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 for the solution of heat conduction problems and was extensively discussed by Carslaw and Jaeger. 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In our terminology, a source is a point, line, surface, or volume at which fluids are withdrawn from the reservoir.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.

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

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

Assume that the initial pressure drop satisfies

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

and we have the condition that

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

On substitution of u = rΔp, Eqs. 1 through 3 become, respectively,

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

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

and

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

The solution of the problem described by Eqs. 4 through 6 is given by

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

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

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

In Eq. 8, 4πα³/3=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, ΔV, which is caused by the change in pressure, Δp_i, then . With the definition of compressibility, c = -(1 / V)(ΔV / Δp_i), we obtain . Then, we can show that

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

Substituting Eq. 9 into Eq. 8, we obtain

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

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

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

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 presented a more general solution for an instantaneous point source of strength acting at t = τ in an infinite, homogeneous, but anisotropic reservoir as

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

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.

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

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

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

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

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

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.

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

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:

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

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

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

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

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

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

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

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

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

Source functions for bounded reservoirs

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

Instantaneous point source near a single linear boundary

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

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

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:

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

• 

  Fig. 1 – Application of the method of images to generate the effect of a linear boundary.

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

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

which, with Poisson’s summation formula given by

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

may be transformed into

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

• 

  Fig. 2 – Application of the method of images to generate the solutions for infinite-slab reservoirs.

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

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

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:

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

• 

  Fig. 3 – Geometry of the well/reservoir system for a partially penetrating vertical well in an infinite-slab reservoir with impermeable boundaries for Example 1.

Nomenclature

a	=	radius of the spherical source, L
c	=	fluid compressibility, atm−1
cf	=	formation compressibility, atm−1
ct	=	total compressibility, atm−1
d	=	distance to a linear boundary, cm
h	=	formation thickness, cm
hp	=	slab thickness, cm
hw	=	well length (penetration), cm
k	=	isotropic permeability, md
M	=	point in space
M′	=	source point in space
Mw	=	point in Γw
M′w	=	source point in Γw
p	=	pressure, atm
pi	=	initial pressure, atm
	=	instantaneous production rate for a point source, cm3/s
r	=	radial coordinate and distance, cm
r′	=	source coordinate in r-direction, cm
S	=	source function
t	=	time, s
u	=	s f(s)
V	=	volume, cm3
x	=	distance in x-direction, cm
x′	=	source coordinate in x-direction, cm
xe	=	distance to the external boundary in x-direction, cm
y	=	distance in y-direction, cm
y′	=	source coordinate in y-direction, cm
ye	=	distance to the external boundary in y-direction, cm
z	=	distance in z-direction, cm
z′	=	source coordinate in z-direction, cm
zw	=	well coordinate in z-direction, cm
Γw	=	length, surface, or volume of the source
Δ	=	difference operator
η	=	diffusivity constant
ηi	=	diffusivity constant in i direction, i = x, y, z, or r
τ	=	time, s
Φ	=	porosity, fraction

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.

External links

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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

Mathematics of fluid flow

Differential calculus refresher

PEH:Mathematics_of_Transient_Analysis