You must log in to edit PetroWiki. Help with editing

Content of PetroWiki is intended for personal use only and to supplement, not replace, engineering judgment. SPE disclaims any and all liability for your use of such content. More information

# PEH:Mathematics of Transient Analysis

**Publication Information**

Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

*Volume I – General Engineering*

John R. Fanchi, Editor

Copyright 2007, Society of Petroleum Engineers

Chapter 3 – Mathematics of Transient Analysis

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.

## Contents

## 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, , and the density of the fluid,*ρ*, as a function of the position (

*x*,

*y*,

*z*) and time,

*t*; that is, = (

*x*,

*y*,

*z*,

*t*) and

*ρ*=

*ρ*(

*x*,

*y*,

*z*,

*t*). Relative to the fixed Cartesian axes, the velocity vector can be written as

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

where

*v*

_{x},

*v*

_{y}, and

*v*

_{z}are the velocity components, and , , and 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

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

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

....................(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, , in a time increment, Δ

*t*, is , and the total mass of the fluid passing through Γ during Δ

*t*is

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

The surface integral in

**Eq. 3.4**accounts for both influx and outflux through the surface of the volume; that is, Δ

*M*

_{g}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

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

By the principle of conservation of mass, equating the right sides of

**Eqs. 3.3**and

**3.5**yields

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

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

*V*; that is,

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

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

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

and

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

With the relation in

**Eq. 3.7**,

**Eq. 3.6**can be recast into

....................(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.

....................(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, , 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.

....................(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

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

where subscript 0 indicates the conditions at the datum. Similarly, the compressibility of the porous rock,

*c*

_{f}, is defined by

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

and the total system compressibility,

*c*

_{t}, is given by

....................(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

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

In

**Eq. 3.16**,

*μ*is the viscosity of the fluid, and

*k*is the permeability tensor of the formation given by

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

where

*α*,

*β*, and

*γ*are the directions, and

*k*

_{ij}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:

....................(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:

....................(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:

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

Then,

**Eq. 3.19**may be written

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

If each coordinate,

*j*=

*x*,

*y*, or

*z*, is multiplied by , where

*k*may be chosen arbitrarily (to preserve the material balance,

*k*is usually chosen to be ),

**Eq. 3.21**may be transformed into the diffusion equation for an isotropic domain:

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

where

*η*is the diffusivity constant defined by

....................(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

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

Here, the pseudopressure is defined by

^{[1]}

....................(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,

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

The most common initial condition is the uniform pressure distribution in the entire porous medium; that is, *f* (*x*, *y*, *z*) = *p*_{i}.

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

....................(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,

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

When used at the inner boundary, this condition represents production at a constant pressure, *p*_{wf}; that is, *h*(*t*) = *p*_{wf}. At the outer boundary, specified pressure, *p*_{e}, 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

....................(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

....................(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

....................(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

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

where *A* and *B* are arbitrary constants, and *J*_{v}(*z*) is the Bessel function of order *v* of the first kind given by

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

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

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

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

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

In **Eq. 3.35**, *Y*_{n}(*z*) is the Bessel function of order *n* of the second kind and is defined by

....................(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

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

where *I*_{v}(*z*) is the modified Bessel function of order *v* of the first kind defined by

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

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

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

where *K*_{n}(*z*) is the modified Bessel function of order *n* of the second kind and is defined by

....................(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:....................(3.41)

and

....................(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]}

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

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

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

where

*γ*= 0.5772…, and

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

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

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

for |

*arg*

*z*| <

*π*/ 2, and

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

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:

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

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

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

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

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

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

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

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

and

....................(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

....................(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,

....................(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.**

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

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

....................(3.62)**Transforms of Integrals.**

....................(3.63)**Substitution.**

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

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

where .**Translation.**

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

where *H*(*t* - *t*_{0}) is Heaviside’s unit step function defined by

....................(3.67)**Convolution.**

....................(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,

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

In this operation,

*p*(

*t*) represents the inverse (transform) of the Laplace domain function, . 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,

*t*

_{i}, where they are defined by

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

and |

*F*(

*t*) | <

*Me*

^{αt}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,

*p*

_{a}(

*T*), of the inverse of the Laplace domain function, , may be obtained at time

*t*=

*T*by

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

where

....................(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 [

*p*

_{a}(

*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,

*p*

_{i}.

*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

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

where ∆

*p*=

*p*

_{i}–

*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

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

which means that the pressure is uniform and equal to

*p*

_{i}initially throughout the reservoir. The outer boundary condition for an infinite reservoir is

....................(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,

*p*

_{i}, has been preserved.

The inner boundary condition depends on the production conditions at the surface of the wellbore (

*r*=

*r*

_{w}). Assuming that the well is produced at a constant rate,

*q*, for all times,

....................(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

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

or, rearranging, we obtain

....................(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.

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

and

....................(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

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

The constants

*C*

_{1}and

*C*

_{2}in

**Eq. 3.81**are obtained from the boundary conditions. The outer boundary condition (

**Eq. 3.79**) indicates that

*C*

_{1}= 0 [because ,

**Eq. 3.79**is satisfied only if

*C*

_{1}= 0]; therefore,

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

From

**Eqs. 3.80**and

**3.82**, we obtain

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

which yields

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

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

....................(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

*r*

_{w}→0 and use the relation given in

**Eq. 3.56**, we obtain

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

Using this relation in

**Eq. 3.85**, we obtain the line-source solution in Laplace domain as

....................(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

....................(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:

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

Making the substitution

*u*= r

^{2}/ (4

*ηt′*) and noting the definition of the exponential integral function,

*Ei*(

*x*), given by

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

we obtain the line-source solution as

....................(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

....................(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,

*p*

_{i}.

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

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

The initial condition corresponding to the uniform pressure distribution equal to

*p*

_{i}is

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

and the inner boundary condition for a constant production rate,

*q*, for all times is

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

The closed outer boundary condition is represented mathematically by zero flux at the outer boundary (

*r*=

*r*

_{e}) that corresponds to

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

The Laplace transforms of

**Eqs. 3.93**through

**3.96**yield, respectively,

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

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

and

....................(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

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

With the outer boundary condition given by

**Eq. 3.99**, we obtain

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

which yields

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

and thus

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

Using the inner boundary condition given by

**Eq. 3.98**yields

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

From

**Eqs. 3.102**and

**3.104**, we obtain the coefficients

*C*

_{1}and

*C*

_{2}as follows:

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

and

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

Substituting

*C*

_{1}and

*C*

_{2}into

**Eq. 3.100**yields

....................(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.

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

In

**Eq. 3.108**,

*β*

_{1},

*β*

_{2}, etc. are the roots of

....................(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

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

where

*q*

_{sf}is the sandface production rate,

*p*(

*r*

_{w}+) denotes the reservoir pressure immediately outside the skin-zone boundary, and

*p*

_{wf}is the flowing wellbore pressure measured inside the wellbore. Rearranging

**Eq. 3.110**, we obtain the following relation for the flowing wellbore pressure.

....................(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,

*q*

_{wb}, and the sandface production rate,

*q*

_{sf}; that is,

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

where

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

and

....................(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.

....................(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:

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

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

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

and

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

The general solution of

**Eq. 3.116**is

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

The condition in

**Eq. 3.117**requires that

*C*

_{1}= 0; therefore,

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

The use of

**Eq. 3.121**in

**Eq. 3.119**yields

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

From

**Eqs. 3.118**,

**3.121**, and

**3.122**, we obtain

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

which yields

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

Substituting

**Eq. 3.124**for

*C*

_{2}in

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

....................(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 Δ

*p*

_{wf}. These derivatives are computed by applying the Laplace transformation property given in

**Eq. 3.60**to

**Eq. 3.125**as follows:

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

Here, we have used Δ

*p*

_{w f}(

*t*= 0) = 0. To obtain the logarithmic derivatives, we simply note that

....................(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

*t*

_{p}, the well is shut in and pressure buildup begins. The system of equations to define this problem is

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

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

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

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

where

*H*(

*t*-

*t*

_{p}) is Heaviside’s unit function (

**Eq. 3.67**), and

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

The right side of the boundary condition in

**Eq. 3.131**accounts for a constant surface production rate,

*q*, for 0 <

*t*<

*t*

_{p}and for shut in (

*q*= 0) for

*t*>

*t*

_{p}. Taking the Laplace transforms of

**Eqs. 3.128**through

**3.132**, we obtain

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

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

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

and

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

The general solution of

**Eq. 3.133**is

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

The condition in

**Eq. 3.134**requires that

*C*

_{1}= 0; therefore,

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

From

**Eqs. 3.138**and

**3.136**, we obtain

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

Substituting the results of

**Eqs. 3.138**and

**3.139**into

**Eq. 3.135**, we have

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

which yields

....................(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.

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

The term contributed by the discontinuity at time

*t*=

*t*

_{p}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

....................(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, *γ*_{f}(*M*, *M′*, *t*, *τ*), of the diffusion equation for fluid flow in porous media satisfies the following differential equation:

....................(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:

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

and

....................(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, Δ*p* = *p*_{i} − *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 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

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

Assume that the initial pressure drop satisfies

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

and we have the condition that

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

On substitution of

*u*=

*r*Δ

*p*,

**Eqs. 3.147**through

**3.149**become, respectively,

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

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

and

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

The solution of the problem described by

**Eqs. 3.150**through

**3.152**is given by

^{[14]}

....................(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

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

In

**Eq. 3.154**, 4

*πα*

^{3}/3=

*V*where

*V*is the volume of the spherical source. If 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*

_{i}, then . With the definition of compressibility,

*c*= -(1 /

*V*)(Δ

*V*/ Δ

*p*

_{i}), we obtain . Then, we can show that

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

Substituting

**Eq. 3.155**into

**Eq. 3.154**, we obtain

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

If we let the radius of the spherical source,

*a*, tend to zero while remains constant,

**Eq. 3.156**yields the point-source solution in spherical coordinates given by

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

This solution may be interpreted as the pressure drop at a distance

*r*because of a volume of fluid, , instantaneously withdrawn at

*r*= 0 and

*t*= 0. Consistent with this interpretation, 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, (see

**Eq. 3.155**).

*. Nisle*

**Instantaneous Point Source in an Infinite Reservoir**^{[21]}presented a more general solution for an instantaneous point source of strength acting at

*t*=

*τ*in an infinite, homogeneous, but anisotropic reservoir as

....................(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, with

*η*

_{x},

*η*

_{y}, and

*η*

_{z}representing the diffusivity constants (defined in

**Eq. 3.23**) in the

*x*,

*y*, and

*z*directions, respectively.

*. If the fluid withdrawal is at a continuous rate, , from time 0 to*

**Continuous Point Source in an Infinite Reservoir***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 over a time interval 0 ≤

*τ*≤

*t*. This is given by

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

where

*S*(

*M*,

*M′*,

*t*−

*τ*) corresponds to a unit-strength , instantaneous point source in an infinite reservoir; that is,

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

*. Similarly, the distribution of instantaneous point sources of strength over a line, surface, or volume, Γ*

**Instantaneous and Continuous Line, Surface, and Volumetric Sources in an Infinite Reservoir**_{w}, 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.

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

In

**Eq. 3.161**,

*M*

_{w}indicates a point on the source (Γ

_{w}) and 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:

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

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

....................(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

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

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

....................(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

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

*. 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*

**Source Functions for Bounded Reservoirs***f*

_{1}and

*f*

_{2}are two linearly independent solutions of a linear PDE and

*c*

_{1}and

*c*

_{2}are two arbitrary constants, then

*f*

_{3}=

*c*

_{1}

*f*

_{1}+

*c*

_{2}

*f*

_{2}is also a solution of the PDE. Examples of source functions in bounded reservoirs are presented here.

*. 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*

**Instantaneous Point Source Near a Single Linear Boundary****Fig. 3.7**), we can apply the method of images to the instantaneous point-source solution given in

**Eq. 3.157**as

....................(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:

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

*. Using the method of images and considering the geometry shown in Col. A of*

**Instantaneous Point Source in an Infinite-Slab Reservoir****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

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

which, with Poisson’s summation formula given by

^{[14]}

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

may be transformed into

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

Following similar lines, if the slab boundaries at

*z*= 0 and h are at a constant pressure equal to

*p*

_{i}, we obtain

....................(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

*p*

_{i}, the following solution may be derived:

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

*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 ≤*

**Instantaneous Point Source in a Closed Parallelepiped.***x*≤

*x*

_{e}, 0 ≤

*y*≤

*y*

_{e}, and 0 ≤

*z*≤

*h*) and the bounding planes are impermeable, then we can use

**Eq. 3.157**and the method of images to write

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

which, with Poisson’s summation formula (

**Eq. 3.170**), may be recast into the following form:

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

*The instantaneous point-source solutions of*

**Instantaneous Infinite-Plane Source in an Infinite-Slab Reservoir With Impermeable Boundaries.****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

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

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

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

*Following similar lines, we can obtain the solution for an instantaneous, infinite-slab source of thickness,*

**Instantaneous Infinite-Slab Source in an Infinite-Slab Reservoir With Impermeable Boundaries.***h*

_{p}, located at

*z*=

*z*

_{w}(

*z*

_{w}is the

*z*-coordinate of the midpoint of the slab source) in an infinite-slab reservoir with impermeable boundaries.

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

If we assume a uniform-flux slab source , then

**Eq. 3.178**yields

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

*Solutions for continuous plane and slab sources can be obtained as indicated by*

**Uniform-Flux, Continuous, Infinite-Slab Source in an Infinite-Slab Reservoir With Impermeable Boundaries.****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

....................(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,

*h*

_{w}, in an infinite, homogeneous, slab reservoir of uniform thickness,

*h*, and initial pressure,

*p*

_{i}, 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 ] for

*S*, we obtain

....................(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., , where

*q*is the constant production rate of the well], then

**Eq. 3.181**yields

....................(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′*, *t* − *τ*) denote the Green’s function, then it should satisfy the diffusion equation; that is,

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

Because *G* is a function of *t* − *τ*, it should also satisfy the adjoint diffusion equation,

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

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

*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, , except at the source location,*M*=*M′*, where it becomes infinite, so that*G*satisfies the delta function property,

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

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

*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:

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

and

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

Then, we can write

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

or

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

where

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

....................(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

....................(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

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

where

*p*

_{i}(

*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, Γ

_{w}, and the outer boundary of the reservoir, Γ

_{e}.

**Eq. 3.193**may be written as

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

As the fourth property of Green’s function noted previously requires, if the outer boundary of the reservoir is impermeable, or at infinity, then

*G*vanishes at the outer boundary; that is,

*G*(Γ

_{e}) = 0. Thus,

**Eq. 3.194**becomes

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

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

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

If the initial pressure,

*p*

_{i}, is uniform over the entire domain (porous medium),

*D*, then, by the third property of Green’s function (

**Eq. 3.186**), we should have

....................(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, Γ

_{w}, at time

*t*be equal to

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

The substitution of

**Eqs. 3.197**and

**3.198**into

**Eq. 3.196**yields

....................(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, , is uniformly distributed on the inner-boundary surface (wellbore), Γ

_{w}. This yields

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

where

....................(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′*,

*t*−

*τ*), may be obtained as the product of two infinite, 1D plane sources, given by

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

as follows

....................(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 (

*η*

_{x}=

*η*

_{y}=

*η*

_{z}),

**Eq. 3.203**yields

....................(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

....................(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

....................(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

*h*

_{f}and horizontal penetration 2

*x*

_{f}in an infinite, homogeneous, slab reservoir of uniform thickness,

*h*, and initial pressure,

*p*

_{i}, 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

*h*

_{f}and length 2

*x*

_{f}. 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*

_{f}at

*x*=

*x′*in an infinite reservoir (Source IV), and an infinite-slab source of thickness

*h*

_{p}=

*h*

_{f}at

*z*=

*z*

_{w}in an infinite-slab reservoir of thickness

*h*(Source VIII). Then, by Newman’s product method, the appropriate source function is given by

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

Assuming that the production is at a constant rate, and using

**Eq. 3.207**for the source function,

*S*, in

**Eq. 3.200**, we obtain

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

If the fracture penetrates the entire thickness of the reservoir (i.e.,

*h*

_{f}=

*h*) as shown in

**Fig. 3.12**, then

**Eq. 3.208**yields

....................(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 2

*x*

_{f}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

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

where

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

and

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

Assuming a constant production rate, , and substituting the source function given by

**Eq. 3.210**in

**Eq. 3.200**, we obtain

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

where erfc (

*z*) is the complementary error function defined by

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

**Example 3.7**

Consider transient flow toward a uniform-flux horizontal well of length

*L*

_{h}located at (

*x′*,

*y′*,

*z*

_{w}) in a closed, homogeneous, rectangular parallelepiped of dimensions 0 ≤

*x*≤

*x*

_{e}, 0 ≤

*y*≤

*y*

_{e}, 0 ≤

*z*≤

*h*and of initial pressure,

*p*

_{i}.

*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*

_{h}, 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*

_{h}at

*x*=

*x′*in an infinite-slab reservoir of thickness

*x*

_{e}with impermeable boundaries (Source VIII), and an infinite-plane source at

*z*=

*z*

_{w}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

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

Assuming that the production is at a constant rate, , and using

**Eq. 3.215**for the source function,

*S*, in

**Eq. 3.200**, we obtain

....................(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

....................(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

....................(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

*L*

_{h}at

*x*=

*x′*in an infinite reservoir (Source IV), and an infinite-plane source at

*z*=

*z*

_{w}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 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

....................(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

....................(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]}

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

where the subscript

*f*indicates the fracture property, and

*t*

_{D}and

*r*

_{D}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

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

The general solutions for

**Eqs. 3.221**and

**3.222**are given, respectively, by

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

and

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

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

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

and

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

Then, it may be shown that

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

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**.

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.

*Consider the flow of a slightly compressible fluid in an infinite, naturally fractured reservoir. We can use the double-porosity model suggested by Barenblatt*

**Point-Source Solution in the Laplace Domain.***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

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

and

....................(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,

*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

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

where

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

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

The definitions of the other variables used in

**Eqs. 3.228**and

**3.229**are

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

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

and

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

where

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

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

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

and

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

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

^{[29]}

....................(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 ≤

*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,

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

For the condition expressed in

**Eq. 3.239**to hold for every

*T*≥ 0, we must have

....................(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

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

The Laplace transform of

**Eqs. 3.228**,

**3.229**,

**3.237**, and

**3.241**yields

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

where

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

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

and

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

In deriving these results, we have used the initial condition given by

**Eq. 3.236**and noted that

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

In

**Eq. 3.245**, the term 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 acting at

*t*= 0:

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

If the source is located at

*x′*

_{D},

*y′*

_{D},

*z′*

_{D}, then, by translation, we can write

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

where

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

and

....................(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,

*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

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

The Laplace transform of

**Eq. 3.251**yields the following continuous point-source solution in an infinite reservoir:

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

where we have substituted

**Eq. 3.249**for

*S*, dropped the subscript

*f*, and defined

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

*The point-source solution in the Laplace domain may be used to obtain the source solutions for different source geometries. If we define*

**Line-, Surface-, and Volumetric-Source Solution in the Laplace Domain.**....................(3.254)

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

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

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

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

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.

*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*

**Solutions for Infinite-Slab Reservoirs.***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

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

where

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

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

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

and

....................(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]}

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

and recast

**Eq. 3.257**as

....................(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

*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.

^{[19]}

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

where

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

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}):

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

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:

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

In

**Eq. 3.267**, is a solution of

**Eq. 3.264**that satisfies

**Eq. 3.266**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 3.6**, 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. 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

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

and

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

According to the boundary condition given by

**Eq. 3.268**, we should choose 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*

_{0}(

*aR*

_{D}) given by

^{[14]}

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

where

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

we can write

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

for

*r*

_{D}<

*r′*

_{D}. If

*r*

_{D}>

*r′*

_{D}, we interchange

*r*

_{D}and

*r′*

_{D}in

**Eq. 3.272**. If we choose in

**Eq. 3.267**as

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

where

*a*

_{k}and

*b*

_{k}are constants, then satisfies the boundary condition given by

**Eq. 3.268**, 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. 3.273**as

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

and

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

then satisfies the impermeable boundary condition at

*r*

_{D}=

*r*

_{eD}given by

**Eq. 3.269**. Thus, the point-source solution for a closed cylindrical reservoir is given by

....................(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

*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. 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′*,

*z*

_{w}is obtained by integrating the corresponding point-source solution given in

**Table 3.7**with respect to

*z′*from

*z*

_{w}–

*h*

_{f}/ 2 to

*z*

_{w}+

*h*

_{f}/ 2 and is given by

....................(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

*x*

_{f}with

*θ*′ =

*α*+

*π*in the third quadrant and with

*θ*′ =

*α*in the first quadrant. This procedure yields

....................(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

*K*

_{0}(

*x*) given by

**Eq. 3.270**, the solution in

**Eq. 3.277**may be written as

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

where

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

and

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

The integration of the partially penetrating vertical well solution given in

**Eq. 3.279**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:

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

where

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

**Example 3.9** 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 3.7** 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

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

*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 (*

**Solutions for Rectangular Parallelepiped Reservoir.***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. 3.252**:

^{[19]}

^{[29]}

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

where

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

and

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

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

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

Ozkan

^{[29]}shows that the triple infinite sums in

**Eq. 3.285**may be reduced to double infinite sums with

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

where

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

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

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

where

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

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

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

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

and

....................(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 *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 3.8**, 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

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

where , , and *ε*_{k} are given respectively by **Eqs. 3.293**, **3.294**, and **3.296**.

**Example 3.11**

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 3.8**, with respect to *x′* from *x*_{w}–*L*_{h} /2 to *x*_{w}+*L*_{h} /2, and is given by

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

where

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

and

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

In **Eq. 3.301**, , , *ε*_{n}, *ε*_{k}, and *ε*_{k, 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

*k*

_{x},

*k*

_{y}, and

*k*

_{z}denoting the corresponding permeabilities. In these solutions, an equivalent isotropic permeability,

*k*, has been defined by

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

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

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

and replace

*k*in the solutions given in this section (

**Sec. 3.4.4**) by

*k*

_{h}. Note that

*k*takes place in the definition of the dimensionless time

*t*

_{D}(

**Eq. 3.230**). Then, if we define a dimensionless time based on

*k*

_{h}, the relation between and

*t*

_{D}is given by

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

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:

....................(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 (

*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 3.6**may be written as

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

In

**Eq. 3.306**,

*s*is the Laplace transform variable with respect to dimensionless time,

*t*

_{D}, based on

*k*and

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

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

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

and

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

If we define the following variables based on

*k*

_{h},

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

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

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

and also note that

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

then, we may rearrange

**Eq. 3.306**in terms of the dimensionless variables based on

*k*

_{h}as

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

where

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

and

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

If we compare

**Eqs. 3.306**and

**3.315**, we can show that

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

where we have used the relation given by

**Eq. 3.305**. If we now define as the Laplace transform variable with respect to , we may write

....................(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

*k*

_{h}:

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

*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.*

**Computational Considerations and Applications.**^{[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:

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

and

....................(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.*

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

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.

^{[19]}

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

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

and

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

The integrals in

**Eqs. 3.324**through

**3.326**may be evaluated with the standard numerical integration algorithms for

*y*

_{D}≠ 0. For

*y*

_{D}= 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**:

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

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

^{[19]}

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

and

....................(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 given, respectively, by

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

where

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

and

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

where

*γ*=0.5772… and

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

It is also useful to note the real inversions of

**Eqs. 3.330**and

**3.332**given, respectively, by

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

and

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

*The Series.*

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

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).

^{[29]}When

*u*is large,

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

and when

*u*+

*a*

^{2}<<

*n*

^{2}

*π*

^{2}/

*h*

^{2}

_{D},

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

*The Series*.

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

Alternative computational forms for the series

*S*

_{2}are given next.

^{[29]}When

*u*is large,

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

and when

*u*+

*a*

^{2}<<

*n*

^{2}

*π*

^{2}/

*h*

^{2}

_{D},

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

*The Series*.

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

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).

^{[29]}When

*u*is large,

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

and when

*u*+

*a*

^{2}<< (2

*n*− 1)

^{2}

*π*

^{2}/(4

*h*

^{2}

_{D}),

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

*The Series*.

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

where

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

The series may be written in the following forms with the use of

**Eqs. 3.324**through

**3.326**.

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

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

and

....................(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]}

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

where

....................(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 . When

*s*is large (small times), may be approximated by

^{[29]}

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

where

*β*is given by

**Eq. 3.331**. If s is sufficiently large, then

**Eq. 3.352**may be further approximated by

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

The inverse Laplace transform of

**Eq. 3.353**yields

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

For small

*s*(large times), depending on the value of

*x*

_{D}, may be approximated by one of the following equations:

^{[29]}

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

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

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

where is given by

**Eq. 3.364**.

*The Series* .

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

where

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

With the relations given in **Eqs. 3.337** and **3.338**, 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).^{[29]} When *u* is large,

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

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

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

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

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

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

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

The inverse Laplace transform of **Eq. 3.362** yields

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

For small *s* (large times), may be approximated by^{[29]}

....................(3.364)*The Ratio* .

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

By elementary considerations, the ratio may be written as^{[29]}

....................(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 *x*_{f} 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 = *x*_{f} and noting that , the solution becomes

....................(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 *x*_{D}.

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

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

and

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

The numerical evaluation of the integrals in **Eqs. 3.368** through **3.370** 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 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,

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

or, in real-time domain,

....................(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:

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

or, in real-time domain,

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

where *γ* = 0.5772… and *σ*(*x*_{D}, *y*_{D}, -1, +1) is given by **Eq. 3.333**.

**Example 3.13**

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 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*_{h} / 2 and noting that , the solution may be written as

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

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

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

with

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

....................(3.377) br>

and

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

The computation of the first term in the right side of **Eq. 3.375** 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 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*_{D} ≤ +1. In this case, from **Eqs. 3.350** and **3.351**, we have

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

where

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

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. 3.375**. To obtain a short-time approximation, we substitute the asymptotic expressions for and as s→∞ given, respectively, by **Eqs. 3.371** and **3.353**. This yields

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

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

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

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

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

where *γ*=0.5772… and *σ*(*x*_{D}, *y*_{D}, -1, +1) is given by **Eq. 3.333**. The inverse Laplace transform of **Eq. 3.383** yields

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

**Example 3.14** 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. 3.278** in **Example 3.8** with *h*_{w} = *h*. Choosing the characteristic length as ℓ = *x*_{f} and noting that , the solution is given by

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

For the computation of the pressure responses at the center of the fracture (*r*_{D} = 0), **Eq. 3.385** simplifies to

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

It is also possible to find a very good approximation for **Eq. 3.385**, especially when *r*_{eD} is large. If we assume^{[19]}

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

and use the following relation^{[4]}

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

then **Eq. 3.385** may be written as

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

Because^{[19]}

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

where

....................(3.391)**Eq. 3.389** may also be written as

....................(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 *r*_{eD} is large. For a fracture at the center of the cylindrical drainage region, **Eq. 3.392** simplifies to

....................(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 (*u*→*s*). As shown in **Sec. 3.2.3**, for small arguments we may approximate the Bessel functions in **Eq. 3.393** by

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

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

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

and

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

where *γ* = 0.5772…. With **Eqs. 3.394** through **3.397** and by neglecting the terms of the order *s*^{3/2}, we may write^{[29]}

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

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

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

where *σ*(*x*_{D}, *y*_{D}, −1, +1) is given by **Eq. 3.333** and

....................(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:

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

**Example 3.15**

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 3.10** and, by choosing ℓ = *x*_{f}, is given by

....................(3.402)

where

....................(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]}

....................(3.404)

where

....................(3.405)

....................(3.406)

and

....................(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

....................(3.408)

where

....................(3.409)

and

....................(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*):

....................(3.411)

where 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 represents the contribution of the lateral boundaries and is given by

....................(3.412)

In Eq. 3.412, , , and 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 term in **Eq. 3.411** becomes negligible compared with the 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]}

....................(3.413)

where

....................(3.414)

and

....................(3.415)

The second equality in Eq. 3.414 results from

....................(3.416)

For small values of *s*, replacing *u* by *s* and *s* + *α* by *α*, and with

....................(3.417)

the term *H* given by **Eq. 3.414** may be approximated by

....................(3.418)

The long-time approximation of the second term in **Eq. 3.413** 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

....................(3.419)

**Example 3.16** 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 3.11** and, by choosing ℓ = *L*_{h} / 2, is given by

....................(3.420)

where is the solution discussed in **Example 3.15**, and is given by

....................(3.421)

In **Eq. 3.421**, and are given by **Eqs. 3.377** and **3.378**, respectively,

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

and

....................(3.422)

The computation and the asymptotic approximations of the term have been discussed in **Example 3.15**. To compute the 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 term in **Eq. 3.421** may be written as

....................(3.423)

where

....................(3.424)

....................(3.425)

....................(3.426)

....................(3.427)

and

....................(3.428)

The computational form of the 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*_{D} ≤ +1 and *y*_{D} = *y*_{wD} given by

....................(3.429)

where

....................(3.430)

which can be written as follows by using the relation given in **Eq. 3.337**:

....................(3.431)

Similarly, for −1 ≤ *x*_{D} ≤ +1 and *y*_{D} = *y*_{wD}, the term given in **Eq. 3.428** may be written as

....................(3.432)

where

....................(3.433)

**Example 3.15** discussed the short- and long-time approximations of the term in **Eq. 3.420**. A small-time approximation for given by **Eq. 3.423** is obtained with *u* = *ωs* and by noting that as *s*→∞, . Then, substituting the short-time approximations for and given, respectively, by **Eqs. 3.371** and **3.353** into **Eq. 3.420**, the following short-time approximation is obtained: ^{[27]}

....................(3.434)

where *β* is given by **Eq. 3.331**. The inverse Laplace transform of **Eq. 3.434** yields

....................(3.435)

The long-time approximation of **Eq. 3.420** is obtained by substituting the long-time approximations of and . The long time-approximation of is obtained in **Example 3.15** (see **Eq. 3.413** through **3.419**). The long-time approximation of is obtained by evaluating the right side of **Eq. 3.421** as *s*→0 (*u*→0) and is given by

....................(3.436)

where

....................(3.437)

and

....................(3.438)

Thus, the long-time approximation **Eq. 3.420** is given by

....................(3.439)

where *p*_{Df} and *F*_{1} are given, respectively, by **Eqs. 3.419** and **3.436**. For computational purposes, however, *F*_{1} may be replaced by

....................(3.440)

In **Eq. 3.440**, *F*, *F*_{b1}, *F*_{b2}, and *F*_{b3} are given, respectively, by

....................(3.441)

....................(3.442)

....................(3.443)

and

....................(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

## Subscripts and Superscripts

## References

- ↑ 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
- ↑
^{2.0}^{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. - ↑ 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.
- ↑
^{4.0}^{4.1}Forchheimer, P.F. 1901. Wasserbewegung durch Boden. Zeitschrift des Vereines deutscher Ingenieure 45 (5): 1781–1788. - ↑
^{5.0}^{5.1}Watson, G. N. 1944. A Treatise on the Theory of Bessel Functions. London: Cambridge University Press. - ↑
^{6.0}^{6.1}^{6.2}^{6.3}Churchill, R.V. 1972. Operational Mathematics, Vol. 2. New York: McGraw-Hill Book Co. - ↑
^{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. - ↑
^{8.0}^{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 - ↑ 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
- ↑
^{10.0}^{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. - ↑ 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
- ↑ 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
- ↑
^{13.0}^{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 - ↑
^{14.0}^{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. - ↑
^{15.0}^{15.1}Stakgold, I. 1979. Green’s Functions and Boundary Value Problems, 86–104. New York: John Wiley & Sons. - ↑
^{16.0}^{16.1}Kelvin, W.T. 1884. Mathematical and Physical Papers, Vol. 2, 41. Cambridge, UK: Cambridge University Press. - ↑
^{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 - ↑
^{18.0}^{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 - ↑
^{19.00}^{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. - ↑
^{20.0}^{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. - ↑ 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.
- ↑
^{22.0}^{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 - ↑ 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.
- ↑ 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
- ↑
^{25.0}^{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 - ↑
^{26.0}^{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 - ↑
^{27.0}^{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 - ↑
^{28.0}^{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 - ↑
^{29.00}^{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 | = | cm^{2} |

°F | (°F−32)/1.8 | = | °C | ||

ft | × | 3.048* | E – 01 | = | m |

*