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# Source function solutions of the diffusion equation

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.

The point-source solution was first introduced by Lord Kelvin^{[1]} for the solution of heat conduction problems and was extensively discussed by Carslaw and Jaeger.^{[2]} 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.

## 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In our terminology, a source is a point, line, surface, or volume at which fluids are withdrawn from the reservoir.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.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}

## Point-source derivation

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

Assume that the initial pressure drop satisfies

and we have the condition that

On substitution of *u* = *r*Δ*p*, **Eqs. 1** through **3** become, respectively,

and

The solution of the problem described by **Eqs. 4** through **6** is given by^{[2]}

If we expand the exponential terms in the integrand in **Eq. 7** in powers of *r′* and neglect the terms with powers higher than four, we obtain

In **Eq. 8**, 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

Substituting **Eq. 9** into **Eq. 8**, we obtain

If we let the radius of the spherical source, *a*, tend to zero while remains constant, **Eq. 10** yields the point-source solution in spherical coordinates given by

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. 9**).

## Instantaneous point source in an infinite reservoir

Nisle^{[10]} presented a more general solution for an instantaneous point source of strength acting at *t* = *τ* in an infinite, homogeneous, but anisotropic reservoir as

In **Eq. 12**, *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. 13**) in the *x*, *y*, and *z* directions, respectively.

## Continuous point source in an infinite reservoir

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

where *S*(*M*, *M′*, *t*−*τ*) corresponds to a unit-strength , instantaneous point source in an infinite reservoir; that is,

## Instantaneous and continuous line, surface, and volumetric sources in an infinite reservoir

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

In **Eq. 16**, *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:

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

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

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

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

## Source functions for bounded reservoirs

The source solutions discussed previously can be extended to bounded reservoirs. The method of images provides a convenient means of generating the bounded reservoir solutions with the use of the infinite reservoir solutions, especially when the reservoir boundaries consist of impermeable and constant pressure planes. The method of images is an application of the principle of superposition, which states that if *f*_{1} and *f*_{2} are two linearly independent solutions of a linear partial differential equation (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.

### Instantaneous point source near a single linear boundary

To generate the effect of an impermeable planar boundary at a distance d from a unit-strength, instantaneous point source in an infinite reservoir (see **Fig. 1**), we can apply the method of images to the instantaneous point-source solution given in **Eq. 11** as

It can be shown from **Eq. 22** 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. 15**) as follows:

### Instantaneous point source in an infinite-slab reservoir

Using the method of images and considering the geometry shown in Col. A of **Fig. 2**, 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

which, with Poisson’s summation formula given by^{[2]}

may be transformed into

Following similar lines, if the slab boundaries at *z* = 0 and h are at a constant pressure equal to *p*_{i}, we obtain

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:

### Instantaneous point source in a closed parallelepiped

The ideas used previously for slab reservoirs may be used to develop solutions for reservoirs bounded by planes in all three directions. For example, if the reservoir is bounded in all three directions (i.e., 0 ≤ *x* ≤ *x*_{e}, 0 ≤ *y* ≤ *y*_{e}, and 0 ≤ *z* ≤ *h*) and the bounding planes are impermeable, then we can use **Eq. 11** and the method of images to write

which, with Poisson’s summation formula (**Eq. 25**), may be recast into the following form:

### Instantaneous infinite-plane source in an infinite-slab reservoir with impermeable boundaries

The instantaneous point-source solutions of **Eqs. 26** through **28** may be extended to different source geometries with **Eq. 16**. 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. 26** for *S* in **Eq. 16**. This yields

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

### Instantaneous infinite-slab source in an infinite-slab reservoir with impermeable boundaries

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

If we assume a uniform-flux slab source , then **Eq. 33** yields

### Uniform-flux, continuous, infinite-slab source in an infinite-slab reservoir with impermeable boundaries

Solutions for continuous plane and slab sources can be obtained as indicated by **Eq. 14** (or **Eq. 21**). 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. 34** for *S* in **Eq. 14** and is given by

The same solution could have been obtained by substituting the unit-strength instantaneous point-source solution given by **Eq. 26** for *S* in **Eq. 21**.

## Example 1

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** 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. 21** and substituting the unit-strength, instantaneous point-source solution in an infinite-slab reservoir with impermeable boundaries [**Eq. 26** with ] for *S*, we obtain

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. 36** yields

## Nomenclature

## References

- ↑
^{1.0}^{1.1}Kelvin, W.T. 1884. Mathematical and Physical Papers, Vol. 2, 41. Cambridge, UK: Cambridge University Press. - ↑
^{2.0}^{2.1}^{2.2}^{2.3}Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, second edition, 353–386. Oxford, UK: Oxford University Press. - ↑ Raghavan, R. 1993. Well Test Analysis, 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.
- ↑ 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
- ↑ Stakgold, I. 1979. Green’s Functions and Boundary Value Problems, 86–104. New York: John Wiley & Sons.
- ↑ 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
- ↑ 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
- ↑ 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.
- ↑ 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.

## Noteworthy papers in OnePetro

Chen, H.Y., Poston, S.W., and Raghavan, R. An Application of the Product Solution Principle for Instantaneous Source and Green's Functions. http://dx.doi.org/10.2118/20801-PA.

Gringarten, A.C. and Ramey, H.J., Jr. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. http://dx.doi.org/10.2118/3818-PA.

Ozkan, E. and Raghavan, R. New Solutions for Well-Test-Analysis Problems: Part 1-Analytical Considerations(includes associated papers 28666 and 29213 ). http://dx.doi.org/10.2118/18615-PA.

Chen, H.Y., Poston, S.W., and Raghavan, R. An Application of the Product Solution Principle for Instantaneous Source and Green's Functions. http://dx.doi.org/10.2118/20801-PA.

Gringarten, A.C. and Ramey, H.J., Jr. The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs. http://dx.doi.org/10.2118/3818-PA.

Ozkan, E. and Raghavan, R. New Solutions for Well-Test-Analysis Problems: Part 1-Analytical Considerations(includes associated papers 28666 and 29213 ). http://dx.doi.org/10.2118/18615-PA.

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## See also

Transient analysis mathematics

Laplace transformation for solving transient flow problems

Green’s function for solving transient flow problems

Solving unsteady flow problems with Green's and source functions

Solving unsteady flow problems with Laplace transform and source functions