A full tensor calculation method of equivalent permeability of fractured media based on EDFM

Abstract

The application discloses a kind of equivalent permeability full tensor calculation method of fractured medium based on EDFM, comprising the following steps: S1: generating random cracks, construct fractured medium physical model;S2: based on EDFM, matrix grid and crack grid are divided, and grid basic parameters are given;S3: based on EDFM, construct fractured medium percolation model;S4: adopt three kinds of boundary condition loading mode, solve the pressure distribution in whole calculation domain;S5: calculate pressure gradient and flow velocity distribution, and calculate the average pressure gradient and average flow velocity in whole calculation domain;S6: calculate equivalent permeability full tensor matrix, and it is symmetrically handled, obtain the calculation result.The application can solve the problem that the crack grid form is complex or the number of cracks is too much in fractured medium, resulting in low calculation efficiency and high calculation cost, and the calculated equivalent permeability full tensor can well represent the heterogeneity and anisotropy of fractured medium, providing technical support for the development of fractured medium.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of fractured medium development, in particular to a full tensor calculation method of equivalent permeability of fractured medium based on EDFM. BACKGROUND

[0002] Fractured medium is difficult to simulate due to containing base rock porosity and rich fracture structure, multiple medium types, large scale difference and complex flow law. The commonly used mathematical model is an equivalent continuous medium model. The equivalent continuous medium model is used to replace the original fractured medium body for simulation by solving the equivalent permeability tensor of the matrix grid block containing fractures.

[0003] Numerical simulation of oil and gas reservoirs is a commonly used means to evaluate reservoir dynamics and can be used to solve seepage or engineering problems in complex situations. However, the simulation scale is large, the grid scale is generally tens to hundreds of meters, and a grid may contain many natural fractures. It is obviously difficult to display and process a large number of natural fractures in the entire oil and gas reservoir, and the calculation cost is too high. Therefore, a certain method is adopted to calculate the comprehensive equivalent permeability of the matrix and the fracture in the unit, which can reduce the calculation cost of numerical simulation of oil and gas reservoirs.

[0004] In the calculation of comprehensive equivalent permeability, the fractures need to be characterized in the unit. Common fracture characterization methods include discrete fracture model, boundary element method, double medium method, etc. Among them, the discrete fracture model method is the most popular. Since the DFM method directly calculates the conduction between the matrix and the fracture when calculating the full tensor of equivalent permeability, fine grid division is required. The DFM method divides the matrix and the fracture by using irregular grids. Especially when dealing with fractures, the fracture needs to be treated as an independent grid unit, resulting in high calculation cost when calculating the full tensor of equivalent permeability. SUMMARY

[0005] In view of the above problems, the present application aims to provide a full tensor calculation method of equivalent permeability of fractured medium based on EDFM.

[0006] The technical scheme of the present application is as follows:

[0007] A full tensor calculation method of equivalent permeability of fractured medium based on EDFM, comprising the following steps:

[0008] S1: generating random fractures to construct a physical model of fractured medium;

[0009] S2: dividing matrix grids and fracture grids based on EDFM on the physical model of fractured medium and assigning grid basic parameters;

[0010] S3: constructing a fractured medium percolation model based on EDFM;

[0011] S4: taking three types of boundary condition loading modes, and respectively carrying out steady percolation numerical simulation according to the fractured medium percolation model to obtain pressure distribution in the whole calculation domain;

[0012] S5: respectively calculating pressure gradient and flow velocity distribution according to the pressure distribution in the whole calculation domain, and respectively calculating average pressure gradient and average flow velocity in the whole calculation domain through volume-weighted average;

[0013] S6: obtaining an equivalent permeability full tensor matrix based on the generalized Darcy law, and performing symmetry processing on the equivalent permeability full tensor matrix to obtain an EDFM-based equivalent permeability full tensor calculation result of the fractured medium.

[0014] Preferably, in step S1, when the random fractures are generated, random generation is performed according to geological statistical analysis of fracture attribute parameters and representative statistical characteristics of the fracture attribute parameters.

[0015] Preferably, in step S3, the fractured medium percolation model includes a flow control equation and a crossflow term calculation equation.

[0016] The flow control equation is:

[0017] (1)

[0018] (2)

[0019] In the formula, φm and φf are matrix and fracture porosities, respectively, and are dimensionless. , ρ is density, t is time, ; represents a divergence operator, and is dimensionless. , vm and vf are percolation velocities in the matrix and the fracture, respectively, ; a is a fracture aperture, ; , , , are crossflow terms between fracture cells and matrix cells and between matrix cells and fracture cells, respectively, ; is a crossflow term between fracture cells , , ;

[0020] The crossflow term calculation equation is:

[0021] (3)

[0022] (4)

[0023] where: is the matrix element is the internal fracture element is the length of the fracture element, ; is the average distance between the matrix element and the fracture element ; ; is the matrix permeability, ; is the viscosity of the fluid within the grid, ; , are the matrix and fracture pressures, respectively, ; , are the interporosity flow coefficients for the fracture element and the fracture element , respectively, dimensionless; , are the pressures for the fracture element and the fracture element , respectively, ;

[0024] The interporosity flow coefficients for the fracture element and the fracture element are calculated by:

[0025] (5)

[0026] where: is the aperture of the fracture element ; ; is the total flow capacity of the fracture element , dimensionless; is the spacing of the fracture element, .

[0027] Preferably, in step S4, the three types of boundary condition loading modes refer to setting boundary conditions on the corresponding two faces in the x, y, and z directions, respectively.

[0028] In the x direction, the corresponding two faces are yz and y'z'; in the y direction, the corresponding two faces are xz and x'z'; and in the z direction, the corresponding two faces are xy and x'y';

[0029] The boundary condition is that a higher pressure I is applied along one direction on one of the corresponding faces, a lower pressure II is applied on the corresponding other face, the matrix mesh of the other four faces serves as a closed boundary, and the pressure II is applied on the other four crack element faces.

[0030] Preferably, in step S5, the velocity distribution is calculated using the following formula:

[0031] (6)

[0032] (7)

[0033] In the formula: subscript When the numbers are equal to 1, 2, or 3, These represent the x, y, and z directions, respectively. , These represent the seepage velocities of the matrix and fracture elements in the x, y, and z directions of the fractured medium in a Cartesian coordinate system. ; For crack permeability, ; , and These represent the pressure gradients of the matrix along the x, y, and z directions of the grid, respectively. ; , and These represent the pressure gradients along the cracks in the x, y, and z directions of the mesh, respectively. ;

[0034] The average pressure gradient and average flow velocity over the entire computational domain are calculated using the following formulas:

[0035] (8)

[0036] (9)

[0037] In the formula: The average pressure gradient over the entire computational domain of the fractured medium. ; The average flow velocity of the fractured medium throughout the entire computational domain. ; The total volume of all grids, ; The summation symbol indicates that all elements within the computational domain are summed. For the volume of each mesh cell within the model, .

[0038] As preferred, in step S6, the equivalent permeability full tensor matrix is calculated by the following formula:

[0039]

[0040] In the formula, the upper indexes I, II, III respectively represent simulation calculation results under loading of three types of boundary conditions; is the equivalent permeability tensor, , the lower indexes or take 1, 2, 3 respectively representing x, y, z three directions;

[0041] The equivalent permeability full tensor matrix is symmetrically processed by the following formula:

[0042] (11)

[0043] (12)

[0044] In the formula, .

[0045] The present application has the following beneficial effects:

[0046] The present application solves the problem of low calculation efficiency and high calculation cost caused by complex grid shape or too many cracks in the fractured medium by solving the equivalent permeability full tensor of the fractured medium based on the EDFM (embedded discrete fracture model) method, dividing the structure grid, avoiding explicit processing of a large number of natural cracks, thereby reducing the calculation cost of the numerical simulation of the oil and gas reservoir, and the calculated equivalent permeability full tensor can well represent the heterogeneity and anisotropy of the fractured medium, thereby providing technical support for the development of the fractured medium. BRIEF DESCRIPTION OF DRAWINGS

[0047] In order to more clearly illustrate the technical solutions of the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments or the prior art description will be briefly introduced. Obviously, the drawings in the following description only some embodiments of the present application, and for those skilled in the art, other drawings can be obtained without creative labor.

[0048] Figure 1 The flowchart of the equivalent permeability full tensor calculation method of the fractured medium based on the EDFM of the present application is shown in the figure.

[0049] Figure 2 The schematic diagram of the boundary condition loading of the present application is shown in the figure.

[0050] Figure 3 The schematic diagram of the grid division result of the physical model of the fractured medium in one specific implementation is shown in the figure.

[0051] Figure 4 Fig. 2 is a schematic diagram of pressure distribution results of a fractured medium under a type I boundary condition in one specific embodiment;

[0052] Figure 5 Fig. 3 is a schematic diagram of pressure distribution results of a fractured medium under a type II boundary condition in one specific embodiment;

[0053] Figure 6 Fig. 4 is a schematic diagram of pressure distribution results of a fractured medium under a type III boundary condition in one specific embodiment;

[0054] Figure 7 Fig. 5 is a schematic diagram of an ellipsoid of the full tensor of equivalent permeability of a fractured medium in one specific embodiment. DETAILED DESCRIPTION

[0055] The application will be further described below in conjunction with the drawings and embodiments. It should be noted that the embodiments in the present application and the technical features in the embodiments can be combined with each other without conflict. It should be noted that all the technical and scientific terms used in the present application have the same meaning as that generally understood by the ordinary skilled in the art to which the present application belongs, unless otherwise specified. The similar words such as “comprise” or “contain” and the like used in the present application mean that the elements or objects before the words cover the elements or objects listed after the words and their equivalents, and do not exclude other elements or objects.

[0056] As shown in Figure 1 , the present application provides a method for calculating the full tensor of equivalent permeability of a fractured medium based on EDFM, comprising the following steps:

[0057] S1: generating random fractures to construct a physical model of a fractured medium.

[0058] In one specific embodiment, when generating random fractures, random generation is performed according to the statistical analysis of the geological properties of the fractures and the representative statistical characteristics of the fracture properties.

[0059] S2: dividing matrix grids and fracture grids on the basis of EDFM in the physical model of the fractured medium and assigning basic parameters to the grids.

[0060] S3: constructing a seepage model of the fractured medium based on EDFM.

[0061] In one specific embodiment, the seepage model of the fractured medium comprises a flow control equation and a calculation equation of a channeling term.

[0062] The flow control equation is:

[0063] (1)

[0064] (2)

[0065] where, , are the matrix and fracture porosities, dimensionless; is the density, ; is the time, ; represents the divergence operator, dimensionless; , are the seepage velocities in the matrix and fracture, ; is the aperture of the fracture, ; , are the crossflow terms between the fracture element and the matrix element and between the matrix element and the fracture element, ; is the crossflow term between the fracture element , ; ;

[0066] The crossflow term calculation equation is:

[0067] (3)

[0068] (4)

[0069] where, is the length of the fracture element within the matrix element , ; is the average distance between the matrix element and the fracture element , ; is the matrix permeability, ; is the viscosity of the fluid within the grid, ; , are the matrix and fracture pressures, ; , are the crossflow coefficients of the fracture element and the fracture element , dimensionless; , are the pressures of the fracture element and the fracture element , ;

[0070] fracture cell and fracture cell The crossflow coefficient of the fracture cell

[0071] (5)

[0072] wherein: is the aperture of the fracture cell ; ; is the total mobility of the fracture cell , dimensionless; is the spacing of the fracture cell .

[0073] It should be noted that the above crossflow term calculation equation only lists the calculation formula of the crossflow term between the matrix cell and the fracture cell is the crossflow term between the fracture cell and the matrix cell, the value of which is opposite to , that is, a negative sign is added in front of the calculation formula of to obtain the calculation formula of .

[0074] S4: Adopt three types of boundary condition loading modes, and respectively carry out steady seepage numerical simulation according to the seepage model of the fractured medium to solve the pressure distribution in the entire calculation domain.

[0075] In a specific embodiment, as shown in Figure 2 , the three types of boundary condition loading modes refer to setting boundary conditions on the corresponding two faces in the x, y and z directions respectively;

[0076] In the x direction, the corresponding two faces are yz and y'z'; in the y direction, the corresponding two faces are xz and x'z'; and in the z direction, the corresponding two faces are xy and x'y';

[0077] The boundary condition is to apply a higher pressure one on the corresponding one of the faces along one direction, and apply a lower pressure two on the corresponding another face, the remaining four matrix grids are closed boundaries, and the remaining four fracture cell faces are applied with the pressure two.

[0078] In a specific embodiment, the pressure one and the pressure two can be determined according to two points of the target formation, that is, determined according to the target formation pressure gradient.

[0079] S5: According to the pressure distribution in the entire calculation domain, respectively calculate the pressure gradient and the flow velocity distribution, and then respectively calculate the average pressure gradient and the average flow velocity in the entire calculation domain through the volume weighted average method.

[0080] In one specific embodiment, the flow rate distribution is calculated by the following formula:

[0081] (6)

[0082] (7)

[0083] wherein the subscripts equal 1, 2, 3, respectively, represent the x, y, z directions, respectively; , are the seepage velocities of the matrix and fracture elements in the x, y, z directions, respectively, of the fractured medium in the Cartesian coordinate system, ; is the fracture permeability, ; , and are the pressure gradients of the matrix along the grid x, y, z directions, respectively, ; , and are the pressure gradients of the fracture along the grid x, y, z directions, respectively, ;

[0084] The average pressure gradient and the average flow rate in the entire calculation domain are calculated by the following formulas, respectively:

[0085] (8)

[0086] (9)

[0087] wherein: is the average pressure gradient in the entire calculation domain of the fractured medium, ; is the average flow rate in the entire calculation domain of the fractured medium, ; is the total volume of all grids, ; is a summation symbol, indicating summation over all elements in the calculation domain; is the volume of each grid element in the model, .

[0088] S6: An equivalent permeability full tensor matrix is calculated based on the generalized Darcy's law, and the equivalent permeability full tensor matrix is symmetrically processed to obtain an equivalent permeability full tensor calculation result of the fractured medium based on EDFM.

[0089] In one specific embodiment, the equivalent permeability full tensor matrix is calculated by the following formula:

[0090]

[0091] wherein: the upper indexes I, II, III respectively represent the simulation calculation results under the loading of three types of boundary conditions; is an equivalent permeability tensor, , the lower indexes or take 1, 2, 3 respectively represent x, y, z three directions;

[0092] The equivalent permeability full tensor matrix is symmetrically processed by the following formula:

[0093] (11)

[0094] (12)

[0095] wherein: .

[0096] In one specific embodiment, taking a certain reservoir as an example, the fracture medium equivalent permeability full tensor calculation method based on EDFM is used to calculate the fracture medium equivalent permeability full tensor of the reservoir, which specifically includes the following steps:

[0097] (1) Random fractures are generated to construct a fracture medium physical model.

[0098] In this embodiment, random fractures are randomly generated according to the geological statistical analysis of fracture attribute parameters and the representative statistical characteristics of fracture attribute parameters.

[0099] (2) Based on EDFM, matrix grids and fracture grids are divided on the fracture medium physical model, and grid basic parameters are assigned.

[0100] In this embodiment, the grid division result is as shown in Figure 3 The grid basic parameters include reservoir matrix permeability 0.1 mD, matrix porosity 0.25, and fracture permeability 1D.

[0101] (3) Based on EDFM, a fracture medium percolation model shown in formulas (1)-(5) is constructed.

[0102] (4) A three-type boundary condition loading mode is adopted, and the steady-state percolation numerical simulation is carried out according to the fracture medium percolation model to solve the pressure distribution in the entire calculation domain.

[0103] In this embodiment, pressure one and pressure two are determined based on the pressure gradient of the target reservoir, with pressure one being 5 MPa and pressure two being 1 MPa. Type I boundary condition loading is performed along the z-direction, Type II boundary condition loading is performed along the x-direction, and Type III boundary condition loading is performed along the y-direction. The simulated pressure distribution of the fractured medium after loading the three types of boundary conditions onto the fractured medium physical model is as follows. Figures 4-6 As shown. From Figures 4-6 It can be seen that the presence of numerous natural fractures within the fractured medium leads to uneven pressure distribution. When the pressure distribution is uneven, the seepage is affected by the fractures, resulting in uneven permeability distribution. Subsequently, the pressure gradient of the entire computational domain can be obtained by calculating the pressure distribution, thereby calculating the equivalent permeability tensor.

[0104] (5) Based on the pressure distribution in the entire computational domain, calculate the pressure gradient and velocity distribution respectively, and then calculate the average pressure gradient and average velocity in the entire computational domain by using the volume weighted average method.

[0105] In this embodiment, the velocity distribution is calculated using equations (6)-(7), and the average pressure gradient and average velocity within the entire computational domain are calculated using equations (8)-(9).

[0106] (6) The equivalent permeability full tensor matrix is ​​calculated by formula (10) based on the generalized Darcy law, and the equivalent permeability full tensor matrix is ​​symmetrically processed by formula (11)-(12) to obtain the calculation result of the equivalent permeability full tensor of the fractured medium based on EDFM.

[0107] In this embodiment, the calculation result of the equivalent permeability tensor of the fractured medium based on EDFM is as follows:

[0108] (13)

[0109] In fractured media, the direction and velocity of fluid flow can be described by the permeability tensor. In this embodiment, the equivalent permeability tensor of the fractured media calculated based on EDFM is a third-order tensor. By solving the eigenvalue equation of the third-order matrix, three permeabilities are obtained, namely the maximum principal permeability. Minimum main penetration rate and average main penetration rate The graph can be drawn based on the three main permeabilities obtained from the calculation. Figure 7 The full tensor ellipsoid plot of equivalent permeability of fractured media shown below provides a visual representation of the calculation results.

[0110] In summary, the application can quickly obtain the full tensor of equivalent permeability of the fractured medium based on the EDFM, and the non-homogeneity and anisotropy of the fractured medium can be represented by the full tensor. Compared with the prior art, the application has significant progress.

[0111] The above is only the preferred embodiment of the present application, and does not limit the present application in any form. Although the present application has been disclosed as the above preferred embodiment, it is not intended to limit the present application. Any person skilled in the art can make some changes or modifications to the above disclosed technical content to obtain equivalent embodiments with equivalent changes, without departing from the technical solution of the present application. Any simple modification, equivalent change and modification of the above embodiments made according to the technical essence of the present application, without departing from the technical solution of the present application, still belong to the scope of the technical solution of the present application.

Claims

1. An EDFM-based full-tensor calculation method for equivalent permeability of fractured media, characterized in that, The method comprises the following steps: S1: generating random cracks to build a fractured medium physical model; S2: dividing matrix grids and crack grids on the fractured medium physical model based on EDFM and assigning basic parameters to the grids; S3: building a fractured medium percolation model based on EDFM; the fractured medium percolation model comprises a flow control equation and a crossflow term calculation equation; the flow control equation is: （1） （2） In the formula: , These are matrix porosity and fracture porosity, respectively, both dimensionless. For density, ; For time, ; The divergence operator is dimensionless. , These represent the seepage velocities in the matrix and within the cracks, respectively. ; The diameter of the crack. ; , These are the flow terms between fracture elements and matrix elements, and the flow terms between matrix elements and fracture elements, respectively. ; For crack elements , Inter-current terms, ; the crossflow term calculation equation is: （3） （4） where: is the length of the matrix element is the internal fracture element ; is the length of the matrix element is the average distance from the matrix element ; is the matrix permeability ; is the viscosity of the fluid within the grid ; , are the matrix and fracture pressures, respectively ; , are the interporosity flow coefficients, dimensionless, for fracture element and fracture element , respectively , are the pressures for fracture element and fracture element , respectively ;​​ fracture element and the fracture element flowing coefficients are calculated by the following equations: （5） wherein: is the aperture of the fracture element , ; is the total flowability of the fracture element , is the spacing of the fracture element ; S4: adopting a three-type boundary condition loading mode to respectively develop steady-state percolation numerical simulation according to the fractured medium percolation model to obtain pressure distribution in the entire calculation domain; S5: calculating pressure gradient and flow velocity distribution according to the pressure distribution in the entire calculation domain, and then respectively calculating average pressure gradient and average flow velocity in the entire calculation domain through volume-weighted average; S6: obtaining an equivalent permeability full tensor matrix based on the generalized Darcy law, and performing symmetric processing on the equivalent permeability full tensor matrix to obtain an EDFM-based fractured medium equivalent permeability full tensor calculation result.

2. The EDFM-based equivalent permeability full tensor calculation method for fractured media according to claim 1, characterized in that, In step S1, when generating random cracks, random generation is performed according to geological statistical analysis of crack attribute parameters and representative statistical characteristics of the crack attribute parameters.

3. The EDFM-based equivalent permeability full tensor calculation method for fractured media according to claim 1, wherein, In step S4, the three-type boundary condition loading mode refers to setting boundary conditions on two corresponding faces in the x, y and z directions. In the x direction, the two corresponding faces are yz and y'z'; in the y direction, the two corresponding faces are xz and x'z'; and in the z direction, the two corresponding faces are xy and x'y'. The boundary condition is to apply a higher pressure one on one corresponding face along one direction and apply a lower pressure two on the other corresponding face, and the matrix grids of the remaining four faces are closed boundaries, and the remaining four crack cell faces are applied with the pressure two.

4. The EDFM-based equivalent permeability full tensor calculation method for fractured media according to claim 1, wherein, In step S5, the flow velocity distribution is calculated by the following formula: （6） （7） wherein: subscript is equal to 1, 2, 3, respectively represent x, y, z three directions; , respectively are the seepage velocity of matrix and fracture unit in x, y, z three directions of the fractured medium under the Cartesian coordinate system, ; is the fracture permeability, ; , and respectively are the pressure gradient of matrix along the grid x, y, z directions, ; , and respectively are the pressure gradient of fracture along the grid x, y, z directions, ; The average pressure gradient and the average flow velocity in the entire calculation domain are calculated by the following formula: （8） （9） where: is the average pressure gradient over the entire computational domain of the fractured medium, ; is the average flow rate over the entire computational domain of the fractured medium, ; is the total volume of all the meshes, ; is the summation symbol, indicating summation over all elements in the computational domain; is the volume of each grid cell in the model, .

5. The EDFM-based equivalent permeability full-tensor calculation method for fractured media according to claim 4, characterized in that, In step S6, the equivalent permeability full tensor matrix is calculated by the following formula: In the formula, the upper indexes I, II, III respectively represent the simulation results under the loading of three types of boundary conditions; is the equivalent permeability tensor, , the lower indexes or take 1, 2, 3 respectively to represent x, y, z three directions; The equivalent permeability full tensor matrix is symmetrically processed by the following formula: （11） （12） In the formulae: .

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