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Green’s function for solving transient flow problems
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.
In 1973, Gringarten and Ramey^{[1]} 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.^{[2]}
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In our terminology, a source is a point, line, surface, or volume at which fluids are withdrawn from the reservoir.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]}^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}
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:
where δ(M, M′, t, τ) is a generalized (symbolic) function^{[5]} called the Dirac delta function and is defined on the basis of its following properties:
and
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this source corresponds to an infinitesimally small well (located at point M′) at which a finite amount of fluid is withdrawn (or injected) instantaneously (at time τ).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. 1 (the fundamental solution) is also known as the instantaneous point-source solution. Formally, the point-source solution corresponds to the pressure drop, Δp = 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.
Green's function
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.^{[1]}^{[4]} If we let G(M, M′, t − τ) denote the Green’s function, then it should satisfy the diffusion equation; that is,
Because G is a function of t − τ, it should also satisfy the adjoint diffusion equation,
Green’s function also has the following properties: ^{[1]}^{[4]}
1. 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. 2. 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,
where D indicates the domain of the porous medium, and φ(M) is any continuous function. 3. Because G corresponds to the pressure because of an instantaneous point source of unit strength, it satisfies
4. 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′→∞.
Nomenclature
D | = | domain |
M | = | point in space |
M′ | = | source point in space |
t | = | time, s |
Γ(x) | = | Gamma function |
γ _{f} | = | fundamental solution of diffusion equation |
δ(x) | = | Dirac delta function |
η | = | diffusivity constant in i direction, i = x, y, z, or r |
τ | = | time, s |
φ(M) | = | any continuous function |
References
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} 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
- ↑ 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
- ↑ Raghavan, R. 1993. Well Test Analysis, 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.
- ↑ ^{4.0} ^{4.1} ^{4.2} Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, second edition, 353–386. Oxford, UK: Oxford University Press.
- ↑ ^{5.0} ^{5.1} Stakgold, I. 1979. Green’s Functions and Boundary Value Problems, 86–104. New York: John Wiley & Sons.
- ↑ Kelvin, W.T. 1884. Mathematical and Physical Papers, Vol. 2, 41. Cambridge, UK: Cambridge University Press.
- ↑ 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.
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See also
Source function solutions of the diffusion equation
Solving unsteady flow problems with Green's and source functions
Transient analysis mathematics
Differential calculus refresher
PEH:Mathematics_of_Transient_Analysis