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# Solving unsteady flow problems with Laplace transform and source functions

There are many advantages of developing transient flow solutions in the Laplace transform domain. For example, in the Laplace transform domain, Duhamel’s theorem^{[1]} provides a convenient means of developing transient flow solutions for variable rate production problems using the solutions for the corresponding constant rate production problem.

## Contents

- 1 Transient flow solutions in the Laplace domain
- 2 Point-source solution in the Laplace domain
- 3 Line-, surface-, and volumetric-source solution in the laplace domain
- 4 Solutions for infinite-slab reservoirs
- 5 Solutions for cylindrical reservoir
- 6 Solutions for rectangular parallelepiped reservoir
- 7 Conversion from 3D to 2D anisotropy
- 8 Computational considerations and applications
- 8.1 The integral
*I* - 8.2 The series
*S*_{1} - 8.3 The series
*S*_{2} - 8.4 The series
*S*_{3} - 8.5 The series
*F* - 8.6 The series
*F*_{1} - 8.7 The ratio
*R*_{1} - 8.8 Example 5 - Fully penetrating uniform flux fracture in an infinite-slab reservoir with closed top and bottom boundaries
- 8.9 Example 6 - Horizontal well in an infinite-slab reservoir with closed top and bottom boundaries
- 8.10 Example 7 - Fully penetrating, uniform-flux fracture in an isotropic and closed cylindrical reservoir
- 8.11 Example 8 - Fully penetrating uniform-flux fracture in an isotropic and closed parallelepiped reservoir
- 8.12 Example 9 - Uniform-flux horizontal well in an isotropic and closed parallelepiped reservoir
- 8.13 Dimensionless fracture pressure

- 8.1 The integral
- 9 Nomenclature
- 10 References
- 11 External links
- 12 See also

## Transient flow solutions in the Laplace domain

Duhamel’s theorem states that if Δ*p* and Δ*p*_{c} denote the pressure drawdown corresponding to the variable production rate, *q*(*t*), and the constant production rate, *q*_{c}, respectively, then

Applying the Laplace transform converts the convolution integral in **Eq. 1** to an algebraic expression, and Duhamel’s theorem is given in the Laplace transform domain as

The simplicity of the expression given in **Eq. 2** 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]}

where the subscript *f* indicates the fracture property, and *t*_{D} and *r*_{D} are the dimensionless time and distance (as defined in **Eqs. 12** and **16**).

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

The general solutions for **Eqs. 3** and **4** are given, respectively, by

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

and

Then, it may be shown that

where the right side of **Eq. 9** 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. 9**.

Obtaining the Laplace transforms of the Green’s and source function solutions developed in the time domain with the methods explained on the Source function solutions of the diffusion equation and Solving unsteady flow problems with Green's and source functions pages 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*.^{[3]} 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.^{[4]}

Ozkan and Raghavan^{[5]}^{[6]} 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*.^{[7]} and Warren and Root^{[8]} to develop the governing flow equations for naturally fractured reservoirs. The results, however, will be applicable to the model suggested by Kazemi^{[9]} and de Swaan-O^{[10]} 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

and

In **Eqs. 10** and **11**, subscripts f and m indicate the property of the fracture and matrix systems, respectively. Initial pressure, *p*_{i}, is assumed to be uniform in the entire system; that is, *p*_{fi} = *p*_{mi} = *p*_{i}. The dimensionless time, *t*_{D}, is defined by

where *ℓ* is a characteristic length in the system, and

The definitions of the other variables used in **Eqs. 10** and **11** are

and

where

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

and

The inner-boundary condition corresponding to the instantaneous withdrawal of an amount of fluid, , 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, , instantaneously withdrawn from the sphere at *t* = 0, we can write^{[11]}

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,

For the condition expressed in **Eq. 21** to hold for every *T* ≥ 0, we must have

where *δ*(*t*) is the Dirac delta function satisfying the properties expressed by **Eqs. 23** and **24**.

Using the results given by **Eqs. 21** and **22** in **Eq. 20**, we obtain

The Laplace transform of **Eqs. 10**, **11**, **19**, and **25** yields

where

and

In deriving these results, we have used the initial condition given by **Eq. 18** and noted that

In **Eq. 29**, the term represents the strength of the source for the naturally fractured porous medium.

The solution of **Eqs. 26**, **28**, and **29** 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 acting at *t* = 0:

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

where

and

The instantaneous point-source solution for the model suggested by Barenblatt *et al*.^{[7]} and Warren and Root^{[8]} can also be used for the model suggested by Kazemi^{[9]} and de Swaan-O,^{[10]} 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*_{f} = 1, and *V*_{m} = 0.

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

The Laplace transform of **Eq. 35** yields the following continuous point-source solution in an infinite reservoir:

where we have substituted **Eq. 33** for *S*, dropped the subscript *f*, and defined

## 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

where Δ*p*_{p} 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_{w}, is given by

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

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

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 1** 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. 36** with the method of images. The result is given by

where

and

The solution given in **Eq. 41** is not very convenient for computational purposes. To obtain a computationally convenient form of the solution, we use the summation formula given by^{[11]}^{[12]}

and recast **Eq. 41** as

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^{[12]} and are given in **Table 1**. The point-source solutions can be used with **Eqs. 38** and **40** to generate the solutions for the other well geometries given in **Table 1**. 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. 47** from *z*_{w} − *h*_{w} / 2 to *z*_{w} + *h*_{w} / 2 with respect to *z′*, where *z*_{w} is the vertical coordinate of the midpoint of the open interval. If *h*_{w} = *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*_{f} and half-length *x*_{f} is obtained if the point-source solution is integrated once with respect to *z′* from *z*_{w} − *h*_{f} / 2 to *z*_{w} + *h*_{f} / 2 and then with respect to *x′* from *x*_{w} – *x*_{f} to *x*_{w} + *x*_{f}, where *x*_{w} and *z*_{w} are the coordinates of the midpoint of the fracture. Similarly, the solution for a horizontal-line source well of length *L*_{h} is obtained by integrating the point-source solution with respect to *x′* from *x*_{w} − *L*_{h} / 2 to *x*_{w} + *L*_{h} / 2, where *x*_{w} 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.^{[6]}

where

....................(49) The point-source solution is also required to satisfy the following flux condition at the source location (*r*_{D} →0+, *θ* = *θ*′, *z*_{D} = *z′*_{D}):

Assuming that the reservoir is bounded by a cylindrical surface at *r*_{D} = *r*_{eD} and by the parallel planes at *z*_{D} = 0 and *h*_{D}, 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:

In **Eq. 51**, is a solution of **Eq. 48** that satisfies **Eq. 50** and the boundary conditions at *z*_{D} = 0 and *h*_{D}. may be chosen as one of the point-source solutions in an infinite-slab reservoir given in **Table 1**, depending on the conditions imposed at the boundaries at *z*_{D} = 0 and *h*_{D}. If is chosen such that it satisfies the boundary conditions at *z*_{D} = 0 and *h*_{D}, its contribution to the flux vanishes at the source location, and + satisfies the appropriate boundary condition at *r*_{D} = *r*_{eD}, then **Eq. 51** 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

and

According to the boundary condition given by **Eq. 52**, we should choose as the point-source solution given in **Table 1** (or by **Eq. 47**). Then, with the addition theorem for the Bessel function *K*_{0}(*aR*_{D}) given by^{[13]}

where

we can write

for *r*_{D} < *r′*_{D}. If *r*_{D} > *r′*_{D}, we interchange *r*_{D} and *r′*_{D} in **Eq. 56**. If we choose in **Eq. 51** as

where *a*_{k} and *b*_{k} are constants, then satisfies the boundary condition given by **Eq. 52**, and the contribution of to the flux at the source location vanishes. If we also choose the constants *a*_{k} and *b*_{k} in **Eq. 57** as

and

then satisfies the impermeable boundary condition at *r*_{D} = *r*_{eD} given by **Eq. 53**. Thus, the point-source solution for a closed cylindrical reservoir is given by

This solution procedure may be extended to the cases in which the boundaries are at constant pressure or of mixed type.^{[6]} **Table 2** 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 2** in **Eq. 39** (or **Eq. 40**).

### Example 1 - Partially penetrating, uniform-flux fracture in an isotropic and closed cylindrical reservoir

Consider a partially penetrating, uniform-flux fracture of height *h*_{f} and half-length *x*_{f} in an isotropic and closed cylindrical reservoir. The center of the fracture is at *r′* = 0, *θ*′ =0, *z′* = *z*_{w}, and the fracture tips extend from (*r′* = *x*_{f}, *θ* = α + *π*) to (*r′* = *x*_{f}, *θ* = *α*).

*Solution.* **Fig. 1** 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′*, *z*_{w} is obtained by integrating the corresponding point-source solution given in **Table 2** with respect to *z′* from *z*_{w} – *h*_{f} / 2 to *z*_{w} + *h*_{f} / 2 and is given by

To generate the solution for a partially penetrating plane source that represents the fracture, the partially penetrating line-source solution given in **Eq. 61** is integrated with respect to *r′* from 0 to *x*_{f} with *θ*′ = *α* + *π* in the third quadrant and with *θ*′ = *α* in the first quadrant. This procedure yields

It is possible to obtain an alternate representation of the solution given in **Eq. 62**. With the addition theorem of the Bessel function *K*_{0}(*x*) given by **Eq. 54**, the solution in **Eq. 61** may be written as

where

and

The integration of the partially penetrating vertical well solution given in **Eq. 63** with respect to *r′* from 0 to *x*_{f} (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:

where

### Example 2 - Uniform-flux, horizontal well in an isotropic and closed cylindrical reservoir

Consider a uniform-flux, horizontal line-source well of length *L*_{h} in an isotropic and closed cylindrical reservoir. The well extends from (*r′* = *L*_{h}/2, *θ* = *α* + *π*) to (*r′* = *L*_{h}/2, *θ* = *α*), and the center of the well is at *r′* = 0, *θ*′ = 0, *z′* = *z*_{w}.

**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 2** with respect to *r′* from 0 to *L*_{h} / 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

## 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*_{e}, 0 < *y* < *y*_{e}, and 0 < *z* < *h* is obtained by applying the method of images to the point-source solution given by **Eq. 36**: ^{[6]}^{[11]}

where

and

Ozkan^{[11]} shows that the triple infinite sums in **Eq. 69** may be reduced to double infinite sums with

where

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

where

and

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.^{[11]}^{[12]} **Table 3** gives these solutions, which may be used to derive the solutions for the other source geometries with **Eq. 39** (or **Eq. 40**). **Examples 3.10** and **3.11** demonstrate the derivation of the solutions for the other source geometries in rectangular reservoirs.

### Example 3 - Fully penetrating vertical fracture in a closed rectangular reservoir

Consider a vertical fracture of half-length *x*_{f} located at *x′* = *x*_{w} and *y′* = *y*_{w} 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**, first with respect to *z′* from 0 to *h* and then with respect to *x′* from *x*_{w} – *x*_{f} to *x*_{w} + *x*_{f}. The result is

where , , and *ε*_{k} are given respectively by **Eqs. 77**, **78**, and **80**.

### Example 4 - Horizontal well in a closed rectangular reservoir

Consider a horizontal well of length *L*_{h} in the *x*-direction located at *x′* = *x*_{w}, *y′* = *y*_{w}, and *z′* = *z*_{w} 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**, with respect to *x′* from *x*_{w}–*L*_{h} /2 to *x*_{w}+*L*_{h} /2, and is given by

where

and

In **Eq. 85**, , , *ε*_{n}, *ε*_{k}, and *ε*_{k, n} are given by **Eqs. 77** through **81**.

## 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*_{x}, *k*_{y}, and *k*_{z} denoting the corresponding permeabilities. In these solutions, an equivalent isotropic permeability, *k*, has been defined by

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

and replace *k* in the solutions by *k*_{h}. Note that *k* takes place in the definition of the dimensionless time *t*_{D} (**Eq. 12**). Then, if we define a dimensionless time based on *k*_{h}, the relation between and *t*_{D} is given by

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

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*_{wD} = 0, *y*_{wD} = 0) and choosing the half-length of the horizontal well as the characteristic length (i.e., ℓ = *L*_{h} / 2), the horizontal-well solution given in **Table 1** may be written as

In **Eq. 90**, *s* is the Laplace transform variable with respect to dimensionless time, *t*_{D}, based on *k* and

and

If we define the following variables based on *k*_{h},

and also note that

then, we may rearrange **Eq. 90** in terms of the dimensionless variables based on *k*_{h} as

where

and

If we compare **Eqs. 90** and **99**, we can show that

where we have used the relation given by **Eq. 90**. If we now define as the Laplace transform variable with respect to , we may write

With the relation given by **Eq. 103** and **Eq. 90**, we obtain the following horizontal-well solution in terms of dimensionless variables based on *k*_{h}:

## 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.^{[5]}^{[6]}^{[11]} 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 Laplace transformation for solving transient flow problems 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. 27** yields the following limiting forms:

and

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 *I*

This integral arises in the computation of many practical transient-pressure solutions and may not be numerically evaluated, especially as *y*_{D}→0; however, the following alternate forms of the integral are numerically computable.^{[6]}

and

The integrals in **Eqs. 108** through **110** may be evaluated with the standard numerical integration algorithms for *y*_{D} ≠ 0. For *y*_{D} = 0, the polynomial approximations given by Luke^{[14]} or the following power series expansion given by Abramowitz and Stegun^{[15]} may be used in the computation of the integrals in **Eqs. 108** through **110**:

For numerical computations and asymptotic evaluations, it may also be useful to note the following relations: ^{[6]}

and

It can be shown from **Eqs. 112** and **113** that, for practical purposes, when *z* ≥ 20, the right sides of **Eqs. 111** and **112** may be approximated by *π*/2 and *π* exp (−|*c*|)/2, respectively.^{[6]}^{[9]}

As a few sources^{[5]}^{[6]}^{[11]} 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 given, respectively, by

where

and

where *γ*=0.5772… and

It is also useful to note the real inversions of **Eqs. 114** and **116** given, respectively, by

and

### The series *S*_{1}

Two alternative expressions for the series *S*_{1} may be convenient for the large and small values of u (i.e., for short and long times).^{[11]} When *u* is large,

and when *u* + *a*^{2} << *n*^{2}*π*^{2}/*h*^{2}_{D},

### The series *S*_{2}

Alternative computational forms for the series *S*_{2} are given next.^{[11]} When *u* is large,

and when *u* + *a*^{2} << *n*^{2}*π*^{2}/*h*^{2}_{D},

### The series *S*_{3}

The following alternative forms for the series may be convenient for the large and small values of *u* (i.e., for short and long times).^{[11]} When *u* is large,

and when *u* + *a*^{2} << (2*n* − 1)^{2} *π*^{2}/(4*h*^{2}_{D}),

### The series *F*

where

The series may be written in the following forms with the use of **Eqs. 108** through **110**.

and

The computation of the series in **Eqs. 131** and **132** should not pose numerical difficulties; however, the series in **Eq. 133** converges slowly. With the relation given in **Eq. 112**, we may write **Eq. 133** as^{[11]}

where

Before discussing the computation of the series given in **Eq. 135**, we first discuss the derivation of the asymptotic approximations for the series . When *s* is large (small times), may be approximated by^{[11]}

where *β* is given by **Eq. 115**. If s is sufficiently large, then **Eq. 136** may be further approximated by

The inverse Laplace transform of **Eq. 137** yields

For small *s* (large times), depending on the value of *x*_{D}, may be approximated by one of the following equations: ^{[11]}

### The series *F*_{1}

where

With the relations given in **Eqs. 121** and **122**, the following alternative forms for the series may be obtained, respectively, for the large and small values of s (i.e., for short and long times).^{[11]} When *u* is large,

and when *u* << *n*^{2}*π*^{2}/*h*^{2}_{D},

It is also possible to derive asymptotic approximations for the series . When *s* is large (small times), may be approximated by^{[11]}

If *s* is sufficiently large, then **Eq. 146** may be further approximated by

The inverse Laplace transform of **Eq. 146** yields

For small *s* (large times), may be approximated by^{[11]}

### The ratio *R*_{1}

By elementary considerations, the ratio may be written as^{[11]}

The expression given in **Eq. 150** provides computational advantages when *s* is small (time is large).

### Example 5 - Fully penetrating uniform flux fracture in an infinite-slab reservoir with closed top and bottom boundaries

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

*Solution.* **Table 1** gives the solution for this problem. For simplicity, assuming an isotropic reservoir, choosing the characteristic length as ℓ = *x*_{f} and noting that , the solution becomes

First consider the numerical evaluation of **Eq. 151**. We note from **Eqs. 108** through **110** that **Eq. 151** may be written in one of the following forms, depending on the value of *x*_{D}.

and

The numerical evaluation of the integrals in **Eqs. 152** through **154** for *y*_{D} ≠ 0 should be straightforward with the use of the standard numerical integration algorithms. For *y*_{D} = 0, the polynomial approximations given by Luke^{[14]} or the power series expansion given by **Eq. 111** should be useful.

The short- and long-time asymptotic approximations of the fracture solution are also obtained by applying the relations given by **Eqs. 114** and **116**, respectively, to the right side of **Eq. 151**. This procedure yields, for short times,

or, in real-time domain,

where *β* is given by **Eq. 115** with *a* = -1 and *b* = +1. At long times, the following asymptotic approximation may be used:

or, in real-time domain,

where *γ* = 0.5772… and *σ*(*x*_{D}, *y*_{D}, -1, +1) is given by **Eq. 117**.

### Example 6 - Horizontal well in an infinite-slab reservoir with closed top and bottom boundaries

Consider a horizontal well of length *L*_{h} located at *x′* = 0, *y′* = 0, and *z′* = *z*_{w} in an infinite-slab reservoir with closed top and bottom boundaries.

*Solution.* **Table 1** gives the horizontal-well solution for an infinite-slab reservoir with impermeable boundaries. Assuming an isotropic reservoir, choosing the characteristic length as ℓ = *L*_{h} / 2 and noting that , the solution may be written as

where is the fracture solution given by the right side of Eq. 151 and is given by

with

and

The computation of the first term in the right side of **Eq. 159** is the same as the computation of the fracture solution given by **Eq. 151** and has been discussed in **Example 5**. The computational form of the second term in the right side of **Eq. 159** is given by **Eqs. 131** through **134**. Of particular interest is the case for −1 ≤ *x*_{D} ≤ +1. In this case, from **Eqs. 134** and **135**, we have

where

The computational considerations for the series have been discussed previously.

Next, we consider the short- and long-time approximations of the horizontal-well solution given by **Eq. 159**. To obtain a short-time approximation, we substitute the asymptotic expressions for and as s→∞ given, respectively, by **Eqs. 155** and **137**. This yields

where *β* is given by **Eq. 115**. The inverse Laplace transform of **Eq. 165** is given by

To obtain the long-time approximation of **Eq. 159**, we substitute the asymptotic expressions for and as s→∞ given, respectively, by **Eq. 158** and **Eqs. 139** through **141**. Of particular interest is the case for −1 ≤ *x*_{D} ≤ +1, where we have

where *γ*=0.5772… and *σ*(*x*_{D}, *y*_{D}, -1, +1) is given by **Eq. 117**. The inverse Laplace transform of **Eq. 167** yields

### Example 7 - Fully penetrating, uniform-flux fracture in an isotropic and closed cylindrical reservoir

Consider a fully penetrating, uniform-flux fracture of half-length *x*_{f} 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*_{f}, *θ* = *α* + *π*) to (*r′* = *x*_{f}, *θ* = *α*).

*Solution.* The solution for this problem has been obtained in **Eq. 62** in **Example 1** with *h*_{w} = *h*. Choosing the characteristic length as ℓ = *x*_{f} and noting that , the solution is given by

For the computation of the pressure responses at the center of the fracture (*r*_{D} = 0), **Eq. 169** simplifies to

It is also possible to find a very good approximation for **Eq. 169**, especially when *r*_{eD} is large. If we assume^{[6]}

and use the following relation^{[16]}

then **Eq. 169** may be written as

Because^{[6]}

where

**Eq. 173** may also be written as

Although the assumption given in **Eq. 171** may not be justified by itself, the solution given in **Eq. 176** is a very good approximation for **Eq. 169**, especially when *r*_{eD} is large. For a fracture at the center of the cylindrical drainage region, **Eq. 176** simplifies to

It is also possible to obtain short- and long-time approximations for the solution given in **Eq. 177**. For short times, *u*→∞ and the second term in the argument of the integral in **Eq. 177** becomes negligible compared with the first term. Then, **Eq. 177** reduces to the solution for an infinite-slab reservoir given by **Eq. 151**, of which the short-time approximation has been discussed in **Example 5**.

To obtain a long-time approximation, we evaluate **Eq. 177** at the limit as *s*→0 (*u*→*s*). As shown in modified bessel functions, for small arguments we may approximate the Bessel functions in **Eq. 177** by

and

where *γ* = 0.5772…. With **Eqs. 178** through **181** and by neglecting the terms of the order *s*^{3/2}, we may write^{[11]}

If we substitute the right side of **Eq. 182** into **Eq. 177**, we obtain

where *σ*(*x*_{D}, *y*_{D}, −1, +1) is given by **Eq. 117** and

The inverse Laplace transform of **Eq. 183** yields the following long-time approximation for a uniform-flux fracture at the center of a closed square:

### Example 8 - Fully penetrating uniform-flux fracture in an isotropic and closed parallelepiped reservoir

Consider a fully penetrating, uniform-flux fracture of half-length *x*_{f} in an isotropic and closed parallelepiped reservoir of dimensions *x*_{e} × *y*_{e} × *h*. The fracture is parallel to the *x* axis and centered at *x*_{w}, *y*_{w}, *z*_{w}.

*Solution.* The solution for this problem has been obtained in **Example 3** and, by choosing ℓ = *x*_{f}, is given by

where

The computation of the ratios of the hyperbolic functions in **Eq. 186** may be difficult, especially when their arguments approach zero or infinity. When *s* is small (long times), **Eq. 150** should be useful to compute the ratios of the hyperbolic functions. When *s* is large (small times), with **Eq. 150** the solution given in **Eq. 186** may be written as^{[11]}

where

and

The last equality in **Eq. 189** follows from the relation given by **Eq. 133**. The expression given in **Eq. 189** may also be written as

where

and

Therefore, the solution given by **Eq. 186** may be written as follows for computation at early times (for large values of *s*):

where is given by **Eq. 193** and corresponds to the solution for a fractured well in an infinite-slab reservoir (see **Eq. 151** in **Example 5**) and represents the contribution of the lateral boundaries and is given by

In Eq. 196, , , and are given, respectively, by **Eqs. 190**, **191**, and **194**. The integrals appearing in **Eqs. 193** and **194** may be evaluated by following the lines outlined by **Eqs. 108** through **110**.

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 term in **Eq. 195** becomes negligible compared with the term for which a short-time approximation has been obtained in **Example 5** (see **Eqs. 155** and **156**).

To obtain a long-time approximation (small values of *s*), the solution given in **Eq. 186** may be written as^{[9]}

where

and

The second equality in Eq. 198 results from^{[17]}

For small values of *s*, replacing *u* by *s* and *s* + *α* by *α*, and with^{[17]}

the term *H* given by **Eq. 198** may be approximated by

The long-time approximation of the second term in **Eq. 197** is obtained by assuming *u* << *k*^{2}*π*^{2}/*x*^{2}_{eD} and taking the inverse Laplace transform of the resulting expressions; therefore, we can obtain the following long-time approximation

### Example 9 - Uniform-flux horizontal well in an isotropic and closed parallelepiped reservoir

Consider a uniform-flux horizontal well of length *L*_{h} in an isotropic and closed parallelepiped reservoir of dimensions *x*_{e} × *y*_{e} × *h*. The center of the well is at *x*_{w}, *y*_{w}, *z*_{w}, and the well is parallel to the x axis.

*Solution.* The solution for this problem was obtained in **Example 4** and, by choosing ℓ = *L*_{h} / 2, is given by

where is the solution discussed in **Example 8**, and is given by

In **Eq. 205**, and are given by **Eqs. 161** and **162**, respectively,

and

The computation and the asymptotic approximations of the term have been discussed in **Example 8**. To compute the term for long times (small *s*), the relation for the ratios of the hyperbolic functions given by **Eq. 150** should be useful. For computations at short times (large values of *s*), following the lines similar to those in **Example 8**, the term in **Eq. 205** may be written as

where

and

The computational form of the term in **Eq. 208** is obtained by applying the relations given by **Eqs. 131** through **134** and **Eq. 112**. Of particular interest is the case for −1 ≤ *x*_{D} ≤ +1 and *y*_{D} = *y*_{wD} given by

where

which can be written as follows by using the relation given in **Eq. 121**:

Similarly, for −1 ≤ *x*_{D} ≤ +1 and *y*_{D} = *y*_{wD}, the term given in **Eq. 212** may be written as

where

### Dimensionless fracture pressure

**Example 8** discussed the short- and long-time approximations of the term in **Eq. 204**. A small-time approximation for given by **Eq. 207** is obtained with *u* = *ωs* and by noting that as *s*→∞, . Then, substituting the short-time approximations for and given, respectively, by **Eqs. 155** and **137** into **Eq. 204**, the following short-time approximation is obtained: ^{[9]}

where *β* is given by **Eq. 115**. The inverse Laplace transform of **Eq. 218** yields

The long-time approximation of **Eq. 204** is obtained by substituting the long-time approximations of and . The long time-approximation of is obtained in **Example 8** (see **Eq. 197** through **203**). The long-time approximation of is obtained by evaluating the right side of **Eq. 205** as *s*→0 (*u*→0) and is given by

where

and

Thus, the long-time approximation **Eq. 204** is given by

where *p*_{Df} and *F*_{1} are given, respectively, by **Eqs. 203** and **220**. For computational purposes, however, *F*_{1} may be replaced by

In **Eq. 224**, *F*, *F*_{b1}, *F*_{b2}, and *F*_{b3} are given, respectively, by

and

When computing the integrals and the trigonometric series, the relations given by **108** through **110** and **129** through **134** are useful.

## Nomenclature

## References

- ↑ _
- ↑
^{2.0}^{2.1}_ - ↑ _
- ↑ _
- ↑
^{5.0}^{5.1}^{5.2}_ - ↑
^{6.00}^{6.01}^{6.02}^{6.03}^{6.04}^{6.05}^{6.06}^{6.07}^{6.08}^{6.09}^{6.10}_ - ↑
^{7.0}^{7.1}_ - ↑
^{8.0}^{8.1}_ - ↑
^{9.0}^{9.1}^{9.2}^{9.3}^{9.4}_ - ↑
^{10.0}^{10.1}_ - ↑
^{11.00}^{11.01}^{11.02}^{11.03}^{11.04}^{11.05}^{11.06}^{11.07}^{11.08}^{11.09}^{11.10}^{11.11}^{11.12}^{11.13}^{11.14}^{11.15}^{11.16}^{11.17}^{11.18}_ - ↑
^{12.0}^{12.1}^{12.2}_ - ↑ _
- ↑
^{14.0}^{14.1}_ - ↑ _
- ↑ _
- ↑
^{17.0}^{17.1}_

==Noteworthy papers in OnePetro==

Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read

Modeling a Fractured Well in a Composite Reservoir" C. Chen and R. Raghavan.http://dx.doi.org/10.2118/28393-PA

## External links

Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro

## See also

Solving unsteady flow problems with Green's and source functions

Source function solutions of the diffusion equation

Laplace transformation for solving transient flow problems

Transient analysis mathematics