CN113868930A - Anisotropic reservoir seepage simulation method based on generalized finite difference method - Google Patents

Anisotropic reservoir seepage simulation method based on generalized finite difference method Download PDF

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CN113868930A
CN113868930A CN202111323669.0A CN202111323669A CN113868930A CN 113868930 A CN113868930 A CN 113868930A CN 202111323669 A CN202111323669 A CN 202111323669A CN 113868930 A CN113868930 A CN 113868930A
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饶翔
赵辉
湛文涛
刘怡娜
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Yangtze University
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Abstract

The invention discloses an anisotropic reservoir seepage simulation method based on a generalized finite difference method, which comprises the following steps of: s1, utilizing a continuous function derivation method to derive a basic control equation of unsteady flow of the two-dimensional porous medium with the permeability field in the form of a continuous function; s2, calculating partial derivatives of other nodes at the node in the influence domain of each node based on a generalized finite difference method; s3, for the anisotropic permeability tensor in the explicit function form, calculating the control equation obtained in S1 and the partial derivatives of other nodes at the node in the influence domain of each node obtained in S2; and S4, calculating the discrete format of the basic control equation of each node in the influence domain of each node for the anisotropic permeability tensor of the implicit function form. The invention provides an anisotropic reservoir seepage simulation method based on a generalized finite difference method, which is used for analyzing the problem of single-phase seepage of an anisotropic stratum for the first time.

Description

Anisotropic reservoir seepage simulation method based on generalized finite difference method
Technical Field
The invention relates to the field of generalized finite difference methods. More particularly, the invention relates to an anisotropic reservoir seepage simulation method based on a generalized finite difference method.
Background
Due to geological factors such as the arrangement and filling mode of rock framework particles, the bedding effect, the ancient water flow direction, the sedimentary structure cracks and the like or the scouring effect of fluid in the oil reservoir exploitation process, the formation permeability distribution often presents anisotropic characteristics. For anisotropic reservoirs, the permeability at each point is no longer a scalar value, but a direction-dependent physical quantity, usually expressed in the form of a tensor. At present, the numerical simulation research on the porous medium seepage in the reservoir is mostly carried out by using a finite difference method based on a Cartesian grid, but the method is difficult to be applied to the reservoir with a complex boundary or complex geological conditions. When other finite volume methods of orthogonal grids or non-orthogonal and non-structural grids, finite element methods or quasi-finite difference methods are adopted, the application of a multipoint flow estimation format is relied on, and the generation difficulty of the matching grids of the complex geometric characteristics of the actual reservoir is also higher, so that the flexible and efficient numerical modeling and calculation method of the anisotropic porous medium flow modeling still needs to be researched.
The Generalized Finite Difference Method (GFDM) is a novel regional grid-free method which is established in recent years, and the method expresses each order partial derivative of unknown quantity in a control equation as linear combination of function values of adjacent nodes in a sub-region based on multivariate function Taylor series expansion and weighted least square fitting in the sub-region, so that the dependence of the traditional Finite Difference Method (FDM) on grids is overcome. J.J.Benito et al propose a self-adaptive generalized finite difference method, which realizes automatic point matching in a local range according to precision requirements. At present, the method is developed rapidly at home and abroad, and is widely applied to solving various scientific and engineering problems, including a coupling thermoelasticity problem, a three-order and four-order partial differential equation, a shallow water equation, transient heat conduction analysis, a seismic wave propagation problem [19], stress analysis, an unsteady Burgers equation, a water wave interaction, an inverse heat source problem, a nonlinear convection diffusion equation, a time fraction diffusion equation and random underground flow. It should be noted that, the advantage and disadvantage of the generalized finite difference method and its engineering application are reviewed by Gavete et al, the influence of various factors on the GFDM value result is analyzed, and it is found that the weight function has little influence on the GFDM value, so that in the practical application of GFDM, the quartic spline function is often used as the weight function. In summary, the generalized finite difference method can realize the accurate solution of the control equation only by arranging a group of nodes in the calculation domain, thereby saving the time-consuming and labor-consuming grid division and numerical integration of the complex geometric characteristics of the calculation domain in the finite element, finite difference and boundary element methods, and being an efficient and high-precision numerical modeling method.
Disclosure of Invention
The invention aims to provide an anisotropic reservoir seepage simulation method based on a generalized finite difference method, which is used for analyzing the problem of anisotropic stratum single-phase seepage by applying a grid-free generalized finite difference method for the first time and provides a method for processing an anisotropic permeability tensor under the condition that a permeability tensor is an explicit functional form and a discrete form respectively.
To achieve these objects and other advantages in accordance with the purpose of the invention, there is provided an anisotropic reservoir seepage simulation method based on a generalized finite difference method, comprising the steps of:
s1, carrying out derivation on the basic control equation of the unsteady flow of the two-dimensional porous medium with the permeability field in the form of a continuous function by using a continuous function derivation method to obtain a derived control equation;
s2, calculating partial derivatives of other nodes at the node in the influence domain of each node based on a generalized finite difference method;
s3, for the anisotropic permeability tensor in the explicit function form, calculating the discrete format of the basic control equation of each node according to the control equation obtained in S1 and the partial derivatives of other nodes at each node in the influence domain of each node obtained in S2, and determining the second type of boundary conditions in the discrete format of the basic control equation of each node based on the discrete format of the generalized finite difference method;
s4, for the anisotropic permeability tensor in the implicit function form, in each node influence domain, calculating the discrete format of the basic control equation of each node in a mode of estimating each order derivative of the central node field variable by adopting a generalized finite difference method and harmonically and averagely coupling the permeability tensor, and determining the second type boundary condition in the discrete format of the basic control equation of each node based on the discrete format of the generalized finite difference method.
Preferably, in the anisotropic reservoir seepage simulation method based on the generalized finite difference method, the derivation of the fundamental control equation of unsteady flow of the two-dimensional porous medium with a permeability field in the form of a continuous function by using the continuous function derivation method in S1 includes the following steps:
s1.1, the basic control equation of unsteady flow of the two-dimensional porous medium is as follows:
Figure BDA0003346160660000021
Figure BDA0003346160660000031
in the above formula: ctIs the compressibility, P is the formation pressure; t is time; k is the permeability, which in two dimensions is expressed as:
Figure BDA0003346160660000032
s1.2, if the permeability field has a continuous function form, transforming the permeability field in the above formulas (1) and (2) by using a continuous function derivation method to obtain a derived control equation:
Figure BDA0003346160660000033
preferably, in the anisotropic reservoir seepage simulation method based on the generalized finite difference method, the step of calculating the partial derivatives of other nodes at each node in the influence domain of the node based on the generalized finite difference method in S2 includes the following steps:
s2.1, node X0＝(x0,y0) Includes another n nodes in the domain of influence, denoted as { X1,X2,X3,…,XNIn which X isi＝(xi,yi) The pressure values at n nodes { P (X) }i) I-1, … n at node X0Taylor unfolding is carried out to obtain:
Figure BDA0003346160660000034
in the above formula: Δ xi＝x0-xi；Δyi＝y0-yi；P0＝P(X0)；
Figure BDA0003346160660000035
Figure BDA0003346160660000036
S2.2, defining an error function B (P) with weight:
Figure BDA0003346160660000037
wherein, ω isi＝ω(Δxi,Δyi) Is a weighting function, which is a quartic spline function:
Figure BDA0003346160660000041
in the above formula: r isiIs node xiEuclidean distance to a node; r ismIs node X0An influence domain radius;
s2.3, to B (P) with respect to DP＝{Px0,Py0,Pxx0,Pyy0,Pxy0The partial derivatives are solved so that the partial derivative for each component is equal to zero:
Figure BDA0003346160660000042
s2.4, rewriting the above formula (7) into a matrix form:
ADP＝b
(8)
s2.5, the right-end term matrix b of the above equation (8) can be decomposed into:
b＝B5×(n+1)P(n+1)×1,
(9)
s2.6, by the above formulas (7) - (9), center node X0The partial derivatives of (a) are:
DP＝[Px0,Py0,Pxx0,Pyy0,Pxy0]T＝A-1b＝A-1BP＝M5×(n+1)P(n+1)×1.
(10)
s2.7, marking the element of the matrix M as MijDeriving the point x from the matrix M0Other node pairs within the influence domain of node x0Partial derivatives of (a):
Figure BDA0003346160660000043
Figure BDA0003346160660000044
Figure BDA0003346160660000051
Figure BDA0003346160660000052
preferably, in the anisotropic reservoir seepage simulation method based on the generalized finite difference method, the calculating the partial derivatives of other nodes at each node in the influence domain of each node from the governing equation calculated in S1 in S3 and from each node obtained in S2 includes:
s3.1, obtaining an approximate functional form of the permeability tensor by interpolation, and obtaining the following discrete format for the node i according to the above equation (3) and equation (11):
Figure BDA0003346160660000053
preferably, in the anisotropic reservoir seepage simulation method based on the generalized finite difference method, the second type of boundary conditions in S3 are as follows:
Figure BDA0003346160660000054
preferably, in the anisotropic reservoir seepage simulation method based on the generalized finite difference method, the calculating the discrete format of the basic control equation of each node in the S4 by using the generalized finite difference method to estimate each order derivative of the central node field variable and harmonically and averagely coupling the permeability tensor comprises:
s4.1, for the node i, the permeability tensor at the node is
Figure BDA0003346160660000061
S4.2, the pressure spread term at node i, i.e., the left side of equation (1) above, can be approximated as:
Figure BDA0003346160660000062
in the above formula (15)
Figure BDA0003346160660000063
Harmonic averaging of node i and node j within its domain of influence:
Figure BDA0003346160660000064
preferably, in the anisotropic reservoir seepage simulation method based on the generalized finite difference method, the second type of boundary conditions in S4 are as follows:
Figure BDA0003346160660000065
the invention provides a method for analyzing seepage problems in anisotropic formations based on a meshless generalized finite difference method, which is used for depicting a computational domain by distributing nodes instead of dividing meshes. The method has the advantages of more flexibility and simplicity when the method is used for processing complex geometric calculation domains and complex boundary conditions. Meanwhile, the invention respectively provides a processing method for anisotropic permeability tensor distribution aiming at two different permeability tensor conditions of an explicit function form and a discrete form. Three numerical examples of single-phase stable seepage of a rectangular calculation domain, single-phase unstable seepage with a complex boundary shape and stable flow with step-type anisotropic permeability distribution in the embodiment also show that the calculation errors of the method are all below 0.4%, and the calculation errors are remarkably reduced along with the increase of the distribution density, so that the method has high precision and good convergence. Compared with the traditional mainstream oil reservoir numerical simulation finite volume method based on the grid, the method can more efficiently and simply process the anisotropic permeability tensor, and avoids the very difficult grid subdivision of a complex geometric calculation domain and the acquisition of a corresponding multipoint flow format by the finite volume method. And unlike the finite volume method which generally only processes closed boundary conditions, the invention can process more complex boundary conditions (including constant pressure boundary conditions and the like) due to the distribution of points on the boundary of a computational domain, so that the work specification GFDM has great application potential in numerical reservoir simulation.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention.
Drawings
FIG. 1 is a diagram of a compute domain in accordance with an embodiment of the present invention;
FIG. 2 is a fine triangulation mesh in an embodiment of the invention;
FIG. 3 is a geometric representation of a computational domain in an embodiment of the invention;
FIG. 4 is a schematic illustration of adding a virtual well point in an embodiment of the present invention;
FIG. 5 is a finite element reference diagram based on a fine triangular mesh in another embodiment of the present invention;
FIG. 6 is a diagram of relative error with a finite element reference solution in another embodiment of the present invention;
FIG. 7 is a homogeneous anisotropic porous medium computational domain in another embodiment of the invention;
FIG. 8 is a fine triangulation mesh in another embodiment of the present invention;
fig. 9 is a node arrangement scheme 1 of adding virtual points in another embodiment of the present invention;
fig. 10 is a node arrangement scheme 2 of adding virtual points in another embodiment of the present invention;
fig. 11 is a node arrangement scheme 3 of adding virtual points in another embodiment of the present invention;
FIG. 12 is a finite element reference diagram based on a fine triangular mesh in another embodiment of the present invention;
FIG. 13 is a graph of relative error versus a finite element reference solution in another embodiment of the present invention;
FIG. 14 is a finite element reference diagram based on a fine triangular mesh in another embodiment of the present invention;
FIG. 15 is a graph showing GFDM pressure distribution at 0.5m spacing in another embodiment of the present invention;
FIG. 16 is a graph of relative error versus a finite element reference solution in another embodiment of the present invention.
Detailed Description
The present invention is further described in detail below with reference to the attached drawings so that those skilled in the art can implement the invention by referring to the description text.
< example 1>
An anisotropic reservoir seepage simulation method based on a generalized finite difference method comprises the following steps:
s1, carrying out derivation on the basic control equation of the unsteady flow of the two-dimensional porous medium with the permeability field in the form of a continuous function by using a continuous function derivation method to obtain a derived control equation;
captive, S1 includes the following steps:
s1.1, the basic control equation of unsteady flow of the two-dimensional porous medium is as follows:
Figure BDA0003346160660000081
in the above formula: ctIs the compression factor; p is the formation pressure; t is time; k is the permeability, which in two dimensions is expressed as:
Figure BDA0003346160660000082
s1.2, if the permeability field has a continuous function form, transforming the permeability field in the above formulas (1) and (2) by using a continuous function derivation method to obtain a derived control equation:
Figure BDA0003346160660000083
s2, calculating partial derivatives of other nodes at the node in the influence domain of each node based on a generalized finite difference method;
specifically, S2 includes the following steps:
s2.1, node X0＝(x0,y0) Includes another n nodes in the domain of influence, denoted as { X1,X2,X3,…,XNIn which X isi＝(xi,yi) The pressure values at n nodes { P (X) }i) I-1, … n at node X0Taylor unfolding is carried out to obtain:
Figure BDA0003346160660000091
in the above formula: Δ xi＝x0-xi；Δyi＝y0-yi；P0＝P(X0)；
Figure BDA0003346160660000092
Figure BDA0003346160660000093
S2.2, defining an error function B (P) with weight:
Figure BDA0003346160660000094
wherein, ω isi＝ω(Δxi,Δyi) Is a weighting function, which is a quartic spline function:
Figure BDA0003346160660000095
in the above formula: r isiIs node xiEuclidean distance to a node; r ismIs node X0An influence domain radius;
s2.3, to B (P) with respect to DP＝{Px0,Py0,Pxx0,Pyy0,Pxy0The partial derivatives are solved so that the partial derivative for each component is equal to zero:
Figure BDA0003346160660000096
s2.4, rewriting the above formula (7) into a matrix form:
ADP＝b
(8)
s2.5, the right-end term matrix b of the above equation (8) can be decomposed into:
b＝B5×(n+1)P(n+1)×1,
(9)
S2.6. from the above formulas (7) to (9), the center node X0The partial derivatives of (a) are:
DP＝[Px0,Py0,Pxx0,Pyy0,Pxy0]T＝A-1b＝A-1BP＝M5×(n+1)P(n+1)×1.
(10)
s2.7, marking the element of the matrix M as MijDeriving the point x from the matrix M0Other node pairs within the influence domain of node x0Partial derivatives of (a):
Figure BDA0003346160660000101
Figure BDA0003346160660000102
Figure BDA0003346160660000103
Figure BDA0003346160660000104
s3, for the anisotropic permeability tensor in the explicit function form, calculating the discrete format of the basic control equation of each node according to the control equation obtained in S1 and the partial derivatives of other nodes at each node in the influence domain of each node obtained in S2, and determining the second type of boundary conditions in the discrete format of the basic control equation of each node based on the discrete format of the generalized finite difference method;
specifically, S3 specifically includes:
s3.1, obtaining an approximate functional form of the permeability tensor by interpolation, and obtaining the following discrete format for the node i according to the above equation (3) and equation (11):
Figure BDA0003346160660000111
the second class of boundary conditions described in S3 is as follows:
Figure BDA0003346160660000112
s4, for the anisotropic permeability tensor in the implicit function form, in each node influence domain, calculating the discrete format of the basic control equation of each node in a mode of estimating each order derivative of the central node field variable by adopting a generalized finite difference method and harmonically and averagely coupling the permeability tensor, and determining a second type of boundary condition in the discrete format of the basic control equation of each node based on the discrete format of the generalized finite difference method;
wherein, S4 specifically includes:
s4.1, for the node i, the permeability tensor at the node is
Figure BDA0003346160660000113
S4.2, the pressure spread term at node i, i.e., the left side of equation (1) above, can be approximated as:
Figure BDA0003346160660000114
in the above formula (15)
Figure BDA0003346160660000115
Harmonic averaging of node i and node j within its domain of influence:
Figure BDA0003346160660000116
Figure BDA0003346160660000121
the second class of boundary conditions described in S4 is as follows:
Figure BDA0003346160660000122
the invention also provides three numerical examples (embodiment 2-embodiment 4), namely three numerical examples of single-phase stable seepage in a rectangular calculation domain, single-phase unsteady seepage with a complex boundary shape and steady-state flow with a step-type anisotropic permeability distribution, and the pressure function distribution P is calculated by respectively adopting different processing methods for anisotropic permeability tensor in embodiment 1 and different distribution modesGFDMAnd using the result of the fine finite element as a reliable reference solution PrefThe accuracy and convergence of the method herein are examined in terms of the average relative error, which is defined as follows:
Figure BDA0003346160660000123
in the above formula, Pref,iIs a reference solution, P, for the ith nodeGFDM,iIs the result of the calculation of the present method for the ith node, npIs the number of points of the problem domain.
< example 2>
As shown in FIG. 1, the computational domain is a rectangular stratum with dimensions of 20m, 10MPa of constant pressure boundary conditions on the left, 5MPa of constant pressure boundary conditions on the right, closed boundary conditions on the upper and lower boundaries, and permeability tensor
Figure BDA0003346160660000124
The problem then satisfies the set of equations:
Figure BDA0003346160660000131
first type boundary conditions:
Figure BDA0003346160660000132
second type boundary conditions:
Figure BDA0003346160660000133
fig. 2 is a fine triangulation diagram of the computational domain, and the pressure value calculated by the fine mesh is used as a reference solution, and fig. 3 is a geometric characterization of the generalized finite difference pair computational domain, and an equidistant point matching scheme is adopted in this section. For practical problems, it is inevitable to process the second type of boundary condition, for example, as shown in fig. 4, a processing manner of adding a virtual node to the second type of boundary point in the external direction thereof is used to improve the estimation accuracy of the derivative type of boundary point pressure function on the spatial coordinate derivative.
In this embodiment, first, under a fine triangulation mesh, a finite element method is used to calculate a pressure distribution as a reference solution, as shown in fig. 5. In addition, three different equidistant point matching schemes are respectively adopted for the calculation domain, the nodes are arranged in a mode that delta x is 0.5m, 1m and 2m, and the radius of the influence domain of each node is taken as
Figure BDA0003346160660000135
The pressure distribution results of GFDM calculations for the above three dosing schemes are compared to a reference solution for finite element calculations. As shown in FIG. 6, we see that the computation error of the generalized finite difference is within 0.2% with respect to the fine finite element reference solution. Of course, as the node density increases, the generalized finite difference method can achieve higher accuracy and has good convergence.
< example 3>
As shown in FIG. 7, example 3 is a reservoir example with a circular boundary1、Γ2Is a left and a right boundary respectively, and is a circular arc with the radius of 100m, gamma3、Γ4The upper and lower boundaries are straight lines having a length of 200m, respectively. Left side is fixedThe pressure boundary condition is 10MPa, the right side is the constant pressure boundary condition of 5MPa, and the upper and lower boundaries are closed boundary conditions.
Considering the problem of unsteady seepage in a two-dimensional porous medium, the basic control equation is as follows:
Figure BDA0003346160660000134
wherein μ is a fluid viscosity, and μ ═ 1mPa · s is not taken herein; ctIs the compression factor, takes 10-3MPa-1(ii) a α is a unit conversion factor; k is the permeability tensor that is,
Figure BDA0003346160660000141
by substituting the parameter values according to the above equation (17), the following equation can be obtained,
Figure BDA0003346160660000142
first type boundary condition (constant pressure boundary):
Figure BDA0003346160660000143
second type boundary conditions (closed boundaries):
Figure BDA0003346160660000144
as shown in fig. 8, a fine triangulation grid diagram of the finite element method is given. Similarly, we present three different node placement schemes with different dot placement densities, such as fig. 9, fig. 10, and fig. 11, and add virtual dots to the second type boundary. In this section, pressure distribution is first calculated by using a finite element method under a fine triangulation mesh as shown in fig. 12. In addition, using the 3-point matching schemes of fig. 9-11, the quasi-steady state results of the GFDM calculated pressure distribution were compared with the quasi-steady state results of the finite elements, as shown in fig. 13, and the calculation errors of the generalized finite differences were all within 0.2% with respect to the fine finite element reference solution. Obviously, as the distribution density of the nodes in the calculation domain increases, the generalized finite difference method can obtain higher precision and has good convergence. For the node arrangement scheme 3, under the sparse point distribution density, the GFDM method can obtain high precision, and the method is very suitable for simple, efficient and high-precision solution of the complex boundary oil reservoir flow problem.
< example 4>
The reservoir model of this example is the same as that of example 2, the reservoir model is a rectangular stratum of 20m × 20m, the left side is a constant pressure boundary condition of 10MPa, the right side is a constant pressure boundary condition of 5MPa, and the upper and lower boundaries are closed boundary conditions, except that a step-like permeability tensor is considered
Figure BDA0003346160660000151
Wherein
Figure BDA0003346160660000152
In this embodiment, first, under a fine triangulation mesh, a finite element method is used to calculate the pressure distribution as shown in fig. 14. In addition, the calculation was performed using the same spotting scheme as in example 2, and fig. 15 shows the pressure field solved for the case of equidistant spotting at a spacing of 0.5 m. As shown in fig. 16, the pressure distribution results of GFDM calculation for three different placement schemes of 0.5m, 1m, and 2m are compared with the results of the fine triangulation, and the calculation errors of the generalized finite difference are all within 0.4% with respect to the fine finite element reference solution, which fully demonstrates that the processing method of estimating each order derivative of the central node field variable by using GFDM and harmonically and evenly coupling the permeability tensor has simple practicability for the permeability problem with the step type, and can ensure high calculation accuracy and good convergence, which directly reveals the huge potential that GFDM can be applied to the actual reservoir numerical simulation.
While embodiments of the invention have been described above, it is not limited to the applications set forth in the description and the embodiments, which are fully applicable to various fields of endeavor for which the invention may be embodied with additional modifications as would be readily apparent to those skilled in the art, and the invention is therefore not limited to the details given herein and to the embodiments shown and described without departing from the generic concept as defined by the claims and their equivalents.

Claims (7)

1. An anisotropic reservoir seepage simulation method based on a generalized finite difference method is characterized by comprising the following steps:
s1, carrying out derivation on the basic control equation of the unsteady flow of the two-dimensional porous medium with the permeability field in the form of a continuous function by using a continuous function derivation method to obtain a derived control equation;
s2, calculating partial derivatives of other nodes at the node in the influence domain of each node based on a generalized finite difference method;
s3, for the anisotropic permeability tensor in the explicit function form, calculating the discrete format of the basic control equation of each node according to the control equation obtained in S1 and the partial derivatives of other nodes at each node in the influence domain of each node obtained in S2, and determining the second type of boundary conditions in the discrete format of the basic control equation of each node based on the discrete format of the generalized finite difference method;
s4, for the anisotropic permeability tensor in the implicit function form, in each node influence domain, calculating the discrete format of the basic control equation of each node in a mode of estimating each order derivative of the central node field variable by adopting a generalized finite difference method and harmonically and averagely coupling the permeability tensor, and determining the second type boundary condition in the discrete format of the basic control equation of each node based on the discrete format of the generalized finite difference method.
2. The method for simulating anisotropic reservoir seepage based on the generalized finite difference method of claim 1, wherein the derivation of the fundamental governing equation of unsteady flow of the two-dimensional porous medium with the permeability field in the form of continuous function by the continuous function derivation method in S1 comprises the following steps:
s1.1, the basic control equation of unsteady flow of the two-dimensional porous medium is as follows:
Figure FDA0003346160650000011
in the above formula: ctIs the compression factor; p is the formation pressure; t is time; k is the permeability, which in two dimensions is expressed as:
Figure FDA0003346160650000012
Figure FDA0003346160650000021
s1.2, if the permeability field has a continuous function form, transforming the permeability field in the above formulas (1) and (2) by using a continuous function derivation method to obtain a derived control equation:
Figure FDA0003346160650000022
3. the method for simulating anisotropic reservoir seepage based on the generalized finite difference method of claim 2, wherein the step of calculating the partial derivatives of other nodes at each node in the influence domain based on the generalized finite difference method in S2 comprises the following steps:
s2.1, node X0＝(x0,y0) Includes another n nodes in the domain of influence, denoted as { X1,X2,X3,…,XNIn which X isi＝(xi,yi) The pressure values at n nodes { P (X) }i) I-1, … n at node X0Taylor unfolding is carried out to obtain:
Figure FDA0003346160650000023
in the above formula: Δ xi＝x0-xi；Δyi＝y0-yi；P0＝P(X0)；
Figure FDA0003346160650000024
Figure FDA0003346160650000025
S2.2, defining an error function B (P) with weight:
Figure FDA0003346160650000026
wherein, ω isi＝ω(Δxi,Δyi) Is a weighting function, which is a quartic spline function:
Figure FDA0003346160650000027
Figure FDA0003346160650000031
in the above formula: r isiIs node xiEuclidean distance to a node; r ismIs node X0An influence domain radius;
s2.3, to B (P) with respect to DP＝{Px0,Py0,Pxx0,Pyy0,Pxy0The partial derivatives are calculated so that for each divisionThe partial derivative of the quantity is equal to zero:
Figure FDA0003346160650000032
s2.4, rewriting the above formula (7) into a matrix form:
ADP＝b
(8)
s2.5, the right-end term matrix b of the above equation (8) can be decomposed into:
b＝B5×(n+1)P(n+1)×1,
(9)
s2.6, by the above formulas (7) - (9), center node X0The partial derivatives of (a) are:
DP＝[Px0,Py0,Pxx0,Pyy0,Pxy0]T＝A-1b＝A-1BP＝M5×(n+1)P(n+1)×1.
(10)
s2.7, marking the element of the matrix M as MijDeriving the point x from the matrix M0Other node pairs within the influence domain of node x0Partial derivatives of (a):
Figure FDA0003346160650000033
Figure FDA0003346160650000034
Figure FDA0003346160650000035
Figure FDA0003346160650000036
Figure FDA0003346160650000041
4. the method for simulating anisotropic reservoir seepage based on the generalized finite difference method of claim 3, wherein the calculating the partial derivatives of other nodes at each node in the influence domain of each node obtained in S1 and S2 in S3 specifically comprises:
s3.1, obtaining an approximate functional form of the permeability tensor by interpolation, and obtaining the following discrete format for the node i according to the above equation (3) and equation (11):
Figure FDA0003346160650000042
5. the method for simulating anisotropic reservoir seepage based on the generalized finite difference method of claim 4, wherein the second type of boundary conditions in S3 are as follows:
Figure FDA0003346160650000043
6. the method for simulating anisotropic reservoir seepage based on the generalized finite difference method of claim 2, wherein the step of calculating the discrete format of the fundamental control equation of each node in the manner of harmonic and average coupling of permeability tensor harmonic and mean derivative estimation of each order of central node field variables by the generalized finite difference method in S4 specifically comprises:
s4.1, for the node i, the permeability tensor at the node is
Figure FDA0003346160650000044
Figure FDA0003346160650000051
S4.2, the pressure spread term at node i, i.e., the left side of equation (1) above, can be approximated as:
Figure FDA0003346160650000052
in the above formula (15)
Figure FDA0003346160650000053
Harmonic averaging of node i and node j within its domain of influence:
Figure FDA0003346160650000054
7. the method for simulating anisotropic reservoir seepage based on the generalized finite difference method of claim 6, wherein the second type of boundary conditions in S4 are as follows:
Figure FDA0003346160650000055
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