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# Fluid flow through permeable media

This article discusses the basic concepts of single-component or constant-composition, single phase fluid flow in homogeneous petroleum reservoirs, which include flow equations for unsteady-state, pseudosteady-state, and steady-state flow of fluids. Various flow geometries are treated, including radial, linear, and spherical flow.

## Ideal reservoir model

Virtually no important applications of fluid flow in permeable media involve single component, single phase 1D, radial or spherical flow in homogeneous systems (multiple phases are almost always involved, which also leads to multidimensional requirements). The applications given in this Chapter are based on a model that includes many simplifying assumptions about the well and reservoir, and are interesting mainly only from a historical perspective  See "Reservoir Simulation" for proper treatment of multi-component, multiphase, multidimensional flow in heterogeneous porous media. The simplifying assumptions are introduced here as needed to combine the law of conservation of mass, Darcy’s law, and equations of state to obtain closed-form solutions for simple cases.

Consider radial flow toward a well in a circular reservoir. Combining the law of conservation of mass and Darcy’s law for the isothermal flow of fluids of small and constant compressibility yields the radial diffusivity equation, [1]

....................(1)

In the derivation of this equation, it is assumed that compressibility of the total system, ct, is small and independent of pressure; permeability, k , is constant and isotropic; viscosity, μ, is independent of pressure; porosity, ϕ, is constant; and that certain terms in the basic differential equation (involving pressure gradients squared) are negligible. The grouping 0.0002637k/ϕμct is called the hydraulic diffusivity and is given the symbol η.  None of these assumptions are valid in general.

## Line-source solution to the diffusivity equation

Assume that a well produces at constant reservoir rate, qB; the well has zero radius; the reservoir is at uniform pressure, pi, before production begins; and the well drains an infinite area (i.e., that ppi as r → ∞). Under these conditions, the solution to Eq. 1 is[1]

....................(2)

where p is the pressure at distance r from the well at time t, and

....................(3)

theEi function or exponential integral.

The Ei-function solution is an accurate approximation to more exact solutions to the diffusivity equation (solutions with finite wellbore radius and finite drainage radius) for 3.79 × 105 ϕμctrw2/k < t < 948 ϕμctre2/k. For smaller times, the assumption of zero well size (line source or sink) limits the accuracy of the equation; for larger times, the reservoir’s boundaries affect the pressure distribution in the reservoir, so that the reservoir is no longer infinite acting.

For the argument, x, of theEi function less than 0.01, theEi function can be approximated with negligible error by

....................(4)

For x > 10, theEi function is zero for practical applications in flow through porous media. For 0.01 < x < 10,Ei functions are determined from tables or subroutines available in appropriate software. [2]

## Altered zone and skin factor

In practice, most wells have reduced permeability (damage) near the wellbore resulting from drilling or completion operations. Many other wells are stimulated by acidization or hydraulic fracturing. Eq. 2 fails to model such wells properly. Its derivation includes the explicit assumption of uniform permeability throughout the drainage area of the well up to the wellbore. Hawkins[3] pointed out that if the damaged or stimulated zone is considered equivalent to an altered zone of uniform permeability. ks, and outer radius, rs, the additional pressure drop, Δps, across this zone can be modeled by the steady-state radial flow equation

....................(5)

Eq. 5 states that the pressure drop in the altered zone is inversely proportional to ks rather than to k and that a correction to the pressure drop in this region must be made. Combining Eqs. 2 and 5, we find that the total pressure drop at the wellbore is

....................(6)

For r = rw, the argument of theEi function is sufficiently small after a short time that we can use the logarithmic approximation; thus, the drawdown is

....................(7)

We can conveniently define a dimensionless skin factor, s, in terms of the properties of the equivalent altered zone:

....................(8)

Thus, the drawdown is

....................(9)

Eq. 9 provides some insight into the physical significance of the algebraic sign of the skin factor. If a well is damaged (ks < k), s will be positive, and the greater the contrast between ks and k and the deeper into the formation the damage extends, the larger the numerical value of s, which has no upper limit. Some newly drilled wells will not flow before stimulation; for these wells, ks = 0 and s → ∞. If a well is stimulated (ks > k), s will be negative, and the deeper the stimulation, the greater the numerical value of s. Rarely does a stimulated well have a skin less than –7, and such skin factors arise only for wells with deeply penetrating, highly conductive hydraulic fractures. If a well is neither damaged nor stimulated (k = ks), s = 0.

The altered zone near a well affects only the pressure near that well; that is, the pressure in the unaltered formation away from the well is not affected by the existence of the altered zone. Thus, use Eq. 9 to calculate pressures at the sandface of a well with an altered zone, and Eq. 2 to calculate pressures beyond the altered zone in the formation surrounding the well. See Fluid flow with formation damage for more information on damage.

## Inertial-turbulent flow and rate-dependent skin

The diffusivity equation, Eq. 1, assumes that Darcy’s law represents the relationship between flow velocity and pressure gradients in the reservoir, an assumption that is adequate for low-velocity or laminar flow. However, at higher flow velocities, deviations from Darcy’s law are observed as a result of inertial effects or even turbulent flow effects. In 1D radial flow, these inertial/turbulent effects (often called non-Darcy flow effects) are confined to the region near the wellbore in which flow velocities are largest. This results in an additional pressure drop similar to that caused by skin, but the additional pressure drop is proportional to flow rate. The apparent skin, s′, for a well with non-Darcy flow near the wellbore is given by[4]

....................(10)

where D is the non-Darcy flow factor for the system. D can be regarded as constant, although, in theory, it depends slightly on near-well pressure. In practice, non-Darcy flow is ordinarily important only for gas wells, which have high-flow velocities near the wellbore, but it can be important for oil wells with high-velocity flow in some situations.

## Radius of investigation and stabilization time

Radius of investigation is the distance a pressure transient has moved into a formation following a rate change in a well. This distance is related to formation rock and fluid properties and time elapsed since a rate change in the well. Consider this concept by visualizing the pressure distributions at increasing times as Fig. 1 shows for a well producing at constant rate from a reservoir initially at uniform pressure. (These pressure distributions were calculated using the Ei-function solution to the diffusivity equation.)

• The pressure in the wellbore, at r = rw, decreases steadily as flow time increases; likewise, pressures at other fixed values of r also decrease as flow time increases.
• The pressure drawdown (or pressure transient) caused by producing the well moves further into the reservoir as flow time increases. For the range of flow times shown, there is always a point beyond which the drawdown in pressure from the original value is negligible. This time-dependent point of "negligible drawdown" can be considered to be a radius of investigation.

Analysis shows that the time, t, at which a pressure disturbance reaches a distance, ri, which is called the radius of investigation, is given by the equation[2]

....................(11)

Investigators differ on the numerical constant in Eq. 11, but this difference is of little practical importance if the radius of investigation is used as a semiquantitative indicator of the distance into the reservoir to which formation properties have influenced the response of a well in a pressure-transient test.

The radius of investigation has several applications in pressure-transient test analysis and design. A qualitative use is to help explain the shape of a pressure buildup or drawdown curve. For example, a buildup test plot may have a complex shape at early times when the radius of investigation is in the altered zone near the wellbore, where the permeability is different from formation permeability. Or a buildup test plot may change shape at long times when the radius of investigation reaches the general vicinity of a reservoir boundary.

The radius-of-investigation concept provides a guide for well-test design. For example, you may want to sample reservoir properties at least 1,000 ft from a test well. The radius of investigation concept allows you to estimate the time required to achieve the desired depth of investigation.

Eq. 11 also provides a means to estimate the time required to achieve "stabilized" flow; that is, the time required for a pressure transient to reach the boundaries of a tested reservoir. For example, if a well is centered in a cylindrical drainage area of radius re, then the time required for stabilization, ts, is

....................(12)

For other drainage shapes, the time to stabilization can be quite different, as discussed later.

The Ei-function solution to the radial diffusivity equation is valid only while a reservoir is infinite-acting; that is, until boundaries begin to affect the pressure drawdown at the well. For the constant rate flow of a well centered in its drainage area of radius, re, and modeled by the Ei-function solution, these effects begin at t = 948 ϕμctre2/k. Before these boundary effects, the regime is called unsteady-state flow. After boundary effects are felt fully, the solution to the radial diffusivity equation for a well centered in a cylindrical drainage area and producing at constant rate is[2]

....................(13)

This equation for calculating pressure in the wellbore becomes valid for t > 948 ϕμctre2/k at the same time at which the Ei-function solution becomes invalid.

Another form of Eq. 13 is useful for some applications. It involves replacing original reservoir pressure, pi, with average pressure, , within the drainage volume of the well. The volumetric average pressure within the drainage volume of the well can be found from material balance. The pressure decrease resulting from removal of qB RB/D of fluid for t hours (a total volume removed of 5.615qBt/24 ft3) is

....................(14)

Substituting in Eq. 13, the time-dependent terms cancel, and the result is

....................(15)

Eqs. 13 and 15 are more useful in practice if they include skin factors to account for damage or stimulation. In Eq. 15,

....................(16)

....................(17)

and ....................(18)

## Productivity index

The productivity index, J, of an oil well is the ratio of the stabilized rate, q, to the pressure drawdown, , required to sustain that rate. For flow from a well centered in a circular drainage area, Eq. 17 allows us to relate productivity index to formation and fluid properties:

....................(19)

Thus, if a well is tested at several different stabilized rates and the stabilized flowing bottomhole pressure (BHP), pwf, is measured at each rate (that is, if pseudosteady-state is attained at each rate), Eq. 19 implies that a plot of test data should produce a straight line with slope J and intercepts q = 0 when pwf = and when pwf = 0. (See Fig. 2.) In practice, actual field data will fall below the theoretical straight line for pressures below the bubblepoint pressure of the oil because of increasing gas saturations and oil viscosities that increase the resistance to flow.

## Generalized drainage area shapes

Eq. 17 is limited to a well centered in a circular drainage area. A similar equation models pseudosteady-state flow in more general reservoir shapes[2]:

....................(20)

where A is the drainage area in square feet, and CA is the dimensionless shape factor for a specific drainage-area shape and configuration. Table 1 gives values of CA.

The productivity index, J, can be expressed for general drainage-area geometry as

....................(21)

Other numerical constants tabulated in Table 1 allow us to calculate the maximum elapsed time during which a reservoir is infinite-acting (so that the Ei-function solution can be used), the time required for the for the pseudosteady-state solution to predict pressure drawdown within 1% accuracy, and time required for the pseudosteady-state solution to be exact. For a given reservoir geometry, the maximum time a reservoir is infinite acting can be determined using the entry in the column "Use Infinite System Solution With Less Than 1% Error for tDA <." This tDA is defined as 0.0002637kt/ϕμctA, so this means that the time in hours is calculated from

....................(22)

Time required for the pseudosteady-state equation to be accurate within 1% can be found from the entry in the column titled "Less Than 1% Error for t DA., " Finally, the time required for the pseudosteady-state equation to be exact is found in the entry in the column "Exact for tDA >."

Figs. 3 and 4 show the flow regimes that occur at various times. These figures show pwf in a well flowing at constant rate, plotted as a function of time on both logarithmic and linear scales. In the transient region, the reservoir is infinite acting and is modeled by Eq. 9, which implies that pwf is a linear function of log t. In the pseudosteady-state region, the reservoir is modeled by Eq. 20 in the general case or Eqs. 15 or 13 for the special case of a well centered in a cylindrical drainage area. Eq. 13 shows a linear relationship between pwf and t during pseudosteady-state flow. This linear relationship also exists in generalized reservoir geometries.

At times between the end of the transient region and the beginning of the pseudosteady-state region, there is a transition region, sometimes called the late-transient region. This region is, for practical purposes, nonexistent for wells centered in circular, square, or hexagonal drainage areas, as Table 1 indicates. However, for a well off-center in its drainage area, the late-transient region can span a significant time region, as Table 1 also indicates.

Pseudosteady-state flow describes production from a closed drainage area (one with no-flow outer boundaries, either permanent and caused by zero-permeability rock or "temporary" and caused by production from offset wells). In pseudosteady-state, reservoir pressure drops at the same rate with time at all points in the reservoir, including at the reservoir boundaries. Ideally, true steady-state flow can occur in the drainage area of a well, but only if pressure at the drainage boundaries of the well can be maintained constant while the well is producing at constant rate. While unlikely, steady-state flow is conceivable for wells with edgewater drive or in repeated flood patterns in a reservoir. The solution to the radial diffusivity equation is based on a constant-pressure outer boundary condition, instead of a no-flow outer boundary condition. The steady-state solution, applicable after boundary effects have been felt, is

....................(23)

## Constant pressure in the well

Both the steady-state solution (constant pressure at the outer boundaries) and the pseudosteady-state solution (no-flow at the outer boundaries) assume constant rate production in the well. A well is actually more likely to be produced at something close to constant flowing BHP than constant rate. When pressure transients reach no-flow drainage area boundaries, the flow regime is not pseudosteady state; instead, it is more correctly called boundary-dominated flow. If the drainage boundaries are maintained at constant pressure, however, steady-state flow is achieved when the pressure transient reaches the reservoir boundaries.

These different flow regimes are clarified with figures showing pressure distributions in the drainage area of wells with constant flow rate and constant-pressure outer boundaries (Fig. 5); constant BHP and constant-pressure outer boundaries (also Fig. 5); constant flow rate and no-flow outer boundaries (Fig. 6); and constant BHP and no-flow outer boundaries (also Fig. 6).

## Wellbore storage

TheEi-function solution to the diffusivity equation assumes constant flow rate in the reservoir, starting at time zero. In practice, only the rate at the surface can be controlled. Under ideal conditions, a constant surface rate can be maintained, but the first fluid produced will be fluid that was stored in the wellbore, and, at first, the flow rate from the reservoir into the wellbore will be zero. As the wellbore is unloaded, the reservoir rate approaches the surface rate (Fig. 7). Only as the reservoir and surface rates become approximately equal does theEi-function solution become valid. This wellbore unloading during flow tests is a special case of a general phenomenon called wellbore storage.

For a pressure buildup test, the surface rate is zero starting at the instant of shut-in. However, fluid continues to flow into the wellbore from the reservoir because of existing pressure gradients. Idealized models of pressure buildup tests assume a reservoir rate of zero starting at the time of shut-in for the test. This assumption is obviously violated because of the afterflow into the wellbore. As the afterflow rate diminishes, the downhole rate approaches the surface rate (zero), and only as the afterflow rate approaches zero closely can the idealized models closely approximate actual well behavior (Fig. 8). Afterflow during buildup tests is another special case of wellbore storage.

The relationship between changes in bottomhole pressure and wellbore unloading or afterflow rates can be modeled with mass balances on the wellbore. There are two special cases of interest: a wellbore completely filled with a single-phase fluid (Fig. 9, usually gas in practice) and a wellbore with a rising or falling liquid/gas interface in the well (Fig. 10).

For the wellbore filled with a single-phase fluid, [2]

....................(24)

For a well with a rising or falling liquid/gas interface, [2]

....................(25)

In most applications, pt is assumed to be constant, a convenient but frequently inaccurate simplification. Both equations can be written in the general form

....................(26)

where, for a fluid-filled wellbore,

....................(27)

and, for a moving liquid/gas interface with unchanging surface pressure,

....................(28)

C is called the wellbore storage coefficient.

For special cases in which, at earliest times for a flowing well, all the production is coming from fluid stored in the wellbore and none is entering the wellbore from the formation (or, for a shut-in well, the rate of afterflow is equal to the rate before shut in), the integration of Eq. 26 yields

....................(29)

where Δp is the pressure change in the time because either the start of flow or shut in and Δt is the elapsed time. On a log-log plot of Δp vs. Δt during these early times, a straight line with a slope of unity will result. For any point on this unit slope line, the wellbore storage coefficient, C, can be found from any point on the line (Δt, Δp) and Eq. 29 (Fig. 11). Alternatively, the slope (qB/24C) of a plot of Δp vs. Δt on Cartesian coordinates also leads to an estimate of the wellbore-storage coefficient.

## Linear flow

Linear flow occurs in some reservoirs with long, highly conductive vertical fractures; in relatively long, relatively narrow reservoirs (channels, such as ancient stream beds); and near horizontal wells during certain times. For unsteady-state linear flow in an unbounded (infinite-acting) reservoir, [2]

....................(30)

## Spherical flow

Spherical flow occurs in wells with limited perforated intervals and into wireline formation test tools. The solution to the spherical/cylindrical, 1D form of the diffusivity equation, subject to the initial condition that pressure is uniform before production and the boundary conditions of constant flow rate and an infinitely large drainage area, is[5]

....................(31)

where ....................(32)

and rsp = the radius of the sphere into which flow converges.

## Superposition

The principle of superposition indicates that the total pressure at any point in a reservoir is the sum of the pressure drops at that point caused by flow in each of the wells in the reservoir. A simple illustration of this principle is the case of three wells in an infinite reservoir. Consider wells A, B, and C, that start to produce at times tA, tB, and tC in an infinite-acting reservoir (Fig. 12). Application of the principle of superposition shows that[2]

....................(33)

For an infinite-acting reservoir, use theEi-function solutions, including the logarithmic approximation at Well A:

....................(34)

where tA, tB, and tC are times at which wells A, B, and C will begin to produce. The skin factor for Well A is included in Eq. 29. The skin factors for other wells are not, because skin factors for individual wells affect only pressures measured inside altered zones for those wells.

Next, consider the use of superposition to model the effects of boundaries in bounded reservoirs. Consider the well in Fig. 13, a distance L from a single no-flow boundary (such as a sealing fault). Mathematically, this problem is identical to the problem of a well at distance 2L from an "image" well; that is, a well that has the same production history as the actual well. The reason that the two-well system simulates the behavior of a well near a boundary is that a line equidistant between the two wells can be shown to be a no-flow boundary. That is, along this line the pressure gradient is zero, which means that there can be no flow. Thus, this problem is a simple problem of two wells in an infinite reservoir:

....................(35)

The drawdown term of the image well does not include a skin factor.

As examples, extend the imaging technique to model wells between boundaries intersecting at 90° (Fig. 14); wells between two parallel boundaries (Fig. 15); wells near single constant-pressure boundaries (Fig. 16); and wells at various locations in closed reservoirs (Fig. 17).

One of the most frequently used applications of superposition is to model variable-rate production. Consider Fig. 18, in which a well in an infinite-acting reservoir produces at rate q1 from time 0 to time t1; q2 from t1 to t2, and q3 for times greater than t2. To model the total drawdown for t > t2, add three drawdowns: the drawdown because of a well producing at rate q1 starting at time zero and continuing to produce to time t; the drawdown because of a well producing at rate (q2q1), starting at time t1 and continuing to time t; and the drawdown because of a well producing at rate (q3q2) starting at time t2 and continuing to time t. The total drawdown is thus

....................(36)

Horner[6] proposed a convenient alternative to superposition to model the many changes in rate in the history of a typical well. With this approximation, the sequence ofEi functions reflecting rate changes can be replaced with a singleEi function that contains a single producing time and a single producing rate. The single rate is the most recent nonzero rate at which the well has produced, qn. The single producing time, called tp, is the ratio of cumulative production, Np, to qn.

....................(37)

This approximation preserves the material balance in the drainage area of the well and properly gives greatest weight to most recent rate (as opposed to average rate), which dominates the pressure distribution near a well out to the radius of investigation achieved while the well was produced at rate qn. The approximation is particularly useful for hand calculations. Given the widespread availability of computer software for analyzing flow and buildup tests on well, the use of more rigorous superposition to model variable-rate production histories is generally more appropriate.

## Semilog methods for flow tests

The logarithmic approximation to theEi-function solution can be used as a basis for analysis of an ideal constant-rate flow test in a well. Written in terms of log10, this equation, which models the BHP for a well in a homogeneous-acting formation with an infinite-acting drainage area and, in absence of wellbore unloading, becomes

....................(38)

This expression has the same form as the equation of a straight line, y = mx + b, with the analogies

....................(39)

....................(40)

....................(41)

....................(42)

These analogies suggest a graphical method of analysis. Eq. 38 indicates that a plot of pwf vs. log10(t) should be a straight line with slope m that will allow an estimate of effective permeability to the single liquid phase flowing. (See Fig. 19.)

....................(43)

From the intercept, b, at t = 1 hr [log10(1) = 0], p1hr, calculate the skin factor.

....................(44)

In these equations, the slope, m, is given by

....................(45)

Eq. 45 indicates that m is most easily determined by choosing values of times t1 and t2 that differ by powers of 10 and is especially easy if t1 and t2 differ by one log cycle. The intercept, p1hr, is the pressure at a time of 1 hour on the best straight line through the data. It may be necessary to extrapolate the straight line to a time of one hour to read the intercept.

## Semilog methods for pressure buildup test

Consider the rate history for an idealized pressure test shown in Fig. 20. A well is produced at constant rate q for a time tp, and then the well shut in (q = 0) for a pressure buildup test. The rate history is modeled as the sum of two constant flow rate periods, one at rate q, beginning at t = 0, and the other at rate –q, beginning at t = tp, at which the time elapsed since shut-in, Δt, is zero. Use the log approximation to theEi-function solution to model the drawdown, and sum them as Fig. 21 shows. Represented mathematically, the superposition process is

....................(46)

This can be simplified to

....................(47)

Like the drawdown equation, Eq. 43 can be interpreted as the equation of a straight line. The analogies are

....................(48)

....................(49)

....................(41)

and ....................(50)

The group [(tp + Δt)/Δt] is called the Horner time ratio (HTR) or sometimes simply the Horner time. Our simple model, which describes a buildup test in a homogeneous, infinite-acting reservoir, a well with one constant rate before shut in and without afterflow (wellbore storage), indicates that a graph of pws vs. the HTR should fall on a straight line. From the slope, m, of this line, the permeability to the single-phase liquid flowing into the wellbore can be estimated. The intercept, b, at log 10 [(tp + Δt)/Δt] = 0 or [(tp + Δt)/Δt] = 1 provides an estimate of original drainage area pressure, pi.

Obtain the slope, m, from

....................(51)

where {[ (tpt)/Δt]1, pws1} and {[ (tpt)/Δt]2, pws2} are any two points on the straight-line (Fig. 22). Normally, choose [tp + Δt)/Δt]1 and [(tp + Δt)/Δt]2 to be powers of 10.

In Fig. 22, which is called a Horner plot, the HTR on the horizontal axis decreases from left to right, so that shut-in time increases from left to right. In some Horner plots, the HTR increases from left to right; in that case shut-in time increases from right to left.

Skin factor can be estimated from a pressure buildup test, even though the skin factor does not appear in the buildup equation, Eq. 47. Simultaneously solve the equation modeling the drawdown at the instant of shut in (at time tp) with Eq. 47, discard terms that are ordinarily negligible, and arrive at the result

....................(52)

The radius-of-investigation concept is also useful for pressure buildup tests, as Fig. 23 illustrates. The approximate position of the point at which the pressure has built up to a uniform level intersects the region in which the pressure is little affected by the shut-in is given by Eq. 11, with elapsed time, t, interpreted as shut-in time, Δt.