CN113033057A - Method, equipment and storage medium for realizing numerical simulation of underground fluid flow based on crack porous medium fluid mathematical model - Google Patents

Method, equipment and storage medium for realizing numerical simulation of underground fluid flow based on crack porous medium fluid mathematical model Download PDF

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CN113033057A
CN113033057A CN202110387105.7A CN202110387105A CN113033057A CN 113033057 A CN113033057 A CN 113033057A CN 202110387105 A CN202110387105 A CN 202110387105A CN 113033057 A CN113033057 A CN 113033057A
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赵烜
芮洪兴
赵继菁
李晓丽
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Abstract

The invention relates to a method, equipment and a storage medium for realizing numerical simulation of underground fluid flow based on a crack porous medium fluid mathematical model, which comprises the following steps: (1) constructing a fracture porous medium fluid mathematical model; according to the characteristics of the porous medium of the crack, a set of partial differential equations describing the fluid flow is provided. (2) And (2) realizing the numerical simulation of the flow of the underground fluid through the fracture porous medium fluid mathematical model constructed in the step (1). The model provided by the invention comprehensively considers the physical phenomena involved in the flowing process, represents the physical phenomena through the mathematical model, more accurately describes the real physical process, and effectively relieves the risk of poor numerical simulation effect caused by inaccurate model.

Description

Method, equipment and storage medium for realizing numerical simulation of underground fluid flow based on crack porous medium fluid mathematical model
Technical Field
The invention relates to a method, equipment and a storage medium for realizing numerical simulation of underground fluid flow based on a crack porous medium fluid mathematical model, and belongs to the technical field of intersection of mathematics and hydromechanics.
Background
In the process of processing the numerical simulation of the fluid flow of the porous medium of the fracture, one technology is to take the fracture area as a common area and not to process the fracture independently, and the technology is called as an 'un-simplified' model technology, so that the calculated amount is more complex and the actual application effect is poorer. Since the width of the crack is usually much smaller than the entire area, dimensionality reduction becomes a more effective technique in dealing with the crack problem, and is also the more popular technique at present. "Modeling fractions and barriers as interfaces for flow in pore media" published by SIAM Journal on Scientific Computing in 2005 and "Modeling fractions as interfaces with non-matching grains" published by n.frih et al in practical Geosciences in 2012 propose treatment models of porous medium fluids for fractures, both of which are subjected to dimension reduction and simplification treatment by the idea of "averaging", and model establishment assumes that the permeability at the fracture is greater than that at the non-fracture. Frih et al, published by Computational Geosciences, "Modeling as interfaces: a model for the forcemeter reactors", in 2008, consider the coupling model of Darcy-forcemeter flow at fractures and Darcy flow at non-fractures. The above studies have only been directed to fracture models of the flow equation. Chave et al, who published in GEM-International Journal on Geomatics, "A hybrid high-order method for passive transport in a fragmented porous medium", studied the passive transport (solute concentration gradient transport) model of solute in the fracture porous medium, discussed the relationship between fluid and concentration, and proposed the coupling conditions of the fracture and the peripheral region, which could catch the concentration discontinuity at the fracture.
However, the equation in the prior art does not relate to viscosity, the concentration equation does not consider time dimension, the viscosity is an important physical quantity in practical problems (especially reservoir numerical calculation), the time dimension is also essential in history fitting and prediction, the flow state of the fluid cannot be accurately described after the two physical quantities are ignored, and certain limitations exist in practical application.
The flow property description of the fracture porous medium fluid is an important research subject in mathematics and fluid mechanics, an accurate mathematical model is built, and a numerical solution is established for solving, so that the numerical simulation precision can be improved. The method has great practical value in the field of underground fluid flow numerical simulation, particularly petroleum exploration, underground water and the like.
Disclosure of Invention
The invention provides a method for realizing underground fluid flow numerical simulation based on a fracture porous medium fluid mathematical model, aiming at describing fluid flow properties more accurately.
Interpretation of terms:
1. a crack region: the fluid flow area contains cracks, and the fluid has different physical properties when flowing through the area, generally expressed as accelerated flow rate and larger numerical simulation difficulty.
2. Non-crack region: and no crack exists in the fluid flow area, and the fluid flows slowly in the area, so that the simulation calculation is easy.
3. And (4) source and sink items: foreign injection or outflow during fluid flow. For example: water injection wells in oil exploration can be considered sources and production wells sink.
4. Permeability tensor: permeability refers to the ability of a rock to allow fluid to pass through, driven by a certain pressure difference, usually expressed in tensor form, which is the ability of the earth or rock itself to transport liquid.
5. Tangential permeability: value of permeability along the tangential direction.
6. Viscosity: when fluid flows under the action of external force, internal friction force is generated between molecules.
7. The interaction between the fractured zone and the non-fractured zone means that when fluid flows in the fractured zone and the non-fractured zone, the fluid affects the interface position of the two zones.
8. Diffusion-diffusion coefficient: indicating the ability of the fluid to diffuse and disperse while flowing.
The technical scheme of the invention is as follows:
a method for realizing numerical simulation of underground fluid flow based on a fracture porous medium fluid mathematical model comprises the following steps:
(1) constructing a fracture porous medium fluid mathematical model; according to the characteristics of the porous medium of the crack, a set of partial differential equations describing the fluid flow is provided.
(2) And (2) realizing the numerical simulation of the flow of the underground fluid through the fracture porous medium fluid mathematical model constructed in the step (1).
Preferably, in step (1), the fracture porous medium fluid mathematical model is constructed by:
the fracture porous medium fluid mathematical model comprises a fracture porous medium fluid mathematical model describing pressure and speed and a fracture porous medium fluid mathematical model describing concentration;
since the flow properties of the non-fracture region are different from the fracture region and the non-fracture boundary is different from the fracture boundary, different models are required for description.
For a non-fracture area, a fracture porous medium fluid mathematical model for describing pressure and speed is shown as a formula (I) and a formula (II):
divui＝fi (1)
Figure BDA0003011672960000021
in the formulae (I) and (II), i is 1 or 2 and denotes the number of the non-fractured region, fiSource and sink terms (usually known quantities), u, representing non-fractured regionsi、piRespectively representing the velocity, pressure, K, of the non-fractured zoneiThe permeability tensor, which represents the non-fractured region, is usually assumed to be a diagonal positive definite matrix,
Figure BDA00030116729600000314
is adhesive; div denotes divergence, which means that the derivatives are taken for each component of the vector and added;
for the fracture area, a fracture porous medium fluid mathematical model for describing pressure and velocity is shown as formula (III) and formula (IV):
Figure BDA0003011672960000031
diVτuτ＝fγ+(u1·n1|γ+u2·n2|γ) (IV)
in the formulae (III) and (IV), urRepresenting the velocity of the fluid in the fracture zone, u1Representing the velocity, u, of the non-fractured zone fluid numbered 12Representing the velocity, p, of the non-fractured zone fluid numbered 2γRepresenting the pressure of the fluid in the fracture zone,
Figure BDA00030116729600000311
denotes the permeability in the crack direction, μγRepresenting the viscosity of the fluid in the fracture zone; f. ofγRepresenting the acting force of the outside on the crack area;
the interaction between fracture and non-fracture zones the interface conditions for the mathematical model coupling of the fracture porous medium fluid describing pressure and velocity are shown in equation (v):
-ξui·ni+αγpi＝αγpγ-(1-ξ)ui+1·ni+1 (V)
in formula (V), i is 1, 2, n1、n2Is the unit external normal vector, u, of the fracture1Representing the velocity, u, of the non-fractured zone fluid numbered 12Representing the velocity, p, of the non-fractured zone fluid numbered 1γThe fluid pressure in the region of the fracture is represented,
Figure BDA00030116729600000312
denotes normal permeability, r denotes the width of the crack, ξ is a constant greater than 1/2;
for non-fractured regions, a fracture porous medium fluid mathematical model describing the concentration is shown as formula (VI):
Figure BDA0003011672960000033
in the formula (VI), ciDenotes the concentration of non-fractured regions, DiThe diffusion dispersion coefficient of the non-fractured region is expressed,
Figure BDA00030116729600000315
negative and positive values of the sink term for the non-fractured region are indicated,
Figure BDA0003011672960000034
represents the porosity of the rock in the non-fractured region, and t represents time;
for fracture zones, a fracture porous medium fluid mathematical model describing the concentration is shown as formula (VII):
Figure BDA0003011672960000035
in the formula (VII), the reaction mixture is,
Figure BDA0003011672960000036
the porosity of the rock in the fracture zone is indicated,
Figure BDA0003011672960000037
denotes diffusion coefficient in crack direction, cγThe concentration of the region of the crack is indicated,
Figure BDA0003011672960000038
negative and positive values of the sink term for the non-fractured region are indicated,
Figure BDA00030116729600000310
the representative number isThe difference between the crack and the non-cracked region 1 and the non-cracked region 2, r represents the width of the crack and is usually a small constant, cγThe concentration of the crack region is shown, D is the diffusion coefficient, and u is the velocity;
the interface conditions for the interaction between the fracture and the non-fracture coupled with respect to the concentration of the fracture porous medium fluid mathematical model are shown as formula (VIII) and formula (IX):
Figure BDA0003011672960000041
Figure BDA0003011672960000042
in the formulae (VIII) and (IX),
Figure BDA0003011672960000046
indicating the normal diffusion coefficient.
The two sets of equations are combined to form the model of the invention, the flow properties of the underground fluid are comprehensively considered, and compared with the prior art, the accuracy and universality of numerical simulation by using the method are higher.
According to the invention, the specific implementation process of the step (2) comprises the following steps:
and (3) discretizing the crack porous medium fluid mathematical model in each small region by using a mass conservation type difference method to finally form a large linear equation set to obtain a discrete solution of the original problem.
According to the position of each physical quantity in each subdivision unit, the invention provides a numerical discrete solving method, which comprises the following steps:
solving a numerical solution of pressure and speed, comprising the steps of:
A. dividing the area;
for a given solving area, carrying out mesh subdivision to form a 40-by-40 rectangular mesh;
B. dispersing a fracture porous medium fluid mathematical model for describing pressure and velocity, wherein the mathematical model comprises a formula (I) to a formula (V);
C. obtaining a system of linear equations related to pressure;
D. solving the linear equation set obtained in the step C;
E. obtaining a pressure numerical solution P and a speed numerical solution U;
solving the numerical solution of the concentration, comprising the following steps:
F. substituting the resulting velocity value solution U into formula (VI) -formula (IX);
G. discretizing a mathematical model formula (VI) to formula (IX) of the fracture porous medium fluid with the describing concentration;
H. forming a system of linear equations with respect to concentration;
I. solving the linear equation set about the concentration obtained in the step H;
J. obtaining a concentration numerical solution C;
and (3) solving a pressure numerical solution P, a speed numerical solution U and a concentration numerical solution C, namely realizing the numerical simulation of the underground fluid flow.
Wherein, the method of differential approximate derivative is adopted:
the temporal layers using backward difference methods, i.e.
Figure BDA0003011672960000051
Wherein n is a positive integer representing the time of the nth layer, cnDenotes t ═ tnThe concentration value at the time.
The spatial region being of a central difference method, i.e.
Figure BDA0003011672960000052
I.e. the physical quantities on the interface are calculated using a linear interpolation formula.
A system for realizing numerical simulation of underground fluid flow comprises a construction unit and a simulation unit, wherein the construction unit is used for realizing the step (1), and the simulation unit is used for realizing the step (2).
A computer apparatus comprising a memory storing a computer program and a processor implementing the steps of a method for numerical simulation of subsurface fluid flow based on a fracture porous medium fluid mathematical model when the computer program is executed.
A computer-readable storage medium having stored thereon a computer program which, when executed by a processor, implements the steps of a method for numerical simulation of subsurface fluid flow based on a fracture porous media fluid mathematical model.
The invention has the beneficial effects that:
1. the model provided by the invention comprehensively considers the physical phenomena involved in the flowing process, represents the physical phenomena through the mathematical model, more accurately describes the real physical process, and effectively relieves the risk of poor numerical simulation effect caused by inaccurate model.
2. The mass conservation type difference method provided by the invention can realize the solution of the invention model, the calculation process is simpler and more efficient, and the coding is easy to realize; and the method can simultaneously calculate three unknown quantities of pressure, speed and concentration, and is more suitable for practical application scenes.
3. The model and the method are combined to form a set of underground fluid flow numerical simulation system, the flow rules of the fluid in a crack area and a non-crack area can be numerically calculated through the system, and the output result of the system can be used for guiding production practice.
Drawings
FIG. 1 is a schematic view of a simplified treatment of a crack
FIG. 2 is a schematic diagram of a numerical approximation solution;
FIG. 3 is a schematic diagram of a numerical simulation of subsurface fluid flow through a fracture porous medium fluid mathematical model constructed in step (1);
FIG. 4 is a graph illustrating the velocity calculation according to the present invention;
FIG. 5 shows the permeability at the crack
Figure BDA0003011672960000053
The calculation result of the pressure is shown schematically;
FIG. 6(a) shows the permeability at the crack
Figure BDA0003011672960000054
The calculation result of the concentration of the invention when t is 1s is shown in a diagram;
FIG. 6(b) shows the permeability at the crack
Figure BDA0003011672960000055
The calculation result of the concentration of the invention when t is 5s is shown in a diagram;
FIG. 6(c) is a graph showing the permeability at the crack
Figure BDA0003011672960000056
The calculation results of the concentration of the invention when t is 10s are shown in the figure.
Detailed Description
The invention is further defined in the following, but not limited to, the figures and examples in the description.
Example 1
A method for realizing numerical simulation of underground fluid flow based on a fracture porous medium fluid mathematical model comprises the following steps:
(1) constructing a fracture porous medium fluid mathematical model; according to the characteristics of the porous medium of the crack, a set of partial differential equations describing the fluid flow is provided.
(2) And (2) realizing the numerical simulation of the flow of the underground fluid through the fracture porous medium fluid mathematical model constructed in the step (1).
The numerical solution method comprises the following steps: according to the partial differential equation set, the solving area is divided, a numerical solving method is established through discretization, and finally a set of large linear equation set is solved to obtain approximate solutions of various unknown quantities.
The prior art is generally based on the following three techniques when dealing with the problem of fractured porous media:
firstly, the crack is taken as an interface, dimension reduction and simplification processing are carried out on the crack, the high-dimension problem is converted into the low-dimension problem, the original physical property can be maintained, and the purpose of simple and convenient calculation can be achieved. As shown in fig. 1. In fig. 1: omegaγDenotes the crack region, Ω1、Ω2Representing the non-fractured regions.
Second, the effect of viscosity is usually ignored for simplicity of analysis and calculation.
Third, when dealing with problems involving concentration, the time dimension is not taken into account, and history fitting and future prediction cannot be performed when numerical simulation is performed.
The invention continues to use the first skill, reducing the computational complexity; however, the second and third techniques limit the application of the model in practice, and cannot be applied to the fields of numerical reservoir simulation and the like.
Example 2
The method for realizing the numerical simulation of the underground fluid flow based on the fracture porous medium fluid mathematical model in the embodiment 1 is characterized by comprising the following steps:
in the step (1), constructing a fracture porous medium fluid mathematical model refers to:
the fracture porous medium fluid mathematical model comprises a fracture porous medium fluid mathematical model for describing pressure and speed and a fracture porous medium fluid mathematical model for describing concentration;
since the flow properties of the non-fracture region are different from the fracture region and the non-fracture boundary is different from the fracture boundary, different models are required for description.
For non-fractured regions omega1And Ω2The mathematical model of the fracture porous medium fluid for describing pressure and speed is shown as the formula (I) and the formula (II):
divui＝fi (Ⅰ)
Figure BDA0003011672960000061
in the formulae (I) and (II), i is 1 or 2 and denotes the number of the non-fractured region, for example, in FIG. 1, Ω1、Ω2，fiIndicating non-crackingThe source and sink term (usually a known quantity), u, of the seam regioni、piRespectively representing the velocity, pressure, K, of the non-fractured zoneiThe permeability tensor, which represents the non-fractured region, is usually assumed to be a diagonal positive definite matrix,
Figure BDA0003011672960000079
is adhesive; div denotes divergence, which means that the derivatives are taken for each component of the vector and added;
for crack regions, e.g. Ω in FIG. 1rThe mathematical model of the fracture porous medium fluid for describing pressure and speed is shown as the formula (III) and the formula (IV):
Figure BDA0003011672960000071
divτuτ＝fγ+(u1·n1|γ+u2·n2|γ) (IV)
in the formulae (III) and (IV), urRepresenting the velocity of the fluid in the fracture zone, u1Representing the velocity, u, of the non-fractured zone fluid numbered 12Representing the velocity, p, of the non-fractured zone fluid numbered 2γRepresenting the pressure of the fluid in the fracture zone,
Figure BDA0003011672960000072
denotes the permeability in the crack direction, μγRepresenting the viscosity of the fluid in the fracture zone; f. ofγRepresenting the acting force of the outside on the crack area; as shown in FIG. 1, n1、n2Is the unit outside normal vector of the fracture; r represents the width of the crack and is typically a small constant.
Figure BDA00030116729600000710
The gradient or divergence in the direction of the crack, if in a cartesian coordinate system,
Figure BDA0003011672960000073
i.e. in the y-direction.
The interaction between fracture and non-fracture zones the interface conditions for the mathematical model coupling of the fracture porous medium fluid describing pressure and velocity are shown in equation (v):
-ξui·ni+αγpi＝αγpγ-(1-ξ)ui+1·ni+1 (Ⅴ)
in formula (V), i is 1, 2, n1、n2Is the unit external normal vector, u, of the fracture1Representing the velocity, u, of the non-fractured zone fluid numbered 12Representing the velocity, p, of the non-fractured zone fluid numbered 1γThe fluid pressure in the region of the fracture is represented,
Figure BDA0003011672960000074
denotes normal permeability, r denotes the width of the crack, ξ is a constant greater than 1/2;
for non-fractured regions, a fracture porous medium fluid mathematical model describing the concentration is shown as formula (VI):
Figure BDA0003011672960000076
in the formula (VI), ciDenotes the concentration of non-fractured regions, DiThe diffusion dispersion coefficient of the non-fractured region is expressed,
Figure BDA00030116729600000711
negative and positive values of the sink term for the non-fractured region are indicated,
Figure BDA0003011672960000077
represents the porosity of the rock in the non-fractured region, and t represents time;
for fracture zones, a fracture porous medium fluid mathematical model describing the concentration is shown as formula (VII):
Figure BDA0003011672960000078
in the formula (VII), the reaction mixture is,
Figure BDA0003011672960000081
the porosity of the rock in the fracture zone is indicated,
Figure BDA0003011672960000082
denotes diffusion coefficient in crack direction, cγThe concentration of the region of the crack is indicated,
Figure BDA0003011672960000083
negative and positive values of the sink term for the non-fractured region are indicated,
Figure BDA0003011672960000085
representing the difference at the crack between the non-cracked region numbered 1 and the non-cracked region numbered 2, r represents the width of the crack, which is usually a small constant, cγThe concentration of the crack region is shown, D is the diffusion coefficient, and u is the velocity;
the interface conditions for the interaction between the fracture and the non-fracture coupled with respect to the concentration of the fracture porous medium fluid mathematical model are shown as formula (VIII) and formula (IX):
Figure BDA0003011672960000086
Figure BDA0003011672960000087
in the formulae (VIII) and (IX),
Figure BDA00030116729600000811
indicating the normal diffusion coefficient.
The two sets of equations are combined to form the model of the invention, the flow properties of the underground fluid are comprehensively considered, and compared with the prior art, the accuracy and universality of numerical simulation by using the method are higher.
As shown in fig. 3, the specific implementation process of step (2) includes:
and (3) discretizing the crack porous medium fluid mathematical model in each small region by using a mass conservation type difference method to finally form a large linear equation set to obtain a discrete solution of the original problem.
According to the position of each physical quantity in each subdivision unit, the invention provides a numerical discrete solving method, which comprises the following steps:
solving a numerical solution of pressure and speed, comprising the steps of:
A. dividing the area;
for a given solving area, carrying out mesh subdivision to form a 40-by-40 rectangular mesh; as shown in fig. 2;
B. dispersing a fracture porous medium fluid mathematical model for describing pressure and velocity, wherein the mathematical model comprises a formula (I) to a formula (V);
C. obtaining a system of linear equations related to pressure;
D. solving the linear equation set obtained in the step C;
E. obtaining a pressure numerical solution P and a speed numerical solution U;
solving the numerical solution of the concentration, comprising the following steps:
F. substituting the resulting velocity value solution U into formula (VI) -formula (IX);
G. discretizing a mathematical model formula (VI) to formula (IX) of the fracture porous medium fluid with the describing concentration;
H. forming a system of linear equations with respect to concentration;
I. solving the linear equation set about the concentration obtained in the step H;
J. obtaining a concentration numerical solution C;
and (3) solving a pressure numerical solution P, a speed numerical solution U and a concentration numerical solution C, namely realizing the numerical simulation of the underground fluid flow.
Wherein, the method of differential approximate derivative is adopted:
the temporal layers using backward difference methods, i.e.
Figure BDA0003011672960000091
Wherein n is a positive integer representing the time of the nth layer, cnDenotes t ═ tnThe concentration value at the time.
The spatial region being of a central difference method, i.e.
Figure BDA0003011672960000092
I.e. the physical quantities on the interface are calculated using a linear interpolation formula.
In order to verify the effectiveness and the practicability of the model and the solving method, a theoretical example and an actual case are provided, and a computer program is utilized to perform simulation calculation. Different permeability rates are selected at the surrounding medium and the crack in the experiment, and calculation results are given for comparison. In all the examples, the calculation region is Ω1＝[-1,0]×[0,1],Ω2＝[0,1]×[0,1],Ωf＝{x＝0}×[0,1]And I is a 2 × 2 identity matrix.
Theoretical calculation example: the permeability of the crack is different from that of the surrounding medium, and K is taken1＝K2＝0.01I,Kf＝0.2I.D1＝D2＝0.01I,DfThe true solution for formation pressure and concentration is 0.4 i:
Figure BDA0003011672960000093
the right-hand term and the boundary condition can be obtained by taking the accurate solution into an equation for calculation, the right-hand term and the boundary condition are calculated by utilizing a programmed computer program, fig. 3 and 4 are respectively comparison graphs of the real solution of the pressure and the concentration and the numerical approximation solution of the invention, and table 1 shows the error between the real solution of the pressure, the speed and the concentration and the numerical approximation solution of the invention when different sections are carried out on a calculation area. It can be seen that the numerical calculation result of the invention can approximate the real solution more accurately.
TABLE 1
Figure BDA0003011672960000094
Figure BDA0003011672960000101
The practical case is as follows: in engineering applications, the permeability at the fracture is usually greater than that of the surrounding medium, at which point the fracture will act as a "pipe" to promote fluid flow, which is calculated using the method of the present invention to give a comparative plot of fluid flow rate, pressure and concentration changes.
In this case, the active term (entrance) is at a point (0.5,0) in the surrounding medium region, and the crack region has no source term; there is a convergence term (exit) at point (0.5, 1); the remaining boundary conditions and the right-hand item are set to 0. To facilitate observation of the effect of the fracture on the fluid flow, the fracture area is set here as a passive term inflow. Then, as the permeability at the fracture is greater than the permeability of the surrounding medium, the fluid will flow from the surrounding medium to the fracture and finally out of the wellhead through the fracture, depending on the actual situation. Get K1＝K2＝0.001I,Kf＝0.1；D1＝D2＝0.001I,Df＝0.005I,T＝[0,100]The slit width r is 10-3It can be seen from fig. 4, 5 and 6(a), 6(b) and 6(c) that the calculated result of the present invention is close to the expected result, the fluid tends to flow toward the crack, and therefore the concentration is changed, compared with the flow of the surrounding medium, the concentration at the crack can rapidly flow from the inlet to the outlet along the crack, and a jump may occur, and the numerical approximation calculated result of the present invention can reflect the property more accurately. Therefore, the model and the solving method can accurately carry out numerical simulation on the flow property of the fracture porous medium fluid and can play a guiding role in oil exploitation or underground fluid numerical simulation.
Fig. 4 depicts the flow trend of the velocity field under the influence of a crack, where the black solid points represent the source term (entrance), the black open points represent the sink term (exit), and the black straight lines represent the location of the crack. Thus, it can be seen that due to the large pore size characteristic of the fracture, fluid flows toward the fracture, which tends to be the desired effect.
As can be seen from fig. 6(a), 6(b), 6(c), the fluid flows toward the fracture while causing a change in concentration so that the concentration changes faster in the direction of the fracture. This further validates the expectation assumption and demonstrates that the model proposed by the present invention can effectively simulate the variation trend of the concentration in the fracture-type porous medium.
Example 3
A system for realizing numerical simulation of underground fluid flow comprises a construction unit and a simulation unit, wherein the construction unit is used for realizing the step (1), and the simulation unit is used for realizing the step (2).
Example 4
A computer apparatus comprising a memory storing a computer program and a processor implementing the steps of a method for numerical simulation of subsurface fluid flow based on a mathematical model of a fractured porous medium fluid when the computer program is executed by the processor.
Example 5
A computer readable storage medium having stored thereon a computer program which, when executed by a processor, performs the steps of a method for performing numerical simulation of subsurface fluid flow based on a fracture porous media fluid mathematical model.

Claims (6)

1. A method for realizing numerical simulation of underground fluid flow based on a fracture porous medium fluid mathematical model is characterized by comprising the following steps:
(1) constructing a fracture porous medium fluid mathematical model;
(2) and (2) realizing the numerical simulation of the flow of the underground fluid through the fracture porous medium fluid mathematical model constructed in the step (1).
2. The method for realizing the numerical simulation of the flow of the underground fluid based on the fracture porous medium fluid mathematical model as claimed in claim 1, wherein in the step (1), the step of constructing the fracture porous medium fluid mathematical model is that:
the fracture porous medium fluid mathematical model comprises a fracture porous medium fluid mathematical model describing pressure and speed and a fracture porous medium fluid mathematical model describing concentration;
for a non-fracture area, a fracture porous medium fluid mathematical model for describing pressure and speed is shown as a formula (I) and a formula (II):
divui＝fi (Ⅰ)
Figure FDA0003011672950000011
in the formulae (I) and (II), i is 1 or 2 and denotes the number of the non-fractured region, fiSource and sink terms, u, representing non-fractured regionsi、piRespectively representing the velocity, pressure, K, of the non-fractured zoneiThe permeability tensor representing the non-fractured region,
Figure FDA0003011672950000015
is adhesive; div denotes divergence, which means that the derivatives are taken for each component of the vector and added;
for the fracture area, a fracture porous medium fluid mathematical model for describing pressure and velocity is shown as formula (III) and formula (IV):
Figure FDA0003011672950000012
divτuτ＝fγ+(u1·n1|γ+u2·n2|γ) (Ⅳ)
in the formulae (III) and (IV), urRepresenting the velocity of the fluid in the fracture zone, u1Representing the velocity, u, of the non-fractured zone fluid numbered 12Representing the velocity, p, of the non-fractured zone fluid numbered 2γRepresenting the pressure of the fluid in the fracture zone,
Figure FDA0003011672950000013
denotes the permeability in the crack direction, μγRepresenting the viscosity of the fluid in the fracture zone; f. ofγRepresenting the acting force of the outside on the crack area;
the interaction between fracture and non-fracture zones the interface conditions for the mathematical model coupling of the fracture porous medium fluid describing pressure and velocity are shown in equation (v):
-ξui·ni+αγpi＝αγpγ-(1-ξ)ui+1·ni+1 (Ⅴ)
in formula (V), i is 1, 2, n1、n2Is the unit external normal vector, u, of the fractureiRepresenting the velocity, u, of the non-fractured zone fluid numbered 12Representing the velocity, p, of the non-fractured zone fluid numbered 1γThe fluid pressure in the region of the fracture is represented,
Figure FDA0003011672950000014
denotes normal permeability, r denotes the width of the crack, ξ is a constant greater than 1/2;
for non-fractured regions, a fracture porous medium fluid mathematical model describing the concentration is shown as formula (VI):
Figure FDA0003011672950000021
in the formula (VI), ciDenotes the concentration of non-fractured regions, DiThe diffusion dispersion coefficient of the non-fractured region is expressed,
Figure FDA00030116729500000210
Figure FDA00030116729500000211
negative and positive values of the sink term for the non-fractured region are indicated,
Figure FDA0003011672950000022
represents the porosity of the rock in the non-fractured region, and t represents time;
for fracture zones, a fracture porous medium fluid mathematical model describing the concentration is shown as formula (VII):
Figure FDA0003011672950000023
in the formula (VII), the reaction mixture is,
Figure FDA0003011672950000024
the porosity of the rock in the fracture zone is indicated,
Figure FDA0003011672950000025
denotes diffusion coefficient in crack direction, cγThe concentration of the region of the crack is indicated,
Figure FDA0003011672950000026
Figure FDA00030116729500000212
negative and positive values of the sink term for the non-fractured region are indicated,
Figure FDA00030116729500000213
representing the difference at the crack between the non-cracked region numbered 1 and the non-cracked region numbered 2, r represents the width of the crack, which is usually a small constant, cγThe concentration of the crack region is shown, D is the diffusion coefficient, and u is the velocity;
the interface conditions for the interaction between the fracture and the non-fracture coupled with respect to the concentration of the fracture porous medium fluid mathematical model are shown as formula (VIII) and formula (IX):
Figure FDA0003011672950000027
in the formulae (VIII) and (IX),
Figure FDA0003011672950000028
Figure FDA00030116729500000214
Figure FDA0003011672950000029
indicating the normal diffusion coefficient.
3. The method for realizing the numerical simulation of the underground fluid flow based on the fracture porous medium fluid mathematical model is characterized in that the specific implementation process of the step (2) comprises the following steps:
solving a numerical solution of pressure and speed, comprising the steps of:
A. dividing the area;
for a given solving area, carrying out mesh subdivision to form a 40-by-40 rectangular mesh;
B. dispersing a fracture porous medium fluid mathematical model for describing pressure and velocity, wherein the mathematical model comprises a formula (I) to a formula (V);
C. obtaining a system of linear equations related to pressure;
D. solving the linear equation set obtained in the step C;
E. obtaining a pressure numerical solution P and a speed numerical solution U;
solving the numerical solution of the concentration, comprising the following steps:
F. substituting the resulting velocity value solution U into formula (VI) -formula (IX);
G. discretizing a mathematical model formula (VI) to formula (IX) of the fracture porous medium fluid with the describing concentration;
H. forming a system of linear equations with respect to concentration;
I. solving the linear equation set about the concentration obtained in the step H;
J. obtaining a concentration numerical solution C;
and (3) solving a pressure numerical solution P, a speed numerical solution U and a concentration numerical solution C, namely realizing the numerical simulation of the underground fluid flow.
4. A system for realizing the method for realizing the numerical simulation of the underground fluid flow based on the fracture porous medium fluid mathematical model as claimed in any one of claims 1 to 3 is characterized by comprising a construction unit and a simulation unit, wherein the construction unit is used for realizing the step (1), and the simulation unit is used for realizing the step (2).
5. A computer apparatus comprising a memory storing a computer program and a processor that when executed performs the steps of the method of performing numerical simulation of subterranean fluid flow based on a mathematical model of a fractured porous media fluid according to any one of claims 1 to 3.
6. A computer-readable storage medium, having stored thereon a computer program which, when executed by a processor, performs the steps of the method of any of claims 1-3 for numerical simulation of subsurface fluid flow based on a mathematical model of a fractured porous media fluid.
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