eq 001.png]]....................(3.310)
<br>
<br>
If we define the following variables based on ''k''<sub>''h''</sub>,
<br>
<br>
[[File:Vol1 page 0145 eq 002.png]]....................(3.311)
<br>
<br>
[[File:Vol1 page 0145 eq 003.png]]....................(3.312)
<br>
<br>
[[File:Vol1 page 0145 eq 004.png]]....................(3.313)
<br>
<br>
and also note that
<br>
<br>
[[File:Vol1 page 0145 eq 005.png]]....................(3.314)
<br>
<br>
then, we may rearrange '''Eq. 3.306''' in terms of the dimensionless variables based on ''k''<sub>''h''</sub> as
<br>
<br>
[[File:Vol1 page 0145 eq 006.png]]....................(3.315)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0145 eq 007.png]]....................(3.316)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0145 eq 008.png]]....................(3.317)
<br>
<br>
If we compare '''Eqs. 3.306''' and '''3.315''', we can show that
<br>
<br>
[[File:Vol1 page 0145 eq 009.png]]....................(3.318)
<br>
<br>
where we have used the relation given by '''Eq. 3.305'''. If we now define [[File:Vol1 page 0145 inline 001.png]] as the Laplace transform variable with respect to [[File:Vol1 page 0144 inline 001.png]], we may write
<br>
<br>
[[File:Vol1 page 0145 eq 010.png]]....................(3.319)
<br>
<br>
With the relation given by '''Eq. 3.319''' and '''Eq. 3.306''', we obtain the following horizontal-well solution in terms of dimensionless variables based on ''k''<sub>''h''</sub>:
<br>
<br>
[[File:Vol1 page 0146 eq 001.png]]....................(3.320)
<br>
<br>
'''''Computational Considerations and Applications.''''' The numerical evaluation of the solutions given previously may be sometimes difficult, inefficient, or even impossible. Alternative computational forms of some of these solutions have been presented in a few sources.<ref name="r18" /><ref name="r19" /><ref name="r29" /> Here, we present a summary of the alternative formulas to be used in the computation of the source functions in the Laplace transform domain. Some of these formulas are for computations at early or late times and may be useful to derive asymptotic approximations of the solutions during the corresponding time periods.
<br>
<br>
As '''Sec. 3.3.1''' notes, the short- and long-time approximations of the solutions correspond to the limiting forms of the solution in the Laplace transform domain as s→∞ and s→0, respectively. In the solutions given in this section, we have defined ''u'' = ''sf''(''s''). From elementary considerations, it is possible to show that the definition of ''f''(''s'') given in '''Eq. 3.243''' yields the following limiting forms:
<br>
<br>
[[File:Vol1 page 0146 eq 002.png]]....................(3.321)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0146 eq 003.png]]....................(3.322)
<br>
<br>
These limiting forms are used in the derivation of the short- and long-time asymptotic approximations. In the following expressions, homogeneous reservoir solutions are obtained by substituting ''ω'' = 1.
<br>
<br>
''The Integral.''
<br>
<br>
[[File:Vol1 page 0146 eq 004.png]]....................(3.323)
<br>
<br>
This integral arises in the computation of many practical transient-pressure solutions and may not be numerically evaluated, especially as ''y''<sub>''D''</sub>→0; however, the following alternate forms of the integral are numerically computable.<ref name="r19" />
<br>
<br>
[[File:Vol1 page 0146 eq 005.png]]....................(3.324)
<br>
<br>
[[File:Vol1 page 0147 eq 001.png]]....................(3.325)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0147 eq 002.png]]....................(3.326)
<br>
<br>
The integrals in '''Eqs. 3.324''' through '''3.326''' may be evaluated with the standard numerical integration algorithms for ''y''<sub>''D''</sub> ≠ 0. For ''y''<sub>''D''</sub> = 0, the polynomial approximations given by Luke<ref name="r30" /> or the following power series expansion given by Abramowitz and Stegun<ref name="r7" /> may be used in the computation of the integrals in '''Eqs. 3.324''' through '''3.326''':
<br>
<br>
[[File:Vol1 page 0147 eq 003.png]]....................(3.327)
<br>
<br>
For numerical computations and asymptotic evaluations, it may also be useful to note the following relations: <ref name="r19" />
<br>
<br>
[[File:Vol1 page 0147 eq 004.png]]....................(3.328)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0147 eq 005.png]]....................(3.329)
<br>
<br>
It can be shown from '''Eqs. 3.328''' and '''3.329''' that, for practical purposes, when ''z'' ≥ 20, the right sides of '''Eqs. 3.327''' and '''3.328''' may be approximated by ''π''/2 and ''π'' exp (−|''c''|)/2, respectively.<ref name="r19" /><ref name="r27" />
<br>
<br>
As a few sources<ref name="r18" /><ref name="r19" /><ref name="r29" /> show, it is possible to derive the following short- and long-time approximations (i.e., the limiting forms as ''s''→∞ and ''s''→0, respectively) for the integral [[File:Vol1 page 0147 inline 001.png]] given, respectively, by
<br>
<br>
[[File:Vol1 page 0147 eq 006.png]]....................(3.330)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0148 eq 001.png]]....................(3.331)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0148 eq 002.png]]....................(3.332)
<br>
<br>
where ''γ''=0.5772… and
<br>
<br>
[[File:Vol1 page 0148 eq 003.png]]....................(3.333)
<br>
<br>
It is also useful to note the real inversions of '''Eqs. 3.330''' and '''3.332''' given, respectively, by
<br>
<br>
[[File:Vol1 page 0148 eq 004.png]]....................(3.334)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0148 eq 005.png]]....................(3.335)
<br>
<br>
''The Series.'' [[File:Vol1 page 0148 inline 001.png]]
<br>
<br>
[[File:Vol1 page 0148 eq 006.png]]....................(3.336)
<br>
<br>
Two alternative expressions for the series ''S''<sub>1</sub> may be convenient for the large and small values of u (i.e., for short and long times).<ref name="r29" /> When ''u'' is large,
<br>
<br>
[[File:Vol1 page 0148 eq 007.png]]....................(3.337)
<br>
<br>
and when ''u'' + ''a''<sup>2</sup> << ''n''<sup>2</sup>''π''<sup>2</sup>/''h''<sup>2</sup><sub>''D''</sub>,
<br>
<br>
[[File:Vol1 page 0149 eq 001.png]]....................(3.338)
<br>
<br>
''The Series'' [[File:Vol1 page 0149 inline 001.png]].
<br>
<br>
[[File:Vol1 page 0149 eq 002.png]]....................(3.339)
<br>
<br>
Alternative computational forms for the series ''S''<sub>2</sub> are given next.<ref name="r29" /> When ''u'' is large,
<br>
<br>
[[File:Vol1 page 0149 eq 003.png]]....................(3.340)
<br>
<br>
and when ''u'' + ''a''<sup>2</sup> << ''n''<sup>2</sup>''π''<sup>2</sup>/''h''<sup>2</sup><sub>''D''</sub>,
<br>
<br>
[[File:Vol1 page 0149 eq 004.png]]....................(3.341)
<br>
<br>
''The Series'' [[File:Vol1 page 0149 inline 003.png]].
<br>
<br>
[[File:Vol1 page 0149 eq 005.png]]....................(3.342)
<br>
<br>
The following alternative forms for the series [[File:Vol1 page 0149 inline 002.png]] may be convenient for the large and small values of ''u'' (i.e., for short and long times).<ref name="r29" /> When ''u'' is large,
<br>
<br>
[[File:Vol1 page 0149 eq 006.png]]....................(3.343)
<br>
<br>
and when ''u'' + ''a''<sup>2</sup> << (2''n'' − 1)<sup>2</sup> ''π''<sup>2</sup>/(4''h''<sup>2</sup><sub>''D''</sub>),
<br>
<br>
[[File:Vol1 page 0150 eq 001.png]]....................(3.344)
<br>
<br>
''The Series [[File:Vol1 page 0150 inline 001.png]]''.
<br>
<br>
[[File:Vol1 page 0150 eq 002.png]]....................(3.345)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0150 eq 003.png]]....................(3.346)
<br>
<br>
The series [[File:Vol1 page 0150 inline 002.png]] may be written in the following forms with the use of '''Eqs. 3.324''' through '''3.326'''.
<br>
<br>
[[File:Vol1 page 0150 eq 004.png]]....................(3.347)
<br>
<br>
[[File:Vol1 page 0150 eq 005.png]]....................(3.348)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0150 eq 006.png]]
[[File:Vol1 page 0151 eq 001.png]]....................(3.349)
<br>
<br>
The computation of the series in '''Eqs. 3.347''' and '''3.348''' should not pose numerical difficulties; however, the series in '''Eq. 3.349''' converges slowly. With the relation given in '''Eq. 3.328''', we may write '''Eq. 3.349''' as<ref name="r29" />
<br>
<br>
[[File:Vol1 page 0151 eq 002.png]]....................(3.350)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0151 eq 003.png]]....................(3.351)
<br>
<br>
Before discussing the computation of the series given in '''Eq. 3.351''', we first discuss the derivation of the asymptotic approximations for the series [[File:Vol1 page 0150 inline 002.png]]. When ''s'' is large (small times), [[File:Vol1 page 0150 inline 002.png]] may be approximated by<ref name="r29" />
<br>
<br>
[[File:Vol1 page 0151 eq 004.png]]....................(3.352)
<br>
<br>
where ''β'' is given by '''Eq. 3.331'''. If s is sufficiently large, then '''Eq. 3.352''' may be further approximated by
<br>
<br>
[[File:Vol1 page 0151 eq 005.png]]....................(3.353)
<br>
<br>
The inverse Laplace transform of '''Eq. 3.353''' yields
<br>
<br>
[[File:Vol1 page 0151 eq 006.png]]....................(3.354)
<br>
<br>
For small ''s'' (large times), depending on the value of ''x''<sub>''D''</sub>, [[File:Vol1 page 0150 inline 002.png]] may be approximated by one of the following equations: <ref name="r29" />
<br>
<br>
[[File:Vol1 page 0152 eq 001.png]]....................(3.355)
<br>
<br>
[[File:Vol1 page 0152 eq 002.png]]....................(3.356)
<br>
<br>
[[File:Vol1 page 0152 eq 003.png]]....................(3.357)
<br>
<br>
where [[File:Vol1 page 0152 inline 001.png]] is given by '''Eq. 3.364'''.


''The Series'' [[File:Vol1 page 0152 inline 002.png]].
''The Series'' [[File:Vol1 page 0152 inline 002.png|RTENOTITLE]].<br/><br/>[[File:Vol1 page 0152 eq 004.png|RTENOTITLE]]....................(3.358)<br/><br/>where<br/><br/>[[File:Vol1 page 0152 eq 005.png|RTENOTITLE]]....................(3.346)<br/><br/>With the relations given in '''Eqs. 3.337''' and '''3.338''', the following alternative forms for the series [[File:Vol1 page 0152 inline 002.png|RTENOTITLE]] may be obtained, respectively, for the large and small values of s (i.e., for short and long times).<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref> When ''u'' is large,<br/><br/>[[File:Vol1 page 0152 eq 006.png|RTENOTITLE]]....................(3.359)<br/><br/>and when ''u'' << ''n''<sup>2</sup>''π''<sup>2</sup>/''h''<sup>2</sup><sub>''D''</sub>,<br/><br/>[[File:Vol1 page 0153 eq 001.png|RTENOTITLE]]....................(3.360)<br/><br/>It is also possible to derive asymptotic approximations for the series [[File:Vol1 page 0152 inline 002.png|RTENOTITLE]]. When ''s'' is large (small times), [[File:Vol1 page 0153 inline 002.png|RTENOTITLE]] may be approximated by<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0153 eq 002.png|RTENOTITLE]]....................(3.361)<br/><br/>If ''s'' is sufficiently large, then '''Eq. 3.361''' may be further approximated by<br/><br/>[[File:Vol1 page 0153 eq 003.png|RTENOTITLE]]....................(3.362)<br/><br/>The inverse Laplace transform of '''Eq. 3.362''' yields<br/><br/>[[File:Vol1 page 0153 eq 004.png|RTENOTITLE]]....................(3.363)<br/><br/>For small ''s'' (large times), [[File:Vol1 page 0152 inline 002.png|RTENOTITLE]] may be approximated by<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0153 eq 005.png|RTENOTITLE]]....................(3.364)<br/><br/>''The Ratio'' [[File:Vol1 page 0153 inline 001.png|RTENOTITLE]].<br/><br/>[[File:Vol1 page 0153 eq 006.png|RTENOTITLE]]....................(3.365)<br/><br/>By elementary considerations, the ratio [[File:Vol1 page 0153 inline 003.png|RTENOTITLE]] may be written as<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0154 eq 001.png|RTENOTITLE]]....................(3.366)<br/><br/>The expression given in '''Eq. 3.366''' provides computational advantages when ''s'' is small (time is large).
<br>
<br>
[[File:Vol1 page 0152 eq 004.png]]....................(3.358)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0152 eq 005.png]]....................(3.346)
<br>
<br>
With the relations given in '''Eqs. 3.337''' and '''3.338''', the following alternative forms for the series [[File:Vol1 page 0152 inline 002.png]] may be obtained, respectively, for the large and small values of s (i.e., for short and long times).<ref name="r29" /> When ''u'' is large,
<br>
<br>
[[File:Vol1 page 0152 eq 006.png]]....................(3.359)
<br>
<br>
and when ''u'' << ''n''<sup>2</sup>''π''<sup>2</sup>/''h''<sup>2</sup><sub>''D''</sub>,
<br>
<br>
[[File:Vol1 page 0153 eq 001.png]]....................(3.360)
<br>
<br>
It is also possible to derive asymptotic approximations for the series [[File:Vol1 page 0152 inline 002.png]]. When ''s'' is large (small times), [[File:Vol1 page 0153 inline 002.png]] may be approximated by<ref name="r29" />
<br>
<br>
[[File:Vol1 page 0153 eq 002.png]]....................(3.361)
<br>
<br>
If ''s'' is sufficiently large, then '''Eq. 3.361''' may be further approximated by
<br>
<br>
[[File:Vol1 page 0153 eq 003.png]]....................(3.362)
<br>
<br>
The inverse Laplace transform of '''Eq. 3.362''' yields
<br>
<br>
[[File:Vol1 page 0153 eq 004.png]]....................(3.363)
<br>
<br>
For small ''s'' (large times), [[File:Vol1 page 0152 inline 002.png]] may be approximated by<ref name="r29" />
<br>
<br>
[[File:Vol1 page 0153 eq 005.png]]....................(3.364)
<br>
<br>
''The Ratio'' [[File:Vol1 page 0153 inline 001.png]].
<br>
<br>
[[File:Vol1 page 0153 eq 006.png]]....................(3.365)
<br>
<br>
By elementary considerations, the ratio [[File:Vol1 page 0153 inline 003.png]] may be written as<ref name="r29" />
<br>
<br>
[[File:Vol1 page 0154 eq 001.png]]....................(3.366)
<br>
<br>
The expression given in '''Eq. 3.366''' provides computational advantages when ''s'' is small (time is large).


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----


'''Example 3.12'''
'''Example 3.12'''<br/><br/>Consider a fully penetrating, uniform-flux fracture of half-length ''x''<sub>''f''</sub> located at ''x′''=0, ''y′''=0 in an infinite-slab reservoir with closed top and bottom boundaries.<br/><br/>''Solution''.'''Table 3.6''' gives the solution for this problem. For simplicity, assuming an isotropic reservoir, choosing the characteristic length as ℓ = ''x''<sub>''f''</sub> and noting that [[File:Vol1 page 0154 inline 001.png|RTENOTITLE]], the solution becomes<br/><br/>[[File:Vol1 page 0154 eq 002.png|RTENOTITLE]]....................(3.367)<br/><br/>First consider the numerical evaluation of '''Eq. 3.367'''. We note from '''Eqs. 3.324''' through '''3.326''' that '''Eq. 3.367''' may be written in one of the following forms, depending on the value of ''x''<sub>''D''</sub>.<br/><br/>[[File:Vol1 page 0154 eq 003.png|RTENOTITLE]]....................(3.368)<br/><br/>[[File:Vol1 page 0154 eq 004.png|RTENOTITLE]]....................(3.369)<br/><br/>and<br/><br/>[[File:Vol1 page 0154 eq 005.png|RTENOTITLE]]....................(3.370)<br/><br/>The numerical evaluation of the integrals in '''Eqs. 3.368''' through '''3.370''' for ''y''<sub>''D''</sub> ≠ 0 should be straightforward with the use of the standard numerical integration algorithms. For ''y''<sub>''D''</sub> = 0, the polynomial approximations given by Luke or the power series expansion given by '''Eq. 3.327''' should be useful.<br/><br/>The short- and long-time asymptotic approximations of the fracture solution are also obtained by applying the relations given by '''Eqs. 3.330''' and '''3.332''', respectively, to the right side of '''Eq. 3.367'''. This procedure yields, for short times,<br/><br/>[[File:Vol1 page 0155 eq 001.png|RTENOTITLE]]....................(3.371)<br/><br/>or, in real-time domain,<br/><br/>[[File:Vol1 page 0155 eq 002.png|RTENOTITLE]]....................(3.372)<br/><br/>where ''β'' is given by '''Eq. 3.331''' with ''a'' = -1 and ''b'' = +1. At long times, the following asymptotic approximation may be used:<br/><br/>[[File:Vol1 page 0155 eq 003.png|RTENOTITLE]]....................(3.373)<br/><br/>or, in real-time domain,<br/><br/>[[File:Vol1 page 0155 eq 004.png|RTENOTITLE]]....................(3.374)<br/><br/>where ''γ'' = 0.5772… and ''σ''(''x''<sub>''D''</sub>, ''y''<sub>''D''</sub>, -1, +1) is given by '''Eq. 3.333'''.
<br>
<br>
Consider a fully penetrating, uniform-flux fracture of half-length ''x''<sub>''f''</sub> located at ''x′''=0, ''y′''=0 in an infinite-slab reservoir with closed top and bottom boundaries.
<br>
<br>
''Solution''.'''Table 3.6''' gives the solution for this problem. For simplicity, assuming an isotropic reservoir, choosing the characteristic length as ℓ = ''x''<sub>''f''</sub> and noting that [[File:Vol1 page 0154 inline 001.png]], the solution becomes
<br>
<br>
[[File:Vol1 page 0154 eq 002.png]]....................(3.367)
<br>
<br>
First consider the numerical evaluation of '''Eq. 3.367'''. We note from '''Eqs. 3.324''' through '''3.326''' that '''Eq. 3.367''' may be written in one of the following forms, depending on the value of ''x''<sub>''D''</sub>.
<br>
<br>
[[File:Vol1 page 0154 eq 003.png]]....................(3.368)
<br>
<br>
[[File:Vol1 page 0154 eq 004.png]]....................(3.369)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0154 eq 005.png]]....................(3.370)
<br>
<br>
The numerical evaluation of the integrals in '''Eqs. 3.368''' through '''3.370''' for ''y''<sub>''D''</sub> ≠ 0 should be straightforward with the use of the standard numerical integration algorithms. For ''y''<sub>''D''</sub> = 0, the polynomial approximations given by Luke<ref name="r30" /> or the power series expansion given by '''Eq. 3.327''' should be useful.
<br>
<br>
The short- and long-time asymptotic approximations of the fracture solution are also obtained by applying the relations given by '''Eqs. 3.330''' and '''3.332''', respectively, to the right side of '''Eq. 3.367'''. This procedure yields, for short times,
<br>
<br>
[[File:Vol1 page 0155 eq 001.png]]....................(3.371)
<br>
<br>
or, in real-time domain,
<br>
<br>
[[File:Vol1 page 0155 eq 002.png]]....................(3.372)
<br>
<br>
where ''β'' is given by '''Eq. 3.331''' with ''a'' = -1 and ''b'' = +1. At long times, the following asymptotic approximation may be used:
<br>
<br>
[[File:Vol1 page 0155 eq 003.png]]....................(3.373)
<br>
<br>
or, in real-time domain,
<br>
<br>
[[File:Vol1 page 0155 eq 004.png]]....................(3.374)
<br>
<br>
where ''γ'' = 0.5772… and ''σ''(''x''<sub>''D''</sub>, ''y''<sub>''D''</sub>, -1, +1) is given by '''Eq. 3.333'''.


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----


'''Example 3.13'''
'''Example 3.13'''<br/><br/>Consider a horizontal well of length ''L''<sub>''h''</sub> located at ''x′'' = 0, ''y′'' = 0, and ''z′'' = ''z''<sub>''w''</sub> in an infinite-slab reservoir with closed top and bottom boundaries.<br/><br/>''Solution''.'''Table 3.6''' gives the horizontal-well solution for an infinite-slab reservoir with impermeable boundaries. Assuming an isotropic reservoir, choosing the characteristic length as ℓ = ''L''<sub>''h''</sub> / 2 and noting that [[File:Vol1 page 0155 inline 001.png|RTENOTITLE]], the solution may be written as<br/><br/>[[File:Vol1 page 0155 eq 005.png|RTENOTITLE]]....................(3.375)<br/><br/>where [[File:Vol1 page 0155 inline 002.png|RTENOTITLE]] is the fracture solution given by the right side of Eq. 3.367 and [[File:Vol1 page 0150 inline 002.png|RTENOTITLE]] is given by<br/><br/>[[File:Vol1 page 0155 eq 006.png|RTENOTITLE]]....................(3.376)<br/><br/>with<br/><br/>[[File:Vol1 page 0155 eq 007.png|RTENOTITLE]]....................(3.346)<br/><br/>[[File:Vol1 page 0155 eq 008.png|RTENOTITLE]]....................(3.377) br><br/>and<br/><br/>[[File:Vol1 page 0155 eq 009.png|RTENOTITLE]]....................(3.378)<br/><br/>The computation of the first term in the right side of '''Eq. 3.375''' [[File:Vol1 page 0156 inline 001.png|RTENOTITLE]] is the same as the computation of the fracture solution given by '''Eq. 3.367''' and has been discussed in '''Example 3.12'''. The computational form of the second term [[File:Vol1 page 0156 inline 002.png|RTENOTITLE]] in the right side of '''Eq. 3.375''' is given by '''Eqs. 3.347''' through '''3.350'''. Of particular interest is the case for −1 ≤ ''x''<sub>''D''</sub> ≤ +1. In this case, from '''Eqs. 3.350''' and '''3.351''', we have<br/><br/>[[File:Vol1 page 0156 eq 001.png|RTENOTITLE]]....................(3.379)<br/><br/>where<br/><br/>[[File:Vol1 page 0156 eq 002.png|RTENOTITLE]]....................(3.380)<br/><br/>The computational considerations for the series [[File:Vol1 page 0156 inline 004.png|RTENOTITLE]] have been discussed previously.<br/><br/>Next, we consider the short- and long-time approximations of the horizontal-well solution given by '''Eq. 3.375'''. To obtain a short-time approximation, we substitute the asymptotic expressions for [[File:Vol1 page 0155 inline 002.png|RTENOTITLE]] and [[File:Vol1 page 0150 inline 002.png|RTENOTITLE]] as s→∞ given, respectively, by '''Eqs. 3.371''' and '''3.353'''. This yields<br/><br/>[[File:Vol1 page 0156 eq 003.png|RTENOTITLE]]....................(3.381)<br/><br/>where ''β'' is given by '''Eq. 3.331'''. The inverse Laplace transform of '''Eq. 3.381''' is given by<br/><br/>[[File:Vol1 page 0156 eq 004.png|RTENOTITLE]]....................(3.382)<br/><br/>To obtain the long-time approximation of '''Eq. 3.375''', we substitute the asymptotic expressions for [[File:Vol1 page 0156 inline 003.png|RTENOTITLE]] and [[File:Vol1 page 0150 inline 001.png|RTENOTITLE]] as s→∞ given, respectively, by '''Eq. 3.374''' and '''Eqs. 3.355''' through '''3.357'''. Of particular interest is the case for −1 ≤ ''x''<sub>''D''</sub> ≤ +1, where we have<br/><br/>[[File:Vol1 page 0156 eq 005.png|RTENOTITLE]] [[File:Vol1 page 0157 eq 001.png|RTENOTITLE]]....................(3.383)<br/><br/>where ''γ''=0.5772… and ''σ''(''x''<sub>''D''</sub>, ''y''<sub>''D''</sub>, -1, +1) is given by '''Eq. 3.333'''. The inverse Laplace transform of '''Eq. 3.383''' yields<br/><br/>[[File:Vol1 page 0157 eq 002.png|RTENOTITLE]]....................(3.384)<br/>
<br>
<br>
Consider a horizontal well of length ''L''<sub>''h''</sub> located at ''x′'' = 0, ''y′'' = 0, and ''z′'' = ''z''<sub>''w''</sub> in an infinite-slab reservoir with closed top and bottom boundaries.
<br>
<br>
''Solution''.'''Table 3.6''' gives the horizontal-well solution for an infinite-slab reservoir with impermeable boundaries. Assuming an isotropic reservoir, choosing the characteristic length as ℓ = ''L''<sub>''h''</sub> / 2 and noting that [[File:Vol1 page 0155 inline 001.png]], the solution may be written as
<br>
<br>
[[File:Vol1 page 0155 eq 005.png]]....................(3.375)
<br>
<br>
where [[File:Vol1 page 0155 inline 002.png]] is the fracture solution given by the right side of Eq. 3.367 and [[File:Vol1 page 0150 inline 002.png]] is given by
<br>
<br>
[[File:Vol1 page 0155 eq 006.png]]....................(3.376)
<br>
<br>
with
<br>
<br>
[[File:Vol1 page 0155 eq 007.png]]....................(3.346)
<br>
<br>
[[File:Vol1 page 0155 eq 008.png]]....................(3.377)
br>
<br>
and
<br>
<br>
[[File:Vol1 page 0155 eq 009.png]]....................(3.378)
<br>
<br>
The computation of the first term in the right side of '''Eq. 3.375''' [[File:Vol1 page 0156 inline 001.png]] is the same as the computation of the fracture solution given by '''Eq. 3.367''' and has been discussed in '''Example 3.12'''. The computational form of the second term [[File:Vol1 page 0156 inline 002.png]] in the right side of '''Eq. 3.375''' is given by '''Eqs. 3.347''' through '''3.350'''. Of particular interest is the case for −1 ≤ ''x''<sub>''D''</sub> ≤ +1. In this case, from '''Eqs. 3.350''' and '''3.351''', we have
<br>
<br>
[[File:Vol1 page 0156 eq 001.png]]....................(3.379)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0156 eq 002.png]]....................(3.380)
<br>
<br>
The computational considerations for the series [[File:Vol1 page 0156 inline 004.png]] have been discussed previously.
<br>
<br>
Next, we consider the short- and long-time approximations of the horizontal-well solution given by '''Eq. 3.375'''. To obtain a short-time approximation, we substitute the asymptotic expressions for [[File:Vol1 page 0155 inline 002.png]] and [[File:Vol1 page 0150 inline 002.png]] as s→∞ given, respectively, by '''Eqs. 3.371''' and '''3.353'''. This yields
<br>
<br>
[[File:Vol1 page 0156 eq 003.png]]....................(3.381)
<br>
<br>
where ''β'' is given by '''Eq. 3.331'''. The inverse Laplace transform of '''Eq. 3.381''' is given by
<br>
<br>
[[File:Vol1 page 0156 eq 004.png]]....................(3.382)
<br>
<br>
To obtain the long-time approximation of '''Eq. 3.375''', we substitute the asymptotic expressions for [[File:Vol1 page 0156 inline 003.png]] and [[File:Vol1 page 0150 inline 001.png]] as s→∞ given, respectively, by '''Eq. 3.374''' and '''Eqs. 3.355''' through '''3.357'''. Of particular interest is the case for −1 ≤ ''x''<sub>''D''</sub> ≤ +1, where we have
<br>
<br>
[[File:Vol1 page 0156 eq 005.png]]
[[File:Vol1 page 0157 eq 001.png]]....................(3.383)
<br>
<br>
where ''γ''=0.5772… and ''σ''(''x''<sub>''D''</sub>, ''y''<sub>''D''</sub>, -1, +1) is given by '''Eq. 3.333'''. The inverse Laplace transform of '''Eq. 3.383''' yields
<br>
<br>
[[File:Vol1 page 0157 eq 002.png]]....................(3.384)
<br>
<br>


----
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'''Example 3.14'''
'''Example 3.14''' Consider a fully penetrating, uniform-flux fracture of half-length ''x''<sub>''f''</sub> in an isotropic and closed cylindrical reservoir. The center of the fracture is at ''r′'' = 0, ''θ''′ = 0 and the fracture tips extend from (''r′'' = ''x''<sub>''f''</sub>, ''θ'' = ''α'' + ''π'') to (''r′'' = ''x''<sub>''f''</sub>, ''θ'' = ''α'').<br/><br/>''Solution''. The solution for this problem has been obtained in '''Eq. 3.278''' in '''Example 3.8''' with ''h''<sub>''w''</sub> = ''h''. Choosing the characteristic length as ℓ = ''x''<sub>''f''</sub> and noting that [[File:Vol1 page 0154 inline 001.png|RTENOTITLE]], the solution is given by<br/><br/>[[File:Vol1 page 0157 eq 003.png|RTENOTITLE]]....................(3.385)<br/><br/>For the computation of the pressure responses at the center of the fracture (''r''<sub>''D''</sub> = 0), '''Eq. 3.385''' simplifies to<br/><br/>[[File:Vol1 page 0157 eq 004.png|RTENOTITLE]]....................(3.386)<br/><br/>It is also possible to find a very good approximation for '''Eq. 3.385''', especially when ''r''<sub>''eD''</sub> is large. If we assume<ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref><br/><br/>[[File:Vol1 page 0158 eq 001.png|RTENOTITLE]]....................(3.387)<br/><br/>and use the following relation<ref name="r4">Forchheimer, P.F. 1901. Wasserbewegung durch Boden. Zeitschrift des Vereines deutscher Ingenieure 45 (5): 1781–1788.</ref><br/><br/>[[File:Vol1 page 0158 eq 002.png|RTENOTITLE]]....................(3.388)<br/><br/>then '''Eq. 3.385''' may be written as<br/><br/>[[File:Vol1 page 0158 eq 003.png|RTENOTITLE]]....................(3.389)<br/><br/>Because<ref name="r19">Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.</ref><br/><br/>[[File:Vol1 page 0158 eq 004.png|RTENOTITLE]]....................(3.390)<br/><br/>where<br/><br/>[[File:Vol1 page 0158 eq 005.png|RTENOTITLE]]....................(3.391)<br/><br/>'''Eq. 3.389''' may also be written as<br/><br/>[[File:Vol1 page 0158 eq 006.png|RTENOTITLE]]....................(3.392)<br/><br/>Although the assumption given in '''Eq. 3.387''' may not be justified by itself, the solution given in '''Eq. 3.392''' is a very good approximation for '''Eq. 3.385''', especially when ''r''<sub>''eD''</sub> is large. For a fracture at the center of the cylindrical drainage region, '''Eq. 3.392''' simplifies to<br/><br/>[[File:Vol1 page 0159 eq 001.png|RTENOTITLE]]....................(3.393)<br/><br/>It is also possible to obtain short- and long-time approximations for the solution given in '''Eq. 3.393'''. For short times, ''u''→∞ and the second term in the argument of the integral in '''Eq. 3.393''' becomes negligible compared with the first term. Then, '''Eq. 3.393''' reduces to the solution for an infinite-slab reservoir given by '''Eq. 3.367''', of which the short-time approximation has been discussed in '''Example 3.12'''.<br/><br/>To obtain a long-time approximation, we evaluate '''Eq. 3.393''' at the limit as ''s''→0 (''u''→''s''). As shown in '''Sec. 3.2.3''', for small arguments we may approximate the Bessel functions in '''Eq. 3.393''' by<br/><br/>[[File:Vol1 page 0159 eq 002.png|RTENOTITLE]]....................(3.394)<br/><br/>[[File:Vol1 page 0159 eq 003.png|RTENOTITLE]]....................(3.395)<br/><br/>[[File:Vol1 page 0159 eq 004.png|RTENOTITLE]]....................(3.396)<br/><br/>and<br/><br/>[[File:Vol1 page 0159 eq 005.png|RTENOTITLE]]....................(3.397)<br/><br/>where ''γ'' = 0.5772…. With '''Eqs. 3.394''' through '''3.397''' and by neglecting the terms of the order ''s''<sup>3/2</sup>, we may write<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0159 eq 006.png|RTENOTITLE]]....................(3.398)<br/><br/>If we substitute the right side of '''Eq. 3.398''' into '''Eq. 3.393''', we obtain<br/><br/>[[File:Vol1 page 0159 eq 007.png|RTENOTITLE]]....................(3.399)<br/><br/>where ''σ''(''x''<sub>''D''</sub>, ''y''<sub>''D''</sub>, −1, +1) is given by '''Eq. 3.333''' and<br/><br/>[[File:Vol1 page 0159 eq 008.png|RTENOTITLE]]....................(3.400)<br/><br/>The inverse Laplace transform of '''Eq. 3.399''' yields the following long-time approximation for a uniform-flux fracture at the center of a closed square:<br/><br/>[[File:Vol1 page 0160 eq 001.png|RTENOTITLE]]....................(3.401)<br/>
Consider a fully penetrating, uniform-flux fracture of half-length ''x''<sub>''f''</sub> in an isotropic and closed cylindrical reservoir. The center of the fracture is at ''r′'' = 0, ''θ''′ = 0 and the fracture tips extend from (''r′'' = ''x''<sub>''f''</sub>, ''θ'' = ''α'' + ''π'') to (''r′'' = ''x''<sub>''f''</sub>, ''θ'' = ''α'').
<br>
<br>
''Solution''. The solution for this problem has been obtained in '''Eq. 3.278''' in '''Example 3.8''' with ''h''<sub>''w''</sub> = ''h''. Choosing the characteristic length as ℓ = ''x''<sub>''f''</sub> and noting that [[File:Vol1 page 0154 inline 001.png]], the solution is given by
<br>
<br>
[[File:Vol1 page 0157 eq 003.png]]....................(3.385)
<br>
<br>
For the computation of the pressure responses at the center of the fracture (''r''<sub>''D''</sub> = 0), '''Eq. 3.385''' simplifies to
<br>
<br>
[[File:Vol1 page 0157 eq 004.png]]....................(3.386)
<br>
<br>
It is also possible to find a very good approximation for '''Eq. 3.385''', especially when ''r''<sub>''eD''</sub> is large. If we assume<ref name="r19" />
<br>
<br>
[[File:Vol1 page 0158 eq 001.png]]....................(3.387)
<br>
<br>
and use the following relation<ref name="r4" />
<br>
<br>
[[File:Vol1 page 0158 eq 002.png]]....................(3.388)
<br>
<br>
then '''Eq. 3.385''' may be written as
<br>
<br>
[[File:Vol1 page 0158 eq 003.png]]....................(3.389)
<br>
<br>
Because<ref name="r19" />
<br>
<br>
[[File:Vol1 page 0158 eq 004.png]]....................(3.390)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0158 eq 005.png]]....................(3.391)
<br>
<br>
'''Eq. 3.389''' may also be written as
<br>
<br>
[[File:Vol1 page 0158 eq 006.png]]....................(3.392)
<br>
<br>
Although the assumption given in '''Eq. 3.387''' may not be justified by itself, the solution given in '''Eq. 3.392''' is a very good approximation for '''Eq. 3.385''', especially when ''r''<sub>''eD''</sub> is large. For a fracture at the center of the cylindrical drainage region, '''Eq. 3.392''' simplifies to
<br>
<br>
[[File:Vol1 page 0159 eq 001.png]]....................(3.393)
<br>
<br>
It is also possible to obtain short- and long-time approximations for the solution given in '''Eq. 3.393'''. For short times, ''u''→∞ and the second term in the argument of the integral in '''Eq. 3.393''' becomes negligible compared with the first term. Then, '''Eq. 3.393''' reduces to the solution for an infinite-slab reservoir given by '''Eq. 3.367''', of which the short-time approximation has been discussed in '''Example 3.12'''.
<br>
<br>
To obtain a long-time approximation, we evaluate '''Eq. 3.393''' at the limit as ''s''→0 (''u''→''s''). As shown in '''Sec. 3.2.3''', for small arguments we may approximate the Bessel functions in '''Eq. 3.393''' by
<br>
<br>
[[File:Vol1 page 0159 eq 002.png]]....................(3.394)
<br>
<br>
[[File:Vol1 page 0159 eq 003.png]]....................(3.395)
<br>
<br>
[[File:Vol1 page 0159 eq 004.png]]....................(3.396)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0159 eq 005.png]]....................(3.397)
<br>
<br>
where ''γ'' = 0.5772…. With '''Eqs. 3.394''' through '''3.397''' and by neglecting the terms of the order ''s''<sup>3/2</sup>, we may write<ref name="r29" />
<br>
<br>
[[File:Vol1 page 0159 eq 006.png]]....................(3.398)
<br>
<br>
If we substitute the right side of '''Eq. 3.398''' into '''Eq. 3.393''', we obtain
<br>
<br>
[[File:Vol1 page 0159 eq 007.png]]....................(3.399)
<br>
<br>
where ''σ''(''x''<sub>''D''</sub>, ''y''<sub>''D''</sub>, −1, +1) is given by '''Eq. 3.333''' and
<br>
<br>
[[File:Vol1 page 0159 eq 008.png]]....................(3.400)
<br>
<br>
The inverse Laplace transform of '''Eq. 3.399''' yields the following long-time approximation for a uniform-flux fracture at the center of a closed square:
<br>
<br>
[[File:Vol1 page 0160 eq 001.png]]....................(3.401)
<br>
<br>


----
----


'''Example 3.15'''
'''Example 3.15'''<br/><br/>Consider a fully penetrating, uniform-flux fracture of half-length ''x''<sub>''f''</sub> in an isotropic and closed parallelepiped reservoir of dimensions ''x''<sub>''e''</sub> × ''y''<sub>''e''</sub> × ''h''. The fracture is parallel to the ''x'' axis and centered at ''x''<sub>''w''</sub>, ''y''<sub>''w''</sub>, ''z''<sub>''w''</sub>.<br/><br/>''Solution''. The solution for this problem has been obtained in '''Example 3.10''' and, by choosing ℓ = ''x''<sub>''f''</sub>, is given by<br/><br/>[[File:Vol1 page 0160 eq 002.png|RTENOTITLE]]....................(3.402)<br/><br/>where<br/><br/>[[File:Vol1 page 0160 eq 003.png|RTENOTITLE]]....................(3.403)<br/><br/>The computation of the ratios of the hyperbolic functions in '''Eq. 3.402''' may be difficult, especially when their arguments approach zero or infinity. When ''s'' is small (long times), '''Eq. 3.366''' should be useful to compute the ratios of the hyperbolic functions. When ''s'' is large (small times), with '''Eq. 3.366''' the solution given in '''Eq. 3.402''' may be written as<ref name="r29">Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.</ref><br/><br/>[[File:Vol1 page 0160 eq 004.png|RTENOTITLE]]....................(3.404)<br/><br/>where<br/><br/>[[File:Vol1 page 0160 eq 005.png|RTENOTITLE]]....................(3.405)<br/><br/>[[File:Vol1 page 0161 eq 001.png|RTENOTITLE]]....................(3.406)<br/><br/>and<br/><br/>[[File:Vol1 page 0161 eq 002.png|RTENOTITLE]]....................(3.407)<br/><br/>The last equality in '''Eq. 3.405''' follows from the relation given by '''Eq. 3.349'''. The expression given in '''Eq. 3.405''' may also be written as<br/><br/>[[File:Vol1 page 0161 eq 003.png|RTENOTITLE]]....................(3.408)<br/><br/>where<br/><br/>[[File:Vol1 page 0161 eq 004.png|RTENOTITLE]]....................(3.409)<br/><br/>and<br/><br/>[[File:Vol1 page 0161 eq 005.png|RTENOTITLE]]....................(3.410)<br/><br/>Therefore, the solution given by '''Eq. 3.402''' may be written as follows for computation at early times (for large values of ''s''):<br/><br/>[[File:Vol1 page 0162 eq 001.png|RTENOTITLE]]....................(3.411)<br/><br/>where [[File:Vol1 page 0162 inline 001.png|RTENOTITLE]] is given by '''Eq. 3.409''' and corresponds to the solution for a fractured well in an infinite-slab reservoir (see '''Eq. 3.367''' in '''Example 3.12''') and [[File:Vol1 page 0162 inline 002.png|RTENOTITLE]] represents the contribution of the lateral boundaries and is given by<br/><br/>[[File:Vol1 page 0162 eq 002.png|RTENOTITLE]]....................(3.412)<br/><br/>In Eq. 3.412, [[File:Vol1 page 0162 inline 003.png|RTENOTITLE]], [[File:Vol1 page 0162 inline 004.png|RTENOTITLE]], and [[File:Vol1 page 0162 inline 005.png|RTENOTITLE]] are given, respectively, by '''Eqs. 3.406''', '''3.407''', and '''3.410'''. The integrals appearing in '''Eqs. 3.409''' and '''3.410''' may be evaluated by following the lines outlined by '''Eqs. 3.324''' through '''3.326'''.<br/><br/>It is also possible to derive short- and long-time approximations for the fracture solution in a closed rectangular parallelepiped. The short-time approximation corresponds to the limit of the solution as ''s''→∞. It can be easily shown that the [[File:Vol1 page 0162 inline 006.png|RTENOTITLE]] term in '''Eq. 3.411''' becomes negligible compared with the [[File:Vol1 page 0162 inline 002.png|RTENOTITLE]] term for which a short-time approximation has been obtained in '''Example 3.12''' (see '''Eqs. 3.371''' and '''3.372''').<br/><br/>To obtain a long-time approximation (small values of ''s''), the solution given in '''Eq. 3.402''' may be written as<ref name="r27">Kazemi, H. 1969. Pressure Transient Analysis of Naturally Fractured Reservoir with Uniform Fracture Distribution. SPE J. 9 (4): 451–462. SPE-2156-PA. http://dx.doi.org/10.2118/2156-PA</ref><br/><br/>[[File:Vol1 page 0162 eq 003.png|RTENOTITLE]]....................(3.413)<br/><br/>where<br/><br/>[[File:Vol1 page 0162 eq 004.png|RTENOTITLE]]....................(3.414)<br/><br/>and<br/><br/>[[File:Vol1 page 0162 eq 005.png|RTENOTITLE]]....................(3.415)<br/><br/>The second equality in Eq. 3.414 results from<br/><br/>[[File:Vol1 page 0162 eq 006.png|RTENOTITLE]]....................(3.416)<br/><br/>For small values of ''s'', replacing ''u'' by ''s'' and ''s'' + ''α'' by ''α'', and with<br/><br/>[[File:Vol1 page 0163 eq 001.png|RTENOTITLE]]....................(3.417)<br/><br/>the term ''H'' given by '''Eq. 3.414''' may be approximated by<br/><br/>[[File:Vol1 page 0163 eq 002.png|RTENOTITLE]]....................(3.418)<br/><br/>The long-time approximation of the second term in '''Eq. 3.413''' is obtained by assuming ''u'' << ''k''<sup>2</sup>''π''<sup>2</sup>/''x''<sup>2</sup><sub>''eD''</sub> and taking the inverse Laplace transform of the resulting expressions; therefore, we can obtain the following long-time approximation<br/><br/>[[File:Vol1 page 0163 eq 003.png|RTENOTITLE]]....................(3.419)<br/>
<br>
<br>
Consider a fully penetrating, uniform-flux fracture of half-length ''x''<sub>''f''</sub> in an isotropic and closed parallelepiped reservoir of dimensions ''x''<sub>''e''</sub> × ''y''<sub>''e''</sub> × ''h''. The fracture is parallel to the ''x'' axis and centered at ''x''<sub>''w''</sub>, ''y''<sub>''w''</sub>, ''z''<sub>''w''</sub>.
<br>
<br>
''Solution''. The solution for this problem has been obtained in '''Example 3.10''' and, by choosing ℓ = ''x''<sub>''f''</sub>, is given by
<br>
<br>
[[File:Vol1 page 0160 eq 002.png]]....................(3.402)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0160 eq 003.png]]....................(3.403)
<br>
<br>
The computation of the ratios of the hyperbolic functions in '''Eq. 3.402''' may be difficult, especially when their arguments approach zero or infinity. When ''s'' is small (long times), '''Eq. 3.366''' should be useful to compute the ratios of the hyperbolic functions. When ''s'' is large (small times), with '''Eq. 3.366''' the solution given in '''Eq. 3.402''' may be written as<ref name="r29" />
<br>
<br>
[[File:Vol1 page 0160 eq 004.png]]....................(3.404)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0160 eq 005.png]]....................(3.405)
<br>
<br>
[[File:Vol1 page 0161 eq 001.png]]....................(3.406)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0161 eq 002.png]]....................(3.407)
<br>
<br>
The last equality in '''Eq. 3.405''' follows from the relation given by '''Eq. 3.349'''. The expression given in '''Eq. 3.405''' may also be written as
<br>
<br>
[[File:Vol1 page 0161 eq 003.png]]....................(3.408)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0161 eq 004.png]]....................(3.409)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0161 eq 005.png]]....................(3.410)
<br>
<br>
Therefore, the solution given by '''Eq. 3.402''' may be written as follows for computation at early times (for large values of ''s''):
<br>
<br>
[[File:Vol1 page 0162 eq 001.png]]....................(3.411)
<br>
<br>
where [[File:Vol1 page 0162 inline 001.png]] is given by '''Eq. 3.409''' and corresponds to the solution for a fractured well in an infinite-slab reservoir (see '''Eq. 3.367''' in '''Example 3.12''') and [[File:Vol1 page 0162 inline 002.png]] represents the contribution of the lateral boundaries and is given by
<br>
<br>
[[File:Vol1 page 0162 eq 002.png]]....................(3.412)
<br>
<br>
In Eq. 3.412, [[File:Vol1 page 0162 inline 003.png]], [[File:Vol1 page 0162 inline 004.png]], and [[File:Vol1 page 0162 inline 005.png]] are given, respectively, by '''Eqs. 3.406''', '''3.407''', and '''3.410'''. The integrals appearing in '''Eqs. 3.409''' and '''3.410''' may be evaluated by following the lines outlined by '''Eqs. 3.324''' through '''3.326'''.
<br>
<br>
It is also possible to derive short- and long-time approximations for the fracture solution in a closed rectangular parallelepiped. The short-time approximation corresponds to the limit of the solution as ''s''→∞. It can be easily shown that the [[File:Vol1 page 0162 inline 006.png]] term in '''Eq. 3.411''' becomes negligible compared with the [[File:Vol1 page 0162 inline 002.png]] term for which a short-time approximation has been obtained in '''Example 3.12''' (see '''Eqs. 3.371''' and '''3.372''').
<br>
<br>
To obtain a long-time approximation (small values of ''s''), the solution given in '''Eq. 3.402''' may be written as<ref name="r27" />
<br>
<br>
[[File:Vol1 page 0162 eq 003.png]]....................(3.413)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0162 eq 004.png]]....................(3.414)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0162 eq 005.png]]....................(3.415)
<br>
<br>
The second equality in Eq. 3.414 results from<ref name="r31" />
<br>
<br>
[[File:Vol1 page 0162 eq 006.png]]....................(3.416)
<br>
<br>
For small values of ''s'', replacing ''u'' by ''s'' and ''s'' + ''α'' by ''α'', and with<ref name="r31" />
<br>
<br>
[[File:Vol1 page 0163 eq 001.png]]....................(3.417)
<br>
<br>
the term ''H'' given by '''Eq. 3.414''' may be approximated by
<br>
<br>
[[File:Vol1 page 0163 eq 002.png]]....................(3.418)
<br>
<br>
The long-time approximation of the second term in '''Eq. 3.413''' is obtained by assuming ''u'' << ''k''<sup>2</sup>''π''<sup>2</sup>/''x''<sup>2</sup><sub>''eD''</sub> and taking the inverse Laplace transform of the resulting expressions; therefore, we can obtain the following long-time approximation
<br>
<br>
[[File:Vol1 page 0163 eq 003.png]]....................(3.419)
<br>
<br>


----
----


'''Example 3.16'''
'''Example 3.16''' Consider a uniform-flux horizontal well of length ''L''<sub>''h''</sub> in an isotropic and closed parallelepiped reservoir of dimensions ''x''<sub>''e''</sub> × ''y''<sub>''e''</sub> × ''h''. The center of the well is at ''x''<sub>''w''</sub>, ''y''<sub>''w''</sub>, ''z''<sub>''w''</sub>, and the well is parallel to the x axis.<br/><br/>''Solution''. The solution for this problem was obtained in '''Example 3.11''' and, by choosing ℓ = ''L''<sub>''h''</sub> / 2, is given by<br/><br/>[[File:Vol1 page 0163 eq 004.png|RTENOTITLE]]....................(3.420)<br/><br/>where [[File:Vol1 page 0163 inline 001.png|RTENOTITLE]] is the solution discussed in '''Example 3.15''', and [[File:Vol1 page 0153 inline 002.png|RTENOTITLE]] is given by<br/><br/>[[File:Vol1 page 0163 eq 005.png|RTENOTITLE]] [[File:Vol1 page 0164 eq 001.png|RTENOTITLE]]....................(3.421)<br/><br/>In '''Eq. 3.421''', [[File:Vol1 page 0164 inline 001.png|RTENOTITLE]] and [[File:Vol1 page 0164 inline 002.png|RTENOTITLE]] are given by '''Eqs. 3.377''' and '''3.378''', respectively,<br/><br/>[[File:Vol1 page 0164 eq 002.png|RTENOTITLE]]....................(3.346)<br/><br/>and<br/><br/>[[File:Vol1 page 0164 eq 003.png|RTENOTITLE]]....................(3.422)<br/><br/>The computation and the asymptotic approximations of the [[File:Vol1 page 0164 inline 003.png|RTENOTITLE]] term have been discussed in '''Example 3.15'''. To compute the [[File:Vol1 page 0152 inline 002.png|RTENOTITLE]] term for long times (small ''s''), the relation for the ratios of the hyperbolic functions given by '''Eq. 3.366''' should be useful. For computations at short times (large values of ''s''), following the lines similar to those in '''Example 3.15''', the [[File:Vol1 page 0153 inline 002.png|RTENOTITLE]] term in '''Eq. 3.421''' may be written as<br/><br/>[[File:Vol1 page 0164 eq 004.png|RTENOTITLE]]....................(3.423)<br/><br/>where<br/><br/>[[File:Vol1 page 0164 eq 005.png|RTENOTITLE]]....................(3.424)<br/><br/>[[File:Vol1 page 0164 eq 006.png|RTENOTITLE]]....................(3.425)<br/><br/>[[File:Vol1 page 0164 eq 007.png|RTENOTITLE]]....................(3.426)<br/><br/>[[File:Vol1 page 0164 eq 008.png|RTENOTITLE]] [[File:Vol1 page 0165 eq 001.png|RTENOTITLE]]....................(3.427)<br/><br/>and<br/><br/>[[File:Vol1 page 0165 eq 002.png|RTENOTITLE]]....................(3.428)<br/><br/>The computational form of the [[File:Vol1 page 0150 inline 002.png|RTENOTITLE]] term in '''Eq. 3.424''' is obtained by applying the relations given by '''Eqs. 3.347''' through '''3.350''' and '''Eq. 3.328'''. Of particular interest is the case for −1 ≤ ''x''<sub>''D''</sub> ≤ +1 and ''y''<sub>''D''</sub> = ''y''<sub>''wD''</sub> given by<br/><br/>[[File:Vol1 page 0165 eq 003.png|RTENOTITLE]]....................(3.429)<br/><br/>where<br/><br/>[[File:Vol1 page 0165 eq 004.png|RTENOTITLE]]....................(3.430)<br/><br/>which can be written as follows by using the relation given in '''Eq. 3.337''':<br/><br/>[[File:Vol1 page 0165 eq 005.png|RTENOTITLE]]....................(3.431)<br/><br/>Similarly, for −1 ≤ ''x''<sub>''D''</sub> ≤ +1 and ''y''<sub>''D''</sub> = ''y''<sub>''wD''</sub>, the [[File:Vol1 page 0165 inline 001.png|RTENOTITLE]] term given in '''Eq. 3.428''' may be written as<br/><br/>[[File:Vol1 page 0165 eq 006.png|RTENOTITLE]] [[File:Vol1 page 0166 eq 001.png|RTENOTITLE]]....................(3.432)<br/><br/>where<br/><br/>[[File:Vol1 page 0166 eq 002.png|RTENOTITLE]]....................(3.433)<br/>
Consider a uniform-flux horizontal well of length ''L''<sub>''h''</sub> in an isotropic and closed parallelepiped reservoir of dimensions ''x''<sub>''e''</sub> × ''y''<sub>''e''</sub> × ''h''. The center of the well is at ''x''<sub>''w''</sub>, ''y''<sub>''w''</sub>, ''z''<sub>''w''</sub>, and the well is parallel to the x axis.
<br>
<br>
''Solution''. The solution for this problem was obtained in '''Example 3.11''' and, by choosing ℓ = ''L''<sub>''h''</sub> / 2, is given by
<br>
<br>
[[File:Vol1 page 0163 eq 004.png]]....................(3.420)
<br>
<br>
where [[File:Vol1 page 0163 inline 001.png]] is the solution discussed in '''Example 3.15''', and [[File:Vol1 page 0153 inline 002.png]] is given by
<br>
<br>
[[File:Vol1 page 0163 eq 005.png]]
[[File:Vol1 page 0164 eq 001.png]]....................(3.421)
<br>
<br>
In '''Eq. 3.421''', [[File:Vol1 page 0164 inline 001.png]] and [[File:Vol1 page 0164 inline 002.png]] are given by '''Eqs. 3.377''' and '''3.378''', respectively,
<br>
<br>
[[File:Vol1 page 0164 eq 002.png]]....................(3.346)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0164 eq 003.png]]....................(3.422)
<br>
<br>
The computation and the asymptotic approximations of the [[File:Vol1 page 0164 inline 003.png]] term have been discussed in '''Example 3.15'''. To compute the [[File:Vol1 page 0152 inline 002.png]] term for long times (small ''s''), the relation for the ratios of the hyperbolic functions given by '''Eq. 3.366''' should be useful. For computations at short times (large values of ''s''), following the lines similar to those in '''Example 3.15''', the [[File:Vol1 page 0153 inline 002.png]] term in '''Eq. 3.421''' may be written as
<br>
<br>
[[File:Vol1 page 0164 eq 004.png]]....................(3.423)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0164 eq 005.png]]....................(3.424)
<br>
<br>
[[File:Vol1 page 0164 eq 006.png]]....................(3.425)
<br>
<br>
[[File:Vol1 page 0164 eq 007.png]]....................(3.426)
<br>
<br>
[[File:Vol1 page 0164 eq 008.png]]
[[File:Vol1 page 0165 eq 001.png]]....................(3.427)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0165 eq 002.png]]....................(3.428)
<br>
<br>
The computational form of the [[File:Vol1 page 0150 inline 002.png]] term in '''Eq. 3.424''' is obtained by applying the relations given by '''Eqs. 3.347''' through '''3.350''' and '''Eq. 3.328'''. Of particular interest is the case for −1 ≤ ''x''<sub>''D''</sub> ≤ +1 and ''y''<sub>''D''</sub> = ''y''<sub>''wD''</sub> given by
<br>
<br>
[[File:Vol1 page 0165 eq 003.png]]....................(3.429)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0165 eq 004.png]]....................(3.430)
<br>
<br>
which can be written as follows by using the relation given in '''Eq. 3.337''':
<br>
<br>
[[File:Vol1 page 0165 eq 005.png]]....................(3.431)
<br>
<br>
Similarly, for −1 ≤ ''x''<sub>''D''</sub> ≤ +1 and ''y''<sub>''D''</sub> = ''y''<sub>''wD''</sub>, the [[File:Vol1 page 0165 inline 001.png]] term given in '''Eq. 3.428''' may be written as
<br>
<br>
[[File:Vol1 page 0165 eq 006.png]]
[[File:Vol1 page 0166 eq 001.png]]....................(3.432)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0166 eq 002.png]]....................(3.433)
<br>
<br>


----
----


'''Example 3.15''' discussed the short- and long-time approximations of the [[File:Vol1 page 0164 inline 003.png]] term in '''Eq. 3.420'''. A small-time approximation for [[File:Vol1 page 0153 inline 002.png]] given by '''Eq. 3.423''' is obtained with ''u'' = ''ωs'' and by noting that as ''s''→∞, [[File:Vol1 page 0166 inline 001.png]]. Then, substituting the short-time approximations for [[File:Vol1 page 0164 inline 003.png]] and [[File:Vol1 page 0150 inline 001.png]] given, respectively, by '''Eqs. 3.371''' and '''3.353''' into '''Eq. 3.420''', the following short-time approximation is obtained: <ref name="r27" />
'''Example 3.15''' discussed the short- and long-time approximations of the [[File:Vol1 page 0164 inline 003.png|RTENOTITLE]] term in '''Eq. 3.420'''. A small-time approximation for [[File:Vol1 page 0153 inline 002.png|RTENOTITLE]] given by '''Eq. 3.423''' is obtained with ''u'' = ''ωs'' and by noting that as ''s''→∞, [[File:Vol1 page 0166 inline 001.png|RTENOTITLE]]. Then, substituting the short-time approximations for [[File:Vol1 page 0164 inline 003.png|RTENOTITLE]] and [[File:Vol1 page 0150 inline 001.png|RTENOTITLE]] given, respectively, by '''Eqs. 3.371''' and '''3.353''' into '''Eq. 3.420''', the following short-time approximation is obtained: <ref name="r27">Kazemi, H. 1969. Pressure Transient Analysis of Naturally Fractured Reservoir with Uniform Fracture Distribution. SPE J. 9 (4): 451–462. SPE-2156-PA. http://dx.doi.org/10.2118/2156-PA</ref><br/><br/>[[File:Vol1 page 0166 eq 003.png|RTENOTITLE]]....................(3.434)<br/><br/>where ''β'' is given by '''Eq. 3.331'''. The inverse Laplace transform of '''Eq. 3.434''' yields<br/><br/>[[File:Vol1 page 0166 eq 004.png|RTENOTITLE]]....................(3.435)<br/><br/>The long-time approximation of '''Eq. 3.420''' is obtained by substituting the long-time approximations of [[File:Vol1 page 0164 inline 003.png|RTENOTITLE]] and [[File:Vol1 page 0152 inline 002.png|RTENOTITLE]]. The long time-approximation of [[File:Vol1 page 0164 inline 003.png|RTENOTITLE]] is obtained in '''Example 3.15''' (see '''Eq. 3.413''' through '''3.419'''). The long-time approximation of [[File:Vol1 page 0153 inline 002.png|RTENOTITLE]] is obtained by evaluating the right side of '''Eq. 3.421''' as ''s''→0 (''u''→0) and is given by<br/><br/>[[File:Vol1 page 0166 eq 005.png|RTENOTITLE]]....................(3.436)<br/><br/>where<br/><br/>[[File:Vol1 page 0167 eq 001.png|RTENOTITLE]]....................(3.437)<br/><br/>and<br/><br/>[[File:Vol1 page 0167 eq 002.png|RTENOTITLE]]....................(3.438)<br/><br/>Thus, the long-time approximation '''Eq. 3.420''' is given by<br/><br/>[[File:Vol1 page 0167 eq 003.png|RTENOTITLE]]....................(3.439)<br/><br/>where ''p''<sub>''Df''</sub> and ''F''<sub>1</sub> are given, respectively, by '''Eqs. 3.419''' and '''3.436'''. For computational purposes, however, ''F''<sub>1</sub> may be replaced by<br/><br/>[[File:Vol1 page 0167 eq 004.png|RTENOTITLE]]....................(3.440)<br/><br/>In '''Eq. 3.440''', ''F'', ''F''<sub>''b''1</sub>, ''F''<sub>''b''2</sub>, and ''F''<sub>''b''3</sub> are given, respectively, by<br/><br/>[[File:Vol1 page 0167 eq 005.png|RTENOTITLE]]....................(3.441)<br/><br/>[[File:Vol1 page 0167 eq 006.png|RTENOTITLE]]....................(3.442)<br/><br/>[[File:Vol1 page 0167 eq 007.png|RTENOTITLE]]....................(3.443)<br/><br/>and<br/><br/>[[File:Vol1 page 0167 eq 008.png|RTENOTITLE]]<br/>[[File:Vol1 page 0168 eq 001.png|RTENOTITLE]]....................(3.444)<br/><br/>When computing the integrals and the trigonometric series, the relations given by '''Eqs. 3.324''' through '''3.326''' and '''3.345''' through '''3.350''' are useful.
<br>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<br>
[[File:Vol1 page 0166 eq 003.png]]....................(3.434)
<br>
<br>
where ''β'' is given by '''Eq. 3.331'''. The inverse Laplace transform of '''Eq. 3.434''' yields
<br>
<br>
[[File:Vol1 page 0166 eq 004.png]]....................(3.435)
<br>
<br>
The long-time approximation of '''Eq. 3.420''' is obtained by substituting the long-time approximations of [[File:Vol1 page 0164 inline 003.png]] and [[File:Vol1 page 0152 inline 002.png]]. The long time-approximation of [[File:Vol1 page 0164 inline 003.png]] is obtained in '''Example 3.15''' (see '''Eq. 3.413''' through '''3.419'''). The long-time approximation of [[File:Vol1 page 0153 inline 002.png]] is obtained by evaluating the right side of '''Eq. 3.421''' as ''s''→0 (''u''→0) and is given by
<br>
<br>
[[File:Vol1 page 0166 eq 005.png]]....................(3.436)
<br>
<br>
where
<br>
<br>
[[File:Vol1 page 0167 eq 001.png]]....................(3.437)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0167 eq 002.png]]....................(3.438)
<br>
<br>
Thus, the long-time approximation '''Eq. 3.420''' is given by
<br>
<br>
[[File:Vol1 page 0167 eq 003.png]]....................(3.439)
<br>
<br>
where ''p''<sub>''Df''</sub> and ''F''<sub>1</sub> are given, respectively, by '''Eqs. 3.419''' and '''3.436'''. For computational purposes, however, ''F''<sub>1</sub> may be replaced by
<br>
<br>
[[File:Vol1 page 0167 eq 004.png]]....................(3.440)
<br>
<br>
In '''Eq. 3.440''', ''F'', ''F''<sub>''b''1</sub>, ''F''<sub>''b''2</sub>, and ''F''<sub>''b''3</sub> are given, respectively, by
<br>
<br>
[[File:Vol1 page 0167 eq 005.png]]....................(3.441)
<br>
<br>
[[File:Vol1 page 0167 eq 006.png]]....................(3.442)
<br>
<br>
[[File:Vol1 page 0167 eq 007.png]]....................(3.443)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0167 eq 008.png]]<br>
[[File:Vol1 page 0168 eq 001.png]]....................(3.444)
<br>
<br>
When computing the integrals and the trigonometric series, the relations given by '''Eqs. 3.324''' through '''3.326''' and '''3.345''' through '''3.350''' are useful.
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
 
== Nomenclature ==
== Nomenclature ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
 
 
{|
{|
|''a''
|=
|radius of the spherical source, ''L''
|-
|-
|''B''  
| ''a''
|=  
| =
|formation volume factor, res cm<sup>3</sup>/std cm<sup>3</sup>
| radius of the spherical source, ''L''
|-
| ''B''
| =
| formation volume factor, res cm<sup>3</sup>/std cm<sup>3</sup>
|-
|-
|''c''  
| ''c''
|=  
| =
|fluid compressibility, atm<sup>−1</sup>
| fluid compressibility, atm<sup>−1</sup>
|-
|-
|''c''<sub>''f''</sub>  
| ''c''<sub>''f''</sub>
|=  
| =
|formation compressibility, atm<sup>−1</sup>
| formation compressibility, atm<sup>−1</sup>
|-
|-
|''c''<sub>''t''</sub>  
| ''c''<sub>''t''</sub>
|=  
| =
|total compressibility, atm<sup>−1</sup>
| total compressibility, atm<sup>−1</sup>
|-
|-
|''C''  
| ''C''
|=  
| =
|wellbore-storage coefficient, cm<sup>3</sup>/atm
| wellbore-storage coefficient, cm<sup>3</sup>/atm
|-
|-
|''d''  
| ''d''
|=  
| =
|distance to a linear boundary, cm
| distance to a linear boundary, cm
|-
|-
|''D''  
| ''D''
|=  
| =
|domain
| domain
|-
|-
|''Ei''(''x'')  
| ''Ei''(''x'')
|=  
| =
|exponential integral function
| exponential integral function
|-
|-
|''f''(''s'')  
| ''f''(''s'')
|=  
| =
|naturally fractured reservoir function
| naturally fractured reservoir function
|-
|-
|[[File:Vol1 page 0168 inline 001.png]]
| [[File:Vol1 page 0168 inline 001.png|RTENOTITLE]]
|=  
| =
|naturally fractured reservoir function based on [[File:Vol1 page 0145 inline 001.png]]
| naturally fractured reservoir function based on [[File:Vol1 page 0145 inline 001.png|RTENOTITLE]]
|-
|-
|[[File:Vol1 page 0168 inline 002.png]]
| [[File:Vol1 page 0168 inline 002.png|RTENOTITLE]]
|=  
| =
|Laplace transform of a function ''f'' (''t'')
| Laplace transform of a function ''f'' (''t'')
|-
|-
|''G''  
| ''G''
|=  
| =
|Green’s function
| Green’s function
|-
|-
|''h''  
| ''h''
|=  
| =
|formation thickness, cm
| formation thickness, cm
|-
|-
|[[File:Vol1 page 0168 inline 003.png]]
| [[File:Vol1 page 0168 inline 003.png|RTENOTITLE]]
|=  
| =
|dimensionless thickness, '''Eq. 3.313'''
| dimensionless thickness, '''Eq. 3.313'''
|-
|-
|''h''<sub>''D''</sub>  
| ''h''<sub>''D''</sub>
|=  
| =
|dimensionless thickness, '''Eq. 3.314'''
| dimensionless thickness, '''Eq. 3.314'''
|-
|-
|''h''<sub>''f''</sub>  
| ''h''<sub>''f''</sub>
|=  
| =
|fracture height (vertical penetration), cm
| fracture height (vertical penetration), cm
|-
|-
|''h''<sub>''p''</sub>  
| ''h''<sub>''p''</sub>
|=  
| =
|slab thickness, cm
| slab thickness, cm
|-
|-
|''h''<sub>''w''</sub>  
| ''h''<sub>''w''</sub>
|=  
| =
|well length (penetration), cm
| well length (penetration), cm
|-
|-
|''H''(''x'' - ''x′'')  
| ''H''(''x'' - ''x′'')
|=  
| =
|Heaviside’s unit step function
| Heaviside’s unit step function
|-
|-
|[[File:Vol1 page 0168 inline 004.png]]
| [[File:Vol1 page 0168 inline 004.png|RTENOTITLE]]
|=  
| =
|unit normal vector in the ''ξ'' direction, ''ξ'' = ''x'', ''y'', ''z'', ''r'', ''θ''
| unit normal vector in the ''ξ'' direction, ''ξ'' = ''x'', ''y'', ''z'', ''r'', ''θ''
|-
|-
|''I''<sub>''v''</sub>(''x'')  
| ''I''<sub>''v''</sub>(''x'')
|=  
| =
|modified Bessel function of the first kind of order ''v''
| modified Bessel function of the first kind of order ''v''
|-
|-
|''I′''<sub>''v''</sub>(''x'')  
| ''I′''<sub>''v''</sub>(''x'')
|=  
| =
|derivative of ''I''<sub>''v''</sub>(''x'')
| derivative of ''I''<sub>''v''</sub>(''x'')
|-
|-
|''J''<sub>''v''</sub>(''x'')  
| ''J''<sub>''v''</sub>(''x'')
|=  
| =
|Bessel function of the first kind of order ''v''
| Bessel function of the first kind of order ''v''
|-
|-
|''k''  
| ''k''
|=  
| =
|isotropic permeability, md
| isotropic permeability, md
|-
|-
|''k''<sub>''f''</sub>  
| ''k''<sub>''f''</sub>
|=  
| =
|fracture permeability, md
| fracture permeability, md
|-
|-
|''k''<sub>''h''</sub>  
| ''k''<sub>''h''</sub>
|=  
| =
|equivalent horizontal permeability, md
| equivalent horizontal permeability, md
|-
|-
|''k''<sub>''i j''</sub>  
| ''k''<sub>''i j''</sub>
|=  
| =
|permeability in i-direction as a result of pressure gradient in ''j''-direction, md
| permeability in i-direction as a result of pressure gradient in ''j''-direction, md
|-
|-
|''k''<sub>''ξ''</sub>  
| ''k''<sub>''ξ''</sub>
|=  
| =
|permeability in ''ξ''-direction, ''ξ'' = ''x'', ''y'', ''z'', md
| permeability in ''ξ''-direction, ''ξ'' = ''x'', ''y'', ''z'', md
|-
|-
|''k''<sub>''ξf''</sub>  
| ''k''<sub>''ξf''</sub>
|=  
| =
|fracture permeability in ''ξ''-direction, ''ξ'' = ''x'', ''y'', ''z'', md
| fracture permeability in ''ξ''-direction, ''ξ'' = ''x'', ''y'', ''z'', md
|-
|-
|''Ki''<sub>1</sub>(''x'')  
| ''Ki''<sub>1</sub>(''x'')
|=  
| =
|first integral of ''K''<sub>0</sub>(''z'')
| first integral of ''K''<sub>0</sub>(''z'')
|-
|-
|''K''<sub>''n''</sub>(''x'')  
| ''K''<sub>''n''</sub>(''x'')
|=  
| =
|modified Bessel function of the second kind of order ''n''
| modified Bessel function of the second kind of order ''n''
|-
|-
|''K′''<sub>''n''</sub>(''x'')  
| ''K′''<sub>''n''</sub>(''x'')
|=  
| =
|derivative of ''K''<sub>''n''</sub>(''x'')
| derivative of ''K''<sub>''n''</sub>(''x'')
|-
|-
|''ℓ''  
| ''ℓ''
|=  
| =
|characteristic length of the system, cm
| characteristic length of the system, cm
|-
|-
|''L''  
| ''L''
|=  
| =
|Laplace transform operator
| Laplace transform operator
|-
|-
|''L''<sup>-1</sup>  
| ''L''<sup>-1</sup>
|=  
| =
|inverse Laplace transform operator
| inverse Laplace transform operator
|-
|-
|''L''<sub>''h''</sub>  
| ''L''<sub>''h''</sub>
|=  
| =
|horizontal-well length, cm
| horizontal-well length, cm
|-
|-
|''m''  
| ''m''
|=  
| =
|pseudopressure, atm<sup>2</sup>/cp
| pseudopressure, atm<sup>2</sup>/cp
|-
|-
|''M''<sub>''g''</sub>  
| ''M''<sub>''g''</sub>
|=  
| =
|mass, g
| mass, g
|-
|-
|''M''  
| ''M''
|=  
| =
|point in space
| point in space
|-
|-
|''M′''  
| ''M′''
|=  
| =
|source point in space
| source point in space
|-
|-
|''M''<sub>''w''</sub>  
| ''M''<sub>''w''</sub>
|=  
| =
|point in Γ<sub>''w''</sub>
| point in Γ<sub>''w''</sub>
|-
|-
|''M′''<sub>''w''</sub>  
| ''M′''<sub>''w''</sub>
|=  
| =
|source point in Γ<sub>''w''</sub>
| source point in Γ<sub>''w''</sub>
|-
|-
|''n''  
| ''n''
|=  
| =
|outward normal direction of the boundary surface
| outward normal direction of the boundary surface
|-
|-
|[[File:Vol1 page 0169 inline 001.png]]
| [[File:Vol1 page 0169 inline 001.png|RTENOTITLE]]
|=  
| =
|normal vector
| normal vector
|-
|-
|''N''  
| ''N''
|=  
| =
|even integer in Stehfest’s algorithm
| even integer in Stehfest’s algorithm
|-
|-
|''p''  
| ''p''
|=  
| =
|pressure, atm
| pressure, atm
|-
|-
|''p''<sub>''c''</sub>  
| ''p''<sub>''c''</sub>
|=  
| =
|pressure for constant production rate, ''q''<sub>''c''</sub>, atm
| pressure for constant production rate, ''q''<sub>''c''</sub>, atm
|-
|-
|[[File:Vol1 page 0169 inline 002.png]]
| [[File:Vol1 page 0169 inline 002.png|RTENOTITLE]]
|=  
| =
|dimensionless fracture pressure
| dimensionless fracture pressure
|-
|-
|''p''<sub>''e''</sub>  
| ''p''<sub>''e''</sub>
|=  
| =
|external boundary pressure, atm
| external boundary pressure, atm
|-
|-
|''p ''<sub>''f''</sub>  
| ''p ''<sub>''f''</sub>
|=  
| =
|fracture pressure, atm
| fracture pressure, atm
|-
|-
|''p''<sub>''f i''</sub>  
| ''p''<sub>''f i''</sub>
|=  
| =
|initial pressure in fracture system, atm
| initial pressure in fracture system, atm
|-
|-
|''p''<sub>''i''</sub>  
| ''p''<sub>''i''</sub>
|=  
| =
|initial pressure, atm
| initial pressure, atm
|-
|-
|''p''<sub>''j''</sub>  
| ''p''<sub>''j''</sub>
|=  
| =
|pressure in medium ''j'', ''j''=''m'', ''f'', atm
| pressure in medium ''j'', ''j''=''m'', ''f'', atm
|-
|-
|''p''<sub>''m''</sub>  
| ''p''<sub>''m''</sub>
|=  
| =
|matrix pressure, atm
| matrix pressure, atm
|-
|-
|''p''<sub>''mi''</sub>  
| ''p''<sub>''mi''</sub>
|=  
| =
|initial pressure in matrix system, atm
| initial pressure in matrix system, atm
|-
|-
|''p''<sub>''w f''</sub>  
| ''p''<sub>''w f''</sub>
|=  
| =
|flowing wellbore pressure, atm
| flowing wellbore pressure, atm
|-
|-
|[[File:Vol1 page 0169 inline 003.png]]  
| [[File:Vol1 page 0169 inline 003.png|RTENOTITLE]]
|=  
| =
|Laplace transform of ''p''(''t'')
| Laplace transform of ''p''(''t'')
|-
|-
|''p''(''t'')  
| ''p''(''t'')
|=  
| =
|inverse of the Laplace domain function
| inverse of the Laplace domain function
|-
|-
|''p''<sub>''a''</sub>(''T'')  
| ''p''<sub>''a''</sub>(''T'')
|=  
| =
|approximate inverse of [[File:Vol1 page 0169 inline 004.png]] at ''t''=''T'', atm
| approximate inverse of [[File:Vol1 page 0169 inline 004.png|RTENOTITLE]] at ''t''=''T'', atm
|-
|-
|''q''  
| ''q''
|=  
| =
|production rate, cm<sup>3</sup>/s
| production rate, cm<sup>3</sup>/s
|-
|-
|[[File:Vol1 page 0102 inline 001.png]]  
| [[File:Vol1 page 0102 inline 001.png|RTENOTITLE]]
|=  
| =
|instantaneous production rate for a point source, cm<sup>3</sup>/s
| instantaneous production rate for a point source, cm<sup>3</sup>/s
|-
|-
|''q''<sub>''c''</sub>  
| ''q''<sub>''c''</sub>
|=  
| =
|constant production rate, cm<sup>3</sup>/s
| constant production rate, cm<sup>3</sup>/s
|-
|-
|''q''<sub>''s f''</sub>  
| ''q''<sub>''s f''</sub>
|=  
| =
|sandface production rate, cm<sup>3</sup>/s
| sandface production rate, cm<sup>3</sup>/s
|-
|-
|''q''<sub>''wb''</sub>  
| ''q''<sub>''wb''</sub>
|=  
| =
|wellbore production rate as a result of storage, cm<sup>3</sup>/s
| wellbore production rate as a result of storage, cm<sup>3</sup>/s
|-
|-
|''r''  
| ''r''
|=  
| =
|radial coordinate and distance, cm
| radial coordinate and distance, cm
|-
|-
|''r′''  
| ''r′''
|=  
| =
|source coordinate in ''r''-direction, cm
| source coordinate in ''r''-direction, cm
|-
|-
|''r''<sub>''e''</sub>  
| ''r''<sub>''e''</sub>
|=  
| =
|external radius of the reservoir, cm
| external radius of the reservoir, cm
|-
|-
|''r''<sub>''w''</sub>  
| ''r''<sub>''w''</sub>
|=  
| =
|wellbore radius, cm
| wellbore radius, cm
|-
|-
|''R''  
| ''R''
|=  
| =
|distance in 3D coordinates, cm
| distance in 3D coordinates, cm
|-
|-
|''R''<sub>''D''</sub>  
| ''R''<sub>''D''</sub>
|=  
| =
|dimensionless radial distance in cylindrical coordinates
| dimensionless radial distance in cylindrical coordinates
|-
|-
|''s''  
| ''s''
|=  
| =
|Laplace transform parameter
| Laplace transform parameter
|-
|-
|[[File:Vol1 page 0145 inline 001.png]]  
| [[File:Vol1 page 0145 inline 001.png|RTENOTITLE]]
|=  
| =
|Laplace transform paraeter based on [[File:Vol1 page 0170 inline 002.png]]
| Laplace transform paraeter based on [[File:Vol1 page 0170 inline 002.png|RTENOTITLE]]
|-
|-
|''s''<sub>''m''</sub>  
| ''s''<sub>''m''</sub>
|=  
| =
|skin factor
| skin factor
|-
|-
|''S''  
| ''S''
|=  
| =
|source function
| source function
|-
|-
|''t''  
| ''t''
|=  
| =
|time, s
| time, s
|-
|-
|[[File:Vol1 page 0170 inline 003.png]]  
| [[File:Vol1 page 0170 inline 003.png|RTENOTITLE]]
|=  
| =
|dimensionless time based on ''k''<sub>''h''</sub>
| dimensionless time based on ''k''<sub>''h''</sub>
|-
|-
|''t''<sub>''AD''</sub>  
| ''t''<sub>''AD''</sub>
|=  
| =
|dimensionless time based on area
| dimensionless time based on area
|-
|-
|''t''<sub>''p''</sub>  
| ''t''<sub>''p''</sub>
|=  
| =
|producing time, s
| producing time, s
|-
|-
|''T''  
| ''T''
|=  
| =
|Temperature, °C
| Temperature, °C
|-
|-
|''u''  
| ''u''
|=  
| =
|''s f''(''s'')
| ''s f''(''s'')
|-
|-
|[[File:Vol1 page 0077 inline 001.png]]
| [[File:Vol1 page 0077 inline 001.png|RTENOTITLE]]
|=  
| =
|velocity vector
| velocity vector
|-
|-
|''v''<sub>''ξ''</sub>  
| ''v''<sub>''ξ''</sub>
|=  
| =
|velocity component in the ''ξ'' direction, ''ξ'' = ''x'', ''y'', ''z'', ''r'', ''θ'', cm/s
| velocity component in the ''ξ'' direction, ''ξ'' = ''x'', ''y'', ''z'', ''r'', ''θ'', cm/s
|-
|-
|''V''  
| ''V''
|=  
| =
|volume, cm<sup>3</sup>
| volume, cm<sup>3</sup>
|-
|-
|''V''<sub>''i''</sub>  
| ''V''<sub>''i''</sub>
|=  
| =
|constant in Stehfest’s algorithm
| constant in Stehfest’s algorithm
|-
|-
|''V''<sub>''f''</sub>  
| ''V''<sub>''f''</sub>
|=  
| =
|fraction of the volume occupied by fractures
| fraction of the volume occupied by fractures
|-
|-
|''V''<sub>''m''</sub>  
| ''V''<sub>''m''</sub>
|=  
| =
|fraction of the volume occupied by matrix
| fraction of the volume occupied by matrix
|-
|-
|''x''  
| ''x''
|=  
| =
|distance in ''x''-direction, cm
| distance in ''x''-direction, cm
|-
|-
|''x′''  
| ''x′''
|=  
| =
|source coordinate in ''x''-direction, cm
| source coordinate in ''x''-direction, cm
|-
|-
|''x''<sub>''e''</sub>  
| ''x''<sub>''e''</sub>
|=  
| =
|distance to the external boundary in ''x''-direction, cm
| distance to the external boundary in ''x''-direction, cm
|-
|-
|''x''<sub>''p''</sub>  
| ''x''<sub>''p''</sub>
|=  
| =
|half slab thickness, cm
| half slab thickness, cm
|-
|-
|''x''<sub>''f''</sub>  
| ''x''<sub>''f''</sub>
|=  
| =
|fracture half-length, cm
| fracture half-length, cm
|-
|-
|[[File:Vol1 page 0170 inline 004.png]]  
| [[File:Vol1 page 0170 inline 004.png|RTENOTITLE]]
|=  
| =
|dimensionless fracture half-length
| dimensionless fracture half-length
|-
|-
|''x''<sub>''w''</sub>  
| ''x''<sub>''w''</sub>
|=  
| =
|well coordinate in ''x''-direction, cm
| well coordinate in ''x''-direction, cm
|-
|-
|''y''  
| ''y''
|=  
| =
|distance in ''y''-direction, cm
| distance in ''y''-direction, cm
|-
|-
|''y′''  
| ''y′''
|=  
| =
|source coordinate in ''y''-direction, cm
| source coordinate in ''y''-direction, cm
|-
|-
|''y''<sub>''e''</sub>  
| ''y''<sub>''e''</sub>
|=  
| =
|distance to the external boundary in ''y''-direction, cm
| distance to the external boundary in ''y''-direction, cm
|-
|-
|''y''<sub>''w''</sub>  
| ''y''<sub>''w''</sub>
|=  
| =
|well coordinate in ''y''-direction, cm
| well coordinate in ''y''-direction, cm
|-
|-
|''Y''<sub>''n''</sub>(''x'')  
| ''Y''<sub>''n''</sub>(''x'')
|=  
| =
|Bessel function of the second kind of order ''n''
| Bessel function of the second kind of order ''n''
|-
|-
|''z''  
| ''z''
|=  
| =
|distance in ''z''-direction, cm
| distance in ''z''-direction, cm
|-
|-
|''z′''  
| ''z′''
|=  
| =
|source coordinate in ''z''-direction, cm
| source coordinate in ''z''-direction, cm
|-
|-
|[[File:Vol1 page 0170 inline 005.png]]  
| [[File:Vol1 page 0170 inline 005.png|RTENOTITLE]]
|=  
| =
|dimensionless distance in ''z''-direction, '''Eq. 3.377'''
| dimensionless distance in ''z''-direction, '''Eq. 3.377'''
|-
|-
|''z''<sub>''w''</sub>  
| ''z''<sub>''w''</sub>
|=  
| =
|well coordinate in ''z''-direction, cm
| well coordinate in ''z''-direction, cm
|-
|-
|[[File:Vol1 page 0170 inline 006.png]]  
| [[File:Vol1 page 0170 inline 006.png|RTENOTITLE]]
|=  
| =
|dimensionless well coordinate in ''z''-direction, '''Eq. 3.378'''
| dimensionless well coordinate in ''z''-direction, '''Eq. 3.378'''
|-
|-
|''Z''  
| ''Z''
|=  
| =
|compressibility factor
| compressibility factor
|-
|-
|''α''  
| ''α''
|=  
| =
|permeability direction, '''Eq. 3.17'''
| permeability direction, '''Eq. 3.17'''
|-
|-
|''β''  
| ''β''
|=  
| =
|permeability direction, '''Eq. 3.17'''
| permeability direction, '''Eq. 3.17'''
|-
|-
|Γ  
| Γ
|=  
| =
|boundary surface, cm<sup>2</sup>
| boundary surface, cm<sup>2</sup>
|-
|-
|Γ<sub>''e''</sub>  
| Γ<sub>''e''</sub>
|=  
| =
|external boundary surface
| external boundary surface
|-
|-
|Γ<sub>''w''</sub>  
| Γ<sub>''w''</sub>
|=  
| =
|length, surface, or volume of the source
| length, surface, or volume of the source
|-
|-
|Γ(''x'')  
| Γ(''x'')
|=  
| =
|Gamma function
| Gamma function
|-
|-
|''γ''  
| ''γ''
|=  
| =
|Euler’s constant (''γ'' = 0.5772...)
| Euler’s constant (''γ'' = 0.5772...)
|-
|-
|''γ''  
| ''γ''
|=  
| =
|permeability direction, '''Eq. 3.17'''
| permeability direction, '''Eq. 3.17'''
|-
|-
|''γ ''<sub>''f''</sub>  
| ''γ ''<sub>''f''</sub>
|=  
| =
|fundamental solution of diffusion equation
| fundamental solution of diffusion equation
|-
|-
|Δ  
| Δ
|=  
| =
|difference operator
| difference operator
|-
|-
|''δ''(''x'')  
| ''δ''(''x'')
|=  
| =
|Dirac delta function
| Dirac delta function
|-
|-
|''η''  
| ''η''
|=  
| =
|diffusivity constant
| diffusivity constant
|-
|-
|''η''<sub>''i''</sub>  
| ''η''<sub>''i''</sub>
|=  
| =
|diffusivity constant in i direction, ''i'' = ''x'', ''y'', ''z'', or ''r''
| diffusivity constant in i direction, ''i'' = ''x'', ''y'', ''z'', or ''r''
|-
|-
|''θ''  
| ''θ''
|=  
| =
|angle from positive x-direction, degrees radian
| angle from positive x-direction, degrees radian
|-
|-
|''θ′''  
| ''θ′''
|=  
| =
|source coordinate in ''θ''-direction, degrees radian
| source coordinate in ''θ''-direction, degrees radian
|-
|-
|''λ''  
| ''λ''
|=  
| =
|transfer coefficient for a naturally fractured reservoir
| transfer coefficient for a naturally fractured reservoir
|-
|-
|[[File:Vol1 page 0171 inline 001.png]]  
| [[File:Vol1 page 0171 inline 001.png|RTENOTITLE]]
|=  
| =
|''λ'' based on ''k''<sub>''h''</sub>
| ''λ'' based on ''k''<sub>''h''</sub>
|-
|-
|''μ''  
| ''μ''
|=  
| =
|viscosity, cp
| viscosity, cp
|-
|-
|''ρ''  
| ''ρ''
|=  
| =
|density, g/cm<sup>3</sup>
| density, g/cm<sup>3</sup>
|-
|-
|''τ''  
| ''τ''
|=  
| =
|time, s
| time, s
|-
|-
|''Φ''  
| ''Φ''
|=  
| =
|porosity, fraction
| porosity, fraction
|-
|-
|''φ''(''M'')  
| ''φ''(''M'')
|=  
| =
|any continuous function
| any continuous function
|-
|-
|''ω''  
| ''ω''
|=  
| =
|storativity ratio for a naturally fractured reservoir
| storativity ratio for a naturally fractured reservoir
|}
|}
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >


</div></div><div class="toccolours mw-collapsible mw-collapsed">
== Subscripts and Superscripts ==
== Subscripts and Superscripts ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
 
 
{|
{|
|D
|=
|dimensionless
|-
|-
|''f''  
| D
|=  
| =
|fracture
| dimensionless
|-
| ''f''
| =
| fracture
|-
|-
|i  
| i
|=  
| =
|initial
| initial
|-
|-
|''m''  
| ''m''
|=  
| =
|matrix
| matrix
|-
|-
|w  
| w
|=  
| =
|wellbore
| wellbore
|-
|-
|[[File:Vol1 page 0171 inline 002.png]]  
| [[File:Vol1 page 0171 inline 002.png|RTENOTITLE]]
|=  
| =
|Laplace transform indicator
| Laplace transform indicator
|}
|}
<br>
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
</div></div><div class="toccolours mw-collapsible mw-collapsed">
== References ==
== References ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/><references />
<references>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<ref name="r1">Al-Hussainy, R., Ramey Jr., H.J., and  Crawford, P.B. 1966. The Flow of Real Gases Through Porous Media. ''J Pet Technol'' '''18''' (5): 624–636. SPE-1243-A-PA. http://dx.doi.org/10.2118/1243-A-PA</ref>
== SI Metric Conversion Factors ==
 
<div class="mw-collapsible-content">
<ref name="r2">Raghavan, R. 1993. ''Well Test Analysis'', 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.</ref>


<ref name="r3">Fancher, G.H., Lewis, J.A., and  Barnes, K.B. 1933. Some Physical Characteristics of Oil Sands. Bulletin 12, Mineral Industries Experimental Station, Pennsylvania State University, University Park, Pennsylvania, 65–167.</ref>


<ref name="r4">Forchheimer, P.F. 1901. Wasserbewegung durch Boden. ''Zeitschrift des Vereines deutscher Ingenieure'' '''45''' (5): 1781–1788. </ref>
<ref name="r5">Watson, G. N. 1944. ''A Treatise on the Theory of Bessel Functions''. London: Cambridge University Press.</ref>
<ref name="r6">Churchill, R.V. 1972. ''Operational Mathematics'', Vol. 2. New York: McGraw-Hill Book Co. </ref>
<ref name="r7">Abramowitz, M. and Stegun, I.A. ed. 1972. ''Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables'', ninth edition, 1020–1029. New York: Dover Publications.</ref>
<ref name="r8">Stehfest, H. 1970. Algorithm 368: Numerical inversion of Laplace transforms. ''Commun. ACM'' '''13''' (1): 47–49. http://dx.doi.org/10.1145/361953.361969</ref>
<ref name="r9">Crump, K.S. 1976. Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation. ''J. ACM'' '''23''' (1): 89-96. http://dx.doi.org/10.1145/321921.321931</ref>
<ref name="r10">van Everdingen, A.F. and Hurst, W. 1953. The Application of the Laplace Transformation to Flow Problems in Reservoirs. In ''Transactions of the American Institute of Mining and Metallurgical Engineers'', Vol. 198, 171. Dallas, Texas: American Institute of Mining and Metallurgical Engineers Inc.</ref>
<ref name="r11">Agarwal, R.G., Al-Hussainy, R., and  Ramey Jr., H.J. 1970. An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment. ''SPE J.'' '''10''' (3): 279-290. SPE-2466-PA. http://dx.doi.org/10.2118/2466-PA</ref>
<ref name="r12">Chen, C.-C. and Raghavan, R. 1996. An Approach To Handle Discontinuities by the Stehfest Algorithm. ''SPE J.'' '''1''' (4): 363-368. SPE-28419-PA. http://dx.doi.org/10.2118/28419-PA</ref>
<ref name="r13">Gringarten, A.C. and Ramey Jr., H.J. 1973. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. ''SPE J.'' '''13''' (5): 285-296. SPE-3818-PA. http://dx.doi.org/10.2118/3818-PA</ref>
<ref name="r14">Carslaw, H.S. and Jaeger, J.C. 1959. ''Conduction of Heat in Solids'', second edition, 353–386. Oxford, UK: Oxford University Press. </ref>
<ref name="r15">Stakgold, I. 1979. ''Green’s Functions and Boundary Value Problems'', 86–104. New York: John Wiley & Sons.</ref>
<ref name="r16">Kelvin, W.T. 1884. ''Mathematical and Physical Papers'', Vol. 2, 41. Cambridge, UK: Cambridge University Press. </ref>
<ref name="r17">Ozkan, E. and Raghavan, R. 1991a. New Solutions for Well-Test-Analysis Problems: Part 1—Analytical Considerations. ''SPE Form Eval'' '''6''' (3): 359–368. SPE-18615-PA. http://dx.doi.org/10.2118/18615-PA</ref>
<ref name="r18">Ozkan, E. and Raghavan, R. 1991b. New Solutions for Well-Test-Analysis Problems: Part 2—Computational Considerations and Applications. ''SPE Form Eval'' '''6''' (3): 369–378. SPE-18616-PA. http://dx.doi.org/10.2118/18616-PA</ref>
<ref name="r19">Raghavan, R. and Ozkan, E. 1994. ''A Method for Computing Unsteady Flows in Porous Media'', No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical. </ref>
<ref name="r20">Raghavan, R. 1993. The Method of Sources and Sinks. In ''Well Test Analysis'', Chap. 10, 336-435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.</ref>
<ref name="r21">Nisle, R.G. 1958. The Effect of Partial Penetration on Pressure Build-Up in Oil Wells. In ''Transactions of the American Institute of Mining, Metallurgical, and Petroleum Engineers'', Vol. 213, Paper 971-G, 85-90. Dallas, Texas: Society of Petroleum Engineers.</ref>
<ref name="r22">Newman, A.B. 1936. Heating and Cooling Rectangular and Cylindrical Solids. ''Ind. Eng. Chem.'' ''28'' (5): 545–548. http://dx.doi.org/10.1021/ie50317a010</ref>
<ref name="r23">Duhamel, J.M.C. 1833. Mémoire sur la méthode générale relative au mouvement de la chaleur dans les corps solides polongé dans les milieux dont la température varie avec le temps. ''Journal de l’École Polytechnique'' '''14''' (22): 20-66.</ref>
<ref name="r24">Chen, H.Y., Poston, S.W., and  Raghavan, R. 1991. An Application of the Product Solution Principle for Instantaneous Source and Green's Functions. ''SPE Form Eval'' '''6''' (2): 161-167. SPE-20801-PA. http://dx.doi.org/10.2118/20801-PA </ref>
<ref name="r25">Barenblatt, G.I., Zheltov, I.P., and  Kochina, I.N. 1960. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. ''J. Appl. Math. Mech.'' '''24''' (5): 1286–1303. http://dx.doi.org/10.1016/0021-8928(60)90107-6</ref>
<ref name="r26">Warren, J.E. and Root, P.J. 1963. The Behavior of Naturally Fractured Reservoirs. ''SPE J.'' '''3''' (3): 245–255. SPE-426-PA. http://dx.doi.org/10.2118/426-PA</ref>
<ref name="r27">Kazemi, H. 1969. Pressure Transient Analysis of Naturally Fractured Reservoir with Uniform Fracture Distribution. ''SPE J.'' '''9''' (4): 451–462. SPE-2156-PA. http://dx.doi.org/10.2118/2156-PA</ref>
<ref name="r28">de Swaan O., A. 1976. Analytical Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing. ''SPE J.'' '''16''' (3): 117–122. SPE-5346-PA. http://dx.doi.org/10.2118/5346-PA</ref>
<ref name="r29">Ozkan, E. 1988. ''Performance of Horizontal Wells''. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.
<ref name="r30">Luke, Y.L. 1962. ''Integrals of Bessel Functions'', 64–66. New York: McGraw-Hill Book Co.</ref>
<ref name="r31">Gradshteyn, I.S. and Ryzhik, I.M. 1980. ''Table of Integrals, Series, and Products'', 40. Orlando, Florida: Academic Press.</ref>
</references>
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
== SI Metric Conversion Factors ==
<div class="mw-collapsible-content">
<br>
{|
{|
|atm
|1.013 250*
|E + 05
|=
|Pa
|-
|-
|cp
| atm
|×  
| ×
|1.0*  
| 1.013 250*
|E – 03
| E + 05
|=  
| =
|Pa•s
| Pa
|-
|-
|in.
| cp
|×  
| ×
|2.54*  
| 1.0*
|E + 00
| E – 03
|=  
| =
|cm
| Pa•s
|-
|-
|in.<sup>2</sup>
| in.
|×  
| ×
|6.451 6*  
| 2.54*
|E + 00  
| E + 00
|=  
| =
|cm<sup>2</sup>
| cm
|-
|-
|°F
| in.<sup>2</sup>
|
| ×
|(°F−32)/1.8
| 6.451 6*
|
| E + 00
|=  
| =
|°C
| cm<sup>2</sup>
|-
|-
|ft  
| °F
|×  
|
|3.048*  
| (°F−32)/1.8
|E – 01  
|
|=  
| =
|m
| °C
|-
| ft
| ×
| 3.048*
| E – 01
| =
| m
|}
|}
<nowiki>*</nowiki>Conversion factor is exact.
</div></div>


[[Category:PEH]]
 
<nowiki>*</nowiki>
Conversion factor is exact.</div></div>[[Category:PEH]]
[[Category:5.6.3 Pressure transient analysis]]

Revision as of 10:53, 24 June 2015

Publication Information

Vol1GECover.png

Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume I – General Engineering

John R. Fanchi, Editor

Chapter 3 – Mathematics of Transient Analysis

Erdal Ozkan, Colorado School of Mines

Pgs. 77-172

ISBN 978-1-55563-108-6
Get permission for reuse



This chapter explains how fluid flow in porous media can be translated into a mathematical statement and how mathematical analysis can be used to answer transient-flow problems. This broad area is common to many other disciplines, such as heat conduction in solids and groundwater hydrology. The objective of this chapter is to introduce the fundamentals of transient analysis, present examples, and guide the interested reader to relevant references.

Introduction


Most physical phenomena in the domain of transient fluid flow in porous media can be described generally by partial differential equations (PDEs). With appropriate boundary conditions and sometimes with simplifying assumptions, the PDE leads to an initial boundary value problem (IBVP) that is solved to find a mathematical statement of the resulting flow in the porous medium. This section briefly discusses the statement of the IBVP for transient fluid flow in porous media.

Equations of Transient Fluid Flow in Porous Media

In essence, fluid motion in porous media can be specified by the knowledge of the velocity vector, RTENOTITLE, and the density of the fluid, ρ, as a function of the position (x, y, z) and time, t; that is, RTENOTITLE = RTENOTITLE (x, y, z, t) and ρ= ρ (x, y, z, t). Relative to the fixed Cartesian axes, the velocity vector can be written as

RTENOTITLE....................(3.1)

where vx, vy, and vz are the velocity components, and RTENOTITLE, RTENOTITLE, and RTENOTITLE are the unit vectors in the x, y, and z directions, respectively.

The physical law governing the macroscopic fluid-flow phenomena in porous media is the conservation of mass, which states that mass is neither created nor destroyed. The mathematical formula of this rule is developed by considering the flow through a fixed arbitrary closed surface, Γ, lying entirely within a porous medium of porosity Φ, which is filled with a fluid of viscosity μ. Fig. 3.1 illustrates an arbitrary closed surface in porous medium.


The conservation of mass principle requires that the difference between the rates at which fluid enters and leaves the volume through its surface must equal the rate at which mass accumulates within the volume. The total mass within the volume at any time is given by

RTENOTITLE....................(3.2)

Then, the time rate of change of mass within Γ is

RTENOTITLE....................(3.3)

which, by the conservation of mass law, is equal to the rate at which mass enters V through the surface.

Consider the differential surface element, dΓ, shown in Fig. 3.1. The mass entering the volume through dΓ at the normal velocity, RTENOTITLE, in a time increment, Δt, is RTENOTITLE, and the total mass of the fluid passing through Γ during Δt is

RTENOTITLE....................(3.4)

The surface integral in Eq. 3.4 accounts for both influx and outflux through the surface of the volume; that is, ΔMg is the difference between the masses entering and leaving the control volume during the time increment, Δt. Then, the mass rate entering the volume, V, through its surface, Γ, can be written as

RTENOTITLE....................(3.5)

By the principle of conservation of mass, equating the right sides of Eqs. 3.3 and 3.5 yields

RTENOTITLE....................(3.6)

A more useful relation is found with the divergence theorem, which states that the flux of RTENOTITLE through the closed surface, Γ, is identical to the volume integral of RTENOTITLE (the divergence of RTENOTITLE) taken throughout V; that is,

RTENOTITLE....................(3.7)

Here, ∇ is the gradient operator, which in 3D Cartesian and cylindrical coordinates is given, respectively, by

RTENOTITLE....................(3.8)

and

RTENOTITLE....................(3.9)

With the relation in Eq. 3.7, Eq. 3.6 can be recast into

RTENOTITLE....................(3.10)

If the functions involved in the argument of the integral in Eq. 3.10 are continuous, then the integral is identically zero if and only if its argument is zero (because the volume integral in Eq. 3.10 is identically zero for any arbitrarily chosen volume). Then, the following continuity equation can be obtained.

RTENOTITLE....................(3.11)

Eq. 3.11 is a PDE that is equivalent to the statement of the conservation of mass for fluid flow in porous media. For practical purposes, however, Eq. 3.11 is expressed in terms of pressure because density and velocity cannot be measured directly. To express density, ρ, and velocity, RTENOTITLE, in terms of pressure, we use an equation of state and a flux law, known as Darcy’s law, respectively.

The following definition of isothermal fluid compressibility, c, is a useful equation of state that relates density to pressure.

RTENOTITLE....................(3.12)

If c is a constant (the compressibility of many reservoir liquids may be considered as constant), then Eq. 3.12 can be integrated to yield

RTENOTITLE....................(3.13)

where subscript 0 indicates the conditions at the datum. Similarly, the compressibility of the porous rock, cf, is defined by

RTENOTITLE....................(3.14)

and the total system compressibility, ct, is given by

RTENOTITLE....................(3.15)

These definitions of compressibility help recast Eq. 3.11 in terms of pressure.

Darcy’s law for fluid flow in porous media is a flux law. Neglecting the gravity effect, it is expressed by

RTENOTITLE....................(3.16)

In Eq. 3.16, μ is the viscosity of the fluid, and k is the permeability tensor of the formation given by

RTENOTITLE....................(3.17)

where α, β, and γ are the directions, and kij is the permeability in the i direction as a result of the pressure gradient in the j direction.

If Eqs. 3.13 through 3.16 are used in Eq. 3.11, an alternative statement of the conservation of mass principle for fluid flow in porous media is obtained:

RTENOTITLE....................(3.18)

Eq. 3.18 is the PDE that governs transient fluid flow in porous media. In the present form, Eq. 3.18 is not very useful in obtaining practical solutions because of the nonlinearity displayed in the second term of the left side. For liquid flow, the viscosity, μ, is constant and Eq. 3.18 can be linearized by assuming that the pressure gradients, ∇p, are small in the reservoir and the compressibility of the reservoir liquids, c, is on the order of 10−5 or smaller. Then, the second term of the left side of Eq. 3.18 may be neglected compared with the remaining terms and the following linear expression is obtained:

RTENOTITLE....................(3.19)

Eq. 3.19 (or Eq. 3.18) is known as the diffusivity equation. As an example in Cartesian coordinates, assuming that the coordinate axes can be chosen in the directions of the principal permeabilities, k, in Eq. 3.19, may be represented by the following diagonal tensor:

RTENOTITLE....................(3.20)

Then, Eq. 3.19 may be written

RTENOTITLE....................(3.21)

If each coordinate, j = x, y, or z, is multiplied by RTENOTITLE, where k may be chosen arbitrarily (to preserve the material balance, k is usually chosen to be RTENOTITLE), Eq. 3.21 may be transformed into the diffusion equation for an isotropic domain:

RTENOTITLE....................(3.22)

where η is the diffusivity constant defined by

RTENOTITLE....................(3.23)

If the same transformation is also applied to the boundary conditions (see Sec. 3.1.2), the problems in anisotropic reservoirs may be transformed into those in isotropic reservoirs provided that the system is infinite or bounded by planes perpendicular to the principal axes of permeability. In all other cases, this transformation distorts the bounding surfaces.

For the flow of gases, the assumptions of small fluid compressibility and pressure gradient may not be appropriate and the c(∇p)2 term in Eq. 3.18 may not be negligible. In these cases, an expression similar to Eq. 3.21 may be obtained from Eq. 3.18 in terms of pseudopressure, m, as

RTENOTITLE....................(3.24)

Here, the pseudopressure is defined by[1]

RTENOTITLE....................(3.25)

where Z is the compressibility factor. To define a complete physical problem, Eq. 3.21 (or 3.24) should be subject to the initial and boundary conditions discussed in Sec. 3.1.2.

Initial and Boundary Conditions

The solution of the diffusivity equation (Eq. 3.19) should satisfy the initial condition in the porous medium. The initial condition is normally expressed in terms of a known pressure distribution at time zero; that is,

RTENOTITLE....................(3.26)

The most common initial condition is the uniform pressure distribution in the entire porous medium; that is, f (x, y, z) = pi.

The boundary conditions are specified at the inner (wellbore) and outer boundaries of the reservoir. These are usually in the form of prescribed flux or pressure at the boundaries. The condition of prescribed flux can be formulated as

RTENOTITLE....................(3.27)

where Γ is the surface of the boundary, and n indicates the outward normal direction of the boundary surface. The prescribed flux condition may be used at the inner and outer boundaries of the reservoir. The most common use of the prescribed flux condition at the inner boundary is for the production at a constant rate. In this case, the function, g(t), is related to a constant production rate, q. At the outer boundary, the prescribed flux condition is usually used to indicate impermeable boundaries [g(t)=0] and leads to a pseudosteady state under the influence of boundaries.

For some applications, pressure may be specified at the inner and outer boundaries. In this case,

RTENOTITLE....................(3.28)

When used at the inner boundary, this condition represents production at a constant pressure, pwf; that is, h(t) = pwf. At the outer boundary, specified pressure, pe, is usually a result of injection or influx from an adjacent aquifer, which usually leads to steady state in the reservoir.

It is also possible to have boundary conditions of mixed type. These usually correspond to interface conditions in porous media. Raghavan[2] contains more details about the common boundary conditions for the diffusion equation.

Assumptions and Limits

Some assumptions have been made in the derivation of the diffusivity equation given by Eq. 3.19. These assumptions determine the limits of applicability of the solutions obtained from Eq. 3.19. One of the most important assumptions involved is the continuity of the properties involved in Eq. 3.19. (This was required to obtain Eq. 3.19 from the more general integral form in Eq. 3.10.) Therefore, sharp changes in the properties of the reservoir rock and fluid (such as faults and fluid banks) should be incorporated in the form of boundary or interface conditions in the solution of Eq. 3.19.

The second important assumption is that Darcy’s law describes the flux in porous media. This assumption is valid at relatively low fluid velocities that may be appropriate to describe liquid flow. At high velocities (when Reynolds number based on average sand grain diameter approaches unity) such as those observed in gas reservoirs, Darcy’s law is not valid.[3] In this case, Forchheimer’s equation,[4] which accounts for the inertial effects, should be used. In petroleum engineering, it is a common practice to consider the additional pressure drop as a result of non-Darcy flow in the form of a pseudoskin because it is usually effective in a small vicinity of the wellbore. Therefore, in this chapter, we do not consider non-Darcy flow in the reservoir.

Bessel Functions


As Sec. 3.3 illustrates, the Laplace transform of the diffusion equation in radial coordinates yields a modified Bessel’s equation, and its solutions are obtained in terms of modified Bessel functions. This section introduces Bessel functions and discusses some of their properties to the extent that they are encountered in the solutions of more common petroleum engineering problems.

Preliminary Definitions

A differential equation of the type

RTENOTITLE....................(3.29)

is called a Bessel’s equation of order v. A solution of Bessel’s equation of order v is called a Bessel function of order v. A differential equation of the type

RTENOTITLE....................(3.30)

is called a modified Bessel’s equation of order v. Eq. 3.30 is obtained by substituting λz for z in Eq. 3.29. Of particular interest is the case in which λ=ki so that Eq. 3.30 becomes

RTENOTITLE....................(3.31)

Eq. 3.31 is called the modified Bessel’s equation of order v. A solution of the modified Bessel’s equation of order v is called a modified Bessel function of order v.

Solutions of Bessel’s Equations and Bessel Functions

There are many methods of obtaining or constructing Bessel functions.[5] Only the final form of the Bessel functions that are of interest are presented here.

If v is not a positive integer, then the general solution of Bessel’s equation of order v (Eq. 3.29) is given by

RTENOTITLE....................(3.32)

where A and B are arbitrary constants, and Jv(z) is the Bessel function of order v of the first kind given by

RTENOTITLE....................(3.33)

In Eq. 3.33, Γ(x) is the gamma function defined by

RTENOTITLE....................(3.34)

If v is a positive integer, n, then Jv and J−v are linearly dependent, and the solution of Eq. 3.29 is written as

RTENOTITLE....................(3.35)

In Eq. 3.35, Yn(z) is the Bessel function of order n of the second kind and is defined by

RTENOTITLE....................(3.36)

Similarly, if v is not a positive integer, the general solution of the modified Bessel’s equation of order v (Eq. 3.31) is given by

RTENOTITLE....................(3.37)

where Iv(z) is the modified Bessel function of order v of the first kind defined by

RTENOTITLE....................(3.38)

If v is a positive integer, n, Iv, and I−v are linearly dependent. The solution for this case is

RTENOTITLE....................(3.39)

where Kn(z) is the modified Bessel function of order n of the second kind and is defined by

RTENOTITLE....................(3.40)

The modified Bessel functions of order zero and one are of special interest, and Sec. 3.2.3 discusses some of their special features.

Modified Bessel Functions of Order Zero and One

Modified Bessel functions of order zero and one are related to each other by the following relations:

RTENOTITLE....................(3.41)

and

RTENOTITLE....................(3.42)

Fig. 3.2 shows these functions graphically.


For small arguments, the following asymptotic expansions may be used for the modified Bessel functions of order zero and one:[5]

RTENOTITLE....................(3.43)

RTENOTITLE....................(3.44)

RTENOTITLE....................(3.45)

where γ = 0.5772…, and

RTENOTITLE....................(3.46)

Also, for large arguments, the following relations may be useful:

RTENOTITLE....................(3.47)

for |argz| < π / 2, and

RTENOTITLE....................(3.48)

for |argz| < 3π / 2. On the basis of the relations given by Eqs. 3.43 through 3.48, the following limiting forms may be written:

RTENOTITLE....................(3.49)

RTENOTITLE....................(3.50)

RTENOTITLE....................(3.51)

RTENOTITLE....................(3.52)

RTENOTITLE....................(3.53)

RTENOTITLE....................(3.54)

RTENOTITLE....................(3.55)

RTENOTITLE....................(3.56)

and

RTENOTITLE....................(3.57)

These relations are useful in the evaluation of the asymptotic behavior of transient-pressure solutions.

Laplace Transformation


Integral transforms are useful in solving differential equations. A special form of the linear integral transforms, known as the Laplace transformation, is particularly useful in the solution of the diffusion equation. The Laplace transformation of a function, F(t), denoted by L{F(t)}, is defined by

RTENOTITLE....................(3.58)

where s is a number whose real part is positive and large enough for the integral in Eq. 3.58 to exist. In this chapter, a bar over the function indicates the image or the Laplace transform of the function; that is,

RTENOTITLE....................(3.59)

Fundamental Properties of the Laplace Transformation

The following fundamental properties of the Laplace transformation are useful in the solution of common transient-flow problems.

Transforms of Derivatives.

RTENOTITLE....................(3.60)

RTENOTITLE....................(3.61)

RTENOTITLE....................(3.62)

Transforms of Integrals.

RTENOTITLE....................(3.63)

Substitution.

RTENOTITLE....................(3.64)

RTENOTITLE....................(3.65)

where RTENOTITLE.

Translation.

RTENOTITLE....................(3.66)

where H(t - t0) is Heaviside’s unit step function defined by

RTENOTITLE....................(3.67)

Convolution.

RTENOTITLE....................(3.68)

Inverse Laplace Transformation and Asymptotic Forms

For the Laplace transform to be useful, the inverse Laplace transformation must be uniquely defined. L−1 denotes the inverse Laplace transform operator; that is,

RTENOTITLE....................(3.69)

In this operation, p(t) represents the inverse (transform) of the Laplace domain function, RTENOTITLE. A uniqueness theorem of the inversion guarantees that no two functions of the class ε have the same Laplace transform.[6] The class ε is defined as the set of sectionally continuous functions F(t) that are continuous on each bounded interval over the half line t > 0 except at a finite number of points, ti, where they are defined by

RTENOTITLE....................(3.70)

and | F(t) | < Meαt for any constants M and α.

The most rigorous technique to find the inverse Laplace transform of a Laplace domain function is the use of the inversion integral,[6] but its discussion is outside the scope of this chapter. For petroleum engineering applications, a simple table look-up procedure is usually the first resort. Table 3.1 shows an example table of Laplace transform pairs that may be used to find the Laplace transforms of real-space functions or the inverse Laplace transforms of the Laplace domain functions. Fairly large tables of Laplace transform pairs can be found in a couple of sources.[6][7] The relations given in the Laplace transform tables may be extended to more complex functions with the fundamental properties of the Laplace transforms noted in Sec. 3.3.1.


When a simple analytical inversion is not possible, numerical inversion of a Laplace domain function is an alternate procedure. Many numerical inversion algorithms have been proposed in the literature. For the inversion of the transient-flow solutions in Laplace domain, the numerical inversion algorithm suggested by Stehfest[8] is the most popular algorithm.

The Stehfest algorithm is based on a stochastic process and suggests that an approximate value, pa(T), of the inverse of the Laplace domain function, RTENOTITLE, may be obtained at time t = T by

RTENOTITLE....................(3.71)

where

RTENOTITLE....................(3.72)

In Eqs. 3.71 and 3.72, N is an even integer. Although, theoretically, the accuracy of the inversion should increase as N tends to infinity [pa (T) should tend to p(T)], the accuracy may be lost because of round-off errors when N becomes large. Normally, the optimum value of N is determined as a result of a numerical experiment. As a reference, however, the range of 6 ≤ N ≤ 18 covers the most common values of N for transient-flow problems. The Stehfest algorithm is not appropriate for the numerical inversion of oscillatory and discontinuous functions. In these cases, a more complex algorithm proposed by Crump[9] may be used.

In some cases, obtaining asymptotic solutions for small and large values of time may be of interest. These asymptotic results may be obtained without inverting the full solution into the real-time domain. The limiting forms of the full solution as s → ∞ and s → 0 correspond to the limiting forms in the time domain for short and long time, respectively. The inversion of the limiting forms may be easier than the inversion of the full solution. Examples 3.1 through 3.4 demonstrate the use of Laplace transformation in the solution of transient-flow problems.
Example 3.1

Consider transient flow toward a fully penetrating vertical well in an infinite homogeneous reservoir of uniform thickness, h, and initial pressure, pi.

Solution. This problem may be formulated most conveniently in the radial coordinates. The diffusivity equation governing fluid flow in porous media is given, in radial coordinates, by

RTENOTITLE....................(3.73)

where ∆p = pip. Eq. 3.73 is the same in absolute (cgs or SI) or Darcy units. (In field units, some conversion coefficients are involved in Eq. 3.73.) The initial condition is

RTENOTITLE....................(3.74)

which means that the pressure is uniform and equal to pi initially throughout the reservoir. The outer boundary condition for an infinite reservoir is

RTENOTITLE....................(3.75)

which physically means that for any given time, t, there is a large enough distance, r, in the reservoir at which the initial pressure, pi, has been preserved.

The inner boundary condition depends on the production conditions at the surface of the wellbore (r = rw). Assuming that the well is produced at a constant rate, q, for all times,

RTENOTITLE....................(3.76)

The inner boundary condition given in Eq. 3.76 is simply a restatement of the flux law (Darcy’s law given by Eq. 3.16) at the surface of the wellbore.

Eqs. 3.73 through 3.76 define the IBVP to be solved to obtain the transient-pressure distribution for the given system. Application of the Laplace transforms to Eq. 3.73 yields

RTENOTITLE....................(3.77)

or, rearranging, we obtain

RTENOTITLE....................(3.78)

In obtaining the right side of Eq. 3.77, the initial condition (Eq. 3.74) has been used. Similarly, Eqs. 3.75 and 3.76 are transformed into the following forms, respectively.

RTENOTITLE....................(3.79)

and

RTENOTITLE....................(3.80)

Comparing Eq. 3.78 with Eq. 3.31, we recognize Eq. 3.78 as the modified Bessel’s equation of order zero. The solution of Eq. 3.78 may be written directly from Eq. 3.39 as

RTENOTITLE....................(3.81)

The constants C1 and C2 in Eq. 3.81 are obtained from the boundary conditions. The outer boundary condition (Eq. 3.79) indicates that C1 = 0 [because RTENOTITLE, Eq. 3.79 is satisfied only if C1 = 0]; therefore,

RTENOTITLE....................(3.82)

From Eqs. 3.80 and 3.82, we obtain

RTENOTITLE....................(3.83)

which yields

RTENOTITLE....................(3.84)

Then, the solution for the transient-pressure distribution is given, in the Laplace transform domain, by

RTENOTITLE....................(3.85)

To complete the solution of the problem, Eq. 3.85 should be inverted into the real-time domain. The real inversion of Eq. 3.85, however, is not available in terms of standard functions. One option is to use Stehfest’s numerical inversion algorithm[8] as discussed in Sec. 3.3.2. The dashed line in Fig. 3.3 represents the numerical inversion of the solution in Eq. 3.85. Another option is to find an approximate inversion. One of these asymptotic forms is known as the line-source solution and commonly used in transient-pressure analysis.


To obtain the line-source approximation of the solution given in Eq. 3.85, we assume that the radius of the wellbore is small compared with the other dimensions of the reservoir. Thus, if we assume rw→0 and use the relation given in Eq. 3.56, we obtain

RTENOTITLE....................(3.86)

Using this relation in Eq. 3.85, we obtain the line-source solution in Laplace domain as

RTENOTITLE....................(3.87)

The inversion of Eq. 3.87 can be accomplished by using a Laplace transform table. From Table 3.1 (or from the tables in two sources[6][7]), we have

RTENOTITLE....................(3.88)

With Eq. 3.88 and the Laplace transform property noted in Eq. 3.63, we obtain the following inversion of Eq. 3.87 in the real-time domain:

RTENOTITLE....................(3.89)

Making the substitution u = r2 / (4ηt′) and noting the definition of the exponential integral function, Ei(x), given by

RTENOTITLE....................(3.90)

we obtain the line-source solution as

RTENOTITLE....................(3.91)

Fig. 3.3 shows a comparison of the results computed from Eq. 3.85 (finite-wellbore radius) and Eq. 3.91 (line source) for the data noted in the figure. The two solutions yield different results at early times but become the same at later times. In fact, it can be shown analytically that the long-time approximation of the finite-wellbore radius solution (Eq. 3.85) is the same as the line-source well solution. To show this, we note that the long-time approximation of the solution in the time domain corresponds to the limiting form of the solution in the Laplace domain as s → 0. Then, with the property of the Bessel function given in Eq. 3.56, we can show that

RTENOTITLE....................(3.92)

Example 3.2

Consider transient flow as a result of constant-rate production from a fully penetrating vertical well in a closed cylindrical reservoir initially at uniform initial pressure, pi.

Solution. Fluid flow in cylindrical porous media is described by the diffusion equation in radial coordinates given by

RTENOTITLE....................(3.93)

The initial condition corresponding to the uniform pressure distribution equal to pi is

RTENOTITLE....................(3.94)

and the inner boundary condition for a constant production rate, q, for all times is

RTENOTITLE....................(3.95)

The closed outer boundary condition is represented mathematically by zero flux at the outer boundary (r = re) that corresponds to

RTENOTITLE....................(3.96)

The Laplace transforms of Eqs. 3.93 through 3.96 yield, respectively,

RTENOTITLE....................(3.97)

RTENOTITLE....................(3.98)

and

RTENOTITLE....................(3.99)

(The initial condition given by Eq. 3.94 has been used to obtain Eq. 3.97.) Because Eq. 3.97 is the modified Bessel’s equation of order zero, its general solution is given by

RTENOTITLE....................(3.100)

With the outer boundary condition given by Eq. 3.99, we obtain

RTENOTITLE....................(3.101)

which yields

RTENOTITLE....................(3.102)

and thus

RTENOTITLE....................(3.103)

Using the inner boundary condition given by Eq. 3.98 yields

RTENOTITLE....................(3.104)

From Eqs. 3.102 and 3.104, we obtain the coefficients C1 and C2 as follows:

RTENOTITLE....................(3.105)

and

RTENOTITLE....................(3.106)

Substituting C1 and C2 into Eq. 3.100 yields

RTENOTITLE....................(3.107)

The inverse of the solution given by Eq. 3.107 may not be found in the Laplace transform tables. van Everdingen and Hurst[10] provided the following analytical inversion of Eq. 3.107 with the inversion integral.

RTENOTITLE RTENOTITLE....................(3.108)

In Eq. 3.108, β1, β2, etc. are the roots of

RTENOTITLE....................(3.109)

The solution given in Eq. 3.107 may also be inverted numerically with the Stehfest algorithm.[8] Fig. 3.4 shows the results of the numerical inversion of Eq. 3.107.


Example 3.3

Consider the flowing wellbore pressure of a fully penetrating vertical well with wellbore storage and skin in an infinite reservoir.

Solution. Revisit the case in Example 3.1 and add the effect of a skin zone around the wellbore. Assume that the constant production rate is specified at the surface so that the storage capacity of the wellbore needs to be taken into account. Before presenting the initial-boundary value problem, skin factor and surface production rate should be defined.

Using van Everdingen and Hurst’s thin-skin concept[10] (vanishingly small skin-zone radius), the skin factor is defined by

RTENOTITLE....................(3.110)

where qsf is the sandface production rate, p(rw +) denotes the reservoir pressure immediately outside the skin-zone boundary, and pwf is the flowing wellbore pressure measured inside the wellbore. Rearranging Eq. 3.110, we obtain the following relation for the flowing wellbore pressure.

RTENOTITLE....................(3.111)

When the production rate is specified at the surface, it is necessary to account for the fact that the wellbore can store and unload fluids. The surface production rate, q, is equal to the sum of the wellbore unloading rate, qwb, and the sandface production rate, qsf; that is,

RTENOTITLE....................(3.112)

where

RTENOTITLE....................(3.113)

and

RTENOTITLE....................(3.114)

In Eq. 3.113, C is the wellbore-storage coefficient. Substituting Eqs. 3.113 and 3.114 into Eq. 3.112, we obtain the following expression for the surface production rate.

RTENOTITLE....................(3.115)

The mathematical statement of the problem under consideration is similar to that in Example 3.1, except that the inner-boundary condition should be replaced by Eq. 3.115, and Eq. 3.111 should be incorporated to account for the skin effect. The IBVP is defined by the following set of equations in the Laplace domain:

RTENOTITLE....................(3.116)

RTENOTITLE....................(3.117)

RTENOTITLE....................(3.118)

and

RTENOTITLE....................(3.119)

The general solution of Eq. 3.116 is

RTENOTITLE....................(3.120)

The condition in Eq. 3.117 requires that C1 = 0; therefore,

RTENOTITLE....................(3.121)

The use of Eq. 3.121 in Eq. 3.119 yields

RTENOTITLE....................(3.122)

From Eqs. 3.118, 3.121, and 3.122, we obtain

RTENOTITLE....................(3.123)

which yields

RTENOTITLE....................(3.124)

Substituting Eq. 3.124 for C2 in Eq. 3.122, we obtain the solution for the transient-pressure distribution in the Laplace transform domain as

RTENOTITLE....................(3.125)

The real inversion of the solution in Eq. 3.125 has been obtained by Agarwal et al.[11] with the inversion integral. It is also possible to invert Eq. 3.125 numerically. Fig. 3.5 shows the results of the numerical inversion of Eq. 3.125 with the Stehfest’s algorithm.[8] Also shown in Fig. 3.5 are the logarithmic derivatives of Δpwf. These derivatives are computed by applying the Laplace transformation property given in Eq. 3.60 to Eq. 3.125 as follows:

RTENOTITLE....................(3.126)

Here, we have used Δpw f(t = 0) = 0. To obtain the logarithmic derivatives, we simply note that

RTENOTITLE....................(3.127)


Example 3.4

Consider pressure buildup with wellbore storage and skin following a drawdown period at a constant rate in an infinite reservoir.

Solution. This example is similar to Example 3.3 except, at time tp, the well is shut in and pressure buildup begins. The system of equations to define this problem is

RTENOTITLE....................(3.128)

RTENOTITLE....................(3.129)

RTENOTITLE....................(3.130)

RTENOTITLE....................(3.131)

where H(t - tp) is Heaviside’s unit function (Eq. 3.67), and

RTENOTITLE....................(3.132)

The right side of the boundary condition in Eq. 3.131 accounts for a constant surface production rate, q, for 0 < t < tp and for shut in (q = 0) for t > tp. Taking the Laplace transforms of Eqs. 3.128 through 3.132, we obtain

RTENOTITLE....................(3.133)

RTENOTITLE....................(3.134)

RTENOTITLE....................(3.135)

and

RTENOTITLE....................(3.136)

The general solution of Eq. 3.133 is

RTENOTITLE....................(3.137)

The condition in Eq. 3.134 requires that C1= 0; therefore,

RTENOTITLE....................(3.138)

From Eqs. 3.138 and 3.136, we obtain

RTENOTITLE....................(3.139)

Substituting the results of Eqs. 3.138 and 3.139 into Eq. 3.135, we have

RTENOTITLE....................(3.140)

which yields

RTENOTITLE....................(3.141)

Substituting Eq. 3.141 into Eq. 3.139, we obtain the following solution in the Laplace transform domain, which covers both the drawdown and buildup periods.

RTENOTITLE....................(3.142)

The RTENOTITLE term contributed by the discontinuity at time t = tp causes difficulties in the numerical inversion of the right side of Eq. 3.142 with the use of the Stehfest algorithm.[8] As suggested by Chen and Raghavan,[12] this problem may be solved by noting that

RTENOTITLE....................(3.143)

and applying the Stehfest algorithm term by term to the right side of Eq. 3.143. Fig. 3.6 shows sample results obtained by the numerical inversion of Eq. 3.142.



Green’s Functions and Source Functions


Green’s function and source functions are used to solve 2D and 3D transient-flow problems that may result from complex well geometries, such as partially penetrating vertical and inclined wells, hydraulically fractured wells, and horizontal wells. Before introducing these techniques, it is useful to clarify the terminology.

In our terminology, a source is a point, line, surface, or volume at which fluids are withdrawn from the reservoir. Strictly speaking, fluid withdrawal should be associated with a sink, and the injection of fluids should be related to a source. Here, however, the term source is used for both production and injection with the convention that a negative withdrawal rate indicates injection.

Green’s functions and source functions are closely related. A Green’s function is defined for a differential equation with specified boundary conditions (prescribed flux or pressure) and corresponds to an instantaneous point-source solution. A source function, on the other hand, is the solution of the given differential equation with specified boundary conditions and source geometry.

The details of the theory and application of Green’s function and source functions for the solution of transient-flow problems in porous media can be found in many sources.[2][13][14][15][16][17][18][19][20] A brief account of the use of these techniques is presented here, as well as an introduction of the fundamental solution and point-source concepts.

Fundamental Solution of the Diffusion Equation

The fundamental solution, γf(M, M′, t, τ), of the diffusion equation for fluid flow in porous media satisfies the following differential equation:

RTENOTITLE....................(3.144)

where δ(M, M′, t, τ) is a generalized (symbolic) function[15] called the Dirac delta function and is defined on the basis of its following properties:

RTENOTITLE....................(3.145)

and

RTENOTITLE....................(3.146)

The delta function is symmetric in M and M′ and also in t and τ. In this formulation, the delta function represents the symbolic density of a unit-strength, concentrated source located at M′ and acting at time τ. In physical terms, this source corresponds to an infinitesimally small well (located at point M′) at which a finite amount of fluid is withdrawn (or injected) instantaneously (at time τ). Therefore, the solution of Eq. 3.144 (the fundamental solution) is also known as the instantaneous point-source solution. Formally, the point-source solution corresponds to the pressure drop, Δp = pip, at a point M and time t in an infinite porous medium (reservoir) because of a point source of unit strength located at point M′ and acting at τ <t.

The Source-Function Solutions of the Diffusion Equation

The point-source solution was first introduced by Lord Kelvin[16] for the solution of heat conduction problems and was extensively discussed by Carslaw and Jaeger.[14] The point-source solution is usually obtained by finding the limiting form of the pressure drop resulting from a spherical source as the source volume vanishes. To demonstrate the derivation of the instantaneous point-source solution, consider the transient flow of a slightly compressible fluid of constant compressibility and viscosity toward a spherical source of radius r = a in an infinite, homogeneous, and isotropic porous medium. Because of the spherical symmetry of the physical problem, we can conveniently express the governing equation of fluid flow in porous media in spherical coordinates as

RTENOTITLE....................(3.147)

Assume that the initial pressure drop satisfies

RTENOTITLE....................(3.148)

and we have the condition that

RTENOTITLE....................(3.149)

On substitution of u = rΔp, Eqs. 3.147 through 3.149 become, respectively,

RTENOTITLE....................(3.150)

RTENOTITLE....................(3.151)

and

RTENOTITLE....................(3.152)

The solution of the problem described by Eqs. 3.150 through 3.152 is given by[14]

RTENOTITLE....................(3.153)

If we expand the exponential terms in the integrand in Eq. 3.153 in powers of r′ and neglect the terms with powers higher than four, we obtain

RTENOTITLE....................(3.154)

In Eq. 3.154, 4πα3/3=V where V is the volume of the spherical source. If RTENOTITLE denotes the volume of the liquid released as a result of the change in the volume of the source, ΔV, which is caused by the change in pressure, Δpi, then RTENOTITLE. With the definition of compressibility, c = -(1 / V)(ΔV / Δpi), we obtain RTENOTITLE. Then, we can show that

RTENOTITLE....................(3.155)

Substituting Eq. 3.155 into Eq. 3.154, we obtain

RTENOTITLE....................(3.156)

If we let the radius of the spherical source, a, tend to zero while RTENOTITLE remains constant, Eq. 3.156 yields the point-source solution in spherical coordinates given by

RTENOTITLE....................(3.157)

This solution may be interpreted as the pressure drop at a distance r because of a volume of fluid, RTENOTITLE, instantaneously withdrawn at r = 0 and t = 0. Consistent with this interpretation, RTENOTITLE is the strength of the source, which is the pressure drop in a unit volume of the porous medium caused by the instantaneous withdrawal of a volume of fluid, RTENOTITLE (see Eq. 3.155).

Instantaneous Point Source in an Infinite Reservoir. Nisle[21] presented a more general solution for an instantaneous point source of strength RTENOTITLE acting at t = τ in an infinite, homogeneous, but anisotropic reservoir as

RTENOTITLE....................(3.158)

In Eq. 3.158, M and M′ indicate the locations of the observation point and the source, respectively. For a 3D Cartesian coordinate system, RTENOTITLE with ηx, ηy, and ηz representing the diffusivity constants (defined in Eq. 3.23) in the x, y, and z directions, respectively.

Continuous Point Source in an Infinite Reservoir. If the fluid withdrawal is at a continuous rate, RTENOTITLE, from time 0 to t, then the pressure drop as a result of a continuous point source in an infinite reservoir is obtained by distributing the point sources of strength RTENOTITLE over a time interval 0 ≤ τt. This is given by

RTENOTITLE....................(3.159)

where S(M, M′, tτ) corresponds to a unit-strength RTENOTITLE, instantaneous point source in an infinite reservoir; that is,

RTENOTITLE....................(3.160)

Instantaneous and Continuous Line, Surface, and Volumetric Sources in an Infinite Reservoir. Similarly, the distribution of instantaneous point sources of strength RTENOTITLE over a line, surface, or volume, Γw, in an infinite reservoir leads to the following solution corresponding to the pressure drop because of production from a line, surface, or volumetric source, respectively.

RTENOTITLE....................(3.161)

In Eq. 3.161, Mw indicates a point on the source (Γw) and RTENOTITLE is the instantaneous withdrawal volume of fluids per unit length, area, or volume of the source, depending on the source geometry. For example, the pressure drop as a result of an infinite line source at x′, y′ and -∞≤ z′ ≤ ∞ may be obtained as follows:

RTENOTITLE....................(3.162)

If we assume that the flux is uniform along the line source and the source strength is unity RTENOTITLE, then we can write the instantaneous, infinite line-source solution in an infinite reservoir as

RTENOTITLE....................(3.163)

As another example, if we consider an instantaneous, infinite plane source at x = x′, -∞ ≤ y′ ≤ ∞, and -∞ ≤ z′ ≤ ∞ in an infinite reservoir, we can write

RTENOTITLE....................(3.164)

which also leads to the following uniform-flux, unit-strength, instantaneous, infinite plane-source solution in an infinite reservoir:

RTENOTITLE....................(3.165)

If the fluid withdrawal is at a continuous rate from time 0 to t, then the continuous line-, surface-, or volumetric-source solution for an infinite reservoir is given by

RTENOTITLE....................(3.166)

Source Functions for Bounded Reservoirs. The source solutions discussed previously can be extended to bounded reservoirs. The method of images provides a convenient means of generating the bounded-reservoir solutions with the use of the infinite reservoir solutions, especially when the reservoir boundaries consist of impermeable and constant-pressure planes. The method of images is an application of the principle of superposition, which states that if f1 and f2 are two linearly independent solutions of a linear PDE and c1 and c2 are two arbitrary constants, then f3 = c1f1 + c2f2 is also a solution of the PDE. Examples of source functions in bounded reservoirs are presented here.

Instantaneous Point Source Near a Single Linear Boundary. To generate the effect of an impermeable planar boundary at a distance d from a unit-strength, instantaneous point source in an infinite reservoir (see Fig. 3.7), we can apply the method of images to the instantaneous point-source solution given in Eq. 3.157 as

RTENOTITLE....................(3.167)

It can be shown from Eq. 3.167 that (∂S/∂x)x=d = 0; that is, the bisector of the distance between the two sources is a no-flow boundary. Similarly, to generate the effect of a constant-pressure boundary, we use the method of images and the unit-strength, instantaneous point-source solution (Eq. 3.160) as follows:

RTENOTITLE....................(3.168)

Instantaneous Point Source in an Infinite-Slab Reservoir. Using the method of images and considering the geometry shown in Col. A of Fig. 3.8, we can generate the solution for a unit-strength, instantaneous point source in an infinite-slab reservoir with impermeable boundaries at z = 0 and h. The result is given by

RTENOTITLE....................(3.169)

which, with Poisson’s summation formula given by[14]

RTENOTITLE....................(3.170)

may be transformed into

RTENOTITLE....................(3.171)


Following similar lines, if the slab boundaries at z = 0 and h are at a constant pressure equal to pi, we obtain

RTENOTITLE....................(3.172)

Similarly, for the case in which the slab boundary at z = 0 is impermeable while the boundary at z = h is at a constant pressure equal to pi, the following solution may be derived:

RTENOTITLE....................(3.173)

Instantaneous Point Source in a Closed Parallelepiped. The ideas used previously for slab reservoirs may be used to develop solutions for reservoirs bounded by planes in all three directions. For example, if the reservoir is bounded in all three directions (i.e., 0 ≤ xxe, 0 ≤ yye, and 0 ≤ zh) and the bounding planes are impermeable, then we can use Eq. 3.157 and the method of images to write

RTENOTITLE RTENOTITLE....................(3.174)

which, with Poisson’s summation formula (Eq. 3.170), may be recast into the following form:

RTENOTITLE....................(3.175)

Instantaneous Infinite-Plane Source in an Infinite-Slab Reservoir With Impermeable Boundaries. The instantaneous point-source solutions of Eqs. 3.171 through 3.173 may be extended to different source geometries with Eq. 3.161. For example, the solution for an instantaneous infinite-plane source at z = z′ in an infinite-slab reservoir with impermeable boundaries is obtained by substituting Eq. 3.171 for S in Eq. 3.161. This yields

RTENOTITLE....................(3.176)

Assuming a unit-strength, uniform-flux source RTENOTITLE, we obtain the following instantaneous infinite-plane source solution in an infinite-slab reservoir with impermeable boundaries:

RTENOTITLE....................(3.177)

Instantaneous Infinite-Slab Source in an Infinite-Slab Reservoir With Impermeable Boundaries. Following similar lines, we can obtain the solution for an instantaneous, infinite-slab source of thickness, hp, located at z = zw (zw is the z-coordinate of the midpoint of the slab source) in an infinite-slab reservoir with impermeable boundaries.

RTENOTITLE....................(3.178)

If we assume a uniform-flux slab source RTENOTITLE, then Eq. 3.178 yields

RTENOTITLE....................(3.179)

Uniform-Flux, Continuous, Infinite-Slab Source in an Infinite-Slab Reservoir With Impermeable Boundaries. Solutions for continuous plane and slab sources can be obtained as indicated by Eq. 3.159 (or Eq. 3.166). For example, the solution for a uniform-flux, continuous, infinite-slab source in an infinite-slab reservoir with impermeable top and bottom boundaries may be obtained by substituting the right side of Eq. 3.179 for S in Eq. 3.159 and is given by

RTENOTITLE....................(3.180)

The same solution could have been obtained by substituting the unit-strength instantaneous point-source solution given by Eq. 3.171 for S in Eq. 3.166.
Example 3.5

Consider transient flow toward a partially penetrating vertical well of penetration length, hw, in an infinite, homogeneous, slab reservoir of uniform thickness, h, and initial pressure, pi, with impermeable top and bottom boundaries.

Solution. Fig. 3.9 shows the geometry of the well and reservoir system of interest. The solution for this problem can be obtained by assuming that the well may be represented by a vertical line source. Then, starting with Eq. 3.166 and substituting the unit-strength, instantaneous point-source solution in an infinite-slab reservoir with impermeable boundaries [Eq. 3.171 with RTENOTITLE] for S, we obtain

RTENOTITLE....................(3.181)

If we assume that the strength of the source is uniformly distributed along its length (this physically corresponds to a uniform-flux distribution) and the production rate is constant over time [i.e., RTENOTITLE, where q is the constant production rate of the well], then Eq. 3.181 yields

RTENOTITLE RTENOTITLE....................(3.182)


The Use of Green’s Functions and Source Functions in Solving Unsteady-Flow Problems

As discussed in Sec. 3.4.2, the conventional development of the source-function solutions uses the instantaneous point-source solution as the building block with the appropriate integration (superposition) in space and time. In 1973, Gringarten and Ramey[13] introduced the use of the source and Green’s function method to the petroleum engineering literature with a more efficient method of developing the source solutions. Specifically, they suggested the use of infinite-plane sources as the building block with Newman’s product method.[22] In this section, we discuss the use of Green’s functions and source functions in solving unsteady-flow problems in reservoirs.

Green’s function for transient flow in a porous medium is defined as the pressure at M (x, y, z) at time t because of an instantaneous point source of unit strength generated at point M′(x′, y′, z′) at time τ < t with the porous medium initially at zero pressure and the boundary of the medium kept at zero pressure or impermeable to flow.[13][14] If we let G(M, M′, tτ) denote the Green’s function, then it should satisfy the diffusion equation; that is,

RTENOTITLE....................(3.183)

Because G is a function of tτ, it should also satisfy the adjoint diffusion equation,

RTENOTITLE....................(3.184)

Green’s function also has the following properties: [13][14]

  1. G is symmetrical in the two points M and M′; that is, Green’s function is invariant as the source and the observation points are interchanged.
  2. As tτ, G vanishes at all points in the porous medium; that is, RTENOTITLE, except at the source location, M = M′, where it becomes infinite, so that G satisfies the delta function property,



RTENOTITLE....................(3.185)

where D indicates the domain of the porous medium, and φ(M) is any continuous function.

  1. Because G corresponds to the pressure because of an instantaneous point source of unit strength, it satisfies



RTENOTITLE....................(3.186)

  1. G or its normal derivative, ∂G/∂n, vanishes at the boundary, Γ, of the porous medium. If the porous medium is infinite, then G vanishes as M or M′→∞.


Let p(M′ , τ) be the pressure in the porous medium and G(M, M′, t - τ) be the Green’s function. Let D denote the domain of the porous medium. Then, p and G satisfy the following differential equations:

RTENOTITLE....................(3.187)

and

RTENOTITLE....................(3.188)

Then, we can write

RTENOTITLE....................(3.189)

or

RTENOTITLE....................(3.190)

where ε is a small positive number. Changing the order of integration and applying the Green’s theorem,

RTENOTITLE....................(3.191)

where D and Γ indicate the volume and boundary surface of the domain, respectively; S denotes the points on the boundary; and /∂n indicates differentiation in the normal direction of the surface Γ; we obtain

RTENOTITLE....................(3.192)

Taking the limit as ε→0 and noting the delta-function property of the Green’s function (Eq. 3.185), Eq. 3.192 yields

RTENOTITLE....................(3.193)

where pi(M) = p(M, t = 0) is the initial pressure at point M.

In Eq. 3.193, the boundary of the porous medium consists of two surfaces: the inner boundary that corresponds to the surface of the wellbore, Γw, and the outer boundary of the reservoir, Γe. Eq. 3.193 may be written as

RTENOTITLE....................(3.194)

As the fourth property of Green’s function noted previously requires, if the outer boundary of the reservoir is impermeable, RTENOTITLE or at infinity, then G vanishes at the outer boundary; that is, Ge) = 0. Thus, Eq. 3.194 becomes

RTENOTITLE....................(3.195)

Similarly, if the flux, RTENOTITLE, is specified at the inner boundary, then the normal derivative of Green’s function, RTENOTITLE, vanishes at that boundary. This yields

RTENOTITLE....................(3.196)

If the initial pressure, pi, is uniform over the entire domain (porous medium), D, then, by the third property of Green’s function (Eq. 3.186), we should have

RTENOTITLE....................(3.197)

Also, the flux law for porous medium (Darcy’s law) requires that the volume of fluid passing through the point, M′w, on the inner-boundary surface, Γw, at time t be equal to

RTENOTITLE....................(3.198)

The substitution of Eqs. 3.197 and 3.198 into Eq. 3.196 yields

RTENOTITLE....................(3.199)

Not surprisingly, Eq. 3.199 is the same as Eq. 3.166 because G in Eq. 3.199 is the instantaneous point-source solution of unit strength denoted by S in Eq. 3.166.

The expression given in Eq. 3.199 may be simplified further by assuming that the flux, RTENOTITLE, is uniformly distributed on the inner-boundary surface (wellbore), Γw. This yields

RTENOTITLE....................(3.200)

where

RTENOTITLE....................(3.201)

The integration in the right side of Eq. 3.201 represents the distribution of instantaneous point sources over the length, area, or volume of the source (well), and S denotes the corresponding instantaneous source function. In Sec. 3.4.2, we discussed the conventional derivation of the source functions starting from the basic instantaneous point-source solution. Here, we discuss the use of infinite-plane sources as the building block with Newman’s product method.[22]

Newman’s product method[22] may be stated for transient-flow problems in porous media as follows: [13] if a well/reservoir system can be visualized as the intersection of 1D or 2D systems, then the instantaneous source or Green’s function for this well/reservoir system can be constructed by the product of the source or Green’s functions for the 1D and/or 2D systems. For example, an infinite line-source well at x = x′, y = y′, and −∞ ≤ z′ ≤ +∞ in an infinite reservoir may be visualized as the intersection of two infinite, 1D plane sources; one at x = x′, −∞ ≤ y′ ≤ +∞, and −∞ ≤ z′ ≤ +∞, and the other at −∞ ≤ x′ ≤ +∞, y = y′, and −∞ ≤ z′ ≤ +∞. Then, the instantaneous source function for this infinite line-source well, S(x, x′, y, y′, tτ), may be obtained as the product of two infinite, 1D plane sources, given by

RTENOTITLE....................(3.202)

as follows

RTENOTITLE....................(3.203)

As expected, this solution is the same as Eq. 3.163, which was obtained by integration of an instantaneous point source in an infinite reservoir. For a radially isotropic reservoir (ηx = ηy = ηz), Eq. 3.203 yields

RTENOTITLE....................(3.204)

where d is the distance between the line source and the observation line in the x-y plane (see Fig. 3.10) and is given by

RTENOTITLE....................(3.205)

Similarly, intersecting three infinite instantaneous plane sources (or a line source and a plane source), we can obtain the instantaneous point-source solution in an infinite reservoir as

RTENOTITLE....................(3.206)

Instantaneous plane sources in slab reservoirs can be generated with the plane sources in infinite reservoirs and the method of images as discussed in Sec. 3.4.2. Similarly, the instantaneous slab sources can be obtained by integrating plane sources over the thickness of the slab source (see Sec. 3.4.2). Table 3.2, compiled from the work of Gringarten and Ramey,[13] presents the basic instantaneous source functions in infinite reservoirs, and Table 3.3 shows the corresponding geometries of the source-reservoir systems. The basic instantaneous source functions given in Table 3.3 may be used to construct the source functions that represent the appropriate well geometry by Newman’s product method.


Gringarten and Ramey[13] have also presented the approximating forms of the instantaneous linear sources and the time ranges for these approximations to be valid. The approximate solutions are very useful in obtaining expressions for pressure distributions at early and late times and identifying the flow regimes during the corresponding time periods. Table 3.4 presents the short- and long-time approximating forms for instantaneous linear sources and their time ranges. Examples 3.6 and 3.7 present some applications of the product-solution method and the derivation of the approximate solutions for pressure distributions.


Example 3.6

Consider transient flow toward a partially penetrating vertical fracture of vertical penetration hf and horizontal penetration 2xf in an infinite, homogeneous, slab reservoir of uniform thickness, h, and initial pressure, pi, with impermeable top and bottom boundaries.

Solution. Fig. 3.11 shows the geometry of the well reservoir system of interest. Approximate the fracture by a vertical plane of height hf and length 2xf. The corresponding source geometry may be visualized as the intersection of an infinite plane source at y = y′ in an infinite reservoir (Source I in Tables 3.2 and 3.3), an infinite-slab source of thickness 2xf at x = x′ in an infinite reservoir (Source IV), and an infinite-slab source of thickness hp = hf at z = zw in an infinite-slab reservoir of thickness h (Source VIII). Then, by Newman’s product method, the appropriate source function is given by

RTENOTITLE RTENOTITLE....................(3.207)

Assuming that the production is at a constant rate, RTENOTITLE and using Eq. 3.207 for the source function, S, in Eq. 3.200, we obtain

RTENOTITLE RTENOTITLE....................(3.208)

If the fracture penetrates the entire thickness of the reservoir (i.e., hf = h) as shown in Fig. 3.12, then Eq. 3.208 yields

RTENOTITLE....................(3.209)

The fully penetrating fracture solution given in Eq. 3.209 also could be obtained by constructing the source function as the product of an infinite plane source at y = y′ in an infinite reservoir (Source I in Tables 3.2 and 3.3) and an infinite-slab source of thickness 2xf at x = x′ in an infinite reservoir (Source IV). This source function then could be used in Eq. 3.200.


Fig. 3.13 presents an example of transient-pressure responses computed from Eq. 3.209. To obtain the results shown in Fig. 3.13, numerical integration has been used to evaluate the right side of Eq. 3.209. It is also of interest to obtain an early-time approximation for the solution given in Eq. 3.209. Substituting the early-time approximating forms for the slab sources in an infinite reservoir (approximations given in Table 3.4 for Source Functions IV and VIII), we obtain

RTENOTITLE....................(3.210)

where

RTENOTITLE....................(3.211)

and

RTENOTITLE....................(3.212)


Assuming a constant production rate, RTENOTITLE, and substituting the source function given by Eq. 3.210 in Eq. 3.200, we obtain

RTENOTITLE....................(3.213)

where erfc (z) is the complementary error function defined by

RTENOTITLE....................(3.214)

Example 3.7

Consider transient flow toward a uniform-flux horizontal well of length Lh located at (x′, y′, zw) in a closed, homogeneous, rectangular parallelepiped of dimensions 0 ≤ xxe, 0 ≤ yye, 0 ≤ zh and of initial pressure, pi.

Solution. Fig. 3.14 shows the sketch of the horizontal-well/reservoir system considered in this example. If we approximate the horizontal well by a horizontal line source of length Lh, then the resulting source/reservoir system may be visualized as the intersection of three sources: an infinite plane source at y = y′ in an infinite-slab reservoir of thickness ye with impermeable boundaries (Source V in Tables 3.2 and 3.3), an infinite-slab source of thickness Lh at x = x′ in an infinite-slab reservoir of thickness xe with impermeable boundaries (Source VIII), and an infinite-plane source at z = zw in an infinite-slab reservoir of thickness h with impermeable boundaries (Source V). Then, by Newman’s product method, the appropriate source function can be obtained as

RTENOTITLE....................(3.215)

Assuming that the production is at a constant rate, RTENOTITLE, and using Eq. 3.215 for the source function, S, in Eq. 3.200, we obtain

RTENOTITLE....................(3.216)

Table 3.5 presents the pressure responses for a uniform-flux horizontal well in a closed square computed from Eq. 3.216. We may obtain a short-time approximation for Eq. 3.216 with the early-time approximations given in Table 3.4 for Source Functions V and VIII. This yields

RTENOTITLE....................(3.217)

where Ei(−x) is the exponential integral function defined by Eq. 3.90. Eq. 3.217 indicates that the early-time flow is radial in the y-z plane around the axis of the horizontal well. This solution corresponds to the time period during which none of the reservoir boundaries influence the pressure response.


It is also possible to obtain another approximation for Eq. 3.216 that covers the intermediate time-flow behavior. If we approximate the source function in the x direction (Source Function VIII) by its early and intermediate time approximation and the source function in the y direction (Source Function V) by its early time approximation given in Table 3.4, we obtain

RTENOTITLE....................(3.218)

This approximation should correspond to the time period during which the influence of the top and/or bottom boundaries may be evident but the lateral boundaries in the x and y directions do not have an influence on the pressure response. This solution also could have been obtained by assuming a laterally infinite reservoir. In this case, the source function would have been constructed as the product of three source functions: an infinite-plane source at y = y′ in an infinite reservoir (Source I in Tables 3.2 and 3.3), an infinite-slab source of thickness Lh at x = x′ in an infinite reservoir (Source IV), and an infinite-plane source at z = zw in an infinite-slab reservoir of thickness h with impermeable boundaries (Source V).

The Use of Source Functions in the Laplace Domain To Solve Unsteady-Flow Problems

There are many advantages of developing transient-flow solutions in the Laplace transform domain. For example, in the Laplace transform domain, Duhamel’s theorem[23] provides a convenient means of developing transient-flow solutions for variable-rate production problems using the solutions for the corresponding constant-rate production problem. Duhamel’s theorem states that if Δp and Δpc denote the pressure drawdown corresponding to the variable production rate, q(t), and the constant production rate, qc, respectively, then

RTENOTITLE....................(3.219)

Applying the Laplace transform converts the convolution integral in Eq. 3.219 to an algebraic expression, and Duhamel’s theorem is given in the Laplace transform domain as

RTENOTITLE....................(3.220)

The simplicity of the expression given in Eq. 3.220 explains our interest in obtaining transient-flow solutions in the Laplace transform domain.

Another example to explain the convenience of the Laplace domain solutions is for the naturally fractured reservoirs. Common transient-flow models of naturally fractured reservoirs lead to the following differential equation in radial coordinates in the Laplace transform domain: [2]

RTENOTITLE....................(3.221)

where the subscript f indicates the fracture property, and tD and rD are the dimensionless time and distance (as defined in Eqs. 3.230 and 3.234).

The naturally fractured reservoir function, f (s), is a function of matrix and fracture properties and depends on the model chosen to represent the naturally fractured reservoir.[2] The corresponding differential equation for a homogeneous reservoir is obtained by setting f (s) = 1 and is given by

RTENOTITLE....................(3.222)

The general solutions for Eqs. 3.221 and 3.222 are given, respectively, by

RTENOTITLE....................(3.223)

and

RTENOTITLE....................(3.224)

To obtain a solution for constant-rate production from an infinite reservoir, for example, the following boundary conditions are imposed:

RTENOTITLE....................(3.225)

and

RTENOTITLE....................(3.226)

Then, it may be shown that

RTENOTITLE....................(3.227)

where the right side of Eq. 3.227 indicates the substitution of sf (s) for s in sΔp(s). This discussion demonstrates that it is possible to derive transient-flow solutions for naturally fractured reservoirs by following the same lines as those for the homogeneous reservoirs. Furthermore, if the solution for the corresponding homogeneous reservoir system is known in the Laplace transform domain, then the solution for the naturally fractured reservoir problem may be directly obtained from Eq. 3.227.

Obtaining the Laplace transforms of the Green’s and source function solutions developed in the time domain with the methods explained in Secs. 3.4.2 and 3.4.3 usually poses a difficult problem. The problems arise mainly because of the use of the product method solution. For a specific class of functions, Chen et al.[24] presented a technique that may be used to apply the Laplace transform to the product-solution technique. For a more general procedure to develop source function solutions in the Laplace transform domain, however, the product solution technique should be avoided.[20]

Ozkan and Raghavan[18][19] have shown that it is more convenient to develop source-function solutions in the Laplace transform domain if the point-source solution is used as a building block. Then, other source geometries may be obtained by the superposition (integration) of the point sources along the length, surface, or volume of the source.

Point-Source Solution in the Laplace Domain. Consider the flow of a slightly compressible fluid in an infinite, naturally fractured reservoir. We can use the double-porosity model suggested by Barenblatt et al.[25] and Warren and Root[26] to develop the governing flow equations for naturally fractured reservoirs. The results, however, will be applicable to the model suggested by Kazemi[27] and de Swaan-O[28] with a simple modification.

Flow around a point source in an infinite porous medium may be expressed conveniently in spherical coordinates. The differential equations governing flow in a naturally fractured reservoir are given in spherical coordinates by

RTENOTITLE....................(3.228)

and

RTENOTITLE....................(3.229)

In Eqs. 3.228 and 3.229, subscripts f and m indicate the property of the fracture and matrix systems, respectively. Initial pressure, pi, is assumed to be uniform in the entire system; that is, pfi = pmi = pi. The dimensionless time, tD, is defined by

RTENOTITLE....................(3.230)

where is a characteristic length in the system, and

RTENOTITLE....................(3.231)

The definitions of the other variables used in Eqs. 3.228 and 3.229 are

RTENOTITLE....................(3.232)

RTENOTITLE....................(3.233)

and

RTENOTITLE....................(3.234)

where

RTENOTITLE....................(3.235)

The initial and outer-boundary conditions are given, respectively, by

RTENOTITLE....................(3.236)

and

RTENOTITLE....................(3.237)

The inner-boundary condition corresponding to the instantaneous withdrawal of an amount of fluid, RTENOTITLE, at t = 0 from a point source is obtained by considering the mass balance on a small sphere. If we require that at any time t = T > 0, the sum of the flux through the surface of a small sphere around the source location must equal the volume of the fluid, RTENOTITLE, instantaneously withdrawn from the sphere at t = 0, we can write[29]

RTENOTITLE....................(3.238)

Although the withdrawal of fluids from the sphere is instantaneous, the resulting flow in the porous medium, and the flux across the surface of the sphere, is continuous. Therefore, if q represents the total flux across the surface of the small sphere during the time interval 0 ≤ tT, then the mass balance requires that the cumulative production (flux across the surface of the small sphere) at time T be equal to the instantaneous withdrawal volume of fluid from the sphere. That is,

RTENOTITLE....................(3.239)

For the condition expressed in Eq. 3.239 to hold for every T ≥ 0, we must have

RTENOTITLE....................(3.240)

where δ(t) is the Dirac delta function satisfying the properties expressed by Eqs. 3.185 and 3.186.

Using the results given by Eqs. 3.239 and 3.240 in Eq. 3.238, we obtain

RTENOTITLE....................(3.241)

The Laplace transform of Eqs. 3.228, 3.229, 3.237, and 3.241 yields

RTENOTITLE....................(3.242)

where

RTENOTITLE....................(3.243)

RTENOTITLE....................(3.244)

and

RTENOTITLE....................(3.245)

In deriving these results, we have used the initial condition given by Eq. 3.236 and noted that

RTENOTITLE....................(3.246)

In Eq. 3.245, the term RTENOTITLE represents the strength of the source for the naturally fractured porous medium.

The solution of Eqs. 3.242, 3.244, and 3.245 yields the following solution for the pressure distribution in the reservoir, except at the source location (the origin), because of an instantaneous point source of strength RTENOTITLE acting at t = 0:

RTENOTITLE....................(3.247)

If the source is located at x′D, y′D, z′D, then, by translation, we can write

RTENOTITLE....................(3.248)

where

RTENOTITLE....................(3.249)

and

RTENOTITLE....................(3.250)

The instantaneous point-source solution for the model suggested by Barenblatt et al.[25] and Warren and Root[26] can also be used for the model suggested by Kazemi[27] and de Swaan-O,[28] provided that the appropriate f(s) function is invoked. To obtain the solution for a homogeneous reservoir, f(s) should be set to unity, Vf = 1, and Vm = 0.

If we consider continuous withdrawal of fluids from the point source, then, by the principle of superposition, we should have

RTENOTITLE....................(3.251)

The Laplace transform of Eq. 3.251 yields the following continuous point-source solution in an infinite reservoir:

RTENOTITLE....................(3.252)

where we have substituted Eq. 3.249 for S, dropped the subscript f, and defined

RTENOTITLE....................(3.253)

Line-, Surface-, and Volumetric-Source Solution in the Laplace Domain. The point-source solution in the Laplace domain may be used to obtain the source solutions for different source geometries. If we define

RTENOTITLE....................(3.254)

where Δpp represents the appropriate point-source solution, then, by the application of the superposition principle, the solution for the withdrawal of fluids from a line, surface, or volume, Γw, is given by

RTENOTITLE....................(3.255)

If we assume a uniform-flux distribution in time and over the length, surface, or volume of the source, then

RTENOTITLE....................(3.256)

The following presentation of the source function approach in the Laplace domain assumes that the flux distribution is uniform, and RTENOTITLE. Also, the constant production rate from the length, area, or the volume of the source, Γw, is denoted by q so that RTENOTITLE.

Only sources in infinite reservoirs have been considered so far. These solutions may be easily extended to bounded reservoirs. The following sections present some useful solutions for transient-flow problems in bounded porous media. The first group of solutions is for laterally infinite reservoirs bounded by parallel planes in the vertical direction (infinite-slab reservoirs). The second and third groups comprise the solutions for cylindrical and rectangular reservoirs, respectively.

Solutions for Infinite-Slab Reservoirs. In this section, we consider one of the most common reservoir geometries used in pressure-transient analysis of wells in porous media. It is assumed that the lateral boundaries of the reservoir are far enough not to influence the pressure response during the time period of interest. The top and bottom boundaries of the reservoir at z = 0 and z = h are parallel planes and may be of impermeable, constant pressure, or mixed type. Table 3.6 presents the solutions for the most common well geometries (point-source, vertical, fractured, and horizontal wells) in infinite-slab reservoirs. Next, we briefly discuss the derivation of these solutions.


Consider a point source in an infinite-slab reservoir with impermeable boundaries at the bottom, z = 0, and the top, z = h. To obtain the point-source solution for this case, we use the point-source solution in an infinite reservoir given by Eq. 3.252 with the method of images. The result is given by

RTENOTITLE....................(3.257)

where

RTENOTITLE....................(3.258)

RTENOTITLE....................(3.259)

RTENOTITLE....................(3.260)

and

RTENOTITLE....................(3.261)

The solution given in Eq. 3.257 is not very convenient for computational purposes. To obtain a computationally convenient form of the solution, we use the summation formula given by[17][29]

RTENOTITLE....................(3.262)

and recast Eq. 3.257 as

RTENOTITLE....................(3.263)

The point-source solutions for infinite-slab reservoirs with constant pressure and mixed boundaries at the top and bottom are obtained in a similar manner[17] and are given in Table 3.6. The point-source solutions can be used with Eqs. 3.254 and 3.256 to generate the solutions for the other well geometries given in Table 3.6. For example, to generate the solution for a partially penetrating vertical line-source well of length hw in an infinite-slab reservoir with impermeable slab boundaries, we integrate the right side of Eq. 3.263 from zwhw / 2 to zw + hw / 2 with respect to z′, where zw is the vertical coordinate of the midpoint of the open interval. If hw = h (i.e., the well penetrates the entire thickness of the slab reservoir), then this procedure yields the solution for a fully penetrating vertical line-source well. The solution for a partially penetrating fracture of height hf and half-length xf is obtained if the point-source solution is integrated once with respect to z′ from zwhf / 2 to zw + hf / 2 and then with respect to x′ from xwxf to xw + xf, where xw and zw are the coordinates of the midpoint of the fracture. Similarly, the solution for a horizontal-line source well of length Lh is obtained by integrating the point-source solution with respect to x′ from xwLh / 2 to xw + Lh / 2, where xw is the x-coordinate of the midpoint of the horizontal well.

Solutions for Cylindrical Reservoir. Solutions for cylindrical reservoirs may also be obtained by starting from the point-source solution in the Laplace transform domain. The Laplace domain solution for a point source located at r′D, θ′, z′D should satisfy the following diffusion equation in cylindrical coordinates.[19]

RTENOTITLE....................(3.264)

where

RTENOTITLE....................(3.265)

The point-source solution is also required to satisfy the following flux condition at the source location (rD →0+, θ = θ′, zD = z′D):

RTENOTITLE....................(3.266)

Assuming that the reservoir is bounded by a cylindrical surface at rD = reD and by the parallel planes at zD = 0 and hD, we should also impose the appropriate physical conditions at these boundaries.

We seek a point-source solution for a cylindrical reservoir in the following form:

RTENOTITLE....................(3.267)

In Eq. 3.267, RTENOTITLE is a solution of Eq. 3.264 that satisfies Eq. 3.266 and the boundary conditions at zD = 0 and hD. RTENOTITLE may be chosen as one of the point-source solutions in an infinite-slab reservoir given in Table 3.6, depending on the conditions imposed at the boundaries at zD = 0 and hD. If RTENOTITLE is chosen such that it satisfies the boundary conditions at zD = 0 and hD, its contribution to the flux vanishes at the source location, and RTENOTITLE + RTENOTITLE satisfies the appropriate boundary condition at rD = reD, then Eq. 3.267 should yield the point-source solution for a cylindrical reservoir with appropriate boundary conditions.

Consider the example of a closed cylindrical reservoir in which the boundary conditions are given by

RTENOTITLE....................(3.268)

and

RTENOTITLE....................(3.269)

According to the boundary condition given by Eq. 3.268, we should choose RTENOTITLE as the point-source solution given in Table 3.6 (or by Eq. 3.263). Then, with the addition theorem for the Bessel function K0(aRD) given by[14]

RTENOTITLE....................(3.270)

where

RTENOTITLE....................(3.271)

we can write

RTENOTITLE....................(3.272)

for rD < r′D. If rD > r′D, we interchange rD and r′D in Eq. 3.272. If we choose RTENOTITLE in Eq. 3.267 as

RTENOTITLE....................(3.273)

where ak and bk are constants, then RTENOTITLE satisfies the boundary condition given by Eq. 3.268, and the contribution of RTENOTITLE to the flux at the source location vanishes. If we also choose the constants ak and bk in Eq. 3.273 as

RTENOTITLE....................(3.274)

and

RTENOTITLE....................(3.275)

then RTENOTITLE satisfies the impermeable boundary condition at rD = reD given by Eq. 3.269. Thus, the point-source solution for a closed cylindrical reservoir is given by

RTENOTITLE....................(3.276)

This solution procedure may be extended to the cases in which the boundaries are at constant pressure or of mixed type.[19] Table 3.7 presents the point-source solutions for cylindrical reservoirs for all possible combinations of boundary conditions. Solutions for other source geometries in cylindrical reservoirs may be obtained by using the point-source solutions in Table 3.7 in Eq. 3.255 (or Eq. 3.256).


Example 3.8

Consider a partially penetrating, uniform-flux fracture of height hf and half-length xf in an isotropic and closed cylindrical reservoir. The center of the fracture is at r′ = 0, θ′ =0, z′ = zw, and the fracture tips extend from (r′ = xf, θ = α + π) to (r′ = xf, θ = α).

Solution. Fig. 3.15 shows the geometry of the fracture/reservoir system considered in this example. The solution for this problem is obtained by first generating a partially penetrating line source and then using this line-source solution to generate the plane source. The solution for a partially penetrating line source at r′D, θ′, zw is obtained by integrating the corresponding point-source solution given in Table 3.7 with respect to z′ from zwhf / 2 to zw + hf / 2 and is given by

RTENOTITLE....................(3.277)


To generate the solution for a partially penetrating plane source that represents the fracture, the partially penetrating line-source solution given in Eq. 3.277 is integrated with respect to r′ from 0 to xf with θ′ = α + π in the third quadrant and with θ′ = α in the first quadrant. This procedure yields

RTENOTITLE RTENOTITLE....................(3.278)

It is possible to obtain an alternate representation of the solution given in Eq. 3.278. With the addition theorem of the Bessel function K0(x) given by Eq. 3.270, the solution in Eq. 3.277 may be written as

RTENOTITLE....................(3.279)

where

RTENOTITLE....................(3.280)

and

RTENOTITLE....................(3.281)

The integration of the partially penetrating vertical well solution given in Eq. 3.279 with respect to r′ from 0 to xf (with θ′ = α + π in the third quadrant and with θ′ = α in the first quadrant) yields the following alternative form of the partially penetrating fracture solution:

RTENOTITLE....................(3.282)

where

RTENOTITLE....................(3.283)

Example 3.9 Consider a uniform-flux, horizontal line-source well of length Lh in an isotropic and closed cylindrical reservoir. The well extends from (r′ = Lh/2, θ = α + π) to (r′ = Lh/2, θ = α), and the center of the well is at r′ = 0, θ′ = 0, z′ = zw.

Solution. The solution for a horizontal line-source well in a closed cylindrical reservoir is obtained by integrating the corresponding point-source solution in Table 3.7 with respect to r′ from 0 to Lh / 2 with θ′ = α + π in the third quadrant and with θ′ = α in the first quadrant. The final form of the solution is given by

RTENOTITLE RTENOTITLE....................(3.284)


Solutions for Rectangular Parallelepiped Reservoir. Solutions for rectangular parallelepiped reservoirs may also be obtained by starting from the point-source solution in the Laplace transform domain in an infinite reservoir and using the method of images to generate the effects of the planar boundaries. Although the formal procedure to obtain the solution is fairly easy, the use of the method of images in three directions (x, y, z) yields triple infinite Fourier series, which may pose computational inconveniences. As an example, the solution for a continuous point source located at x′, y′, z′ in a rectangular porous medium occupying the region 0 < x < xe, 0 < y < ye, and 0 < z < h is obtained by applying the method of images to the point-source solution given by Eq. 3.252: [19][29]

RTENOTITLE....................(3.285)

where

RTENOTITLE....................(3.286)

and

RTENOTITLE....................(3.287)

RTENOTITLE....................(3.288)

RTENOTITLE....................(3.289)

Ozkan[29] shows that the triple infinite sums in Eq. 3.285 may be reduced to double infinite sums with

RTENOTITLE....................(3.290)

where

RTENOTITLE....................(3.291)

The resulting continuous point-source solution for a closed rectangular reservoir is given by

RTENOTITLE....................(3.292)

where

RTENOTITLE....................(3.293)

RTENOTITLE....................(3.294)

RTENOTITLE....................(3.295)

RTENOTITLE....................(3.296)

and

RTENOTITLE....................(3.297)

Following a procedure similar to the one explained here, it is possible to derive the point-source solutions in rectangular parallelepiped reservoirs for different combinations of boundary conditions.[17][29] Table 3.8 gives these solutions, which may be used to derive the solutions for the other source geometries with Eq. 3.255 (or Eq. 3.256). Examples 3.10 and 3.11 demonstrate the derivation of the solutions for the other source geometries in rectangular reservoirs.


Example 3.10

Consider a fully penetrating vertical fracture of half-length xf located at x′ = xw and y′ = yw in a closed rectangular reservoir.

Solution. Assuming uniform-flux distribution along the fracture surface, the solution for this problem is obtained by integrating the corresponding point-source solution in Table 3.8, first with respect to z′ from 0 to h and then with respect to x′ from xwxf to xw + xf. The result is

RTENOTITLE....................(3.298)

where RTENOTITLE, RTENOTITLE, and εk are given respectively by Eqs. 3.293, 3.294, and 3.296.


Example 3.11

Consider a horizontal well of length Lh in the x-direction located at x′ = xw, y′ = yw, and z′ = zw in a closed rectangular reservoir.

Solution. The solution for a horizontal line-source well is obtained by integrating the corresponding point-source solution in Table 3.8, with respect to x′ from xwLh /2 to xw+Lh /2, and is given by

RTENOTITLE....................(3.299)

where

RTENOTITLE....................(3.300)

and

RTENOTITLE RTENOTITLE....................(3.301)

In Eq. 3.301, RTENOTITLE, RTENOTITLE, εn, εk, and εk, n are given by Eqs. 3.293 through 3.297.


Conversion From 3D to 2D Anisotropy. The solutions previously presented assume that the reservoir is anisotropic in all three principal directions, x, y, and z with kx, ky, and kz denoting the corresponding permeabilities. In these solutions, an equivalent isotropic permeability, k, has been defined by

RTENOTITLE....................(3.302)

For some applications, it may be more convenient to define an equivalent horizontal permeability by

RTENOTITLE....................(3.303)

and replace k in the solutions given in this section (Sec. 3.4.4) by kh. Note that k takes place in the definition of the dimensionless time tD (Eq. 3.230). Then, if we define a dimensionless time RTENOTITLE based on kh, the relation between RTENOTITLE and tD is given by

RTENOTITLE....................(3.304)

Because in the solutions given in this section the Laplace transformation is with respect to tD, conversion from 3D to 2D anisotropy requires the use of the following property of the Laplace transforms:

RTENOTITLE....................(3.305)

As an example, consider the solution for a horizontal well in an infinite-slab reservoir. Assuming that the midpoint of the well is the origin (xwD = 0, ywD = 0) and choosing the half-length of the horizontal well as the characteristic length (i.e., ℓ = Lh / 2), the horizontal-well solution given in Table 3.6 may be written as

RTENOTITLE....................(3.306)

In Eq. 3.306, s is the Laplace transform variable with respect to dimensionless time, tD, based on k and

RTENOTITLE....................(3.307)

RTENOTITLE....................(3.308)

RTENOTITLE....................(3.309)

and

RTENOTITLE....................(3.310)

If we define the following variables based on kh,

RTENOTITLE....................(3.311)

RTENOTITLE....................(3.312)

RTENOTITLE....................(3.313)

and also note that

RTENOTITLE....................(3.314)

then, we may rearrange Eq. 3.306 in terms of the dimensionless variables based on kh as

RTENOTITLE....................(3.315)

where

RTENOTITLE....................(3.316)

and

RTENOTITLE....................(3.317)

If we compare Eqs. 3.306 and 3.315, we can show that

RTENOTITLE....................(3.318)

where we have used the relation given by Eq. 3.305. If we now define RTENOTITLE as the Laplace transform variable with respect to RTENOTITLE, we may write

RTENOTITLE....................(3.319)

With the relation given by Eq. 3.319 and Eq. 3.306, we obtain the following horizontal-well solution in terms of dimensionless variables based on kh:

RTENOTITLE....................(3.320)

Computational Considerations and Applications. The numerical evaluation of the solutions given previously may be sometimes difficult, inefficient, or even impossible. Alternative computational forms of some of these solutions have been presented in a few sources.[18][19][29] Here, we present a summary of the alternative formulas to be used in the computation of the source functions in the Laplace transform domain. Some of these formulas are for computations at early or late times and may be useful to derive asymptotic approximations of the solutions during the corresponding time periods.

As Sec. 3.3.1 notes, the short- and long-time approximations of the solutions correspond to the limiting forms of the solution in the Laplace transform domain as s→∞ and s→0, respectively. In the solutions given in this section, we have defined u = sf(s). From elementary considerations, it is possible to show that the definition of f(s) given in Eq. 3.243 yields the following limiting forms:

RTENOTITLE....................(3.321)

and

RTENOTITLE....................(3.322)

These limiting forms are used in the derivation of the short- and long-time asymptotic approximations. In the following expressions, homogeneous reservoir solutions are obtained by substituting ω = 1.

The Integral.

RTENOTITLE....................(3.323)

This integral arises in the computation of many practical transient-pressure solutions and may not be numerically evaluated, especially as yD→0; however, the following alternate forms of the integral are numerically computable.[19]

RTENOTITLE....................(3.324)

RTENOTITLE....................(3.325)

and

RTENOTITLE....................(3.326)

The integrals in Eqs. 3.324 through 3.326 may be evaluated with the standard numerical integration algorithms for yD ≠ 0. For yD = 0, the polynomial approximations given by Luke or the following power series expansion given by Abramowitz and Stegun[7] may be used in the computation of the integrals in Eqs. 3.324 through 3.326:

RTENOTITLE....................(3.327)

For numerical computations and asymptotic evaluations, it may also be useful to note the following relations: [19]

RTENOTITLE....................(3.328)

and

RTENOTITLE....................(3.329)

It can be shown from Eqs. 3.328 and 3.329 that, for practical purposes, when z ≥ 20, the right sides of Eqs. 3.327 and 3.328 may be approximated by π/2 and π exp (−|c|)/2, respectively.[19][27]

As a few sources[18][19][29] show, it is possible to derive the following short- and long-time approximations (i.e., the limiting forms as s→∞ and s→0, respectively) for the integral RTENOTITLE given, respectively, by

RTENOTITLE....................(3.330)

where

RTENOTITLE....................(3.331)

and

RTENOTITLE....................(3.332)

where γ=0.5772… and

RTENOTITLE....................(3.333)

It is also useful to note the real inversions of Eqs. 3.330 and 3.332 given, respectively, by

RTENOTITLE....................(3.334)

and

RTENOTITLE....................(3.335)

The Series. RTENOTITLE

RTENOTITLE....................(3.336)

Two alternative expressions for the series S1 may be convenient for the large and small values of u (i.e., for short and long times).[29] When u is large,

RTENOTITLE....................(3.337)

and when u + a2 << n2π2/h2D,

RTENOTITLE....................(3.338)

The Series RTENOTITLE.

RTENOTITLE....................(3.339)

Alternative computational forms for the series S2 are given next.[29] When u is large,

RTENOTITLE....................(3.340)

and when u + a2 << n2π2/h2D,

RTENOTITLE....................(3.341)

The Series RTENOTITLE.

RTENOTITLE....................(3.342)

The following alternative forms for the series RTENOTITLE may be convenient for the large and small values of u (i.e., for short and long times).[29] When u is large,

RTENOTITLE....................(3.343)

and when u + a2 << (2n − 1)2 π2/(4h2D),

RTENOTITLE....................(3.344)

The Series RTENOTITLE.

RTENOTITLE....................(3.345)

where

RTENOTITLE....................(3.346)

The series RTENOTITLE may be written in the following forms with the use of Eqs. 3.324 through 3.326.

RTENOTITLE....................(3.347)

RTENOTITLE....................(3.348)

and

RTENOTITLE RTENOTITLE....................(3.349)

The computation of the series in Eqs. 3.347 and 3.348 should not pose numerical difficulties; however, the series in Eq. 3.349 converges slowly. With the relation given in Eq. 3.328, we may write Eq. 3.349 as[29]

RTENOTITLE....................(3.350)

where

RTENOTITLE....................(3.351)

Before discussing the computation of the series given in Eq. 3.351, we first discuss the derivation of the asymptotic approximations for the series RTENOTITLE. When s is large (small times), RTENOTITLE may be approximated by[29]

RTENOTITLE....................(3.352)

where β is given by Eq. 3.331. If s is sufficiently large, then Eq. 3.352 may be further approximated by

RTENOTITLE....................(3.353)

The inverse Laplace transform of Eq. 3.353 yields

RTENOTITLE....................(3.354)

For small s (large times), depending on the value of xD, RTENOTITLE may be approximated by one of the following equations: [29]

RTENOTITLE....................(3.355)

RTENOTITLE....................(3.356)

RTENOTITLE....................(3.357)

where RTENOTITLE is given by Eq. 3.364.

The Series RTENOTITLE.

RTENOTITLE....................(3.358)

where

RTENOTITLE....................(3.346)

With the relations given in Eqs. 3.337 and 3.338, the following alternative forms for the series RTENOTITLE may be obtained, respectively, for the large and small values of s (i.e., for short and long times).[29] When u is large,

RTENOTITLE....................(3.359)

and when u << n2π2/h2D,

RTENOTITLE....................(3.360)

It is also possible to derive asymptotic approximations for the series RTENOTITLE. When s is large (small times), RTENOTITLE may be approximated by[29]

RTENOTITLE....................(3.361)

If s is sufficiently large, then Eq. 3.361 may be further approximated by

RTENOTITLE....................(3.362)

The inverse Laplace transform of Eq. 3.362 yields

RTENOTITLE....................(3.363)

For small s (large times), RTENOTITLE may be approximated by[29]

RTENOTITLE....................(3.364)

The Ratio RTENOTITLE.

RTENOTITLE....................(3.365)

By elementary considerations, the ratio RTENOTITLE may be written as[29]

RTENOTITLE....................(3.366)

The expression given in Eq. 3.366 provides computational advantages when s is small (time is large).


Example 3.12

Consider a fully penetrating, uniform-flux fracture of half-length xf located at x′=0, y′=0 in an infinite-slab reservoir with closed top and bottom boundaries.

Solution.Table 3.6 gives the solution for this problem. For simplicity, assuming an isotropic reservoir, choosing the characteristic length as ℓ = xf and noting that RTENOTITLE, the solution becomes

RTENOTITLE....................(3.367)

First consider the numerical evaluation of Eq. 3.367. We note from Eqs. 3.324 through 3.326 that Eq. 3.367 may be written in one of the following forms, depending on the value of xD.

RTENOTITLE....................(3.368)

RTENOTITLE....................(3.369)

and

RTENOTITLE....................(3.370)

The numerical evaluation of the integrals in Eqs. 3.368 through 3.370 for yD ≠ 0 should be straightforward with the use of the standard numerical integration algorithms. For yD = 0, the polynomial approximations given by Luke or the power series expansion given by Eq. 3.327 should be useful.

The short- and long-time asymptotic approximations of the fracture solution are also obtained by applying the relations given by Eqs. 3.330 and 3.332, respectively, to the right side of Eq. 3.367. This procedure yields, for short times,

RTENOTITLE....................(3.371)

or, in real-time domain,

RTENOTITLE....................(3.372)

where β is given by Eq. 3.331 with a = -1 and b = +1. At long times, the following asymptotic approximation may be used:

RTENOTITLE....................(3.373)

or, in real-time domain,

RTENOTITLE....................(3.374)

where γ = 0.5772… and σ(xD, yD, -1, +1) is given by Eq. 3.333.


Example 3.13

Consider a horizontal well of length Lh located at x′ = 0, y′ = 0, and z′ = zw in an infinite-slab reservoir with closed top and bottom boundaries.

Solution.Table 3.6 gives the horizontal-well solution for an infinite-slab reservoir with impermeable boundaries. Assuming an isotropic reservoir, choosing the characteristic length as ℓ = Lh / 2 and noting that RTENOTITLE, the solution may be written as

RTENOTITLE....................(3.375)

where RTENOTITLE is the fracture solution given by the right side of Eq. 3.367 and RTENOTITLE is given by

RTENOTITLE....................(3.376)

with

RTENOTITLE....................(3.346)

RTENOTITLE....................(3.377) br>
and

RTENOTITLE....................(3.378)

The computation of the first term in the right side of Eq. 3.375 RTENOTITLE is the same as the computation of the fracture solution given by Eq. 3.367 and has been discussed in Example 3.12. The computational form of the second term RTENOTITLE in the right side of Eq. 3.375 is given by Eqs. 3.347 through 3.350. Of particular interest is the case for −1 ≤ xD ≤ +1. In this case, from Eqs. 3.350 and 3.351, we have

RTENOTITLE....................(3.379)

where

RTENOTITLE....................(3.380)

The computational considerations for the series RTENOTITLE have been discussed previously.

Next, we consider the short- and long-time approximations of the horizontal-well solution given by Eq. 3.375. To obtain a short-time approximation, we substitute the asymptotic expressions for RTENOTITLE and RTENOTITLE as s→∞ given, respectively, by Eqs. 3.371 and 3.353. This yields

RTENOTITLE....................(3.381)

where β is given by Eq. 3.331. The inverse Laplace transform of Eq. 3.381 is given by

RTENOTITLE....................(3.382)

To obtain the long-time approximation of Eq. 3.375, we substitute the asymptotic expressions for RTENOTITLE and RTENOTITLE as s→∞ given, respectively, by Eq. 3.374 and Eqs. 3.355 through 3.357. Of particular interest is the case for −1 ≤ xD ≤ +1, where we have

RTENOTITLE RTENOTITLE....................(3.383)

where γ=0.5772… and σ(xD, yD, -1, +1) is given by Eq. 3.333. The inverse Laplace transform of Eq. 3.383 yields

RTENOTITLE....................(3.384)


Example 3.14 Consider a fully penetrating, uniform-flux fracture of half-length xf in an isotropic and closed cylindrical reservoir. The center of the fracture is at r′ = 0, θ′ = 0 and the fracture tips extend from (r′ = xf, θ = α + π) to (r′ = xf, θ = α).

Solution. The solution for this problem has been obtained in Eq. 3.278 in Example 3.8 with hw = h. Choosing the characteristic length as ℓ = xf and noting that RTENOTITLE, the solution is given by

RTENOTITLE....................(3.385)

For the computation of the pressure responses at the center of the fracture (rD = 0), Eq. 3.385 simplifies to

RTENOTITLE....................(3.386)

It is also possible to find a very good approximation for Eq. 3.385, especially when reD is large. If we assume[19]

RTENOTITLE....................(3.387)

and use the following relation[4]

RTENOTITLE....................(3.388)

then Eq. 3.385 may be written as

RTENOTITLE....................(3.389)

Because[19]

RTENOTITLE....................(3.390)

where

RTENOTITLE....................(3.391)

Eq. 3.389 may also be written as

RTENOTITLE....................(3.392)

Although the assumption given in Eq. 3.387 may not be justified by itself, the solution given in Eq. 3.392 is a very good approximation for Eq. 3.385, especially when reD is large. For a fracture at the center of the cylindrical drainage region, Eq. 3.392 simplifies to

RTENOTITLE....................(3.393)

It is also possible to obtain short- and long-time approximations for the solution given in Eq. 3.393. For short times, u→∞ and the second term in the argument of the integral in Eq. 3.393 becomes negligible compared with the first term. Then, Eq. 3.393 reduces to the solution for an infinite-slab reservoir given by Eq. 3.367, of which the short-time approximation has been discussed in Example 3.12.

To obtain a long-time approximation, we evaluate Eq. 3.393 at the limit as s→0 (us). As shown in Sec. 3.2.3, for small arguments we may approximate the Bessel functions in Eq. 3.393 by

RTENOTITLE....................(3.394)

RTENOTITLE....................(3.395)

RTENOTITLE....................(3.396)

and

RTENOTITLE....................(3.397)

where γ = 0.5772…. With Eqs. 3.394 through 3.397 and by neglecting the terms of the order s3/2, we may write[29]

RTENOTITLE....................(3.398)

If we substitute the right side of Eq. 3.398 into Eq. 3.393, we obtain

RTENOTITLE....................(3.399)

where σ(xD, yD, −1, +1) is given by Eq. 3.333 and

RTENOTITLE....................(3.400)

The inverse Laplace transform of Eq. 3.399 yields the following long-time approximation for a uniform-flux fracture at the center of a closed square:

RTENOTITLE....................(3.401)


Example 3.15

Consider a fully penetrating, uniform-flux fracture of half-length xf in an isotropic and closed parallelepiped reservoir of dimensions xe × ye × h. The fracture is parallel to the x axis and centered at xw, yw, zw.

Solution. The solution for this problem has been obtained in Example 3.10 and, by choosing ℓ = xf, is given by

RTENOTITLE....................(3.402)

where

RTENOTITLE....................(3.403)

The computation of the ratios of the hyperbolic functions in Eq. 3.402 may be difficult, especially when their arguments approach zero or infinity. When s is small (long times), Eq. 3.366 should be useful to compute the ratios of the hyperbolic functions. When s is large (small times), with Eq. 3.366 the solution given in Eq. 3.402 may be written as[29]

RTENOTITLE....................(3.404)

where

RTENOTITLE....................(3.405)

RTENOTITLE....................(3.406)

and

RTENOTITLE....................(3.407)

The last equality in Eq. 3.405 follows from the relation given by Eq. 3.349. The expression given in Eq. 3.405 may also be written as

RTENOTITLE....................(3.408)

where

RTENOTITLE....................(3.409)

and

RTENOTITLE....................(3.410)

Therefore, the solution given by Eq. 3.402 may be written as follows for computation at early times (for large values of s):

RTENOTITLE....................(3.411)

where RTENOTITLE is given by Eq. 3.409 and corresponds to the solution for a fractured well in an infinite-slab reservoir (see Eq. 3.367 in Example 3.12) and RTENOTITLE represents the contribution of the lateral boundaries and is given by

RTENOTITLE....................(3.412)

In Eq. 3.412, RTENOTITLE, RTENOTITLE, and RTENOTITLE are given, respectively, by Eqs. 3.406, 3.407, and 3.410. The integrals appearing in Eqs. 3.409 and 3.410 may be evaluated by following the lines outlined by Eqs. 3.324 through 3.326.

It is also possible to derive short- and long-time approximations for the fracture solution in a closed rectangular parallelepiped. The short-time approximation corresponds to the limit of the solution as s→∞. It can be easily shown that the RTENOTITLE term in Eq. 3.411 becomes negligible compared with the RTENOTITLE term for which a short-time approximation has been obtained in Example 3.12 (see Eqs. 3.371 and 3.372).

To obtain a long-time approximation (small values of s), the solution given in Eq. 3.402 may be written as[27]

RTENOTITLE....................(3.413)

where

RTENOTITLE....................(3.414)

and

RTENOTITLE....................(3.415)

The second equality in Eq. 3.414 results from

RTENOTITLE....................(3.416)

For small values of s, replacing u by s and s + α by α, and with

RTENOTITLE....................(3.417)

the term H given by Eq. 3.414 may be approximated by

RTENOTITLE....................(3.418)

The long-time approximation of the second term in Eq. 3.413 is obtained by assuming u << k2π2/x2eD and taking the inverse Laplace transform of the resulting expressions; therefore, we can obtain the following long-time approximation

RTENOTITLE....................(3.419)


Example 3.16 Consider a uniform-flux horizontal well of length Lh in an isotropic and closed parallelepiped reservoir of dimensions xe × ye × h. The center of the well is at xw, yw, zw, and the well is parallel to the x axis.

Solution. The solution for this problem was obtained in Example 3.11 and, by choosing ℓ = Lh / 2, is given by

RTENOTITLE....................(3.420)

where RTENOTITLE is the solution discussed in Example 3.15, and RTENOTITLE is given by

RTENOTITLE RTENOTITLE....................(3.421)

In Eq. 3.421, RTENOTITLE and RTENOTITLE are given by Eqs. 3.377 and 3.378, respectively,

RTENOTITLE....................(3.346)

and

RTENOTITLE....................(3.422)

The computation and the asymptotic approximations of the RTENOTITLE term have been discussed in Example 3.15. To compute the RTENOTITLE term for long times (small s), the relation for the ratios of the hyperbolic functions given by Eq. 3.366 should be useful. For computations at short times (large values of s), following the lines similar to those in Example 3.15, the RTENOTITLE term in Eq. 3.421 may be written as

RTENOTITLE....................(3.423)

where

RTENOTITLE....................(3.424)

RTENOTITLE....................(3.425)

RTENOTITLE....................(3.426)

RTENOTITLE RTENOTITLE....................(3.427)

and

RTENOTITLE....................(3.428)

The computational form of the RTENOTITLE term in Eq. 3.424 is obtained by applying the relations given by Eqs. 3.347 through 3.350 and Eq. 3.328. Of particular interest is the case for −1 ≤ xD ≤ +1 and yD = ywD given by

RTENOTITLE....................(3.429)

where

RTENOTITLE....................(3.430)

which can be written as follows by using the relation given in Eq. 3.337:

RTENOTITLE....................(3.431)

Similarly, for −1 ≤ xD ≤ +1 and yD = ywD, the RTENOTITLE term given in Eq. 3.428 may be written as

RTENOTITLE RTENOTITLE....................(3.432)

where

RTENOTITLE....................(3.433)


Example 3.15 discussed the short- and long-time approximations of the RTENOTITLE term in Eq. 3.420. A small-time approximation for RTENOTITLE given by Eq. 3.423 is obtained with u = ωs and by noting that as s→∞, RTENOTITLE. Then, substituting the short-time approximations for RTENOTITLE and RTENOTITLE given, respectively, by Eqs. 3.371 and 3.353 into Eq. 3.420, the following short-time approximation is obtained: [27]

RTENOTITLE....................(3.434)

where β is given by Eq. 3.331. The inverse Laplace transform of Eq. 3.434 yields

RTENOTITLE....................(3.435)

The long-time approximation of Eq. 3.420 is obtained by substituting the long-time approximations of RTENOTITLE and RTENOTITLE. The long time-approximation of RTENOTITLE is obtained in Example 3.15 (see Eq. 3.413 through 3.419). The long-time approximation of RTENOTITLE is obtained by evaluating the right side of Eq. 3.421 as s→0 (u→0) and is given by

RTENOTITLE....................(3.436)

where

RTENOTITLE....................(3.437)

and

RTENOTITLE....................(3.438)

Thus, the long-time approximation Eq. 3.420 is given by

RTENOTITLE....................(3.439)

where pDf and F1 are given, respectively, by Eqs. 3.419 and 3.436. For computational purposes, however, F1 may be replaced by

RTENOTITLE....................(3.440)

In Eq. 3.440, F, Fb1, Fb2, and Fb3 are given, respectively, by

RTENOTITLE....................(3.441)

RTENOTITLE....................(3.442)

RTENOTITLE....................(3.443)

and

RTENOTITLE
RTENOTITLE....................(3.444)

When computing the integrals and the trigonometric series, the relations given by Eqs. 3.324 through 3.326 and 3.345 through 3.350 are useful.

Nomenclature


a = radius of the spherical source, L
B = formation volume factor, res cm3/std cm3
c = fluid compressibility, atm−1
cf = formation compressibility, atm−1
ct = total compressibility, atm−1
C = wellbore-storage coefficient, cm3/atm
d = distance to a linear boundary, cm
D = domain
Ei(x) = exponential integral function
f(s) = naturally fractured reservoir function
RTENOTITLE = naturally fractured reservoir function based on RTENOTITLE
RTENOTITLE = Laplace transform of a function f (t)
G = Green’s function
h = formation thickness, cm
RTENOTITLE = dimensionless thickness, Eq. 3.313
hD = dimensionless thickness, Eq. 3.314
hf = fracture height (vertical penetration), cm
hp = slab thickness, cm
hw = well length (penetration), cm
H(x - x′) = Heaviside’s unit step function
RTENOTITLE = unit normal vector in the ξ direction, ξ = x, y, z, r, θ
Iv(x) = modified Bessel function of the first kind of order v
I′v(x) = derivative of Iv(x)
Jv(x) = Bessel function of the first kind of order v
k = isotropic permeability, md
kf = fracture permeability, md
kh = equivalent horizontal permeability, md
ki j = permeability in i-direction as a result of pressure gradient in j-direction, md
kξ = permeability in ξ-direction, ξ = x, y, z, md
kξf = fracture permeability in ξ-direction, ξ = x, y, z, md
Ki1(x) = first integral of K0(z)
Kn(x) = modified Bessel function of the second kind of order n
K′n(x) = derivative of Kn(x)
= characteristic length of the system, cm
L = Laplace transform operator
L-1 = inverse Laplace transform operator
Lh = horizontal-well length, cm
m = pseudopressure, atm2/cp
Mg = mass, g
M = point in space
M′ = source point in space
Mw = point in Γw
M′w = source point in Γw
n = outward normal direction of the boundary surface
RTENOTITLE = normal vector
N = even integer in Stehfest’s algorithm
p = pressure, atm
pc = pressure for constant production rate, qc, atm
RTENOTITLE = dimensionless fracture pressure
pe = external boundary pressure, atm
p f = fracture pressure, atm
pf i = initial pressure in fracture system, atm
pi = initial pressure, atm
pj = pressure in medium j, j=m, f, atm
pm = matrix pressure, atm
pmi = initial pressure in matrix system, atm
pw f = flowing wellbore pressure, atm
RTENOTITLE = Laplace transform of p(t)
p(t) = inverse of the Laplace domain function
pa(T) = approximate inverse of RTENOTITLE at t=T, atm
q = production rate, cm3/s
RTENOTITLE = instantaneous production rate for a point source, cm3/s
qc = constant production rate, cm3/s
qs f = sandface production rate, cm3/s
qwb = wellbore production rate as a result of storage, cm3/s
r = radial coordinate and distance, cm
r′ = source coordinate in r-direction, cm
re = external radius of the reservoir, cm
rw = wellbore radius, cm
R = distance in 3D coordinates, cm
RD = dimensionless radial distance in cylindrical coordinates
s = Laplace transform parameter
RTENOTITLE = Laplace transform paraeter based on RTENOTITLE
sm = skin factor
S = source function
t = time, s
RTENOTITLE = dimensionless time based on kh
tAD = dimensionless time based on area
tp = producing time, s
T = Temperature, °C
u = s f(s)
RTENOTITLE = velocity vector
vξ = velocity component in the ξ direction, ξ = x, y, z, r, θ, cm/s
V = volume, cm3
Vi = constant in Stehfest’s algorithm
Vf = fraction of the volume occupied by fractures
Vm = fraction of the volume occupied by matrix
x = distance in x-direction, cm
x′ = source coordinate in x-direction, cm
xe = distance to the external boundary in x-direction, cm
xp = half slab thickness, cm
xf = fracture half-length, cm
RTENOTITLE = dimensionless fracture half-length
xw = well coordinate in x-direction, cm
y = distance in y-direction, cm
y′ = source coordinate in y-direction, cm
ye = distance to the external boundary in y-direction, cm
yw = well coordinate in y-direction, cm
Yn(x) = Bessel function of the second kind of order n
z = distance in z-direction, cm
z′ = source coordinate in z-direction, cm
RTENOTITLE = dimensionless distance in z-direction, Eq. 3.377
zw = well coordinate in z-direction, cm
RTENOTITLE = dimensionless well coordinate in z-direction, Eq. 3.378
Z = compressibility factor
α = permeability direction, Eq. 3.17
β = permeability direction, Eq. 3.17
Γ = boundary surface, cm2
Γe = external boundary surface
Γw = length, surface, or volume of the source
Γ(x) = Gamma function
γ = Euler’s constant (γ = 0.5772...)
γ = permeability direction, Eq. 3.17
γ f = fundamental solution of diffusion equation
Δ = difference operator
δ(x) = Dirac delta function
η = diffusivity constant
ηi = diffusivity constant in i direction, i = x, y, z, or r
θ = angle from positive x-direction, degrees radian
θ′ = source coordinate in θ-direction, degrees radian
λ = transfer coefficient for a naturally fractured reservoir
RTENOTITLE = λ based on kh
μ = viscosity, cp
ρ = density, g/cm3
τ = time, s
Φ = porosity, fraction
φ(M) = any continuous function
ω = storativity ratio for a naturally fractured reservoir


Subscripts and Superscripts


D = dimensionless
f = fracture
i = initial
m = matrix
w = wellbore
RTENOTITLE = Laplace transform indicator


References


  1. Al-Hussainy, R., Ramey Jr., H.J., and Crawford, P.B. 1966. The Flow of Real Gases Through Porous Media. J Pet Technol 18 (5): 624–636. SPE-1243-A-PA. http://dx.doi.org/10.2118/1243-A-PA
  2. 2.0 2.1 2.2 2.3 Raghavan, R. 1993. Well Test Analysis, 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.
  3. Fancher, G.H., Lewis, J.A., and Barnes, K.B. 1933. Some Physical Characteristics of Oil Sands. Bulletin 12, Mineral Industries Experimental Station, Pennsylvania State University, University Park, Pennsylvania, 65–167.
  4. 4.0 4.1 Forchheimer, P.F. 1901. Wasserbewegung durch Boden. Zeitschrift des Vereines deutscher Ingenieure 45 (5): 1781–1788.
  5. 5.0 5.1 Watson, G. N. 1944. A Treatise on the Theory of Bessel Functions. London: Cambridge University Press.
  6. 6.0 6.1 6.2 6.3 Churchill, R.V. 1972. Operational Mathematics, Vol. 2. New York: McGraw-Hill Book Co.
  7. 7.0 7.1 7.2 Abramowitz, M. and Stegun, I.A. ed. 1972. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, ninth edition, 1020–1029. New York: Dover Publications.
  8. 8.0 8.1 8.2 8.3 8.4 Stehfest, H. 1970. Algorithm 368: Numerical inversion of Laplace transforms. Commun. ACM 13 (1): 47–49. http://dx.doi.org/10.1145/361953.361969
  9. Crump, K.S. 1976. Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation. J. ACM 23 (1): 89-96. http://dx.doi.org/10.1145/321921.321931
  10. 10.0 10.1 van Everdingen, A.F. and Hurst, W. 1953. The Application of the Laplace Transformation to Flow Problems in Reservoirs. In Transactions of the American Institute of Mining and Metallurgical Engineers, Vol. 198, 171.
  11. Agarwal, R.G., Al-Hussainy, R., and Ramey Jr., H.J. 1970. An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment. SPE J. 10 (3): 279-290. SPE-2466-PA. http://dx.doi.org/10.2118/2466-PA
  12. Chen, C.-C. and Raghavan, R. 1996. An Approach To Handle Discontinuities by the Stehfest Algorithm. SPE J. 1 (4): 363-368. SPE-28419-PA. http://dx.doi.org/10.2118/28419-PA
  13. 13.0 13.1 13.2 13.3 13.4 13.5 13.6 Gringarten, A.C. and Ramey Jr., H.J. 1973. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. SPE J. 13 (5): 285-296. SPE-3818-PA. http://dx.doi.org/10.2118/3818-PA
  14. 14.0 14.1 14.2 14.3 14.4 14.5 14.6 Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, second edition, 353–386. Oxford, UK: Oxford University Press.
  15. 15.0 15.1 Stakgold, I. 1979. Green’s Functions and Boundary Value Problems, 86–104. New York: John Wiley & Sons.
  16. 16.0 16.1 Kelvin, W.T. 1884. Mathematical and Physical Papers, Vol. 2, 41. Cambridge, UK: Cambridge University Press.
  17. 17.0 17.1 17.2 17.3 Ozkan, E. and Raghavan, R. 1991a. New Solutions for Well-Test-Analysis Problems: Part 1—Analytical Considerations. SPE Form Eval 6 (3): 359–368. SPE-18615-PA. http://dx.doi.org/10.2118/18615-PA
  18. 18.0 18.1 18.2 18.3 Ozkan, E. and Raghavan, R. 1991b. New Solutions for Well-Test-Analysis Problems: Part 2—Computational Considerations and Applications. SPE Form Eval 6 (3): 369–378. SPE-18616-PA. http://dx.doi.org/10.2118/18616-PA
  19. 19.00 19.01 19.02 19.03 19.04 19.05 19.06 19.07 19.08 19.09 19.10 19.11 Raghavan, R. and Ozkan, E. 1994. A Method for Computing Unsteady Flows in Porous Media, No. 318. Essex, England: Pitman Research Notes in Mathematics Series, Longman Scientific & Technical.
  20. 20.0 20.1 Raghavan, R. 1993. The Method of Sources and Sinks. In Well Test Analysis, Chap. 10, 336-435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.
  21. Nisle, R.G. 1958. The Effect of Partial Penetration on Pressure Build-Up in Oil Wells. In Transactions of the American Institute of Mining, Metallurgical, and Petroleum Engineers, Vol. 213, Paper 971-G, 85-90. Dallas, Texas: Society of Petroleum Engineers.
  22. 22.0 22.1 22.2 Newman, A.B. 1936. Heating and Cooling Rectangular and Cylindrical Solids. Ind. Eng. Chem. 28 (5): 545–548. http://dx.doi.org/10.1021/ie50317a010
  23. Duhamel, J.M.C. 1833. Mémoire sur la méthode générale relative au mouvement de la chaleur dans les corps solides polongé dans les milieux dont la température varie avec le temps. Journal de l’École Polytechnique 14 (22): 20-66.
  24. Chen, H.Y., Poston, S.W., and Raghavan, R. 1991. An Application of the Product Solution Principle for Instantaneous Source and Green's Functions. SPE Form Eval 6 (2): 161-167. SPE-20801-PA. http://dx.doi.org/10.2118/20801-PA
  25. 25.0 25.1 Barenblatt, G.I., Zheltov, I.P., and Kochina, I.N. 1960. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24 (5): 1286–1303. http://dx.doi.org/10.1016/0021-8928(60)90107-6
  26. 26.0 26.1 Warren, J.E. and Root, P.J. 1963. The Behavior of Naturally Fractured Reservoirs. SPE J. 3 (3): 245–255. SPE-426-PA. http://dx.doi.org/10.2118/426-PA
  27. 27.0 27.1 27.2 27.3 27.4 Kazemi, H. 1969. Pressure Transient Analysis of Naturally Fractured Reservoir with Uniform Fracture Distribution. SPE J. 9 (4): 451–462. SPE-2156-PA. http://dx.doi.org/10.2118/2156-PA
  28. 28.0 28.1 de Swaan O., A. 1976. Analytical Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing. SPE J. 16 (3): 117–122. SPE-5346-PA. http://dx.doi.org/10.2118/5346-PA
  29. 29.00 29.01 29.02 29.03 29.04 29.05 29.06 29.07 29.08 29.09 29.10 29.11 29.12 29.13 29.14 29.15 29.16 29.17 29.18 Ozkan, E. 1988. Performance of Horizontal Wells. PhD dissertation, University of Tulsa, Tulsa, Oklahoma.

SI Metric Conversion Factors


atm × 1.013 250* E + 05 = Pa
cp × 1.0* E – 03 = Pa•s
in. × 2.54* E + 00 = cm
in.2 × 6.451 6* E + 00 = cm2
°F (°F−32)/1.8 = °C
ft × 3.048* E – 01 = m


*

Conversion factor is exact.