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Green’s function for solving transient flow problems: Difference between revisions
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Green’s function and [[ | Green’s function and [[Source_function_solutions_of_the_diffusion_equation|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<ref name="r1" /> 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.<ref name="r2" /> | In 1973, Gringarten and Ramey<ref name="r1">_</ref> 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.<ref name="r2">_</ref> | ||
== 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In our terminology, a source is a point, line, surface, or volume at which fluids are withdrawn from the reservoir.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. | In our terminology, a source is a point, line, surface, or volume at which fluids are withdrawn from the reservoir.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. | 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.<ref name="r1" /><ref name="r3" /><ref name="r4" /><ref name="r5" /><ref name="r6" /><ref name="r7" /><ref name="r8" /><ref name="r9" /><ref name="r10" /> | 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.<ref name="r1">_</ref><ref name="r3">_</ref><ref name="r4">_</ref><ref name="r5">_</ref><ref name="r6">_</ref><ref name="r7">_</ref><ref name="r8">_</ref><ref name="r9">_</ref><ref name="r10">_</ref> | ||
== Fundamental solution of the diffusion equation == | |||
The fundamental solution, ''γ''<sub>''f''</sub>(''M'', ''M′'', ''t'', ''τ''), of the diffusion equation for fluid flow in porous media satisfies the following differential equation: | The fundamental solution, ''γ''<sub>''f''</sub>(''M'', ''M′'', ''t'', ''τ''), of the diffusion equation for fluid flow in porous media satisfies the following differential equation: | ||
[[File:Vol1 page 0101 eq 001.png]]....................(1) | [[File:Vol1 page 0101 eq 001.png|RTENOTITLE]]....................(1) | ||
where ''δ''(''M'', ''M′'', ''t'', ''τ'') is a generalized (symbolic) function<ref name="r5" /> called the Dirac delta function and is defined on the basis of its following properties: | where ''δ''(''M'', ''M′'', ''t'', ''τ'') is a generalized (symbolic) function<ref name="r5">_</ref> called the Dirac delta function and is defined on the basis of its following properties: | ||
[[File:Vol1 page 0101 eq 002.png]]....................(2) | [[File:Vol1 page 0101 eq 002.png|RTENOTITLE]]....................(2) | ||
and | and | ||
[[File:Vol1 page 0101 eq 003.png]]....................(3) | [[File:Vol1 page 0101 eq 003.png|RTENOTITLE]]....................(3) | ||
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, Δ''p'' = ''p''<sub>''i''</sub> − ''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 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, Δ''p'' = ''p''<sub>''i''</sub> − ''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 == | ||
[[File:Vol1 page 0110 eq 002.png]]....................(4) | 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.<ref name="r1">_</ref><ref name="r4">_</ref> If we let ''G''(''M'', ''M′'', ''t'' − ''τ'') denote the Green’s function, then it should satisfy the diffusion equation; that is, | ||
[[File:Vol1 page 0110 eq 002.png|RTENOTITLE]]....................(4) | |||
Because ''G'' is a function of ''t'' − ''τ'', it should also satisfy the adjoint diffusion equation, | Because ''G'' is a function of ''t'' − ''τ'', it should also satisfy the adjoint diffusion equation, | ||
[[File:Vol1 page 0110 eq 003.png]]....................(5) | [[File:Vol1 page 0110 eq 003.png|RTENOTITLE]]....................(5) | ||
Green’s function also has the following properties: <ref name="r1" /><ref name="r4" /> | Green’s function also has the following properties: <ref name="r1">_</ref><ref name="r4">_</ref> | ||
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. | 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, [[File:Vol1 page 0110 inline 001.png|RTENOTITLE]], except at the source location, ''M'' = ''M′'', where it becomes infinite, so that ''G'' satisfies the delta function property, | ||
2. As ''t'' → ''τ'', ''G'' vanishes at all points in the porous medium; that is, [[File:Vol1 page 0110 inline 001.png]], except at the source location, ''M'' = ''M′'', where it becomes infinite, so that ''G'' satisfies the delta function property, | |||
[[File:Vol1 page 0110 eq 004.png]]....................(6) | [[File:Vol1 page 0110 eq 004.png|RTENOTITLE]]....................(6) | ||
where ''D'' indicates the domain of the porous medium, and ''φ''(''M'') is any continuous function. | 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 | ||
3. Because ''G'' corresponds to the pressure because of an instantaneous point source of unit strength, it satisfies | |||
[[File:Vol1 page 0110 eq 005.png]]....................(7) | [[File:Vol1 page 0110 eq 005.png|RTENOTITLE]]....................(7) | ||
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′''→∞. | 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== | == Nomenclature == | ||
{| | {| | ||
|- | |- | ||
|'' | | ''D'' | ||
|= | | = | ||
| | | domain | ||
|- | |- | ||
|'' | | ''M'' | ||
|= | | = | ||
| | | point in space | ||
|- | |- | ||
|'' | | ''M′'' | ||
|= | | = | ||
| | | source point in space | ||
|- | |- | ||
| | | ''t'' | ||
|= | | = | ||
| | | time, s | ||
|- | |- | ||
|'' | | Γ(''x'') | ||
|= | | = | ||
| | | Gamma function | ||
|- | |- | ||
|'' | | ''γ ''<sub>''f''</sub> | ||
|= | | = | ||
| | | 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== | == References == | ||
<references | |||
<references /> | |||
== Noteworthy papers in OnePetro == | |||
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read | Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read | ||
==External links== | == External links == | ||
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro | Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro | ||
==See also== | == See also == | ||
[[Source function solutions of the diffusion equation]] | |||
[[Source_function_solutions_of_the_diffusion_equation|Source function solutions of the diffusion equation]] | |||
[[Solving_unsteady_flow_problems_with_Green's_and_source_functions|Solving unsteady flow problems with Green's and source functions]] | |||
[[ | [[Transient_analysis_mathematics|Transient analysis mathematics]] | ||
[[ | [[Mathematics_of_fluid_flow|Mathematics of fluid flow]] | ||
[[ | [[Differential_calculus_refresher|Differential calculus refresher]] | ||
[[ | [[PEH:Mathematics_of_Transient_Analysis]] | ||
[[ | [[Category:5.6.3 Pressure Transient analysis]] | ||
Revision as of 15:50, 3 June 2015
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, γf(M, M′, t, τ), of the diffusion equation for fluid flow in porous media satisfies the following differential equation:
....................(1)
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:
....................(2)
and
....................(3)
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 = pi − 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,
....................(4)
Because G is a function of t − τ, it should also satisfy the adjoint diffusion equation,
....................(5)
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,
....................(6)
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
....................(7)
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
Noteworthy papers in OnePetro
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read
External links
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro
See also
Source function solutions of the diffusion equation
Solving unsteady flow problems with Green's and source functions
Transient analysis mathematics