CN118410260A - Coupling solving method and device for two-phase flow mixed dimension crack model - Google Patents

Abstract

The present invention provides a coupling solution method and device for a two-phase flow mixed-dimensional fracture model, and relates to the technical field of porous media seepage mechanics numerical simulation and electrical digital data processing. The method comprises: establishing an initial model according to the saturation equation and Darcy's law; introducing the total velocity and global pressure, and converting the initial model into a global pressure phase saturation equation; dividing the bounded domains of the rock matrix and the fracture in the coupling matrix, reducing the dimension of the fracture area, and solving the two-phase flow mixed-dimensional fracture model of the porous medium according to the mixed element method and the discontinuous element method and the backward Euler method to obtain a discrete solution. The present invention reduces the dimension of the fracture area, breaks the limitation of meshing, reduces the number of unknowns, and improves the calculation efficiency.

Description

A coupling solution method and device for two-phase flow mixed-dimensional fracture model

Technical Field

The invention relates to the technical field of numerical simulation of porous medium seepage mechanics, and in particular to a coupling solution method and device for a two-phase flow mixed-dimensional fracture model.

Background technique

There are generally various geological structures and fractures with different physical properties in the formation, including rock matrices with mature fractures, which are called fractured porous media. Understanding the flow phenomenon of fluids in fractured porous media, especially two-phase flow, is of great significance in practical applications, including environmental and engineering applications, such as prevention and control of groundwater pollution, disposal of underground nuclear waste, underground storage of carbon dioxide, and improvement of oil recovery. Fractures have very different hydraulic properties compared to rock matrices. Depending on the formation background, fractures can have completely different effects on fluids. They can be barriers to fluid flow or main flow channels for fluids. The physical parameters in the fractures and their surrounding matrices may change by several orders of magnitude. Therefore, the role of fractures in porous media is extremely complex and cannot be simply simulated by homogenization. In addition, the thickness of the fractures is actually very small compared to the size of the study area. Therefore, if the fractures are meshed with explicit same-dimensionality, especially two-phase flow, the meshing units will be very small, resulting in a large-scale solution system and high solution cost.

Summary of the invention

In order to solve the technical problem that it is difficult to solve the digital model of fluid in porous media in the prior art, the embodiment of the present invention provides a coupled solution method and device for a two-phase flow mixed-dimensional fracture model. The technical solution is as follows:

On the one hand, a coupled solution method of a two-phase flow mixed-dimensional fracture model is provided, the method is implemented by a coupled solution device of the two-phase flow mixed-dimensional fracture model, and the method comprises:

According to the saturation equation and Darcy's law, an initial model is established;

Introducing total velocity and global pressure, transforming the initial model into a global pressure-phase saturation equation;

Dividing bounded domains of coupled rock matrix and fractures, reducing and compressing the fracture region, and defining related physical parameters to obtain a mixed dimensional fracture model of two-phase flow in porous media;

The two-phase flow mixed dimension fracture model is spatially discretized according to the mixed element and discontinuous element methods;

The two-phase flow mixed dimensional fracture model is time discretized according to the backward Euler method;

By combining spatial discretization with temporal discretization, a fully discrete format of the two-phase flow mixed-dimensional fracture model is obtained, and its numerical solution is obtained.

Optionally, the initial model is:

;

in, , represents the wetting phase, represents the non-wetting phase, Indicates phase The saturation of represents the Darcy speed, represents the source and sink terms, represents the absolute permeability tensor, represents the migration rate, Indicates pressure, represents density, represents the acceleration due to gravity, represents the porosity, represents the gradient operator, Indicates the crack depth, Represents capillary force.

Optionally, the global pressure phase saturation equation is:

;

in, Indicates the total speed, ; is the wetting phase velocity, is the non-wetting phase velocity, represents the source and sink of the moist phase, represents the non-lubricating source and sink terms and represents the source and sink terms, , ; , represents the integral variable, , ,

, , represents the lubrication pressure, represents the non-lubricating phase pressure, represents the lubrication phase mobility, represents the non-wetting phase mobility, represents the density of the lubricating phase, represents the density of the non-lubricating phase, Indicates saturation, , , , Both represent intermediate variables of operations.

Optionally, dividing the rock matrix region and the fracture region in the porous medium comprises:

Delimiting bounded domains of porous media;

Dividing the bounded domain of the porous medium into a plurality of non-overlapping and connected subdomains;

The subdomain includes a fracture region and a rock matrix region, and the rock matrix region is located on both sides of the fracture region.

Optionally, the two-phase flow mixed-dimensional fracture model of porous media includes:

Rock matrix area The saturation equation is:

;

Rock matrix area The pressure equation is:

;

The crack region dimension reduction compression surface The saturation equation is:

;

The crack area is compressed by reducing the dimension The pressure equation is:

;

In the interface Transmission conditions:

;

;

Saturation initial conditions:

, , , , , ;

in, represents the divergence operator, represents overlapping and interconnected subdomains in the rock matrix, , represents the crack area, ; , Indicates perpendicular to At the point The unit component in the direction is given by direction , Indicates cracks The dimension reduction surface of express The external normal component of express The external normal component of , Indicates the boundary, Representing Boundaries and The intersection of .

Optionally, the spatial discretization of the two-phase flow mixed dimensional fracture model according to the mixed element and discontinuous element method to obtain the spatial discretization form of the two-phase flow mixed dimensional fracture model includes:

The interface transport condition is taken as the homogeneous boundary of the mixed-dimensional fracture model of two-phase flow in porous media;

The rock matrix region and the fracture region are respectively meshed to form mesh units;

Obtaining a set of points and edges contained in the grid unit, thereby defining a Sobolev space;

Set the homogeneous boundary restrictions to Dirichlet boundary conditions;

According to the Sobolev space under the boundary restriction, the weak form of the two-phase flow mixing dimensional fracture model is derived;

The pressure and velocity in the weak form are numerically approximated using the Raviart-Thomas mixed element space, and the saturation is numerically approximated using the discontinuous piecewise polynomial space, thereby obtaining the spatial discrete form of the mixed-dimensional fracture model of two-phase flow in porous media.

Optionally, the rock matrix region and the fracture region are meshed using the same grid or completely independent grids.

On the other hand, a coupling device of a two-phase flow mixed dimensional fracture model is provided, and the device is applied to a coupling solution method of a two-phase flow mixed dimensional fracture model, and the device comprises:

An initial module, used to establish an initial model based on the saturation equation and Darcy's law;

An equation calculation module, used for introducing total velocity and global pressure, and converting the initial model into a global pressure phase saturation equation;

A model generation module is used to divide the rock matrix region and the fracture region in the porous medium, and reduce the dimension of the fracture region to obtain the multidimensional physical function of the rock matrix and the reduced dimension physical function of the fracture region;

A spatial discretization module is used to perform spatial discretization on the two-phase flow mixed dimensional fracture model according to a mixed element and a discontinuous element method to obtain a spatial discretization form of the two-phase flow mixed dimensional fracture model;

A time discretization module is used to perform time discretization on the two-phase flow mixed dimensional fracture model according to the backward Euler method to obtain a spatial discretization form of the two-phase flow mixed dimensional fracture model;

The solution module combines the spatial discrete form and the temporal discrete form of the two-phase flow mixed dimensional fracture model to obtain the full discrete format of the two-phase flow mixed dimensional fracture model and obtain the numerical solution of the two-phase flow mixed dimensional fracture model.

On the other hand, a coupling device for a two-phase flow mixed dimensional fracture model is provided, and the coupling device for the two-phase flow mixed dimensional fracture model comprises: a processor; a memory, wherein the memory stores computer-readable instructions, and when the computer-readable instructions are executed by the processor, any one of the coupling solution methods for the two-phase flow mixed dimensional fracture model described above is implemented.

On the other hand, a computer-readable storage medium is provided, wherein at least one instruction is stored in the storage medium, and the at least one instruction is loaded and executed by a processor to implement any one of the above-mentioned coupled solution methods of the two-phase flow mixed-dimensional fracture model.

The beneficial effects brought about by the technical solution provided by the embodiment of the present invention include at least:

The coupling solution method of the two-phase flow mixed-dimensional fracture model provided by the present invention can adopt a mixed-dimensional fracture model. In this type of model, the fracture and the surrounding matrix can be represented separately, and the transmission of fluid between different regions can be described by appropriate coupling conditions. More importantly, the fracture region is reduced in dimensionality, that is, the fracture is regarded as an (n-1)-dimensional interface immersed in an n-dimensional surrounding matrix. This dimensionality reduction method avoids the use of very fine grids in numerical simulations to capture a very thin n-dimensional region, breaks the restrictions on grid division, reduces the number of unknowns, and greatly improves calculation efficiency.

The coupled solution method of the two-phase flow mixed-dimensional fracture model provided by the present invention adopts a coupled algorithm of finite element and saturation finite volume method, considers discontinuous element method, and improves the stability and convergence of discrete solution.

The present invention numerically simulates an incompressible two-phase flow mixed-dimensional fracture model. The model is not only about multi-region coupling of fractures and surrounding matrix, but also about multi-equation coupling of pressure and saturation in each region, filling the gap in the technical field of numerical simulation of incompressible two-phase flow mixed-dimensional fracture models.

The coupled solution method of the two-phase flow mixed-dimensional fracture model provided by the present invention uses mixed element and discontinuous element methods to discretize the pressure equation and the saturation equation respectively, and uses backward Euler time discretization to form a full discrete format, and analyzes the stability and error estimation of the numerical solution, thereby improving the calculation flexibility.

The coupled solution method of the two-phase flow mixed-dimensional fracture model provided by the present invention can be extended to a non-matching grid by using a coupled algorithm of a mixed element and a discontinuous element method, wherein the fracture and the surrounding matrix can be divided separately, that is, a non-matching grid can be used at the interface between the fracture and the matrix. That is, by using a non-matching grid, the stability and optimal order convergence of the algorithm remain unchanged, and the application range is wider.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings required for use in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

FIG1 is a flow chart of a coupled solution method for a two-phase flow mixed-dimensional fracture model provided by an embodiment of the present invention;

FIG2 is a schematic diagram of bounded domain division of rock matrix and fractures in a coupling matrix provided by an embodiment of the present invention;

FIG3 is a schematic diagram of a mixed dimensional model of rock matrix and fractures in a coupled matrix provided by an embodiment of the present invention.

FIG4 is a block diagram of a coupling device for a two-phase flow mixed-dimensional fracture model provided by an embodiment of the present invention;

FIG5 is a schematic diagram of the structure of a coupling device for a two-phase flow mixed-dimensional fracture model provided by an embodiment of the present invention.

Detailed ways

The technical solution of the present invention is described below in conjunction with the accompanying drawings.

In the embodiments of the present invention, words such as "exemplarily" and "for example" are used to indicate examples, illustrations or explanations. Any embodiment or design described as "example" in the present invention should not be interpreted as being more preferred or more advantageous than other embodiments or designs. Specifically, the use of the word "example" is intended to present the concept in a specific way. In addition, in the embodiments of the present invention, the meaning expressed by "and/or" can be both, or it can be either of the two.

In the embodiments of the present invention, "image" and "picture" can sometimes be used interchangeably. It should be noted that when the difference between them is not emphasized, the meanings they intend to express are the same. "of", "corresponding, relevant" and "corresponding" can sometimes be used interchangeably. It should be noted that when the difference between them is not emphasized, the meanings they intend to express are the same.

In the embodiments of the present invention, sometimes a subscript such as _W1 may be expressed in a non-subscript form such as W1. When the difference is not emphasized, the meanings to be expressed are consistent.

In order to make the technical problems, technical solutions and advantages to be solved by the present invention more clear, a detailed description will be given below with reference to the accompanying drawings and specific embodiments.

The embodiment of the present invention provides a coupling solution method for a two-phase flow mixed dimensional fracture model, which can be implemented by a coupling device for the two-phase flow mixed dimensional fracture model, and the coupling device for the two-phase flow mixed dimensional fracture model can be a terminal or a server. As shown in FIG1 , the flow chart of the coupling solution method for the two-phase flow mixed dimensional fracture model, the processing flow of the method may include the following steps:

Step S110, establish an initial model according to the saturation equation and Darcy's law; the incompressible two-phase flow model in porous media is composed of the saturation equation and Darcy's law, as well as the saturation constraint and capillary force relationship, which is called the "phase pressure, phase saturation" formula. The specific porous initial model is:

(1);

in, , represents the wetting phase, represents the non-wetting phase, Indicates phase The saturation of represents the Darcy speed, represents the source and sink terms, represents the absolute permeability tensor, represents the migration rate, Indicates pressure, represents density, represents the acceleration due to gravity, represents the porosity, represents the gradient operator, Indicates the crack depth, In this embodiment, the incompressible two-phase flow model in porous media is mainly described.

Step S120, introduce the total velocity and global pressure, and convert the initial model into the global pressure phase saturation equation; that is, the "global pressure, phase saturation" formula. The specific conversion process is: select the saturation of the wetting phase As the main saturation unknown, the total speed is introduced , which is the sum of the two Darcy velocities of the wetting phase and the non-wetting phase:

(2);

From equation (1) in step S110, i.e. the first and third equations in the initial model, we can obtain

(3);

By direct calculation The expression can be written as:

(4);

in, , , , , .

The global pressure is defined as:

,in, (5);

The global pressure is not a physical quantity, but is used to form a Darcy-type equation with the total velocity:

,in, (6);

According to the above definition, the equivalent global pressure and phase saturation equation is:

(7);

in, Indicates the total speed, ; is the wetting phase velocity, is the non-wetting phase velocity, represents the source and sink of the moist phase, represents the non-lubricating source and sink terms and represents the source and sink terms, , ; , represents the integral variable, , , , , represents the lubrication pressure, represents the non-lubricating phase pressure, represents the lubrication phase mobility, represents the non-wetting phase mobility, represents the density of the lubricating phase, represents the density of the non-lubricating phase, Indicates saturation, , , , Both represent intermediate variables of operations.

Step S130, dividing the rock matrix region and the fracture region in the porous medium, and reducing the dimension of the fracture region to obtain the multidimensional physical function of the rock matrix and the reduced-dimensional physical function of the fracture region; bringing the multidimensional physical function of the rock matrix and the reduced-dimensional physical function of the fracture region into the global pressure phase saturation equation to obtain a two-phase flow mixed-dimensional fracture model in the porous medium;

As shown in FIG. 2 , the bounded domains of the rock matrix and the fracture in the coupled matrix are divided. The embodiment of the present invention mainly considers solving the incompressible two-phase flow fracture model in the porous medium, including the following steps:

Step S131, dividing the boundaries of the bounded domains of the rock matrix and the fractures in the coupled matrix; defining the bounded domains Its boundaries ,in, .

Step S132: Divide the bounded domain into multiple non-overlapping and connected regions, including the crack region and the regions on both sides of the crack; specifically, assume that the domain is divided into three non-overlapping and connected sub-domains. , and , where subdomain and represents the rock matrix, Represents a crack.

and , ,

set up express and The intersection of .

Step S133, reducing the dimension of the crack area, specifically including: compared with the size of the entire area, the crack Width is very small. The cracks in this model have a dimension of Surface As shown in Figure 3, can be rewritten as ,in , is perpendicular to At the point The unit vector in the direction, express The external normal component of express The mixed-dimensional fracture model requires the original definition The series of equations on the On, with subdomain and cracks All related physical functions and unknown variables will be expressed as , and To express.

Step S134: Substitute the physical function into the global pressure phase saturation equation to generate a two-phase flow mixed dimensional fracture model for porous media. dimensional equations and cracks The two-phase flow mixed-dimensional fracture model of the dimensional equation can be expressed as: ; : The saturation system includes the following equations:

Rock matrix area The saturation equation is:

(8);

Rock matrix area The pressure equation is:

(9);

Dimensionality reduction compression surface of crack area The saturation equation is:

(10);

Dimensionality reduction and compression of crack area The pressure equation is:

(11);

In the interface Transmission conditions:

;

(12);

Saturation initial condition

, , , , , (13);

in, represents the divergence operator, represents overlapping and interconnected subdomains in the rock matrix, , represents the crack area, ; , is perpendicular to At the point The unit component in the direction is given by direction , Indicates cracks The dimension reduction surface of The above definition and for and The outward normal unit vector of , Indicates a boundary. Symbol ,and represents the tangential component of the gradient operator and the divergence operator; and , and They are The original absolute permeability The fracture equation receives the matrix equation input through the additional source-sink term in the first equation of (10) and (11), which represents the matrix contribution to the fluid in the fracture, while the matrix equation is applied to The transport condition (12) receives the input of the equations on the fracture, which provides the Robin boundary conditions for solving the matrix equations. It is also assumed that, although both the matrix and the fracture are porous media, they belong to different rock types, so not only the porosity and absolute permeability may be different, but also nonlinear functions of saturation (such as capillary forces) may be discontinuous at the interface.

In another embodiment, some assumptions are made for the mixed-dimensional fracture model, which will be used for theoretical analysis:

(H1) Permeability tensor and is a bounded symmetric uniformly positive definite matrix that satisfies the following conditions: there are two constants Make

, , , , , , ,

(H2) Porosity and Uniformly bounded:

;

(H3) Total mobility and , and , uniformly bounded:

;

and and Respectively and Lipschitz bounded, that is, there is a constant that is independent of the variables Make

, ;

, ;

, .

(H4) Function and Uniformly bounded and Lipschitz continuous, that is, there is a constant Independent of variables Make , , , ,

(H5) Function , and , It is a bounded function and a Lipschitz continuous function, which means that there is a constant that is independent of the variable and Make , , , , ,

(H6) and is globally Lipschitz continuous, that is, there is a constant that is independent of the variables Make , ,

The same assumption holds for and This also holds true, that is, there exists a constant , and Make

, , , ,

, .

Step S140, spatially discretizing the two-phase flow mixed-dimensional fracture model of the porous medium according to the mixed element and discontinuous element method to obtain the spatial discretization form of the two-phase flow mixed-dimensional fracture model; specifically, the steps include:

Step S141, the interface transport condition is used as the homogeneous boundary of the two-phase flow mixed dimensional fracture model of the porous medium; specifically, the two-phase flow mixed fracture model (8)-(13) is designed with a full discrete format, in which the pressure equation and the saturation equation are discretized using the mixed element method and the discontinuous element method in space, respectively, and the general notation of Lebesgue space is used. , Sobolev space is the notation for Sobolev space ,in represents the domain, In particular, express , , and the corresponding full norms are expressed as , and , and the semi-norm is expressed as and For simplicity, , Denoted as , , Denoted as and .

definition

, is a Hilbert space, whose norm ;

For the crack model under consideration, the Hilbert space defined under the Robin boundary condition is and :

, (14);

in, , ,

,and , , ,

Its norm is defined as: and ;

;

(15);

Step S142, meshing the rock matrix region to form grid units; performing global division according to the parameters of the grid units, and defining a set of points or edges contained in the grid units of the rock matrix region and the fracture region, and further defining a Sobolev space;

Assumptions and It is a region and Non-degenerate triangular and tetrahedral meshes of is the maximum diameter of the unit, they are at the interface Overlapping, resulting in The only partition of .make It is a global partition of the entire domain. The set of all inner edges or faces of The set of all internal points or edges of and ,and and yes External edges or faces on The set of external points or edges on the The set of all edges or faces on and cracks The set of all points or edges on .

when When, the discontinuous Sobolev space is defined as

;

,

.

The norm is as follows: ,

,

, ;

The whole area The norm is defined as:

;

Define a series of spaces related to time in the domain. Let is the time interval, For an arbitrarily defined space:

,

,

.

set up or For two units and The common inner edge of ,and is defined in Up The external normal vector of exist The average and jump values on .

(16);

or , and or The external normal vectors on them coincide with each other.

Step S143, setting the homogeneous boundary restriction to a homogeneous Dirichlet boundary condition, i.e., a Dirichlet boundary condition, and deriving a weak form of a two-phase flow mixed dimensional fracture model according to the Sobolev space under the restricted boundary condition;

Step S144: numerically approximate the pressure and velocity in the weak form using the Raviart-Thomas mixed element space, and approximate the saturation using discontinuous piecewise polynomials to obtain a two-phase flow mixed dimensional fracture in the porous medium.

The weak form of the two-phase flow mixing dimensional fracture model is derived as follows: without loss of generality, the influence of gravity is ignored and the boundary conditions are restricted to homogeneous Dirichlet boundary conditions, namely, Dirichlet boundary conditions.

First, for the region Multiply the second equation in (9) by the check function , in each unit Integrating above, we get from Green's formula:

.

From the first interface transmission condition of formula (12) and The homogeneous boundary conditions on the surface can be rewritten as:

(17)

Multiply the first equation in (9) by the check function get

(18);

Likewise, in the cracks Above, the following formula is established.

;

(19)

According to formulas (17)-(19), the following forms of velocity and pressure are introduced: , , and ,

(20)

Then, for The saturation equation in (8) is multiplied by the check function , in each unit Inner integral, and using Green's formula, we get

(twenty one);

in, yes The external normal vector of . Based on the two interface transmission conditions in equation (12), we can obtain superior and The relationship between.

(twenty two);

Using equations (8) and the second equation of (22), equation (21) can be rewritten as:

.

right Sum all the units in , and we get ;

(twenty three)

Likewise, in the cracks Above, for any ,

(twenty four)

In the above formula, if the crack is a one-dimensional region, then is the split point. In this case, the fourth term It should be defined in Up , but for simplicity, the previous notation is retained to maintain consistency in this paper.

Based on equations (23)-(24), the following notation is introduced: , , .

(25)

In the above form, take , is a positive constant, and the penalty function is:

;in, ,and

;

Based on equations (17)-(19), (23)-(24), (20) and (25), the weak form of the mixed-dimensional fracture model of two-phase flow can be written as: So that:

(26)

Arbitrary .

Step S150: time discretize the two-phase flow mixed dimensional fracture model of the porous medium according to the backward Euler method to obtain the time discretization form of the two-phase flow mixed dimensional fracture model; in the time discretization process, the time interval is divided into M time intervals of length Let , Indicates at time The full discrete format of equations (9)-(12) is obtained by applying the backward Euler method in time and the MFE-DG method in space.

Step S160: Obtain a discrete solution to the two-phase flow mixed dimensional fracture model in porous media. The solution process uses the MFE-DG algorithm, which uses the Raviart-Thomas mixed element space to numerically approximate the pressure and velocity, and uses a discontinuous piecewise polynomial to approximate the saturation. Assume and Separate and Up Raviart-Thomas space of order. Definition

.

use represents the space of polynomials with a total degree less than or equal to r on the unit K element, then the discontinuous finite element space can be defined as

Specifically, Algorithm 1 (full discrete format: MFE-DG format) is applied to ,given ,

So that:

(27)

Arbitrary .

In a preferred embodiment, the stability of the fully discrete solution is also included, specifically including: assuming that the orders of the Raviart-Thomas finite element and the discontinuous finite element space are and Now, we consider the stability of the proposed fully discrete solution.

Lemma 1. There exists a positive number and The following is established:

, ,in, ;

In addition, there is a constant , so that

, .

Lemma 2. Let and , existence and and Irrelevant constants , so that .

Lemma 3. Let and , existence and and Irrelevant constants , so that

.

Lemma 4. For any , existence and Irrelevant constants , so that

and .

Lemma 5. If Large enough for and , the following formula holds

(28)

Theorem 1. If the penalty parameter big enough, Small enough, then there is

in, and and Same reason.

In a preferred embodiment, this embodiment derives the error estimation of the fully discrete MFE-DG algorithm. Specifically, for pressure and speed , a mapping is introduced in the case of time continuity. Assume that By mapping Projection into MFE space:

(29)

The following error inequality holds

(30)

Lemma 6. There exists a Irrelevant constants , so that

(31)

Lemma 7. If Big enough, yes ,have

(32)

Theorem 2. Assume that in the matrix middle,

Assume that in the crack middle,

Then, if big enough, and is small enough, then there exists a constant , so that

(33)

In a preferred embodiment, the rock matrix region is meshed using the same or completely independent meshes, and the MFE-DG algorithm is extended to non-matching grids, where completely independent meshes are used for the matrix and fractures. Still expressed as and The finite element mesh of , but there is no matching requirement between these regions. Using the same notation as before, the full discretization scheme based on the MFE-DG method on non-matching meshes combined with the backward Euler method can be similarly represented by the same notation as the algorithm for matching meshes. Some integrals on , the integral function in different discretization spaces, such as (Divide the area and Region ), (Divide the area and Region ). The calculation process of the full discrete format is:

Algorithm 2 (Fully Discrete Format: MFE-DG Format for Unmatched Grids)

right ,given ,try to find Make

(34)

For any .

Next, we will conduct a corresponding theoretical analysis of the above algorithm. of - Elliptical form and The theoretical proof of the imf-sup condition, that is, Lemma 1 still holds for mismatched grids. In addition, the conclusion of Lemma 5 can also be obtained in the same way. On this basis, the stability and error estimation of the discrete solution of the MFE-DG algorithm for mismatched grids are as follows.

Theorem 3. If the penalty function big enough, is small enough, then the discrete solution of Algorithm 2 satisfy

(35)

Theorem 4. Under the same assumptions as Theorem 2, we can get the discrete solution The error estimate is as follows:

(36)

FIG4 is a block diagram of a coupling device for a two-phase flow mixed dimensional fracture model according to an exemplary embodiment, and the device is used for a coupling solution method for a two-phase flow mixed dimensional fracture model. Referring to FIG4 , the device includes an initial module 310, an equation calculation module 320, a model generation module 330, a space discretization module 340, a time discretization module 350 and a solution module 360. Among them:

The initial module 310 is used to establish an initial model according to the saturation equation and Darcy's law;

The equation calculation module 320 is used to introduce the total velocity and global pressure to convert the initial model into a global pressure phase saturation equation;

The model generation module 330 is used to divide the bounded domains of the rock matrix and the fractures in the coupled matrix and define the relevant physical parameters to obtain a two-phase flow mixed dimensional fracture model of the porous medium;

The spatial discretization module 340 is used to perform spatial discretization on the two-phase flow mixed dimensional fracture model of the porous medium according to the mixed element and discontinuous element methods;

The time discretization module 350 is used to perform time discretization on the two-phase flow mixed dimensional fracture model of the porous medium according to the backward Euler method;

The solution module 360 is used to solve and obtain a discrete solution of the two-phase flow mixed-dimensional fracture model of the porous medium.

FIG5 is a schematic diagram of the structure of a coupling device of a two-phase flow mixed dimensional fracture model provided by an embodiment of the present invention. As shown in FIG5, the coupling device of the two-phase flow mixed dimensional fracture model may include the coupling device of the two-phase flow mixed dimensional fracture model shown in FIG3. Optionally, the coupling device 410 of the two-phase flow mixed dimensional fracture model may include a first processor 2001.

Optionally, the coupling device 410 of the two-phase flow mixed-dimensional fracture model may further include a memory 2002 and a transceiver 2003 .

The first processor 2001, the memory 2002 and the transceiver 2003 may be connected via a communication bus.

The following is a detailed introduction to the various components of the coupling device 410 of the two-phase flow mixed dimensional fracture model in conjunction with FIG. 5 :

The first processor 2001 is the control center of the coupling device 410 of the two-phase flow mixed dimensional fracture model, and can be a processor or a general term for multiple processing elements. For example, the first processor 2001 is one or more central processing units (CPUs), or an application specific integrated circuit (ASIC), or one or more integrated circuits configured to implement the embodiment of the present invention, such as one or more microprocessors (digital signal processors, DSPs), or one or more field programmable gate arrays (field programmable gate arrays, FPGAs).

Optionally, the first processor 2001 may execute various functions of the coupling device 410 of the two-phase flow mixed dimensional fracture model by running or executing a software program stored in the memory 2002 and calling data stored in the memory 2002 .

In a specific implementation, as an embodiment, the first processor 2001 may include one or more CPUs, such as CPU0 and CPU1 shown in FIG. 5 .

In a specific implementation, as an embodiment, the coupling device 410 of the two-phase flow mixed dimensional fracture model may also include multiple processors, such as the first processor 2001 and the second processor 2004 shown in FIG5 . Each of these processors may be a single-core processor (single-CPU) or a multi-core processor (multi-CPU). The processor here may refer to one or more devices, circuits, and/or processing cores for processing data (such as computer program instructions).

The memory 2002 is used to store the software program for executing the solution of the present invention, and is controlled to be executed by the first processor 2001. The specific implementation method can refer to the above method embodiment, which will not be repeated here.

Optionally, the memory 2002 may be a read-only memory (ROM) or other types of static storage devices capable of storing static information and instructions, a random access memory (RAM) or other types of dynamic storage devices capable of storing information and instructions, or an electrically erasable programmable read-only memory (EEPROM), a compact disc read-only memory (CD-ROM) or other optical disc storage, optical disc storage (including compressed optical disc, laser disc, optical disc, digital versatile disc, Blu-ray disc, etc.), a magnetic disk storage medium or other magnetic storage device, or any other medium capable of carrying or storing the desired program code in the form of an instruction or data structure and capable of being accessed by a computer, but not limited thereto. The memory 2002 may be integrated with the first processor 2001, or may exist independently, and be coupled to the first processor 2001 through an interface circuit (not shown in FIG. 5 ) of the coupling device 410 of the two-phase flow mixed dimensional fracture model, which is not specifically limited in the embodiment of the present invention.

The transceiver 2003 is used to communicate with a network device or a terminal device.

Optionally, the transceiver 2003 may include a receiver and a transmitter (not shown separately in FIG5 ), wherein the receiver is used to implement a receiving function, and the transmitter is used to implement a sending function.

Optionally, the transceiver 2003 can be integrated with the first processor 2001, or can exist independently and be coupled to the first processor 2001 through the interface circuit (not shown in FIG. 5 ) of the coupling device 410 of the two-phase flow mixed dimensional fracture model, which is not specifically limited in this embodiment of the present invention.

It should be noted that the structure of the coupling device 410 of the two-phase flow mixed dimensional fracture model shown in Figure 5 does not constitute a limitation on the router. The actual knowledge structure identification device may include more or fewer components than shown in the figure, or combine certain components, or arrange the components differently.

In addition, the technical effects of the coupling device 410 for the two-phase flow mixed dimensional fracture model can refer to the technical effects of the coupling method for the two-phase flow mixed dimensional fracture model described in the above method embodiment, which will not be repeated here.

It should be understood that the first processor 2001 in the embodiment of the present invention may be a central processing unit (CPU), and the processor may also be other general-purpose processors, digital signal processors (DSP), application specific integrated circuits (ASIC), field programmable gate arrays (FPGA) or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor may be a microprocessor or the processor may also be any conventional processor, etc.

It should also be understood that the memory in the embodiments of the present invention may be a volatile memory or a non-volatile memory, or may include both volatile and non-volatile memories. Among them, the non-volatile memory may be a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), or a flash memory. The volatile memory may be a random access memory (RAM), which is used as an external cache. By way of example and not limitation, many forms of random access memory (RAM) are available, such as static RAM (SRAM), dynamic random access memory (DRAM), synchronous DRAM (SDRAM), double data rate synchronous dynamic random access memory (DDR SDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synchronous link DRAM (SLDRAM), and direct rambus RAM (DR RAM).

The above embodiments can be implemented in whole or in part by software, hardware (such as circuits), firmware or any other combination. When implemented by software, the above embodiments can be implemented in whole or in part in the form of a computer program product. The computer program product includes one or more computer instructions or computer programs. When the computer instructions or computer programs are loaded or executed on a computer, the process or function described in the embodiment of the present invention is generated in whole or in part. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium, or transmitted from one computer-readable storage medium to another computer-readable storage medium. For example, the computer instructions can be transmitted from one website, computer, server or data center to another website, computer, server or data center by wired (such as infrared, wireless, microwave, etc.). The computer-readable storage medium can be any available medium that can be accessed by a computer or a data storage device such as a server or data center that contains one or more available media sets. The available medium can be a magnetic medium (for example, a floppy disk, a hard disk, a tape), an optical medium (for example, a DVD), or a semiconductor medium. The semiconductor medium can be a solid-state hard disk.

It should be understood that the term "and/or" in this article is only a description of the association relationship of associated objects, indicating that there can be three relationships. For example, A and/or B can represent: A exists alone, A and B exist at the same time, and B exists alone. A and B can be singular or plural. In addition, the character "/" in this article generally indicates that the associated objects before and after are in an "or" relationship, but it may also indicate an "and/or" relationship. Please refer to the context for specific understanding.

In the present invention, "at least one" means one or more, and "plurality" means two or more. "At least one of the following" or similar expressions refers to any combination of these items, including any combination of single items or plural items. For example, at least one of a, b, or c can mean: a, b, c, a-b, a-c, b-c, or a-b-c, where a, b, and c can be single or multiple.

It should be understood that in various embodiments of the present invention, the size of the serial numbers of the above-mentioned processes does not mean the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of the present invention.

Those of ordinary skill in the art will appreciate that the units and algorithm steps of each example described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Professional and technical personnel can use different methods to implement the described functions for each specific application, but such implementation should not be considered to be beyond the scope of the present invention.

Those skilled in the art can clearly understand that, for the convenience and brevity of description, the specific working processes of the above-described equipment, devices and units can refer to the corresponding processes in the aforementioned method embodiments and will not be repeated here.

In the several embodiments provided by the present invention, it should be understood that the disclosed devices, apparatuses and methods can be implemented in other ways. For example, the device embodiments described above are only schematic. For example, the division of the units is only a logical function division. There may be other division methods in actual implementation, such as multiple units or components can be combined or integrated into another device, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some interfaces, indirect coupling or communication connection of devices or units, which can be electrical, mechanical or other forms.

The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed on multiple network units. Some or all of the units may be selected according to actual needs to achieve the purpose of the solution of this embodiment.

In addition, each functional unit in each embodiment of the present invention may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit.

If the functions are implemented in the form of software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art or the part of the technical solution, can be embodied in the form of a software product. The computer software product is stored in a storage medium, including several instructions for a computer device (which can be a personal computer, a server, or a network device, etc.) to perform all or part of the steps of the methods described in various embodiments of the present invention. The aforementioned storage medium includes: U disk, mobile hard disk, read-only memory (ROM), random access memory (RAM), disk or optical disk, etc., various media that can store program codes.

The above is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art who is familiar with the technical field can easily think of changes or substitutions within the technical scope disclosed by the present invention, which should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims (10)

1. A method for solving a coupling of a two-phase flow mixed dimension fracture model, the method comprising:

establishing an initial model according to a saturation equation and Darcy's law;

introducing a total speed and global pressure, and converting the initial model into a global pressure phase saturation equation;

Dividing a rock matrix region and a crack region in a porous medium, and performing dimension reduction compression on the crack region to obtain a multidimensional physical function of the rock matrix and a dimension reduction physical function of the crack region;

bringing the multidimensional physical function of the rock matrix and the dimensionality reduction physical function of the fracture region into the global pressure phase saturation equation to obtain a two-phase flow mixed dimensionality fracture model in a porous medium;

Performing spatial dispersion on the two-phase flow mixing dimension crack model according to a mixing element and intermittent element method to obtain a spatial dispersion form of the two-phase flow mixing dimension crack model;

performing time dispersion on the two-phase flow mixing dimension crack model according to a backward Euler method to obtain a time dispersion form of the two-phase flow mixing dimension crack model;

And combining the space discrete form and the time discrete form of the two-phase flow mixing dimension crack model to obtain a full discrete format of the two-phase flow mixing dimension crack model, and obtaining a numerical solution of the two-phase flow mixing dimension crack model.

2. The method for solving the coupling of the two-phase flow mixing dimension fracture model according to claim 1, wherein the initial model is:

；

Wherein, ，Indicating the wetting phase of the liquid phase,Indicating that the non-wetting phase is present,Representing a phaseIs used for the saturation level of (a),The speed of the darcy is indicated,Representing the source sink item(s),Representing the absolute permeability tensor,The mobility is indicated as being the ratio of the mobility,The pressure is indicated as such and is,The density is indicated by the term "density",Indicating the acceleration of gravity and,The porosity is indicated by the fact that,The gradient operator is represented by a gradient operator,Indicating the depth of the crack,Representing capillary force.

3. The coupling solution method of the two-phase flow mixed dimension fracture model according to claim 2, wherein the global pressure phase saturation equation is:

；

Wherein, Which is indicative of the overall speed of the vehicle,；Indicating the velocity of the wetting phase,Indicating the non-wetting phase velocity of the liquid,Represents the source and sink items of the moist phase,Representing non-wetting source sink itemsRepresenting the source sink item(s),，；，The integral variable is represented by a value of the integral variable,,,

,，The pressure of the moist phase is expressed,The non-wetting phase pressure is indicated by the expression,Indicating the mobility of the wet phase,Indicating the mobility of the non-wetting phase,The density of the wet phase is expressed,The density of the non-wetting phase is indicated,The saturation level is indicated and the saturation level,、、、All represent operational intermediate variables.

4. The method of claim 3, wherein the dividing the rock matrix region and the fracture region in the porous medium comprises:

dividing a bounded domain of a porous medium;

dividing the bounded domain of the porous medium into a plurality of non-overlapping and communicating sub-domains;

The subdomain includes a fracture region and a rock matrix region, the rock matrix region being located on either side of the fracture region.

5. The method of claim 4, wherein the two-phase flow mixing dimension fracture model of the porous medium comprises:

Rock matrix region The saturation equation for (2) is:

；

Rock matrix region The pressure equation of (2) is:

；

dimension-reducing compression curved surface of crack area The saturation equation for (2) is:

；

dimension reduction compression of the fracture region The pressure equation of (2) is:

；

At the interface Transmission conditions:

；

；

Saturation initial conditions:

，，，，，;

Wherein, The operator of the degree of divergence is represented,Representing overlapping and communicating sub-fields in the rock matrix,，Indicating the area of the crack(s),；，The representation being perpendicular toAt the pointA unit component in a direction defined byPointing to，Representing cracksIs characterized by that on the curved surface of said curved surface,Representation ofIs added to the outer normal component of (c),Representation ofIs characterized by an external phase component of (2),，The boundary is represented by a representation of the boundary,Representing boundariesAnd Is a complex of the two.

6. The method for solving the coupling of two-phase flow mixed dimension fracture model according to claim 5, wherein the performing spatial dispersion on the two-phase flow mixed dimension fracture model according to the mixed element and intermittent element method to obtain a spatial dispersion form of the two-phase flow mixed dimension fracture model comprises:

Taking the interface transmission condition as a homogeneous boundary of a two-phase flow mixing dimension crack model of the porous medium;

Respectively carrying out grid subdivision on the rock matrix area and the crack area to form grid units;

Acquiring a set of points and edges contained in the grid unit, and further defining a Sobber's space;

setting homogeneous boundary limit as Dirichlet boundary condition;

deducing a weak form of the two-phase flow mixed dimension fracture model according to the Sobber space under the boundary limit;

And (3) performing numerical approximation on the pressure and the speed in the weak form by utilizing Raviart-Thomas mixed element space, and performing numerical approximation on the saturation by utilizing discontinuous slicing polynomial space to obtain a space discrete form of the two-phase flow mixed dimension crack model in the porous medium.

7. The method of claim 6, wherein the rock matrix region and the fracture region are meshing with the same mesh or meshing with completely independent meshes.

8. A coupling device of a two-phase flow mixed dimension fracture model, the coupling device of the two-phase flow mixed dimension fracture model being used for realizing the coupling solving method of the two-phase flow mixed dimension fracture model according to any one of claims 1 to 7, the device comprising:

The initial module is used for establishing an initial model according to a saturation equation and Darcy's law;

The equation calculation module is used for introducing the total speed and the global pressure and converting the initial model into a global pressure phase saturation equation;

The model generation module is used for dividing a rock matrix area and a crack area in the porous medium, and carrying out dimension reduction compression on the crack area to obtain a multidimensional physical function of the rock matrix and a dimension reduction physical function of the crack area;

The space discrete module is used for performing space discrete on the two-phase flow mixing dimension crack model according to a mixing element and intermittent element method to obtain a space discrete form of the two-phase flow mixing dimension crack model;

the time discrete module is used for performing time discrete on the two-phase flow mixed dimension crack model according to a backward Euler method to obtain a space discrete form of the two-phase flow mixed dimension crack model;

And the solving module is used for combining the space discrete form and the time discrete form of the two-phase flow mixing dimension crack model to obtain a full discrete format of the two-phase flow mixing dimension crack model and solving a numerical solution of the two-phase flow mixing dimension crack model.

9. A coupling device for a two-phase flow mixed-dimensional fracture model, the coupling device comprising:

A processor;

a memory having stored thereon computer readable instructions which, when executed by the processor, implement the method of any of claims 1 to 7.

10. A computer readable storage medium having stored therein program code which is callable by a processor to perform the method of any one of claims 1 to 7.