Method and system for realizing porous medium fluid radial flow numerical simulation by acidification model based on DBF framework

Abstract

The invention relates to a method and a system for realizing porous medium fluid radial flow numerical simulation based on an acidification model of a DBF framework. The method comprises the following steps: step 1, constructing an acidification model for describing fluid flow, solute reaction transportation and rock property change; step 2, carrying out mesh generation on simulation time and a circular ring type solving region; defining different variables of the acidification model at different positions of the grid units; 3, performing numerical discretization on the acidification model constructed in the step 1 to form a large linear equation set; and finally, solving by combining set parameters to realize numerical simulation of the acid etched wormhole. The invention provides a staggered mesh finite difference method for realizing porous medium fluid radial flow numerical simulation based on an acidification model of a DBF framework, numerical discretization and efficient solution are performed on the acidification model, accurate numerical simulation of a wormhole acid etching process is realized, and an important guiding effect is played on oil and gas exploitation.

Description

Technical Field

[0001] The present invention relates to a method and system for numerically simulating the radial flow of porous media fluid based on an acidification model of a DBF framework, belonging to the technical field of numerical simulation of acid-etched wormholes in carbonate matrix acidification. Background Art

[0002] Acidification treatment is an enhanced oil recovery technology widely used in the petroleum industry, aiming to increase the effective connectivity between the wellbore and the carbonate reservoir, and then efficiently extract the oil in the reservoir to the surface. Matrix acidification is considered an effective and efficient acidification technology for carbonate formations, which can increase the permeability and porosity of carbonate rocks, improve the acidification efficiency, and at the same time increase the oil and gas production.

[0003] Acid-etched wormholes refer to injecting acid fluid into the wellbore. After the acid flows into the carbonate reservoir, it reacts with the minerals in the formation, changes the rock structure, increases the permeability and porosity of the carbonate rock, and then generates a series of irregular holes in the shape of earthworms. As high-porosity diversion channels, acid-etched wormholes can change the flow characteristics of porous media fluid, improve the acidification efficiency, and at the same time increase the oil and gas production. In addition, the growth trajectory and trend of the wormholes are closely related to the effect of carbonate matrix acidification. Therefore, in order to achieve a good acidification effect, it is crucial to numerically simulate the process of acid-etched wormholes, and thus guide the oil and gas production work to a certain extent.

[0004] At present, most of the numerical simulations of acid-etched wormholes are achieved by solving the acidification model based on the Darcy framework in the Cartesian coordinate system. Kou, Sun and others proposed a globally conservative mixed finite element method for solving the acidification model based on the Darcy-Forchheimer framework in the Cartesian coordinate system in [Comput. Methods Appl. Mech. Engrg., 2016, 298: 279-302]. Li and Rui proposed a block-centered finite difference method for solving the compressible acidification model based on the Darcy framework in the Cartesian coordinate system in [J. Sci. Comput., 2018, 74:1115-1145] and a globally conservative finite difference method for simulating the Darcy-Brinkman-Forchheimer acidification model in the Cartesian coordinate system in [J. Fluid Mech., 2019, 872: 438-471]. Yang et al. proposed a non-linear complementary simulator with CPR preconditioner for simulating the acid-etched wormhole process based on the Darcy-Brinkman framework in the Cartesian coordinate system in [J. Comput. Phys., 2023, 473]. In addition, Guo et al. proposed a high-order boundary-preserving finite difference method for solving the incompressible acidification model based on the Darcy framework in the Cartesian coordinate system in [J. Sci. Comput., 2021, 89(1)]. Xia Ning et al. gave an algorithmic study of the compressible acidification model based on the Darcy framework in two-dimensional polar coordinates in [Science China Mathematics, 2024, 1-30].

[0005] The acidification model based on the Darcy framework can well describe the acid-etched wormhole process when the porosity does not change significantly; however, once the porosity changes significantly with time evolution, the flow velocity of the fluid in the high-porosity region will increase, and at this time, the acidification model under the Darcy framework is not sufficient to characterize the above acid-etched wormhole process. In order to better achieve the numerical simulation of acid-etched wormholes, it is necessary to establish an acidification model based on the Darcy-Brinkman-Forchheimer framework. At the same time, the actual matrix acidification is to inject acid into a circular wellbore, and the flow of acid from the wellbore to the carbonate reservoir during this process is radial. Therefore, an acidification model of Darcy-Brinkman-Forchheimer in polar coordinates is further established to more accurately simulate the acid-etched wormhole process in actual industrial applications.

[0006] The characteristic of the staggered grid is that different variables can be stored at different positions of the grid. That is, for the established acidification model regarding pressure, velocity, concentration, and porosity, the pressure, concentration, and porosity are defined at the center of the grid cell, while the velocity is defined at the midpoints of the four sides of the grid cell. Using the staggered grid finite difference method to numerically discretize the acidification model in polar coordinates can well maintain the original physical properties of the model, such as mass conservation, momentum conservation, etc., and the numerical solution process is simple and efficient.

[0007] The acid-etched wormhole process can effectively describe the matrix acidification effect. For the radial flow of porous medium fluid in carbonate reservoirs, a practical acidification model is constructed. Using the simple and efficient staggered grid finite difference method to numerically discretize and solve it can effectively improve the numerical simulation accuracy and has certain guiding significance for oil and gas exploitation. Summary of the Invention

[0008] The first aspect of the present invention proposes a method for numerically simulating the radial flow of porous medium fluid based on an acidification model of the DBF framework, including: Step 1: Construct an acidification model for describing fluid flow, solute reaction transport, and rock property changes; Step 2: Perform grid meshing on the simulation time and the circular solution region; then define different variables of the acidification model at different positions of the grid cell; Step 3: Numerically discretize the acidification model constructed in Step 1 to form a large linear equation system; finally, solve it in combination with the set parameters to realize the numerical simulation of acid-etched wormholes.

[0009] Preferably according to the present invention, constructing an acidification model for describing fluid flow, solute reaction transport, and rock property changes includes: Establish a set of acidification models based on the DBF framework in two-dimensional polar coordinates as follows: ; (1) ; (2)

[0010] ; (3) ; (4) Among them, Formulas (1) and (2) represent the fluid flow process based on the DBF framework, that is, the acidification model for describing fluid flow; Formula (1) is a vector equation representing the momentum conservation equation, and the right end of Formula (1): represents the Darcy term, which is used to characterize the Darcy seepage phenomenon of porous media; the second term on the left end of Formula (1): Denotes the Brinkman term, which is used to characterize the transitional flow between boundaries; the last term on the left side of Equation (1): Denotes the Forchheimer term, also known as the inertia term, which is used to describe the significant inertial effect of the fluid when the flow velocity is high; Equation (2) is the mass conservation equation; Equation (3) is the acid concentration reaction transport equation, that is, the acidification model used to describe the solute reaction transport; Equation (4) reflects the change of rock porosity with time evolution, that is, the acidification model used to describe the change of rock properties; Among them, , is the flow radius, with the unit of ; is an annular region; is time, is the final moment, with the unit of ; is the fluid velocity vector, are the velocities of the fluid in the direction of the polar radius and the direction of the polar angle respectively, with the unit of ; , denote the boundaries in the direction of the polar radius ; is the pressure of the fluid, with the unit of ; is the concentration of the acid, with the unit of ; is the porosity of the rock, a dimensionless quantity; the four variables of velocity , pressure , concentration , and porosity are all unknowns; is the density of the fluid, with the unit of ; is the viscosity of the fluid, with the unit of ; is a pseudo-compression coefficient, with the unit of , causes a slight change in the fluid density during the solute dissolution process, denotes a positive number to ensure the invertibility of the coefficient matrix; is the local mass transfer coefficient, with the unit of ; is the injection concentration, with the unit of ; is the acid dissolution capacity, with the unit of ; is the density of the rock, with the unit of ; is the Forchheimer coefficient, a dimensionless quantity; is the permeability of the rock, with the unit of ; , where is the injection rate, is the production rate, with the unit of ; positive definite matrix is the diffusion coefficient of acid in the porous medium, with the unit of , and are in the and directions; for simplicity, assume is a diagonal matrix, where is the molecular diffusion rate, is the identity matrix, represents the diagonal matrix with as the diagonal elements; are all given functions, while is a function of the rock porosity; is the concentration of acid at the interface between the fluid (acid) and the solid (rock), with the unit of . According to the first-order kinetic reaction, the concentration of acid at the fluid-solid interface and the concentration of acid in the fluid have the following relationship: ; (5) where is the surface reaction rate constant, with the unit of ; The pore-scale model depicting the change in rock properties is as follows: ; (6) ; (7) where formula (6) was established by Carman–Kozeny to depict the relationship between the rock porosity and permeability, represents the permeability, and are the initial porosity and initial permeability of the rock, respectively; the porosity and permeability are calculated to obtain , where is the interfacial area (specific surface area) per unit volume of the medium used for the reaction, with the unit of , is the initial interfacial area; The boundary and initial conditions are as follows: (8) Among them, the two conditions in the first line: is the boundary condition, indicating that the gradients of velocity and concentration are both 0 at the inlet and outlet boundaries; the following four are initial conditions. 、 、 、 Represents velocity, pressure, concentration and porosity in the region The initial distribution function within ; The acidification model of the present invention is constructed by combining the acidification model formulas (1)-(4) based on the DBF framework in two-dimensional polar coordinates with the pore-scale model formulas (6)-(7) and the initial boundary condition formula (8). The flow properties of the non-Darcy seepage of porous media in the acidification process of carbonate rock matrix are comprehensively considered. Compared with the existing technology, the acidification model established by the present invention can more accurately describe the acid wormhole process. In order to simplify the equation and the following discrete format writing, define an auxiliary variable ,in, , combined with formulas (2), (5)-(7), the acid concentration reaction transmission formula (3) is simplified and converted into the following: (9).

[0011] According to the present invention, preferably, the simulation time and annular solution region Perform grid division and define different variables of the acidification model at different locations of the grid cells, including: The total simulation time (Final moment) average score The time step is And the first moment ; In order to better simulate the real situation, for a given simulation area That is, the circular solution area is divided into non-uniform grids. The length of the region in the direction is divided into share, The length of the region in the direction is divided into parts, form grid cells, the grid points are: And there is ;remember Direction and The midpoint of the grid cell edge in the direction 、 and the segmentation step size 、 、 、 They are respectively:

[0012] The characteristics of the staggered grid are that different variables can be stored in different positions of the grid cells; for the non - linear strongly coupled acidification model of pressure , velocity , concentration , porosity established in step 1, namely formula (1) - (4), the numerical solutions of pressure , concentration , porosity are defined at the center of the grid cells; the numerical solution of velocity is defined at the mid - points of the four sides of the grid cells. Among them, the component of velocity in the direction, that is , the velocity in the direction is defined at the mid - point of the tangential side of the grid cell, and the component of velocity in the

[0013] direction is defined at the mid - point of the radial side of the grid cell. According to the preference of the present invention, the acidification model constructed in step 1 is numerically discretized to form a large - scale linear equation set; finally, it is solved by combining the set parameters to realize the numerical simulation of acid - etched wormholes; including: <> and having function values at appropriate discrete nodes is represented as , represents at time , the value of the function at the grid point are respectively the coordinates of the grid point in the direction and direction. takes , takes ; denote , where the superscript represents the th time , and the subscript represents the spatial position . In the case of no ambiguity, the superscript is usually omitted; if is a vector function, then on this basis, the superscript is also needed to distinguish which specific direction the component is in; the abbreviations of the remaining variables are similar; gives the symbol definition of replacing the derivative with the difference quotient. Denote , , , , is a function with respect to , , at a certain grid point, and its definition is as follows:

[0014] where , represents approximating the derivative at the midpoint of the grid cell edge with the difference quotient of the function values at the center of the grid cell; , represents approximating the derivative at the center of the grid cell with the difference quotient of the function values at the midpoint of the grid cell edge; The definitions of the interpolation operator and the root mean square operator, including: For the point , assume . It should be noted that the direction is a periodic boundary, , and use to define the following bilinear interpolation operator : When , there is the following two-point extrapolation formula:

[0015] where ; For the discrete function , define a piecewise constant function on ; For the vector function , its component is a pair of discrete functions , and define the interpolation operator as: ; where respectively represent the components of the interpolation operator in the direction and the direction, represents at the point function value, represents at the point function value; Let be the norm function of the vector , and there is ; Define the square root mean operator and , as follows:

[0016] Specifically expressed as:

[0017] Use to represent the corresponding numerical solution, that is, P is the numerical solution of the pressure p, W is the numerical solution of the fluid velocity u, is the concentration of the acid 's numerical solution, Q is the numerical solution of the auxiliary variable q, T is the rock porosity 's numerical solution; where the superscript represents the time level; the velocity and the auxiliary variable are vectors, so the superscript is used to distinguish directions.

[0018] According to the preference of the present invention, numerically discretize the acidification model constructed in step 1 using backward Euler in time and staggered grid finite difference method in space to obtain the staggered grid finite difference format in polar coordinates, namely formulas (10)-(16), including: A. Given and , representing the th moment , the concentration , the porosity at the grid point . According to the equation formula (4) of the porosity changing with time, obtain , as follows: ; (10) where , ; represents the initial moment, and the rock porosity at the grid point ; B. Given , where represents the th moment , the component of the velocity in the direction at the grid point , represents the th moment , the component of the velocity in the direction at the grid point The value at represents the th moment , and the pressure of the fluid at the grid point ; According to the momentum conservation equation formula (1) and mass conservation equation formula (2) describing fluid flow, we obtain as follows: (11) (12) ; (13) where represents the interpolation operator, represents the permeability of the rock, represents the Forchheimer coefficient; C. Given , according to the acid concentration reaction transport equation formula (9) and the definition of the auxiliary variable , we obtain as follows: (14) ; (15) ; (16) where , , are the numerical approximations of respectively; D. Initial and boundary conditions: (17) Combined with the above numerical format, using the initial and boundary conditions given in step D, i.e., formula (17), starting from the time loop is initiated, successively solve the explicit equation, i.e., formula (10), and the large linear equations, i.e., formulas (11)-(13) and (14)-(16), repeat steps A, B, and C these 3 steps until the numerical solutions of the four unknown variables at the final moment are obtained, namely the porosity , pressure , velocity , and concentration . By solving the acidification model using the staggered grid finite difference method to obtain the numerical solutions, the distribution of rock porosity at any discrete time point can be simulated, and then the evolution of rock porosity over time can be plotted.

[0019] A computer device includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, the steps of a numerical simulation method for the radial flow of fluid in a porous medium based on an acidification model of the DBF framework are implemented.

[0020] A computer-readable storage medium stores a computer program. When the computer program is executed by a processor, the steps of a numerical simulation method for the radial flow of fluid in a porous medium based on an acidification model of the DBF framework are implemented.

[0021] The second aspect of the present invention provides a numerical simulation system for the radial flow of fluid in a porous medium based on an acidification model of the DBF framework, including: An acidification model establishment module, configured to: construct an acidification model for describing fluid flow, solute reaction transport, and rock property changes; A grid meshing module, configured to: perform grid meshing on the simulation time and the toroidal solution region; and then define different variables of the acidification model at different positions of the grid cells; A numerical solution module, configured to: numerically discretize the acidification model constructed by the acidification model establishment module to form a large linear equation system; and finally solve it in combination with set parameters to achieve numerical simulation of acid-etched wormholes.

[0022] The beneficial effects of the present invention are as follows: 1. Most existing acidification models are mathematical models based on the Darcy framework in the Cartesian coordinate system. However, considering that during the carbonate matrix acidification process, the porosity of the rock continuously changes with time evolution and the flow direction of the acid fluid flowing into the carbonate matrix through the wellbore is radial; the present invention establishes an acidification model based on the Darcy-Brinkman-Forchheimer framework in the polar coordinate system, which can more accurately depict the flow phenomenon of non-Darcy seepage in the porous medium during the carbonate matrix acidification process, and thus achieve more accurate numerical simulation of acid-etched wormholes, playing an important guiding role in oil and gas exploitation.

[0023] 2. The present invention proposes to use the staggered grid finite difference method to discretely solve the acidification model in polar coordinates. One of its major features is that different variables can be stored at different positions of the grid. This method can not only well maintain the original physical properties of the model, such as mass conservation, momentum conservation, etc., but also maintain the high precision of the four unknowns of porosity, pressure, velocity, and concentration under non-uniform grids, and the numerical solution process is simple and efficient. Description of the Drawings

[0024] Figure 1 It is a framework diagram of the numerical simulation method for the radial flow of fluid in a porous medium based on the acidification model of the DBF framework of the present invention; Figure 2 The non-uniform grid of the present invention when ; Figure 3 When the initial porosity and permeability of the matrix of the present invention follow a uniform distribution and acid is continuously injected into the matrix through a circular wellbore, the distribution diagram of rock porosity at moment; Figure 4 When the initial porosity and permeability of the matrix of the present invention are only large at individual points and acid is continuously injected into the matrix through a circular wellbore, the distribution diagram of rock porosity at moment; Figure 5 When the initial porosity and permeability of the matrix of the present invention follow a uniform distribution and acid is continuously injected into the matrix only through a circular wellbore, the distribution diagram of rock porosity at moment; Figure 6 When the initial porosity and permeability of two small continuous regions of the matrix are very large only along the radial direction and acid is continuously injected into the matrix only through a small-range wellbore near it, the distribution diagram of rock porosity at moment. Detailed implementation manners

[0025] The present invention will be further described below by way of examples in conjunction with the accompanying drawings, but is not limited thereto.

[0026] Example 1 A numerical simulation method for radial flow of fluid in porous media by implementing an acidification model based on the DBF framework, as Figure 1 shown, includes: Step 1: Construct an acidification model for describing fluid flow, solute reaction transport, and rock property changes; Step 2: Perform grid discretization on the simulation time and the circular ring-shaped solution region; then define different variables of the acidification model at different positions of the grid cells; Step 3: Numerically discretize the acidification model constructed in Step 1 to form a large linear equation system; finally, solve it in combination with the set parameters to achieve numerical simulation of acid-etched wormholes.

[0027] Example 2 A numerical simulation method for radial flow of fluid in porous media by implementing an acidification model based on the DBF framework according to Example 1, which is characterized in that: Establish an acidification model for describing fluid flow, solute reaction transport, and rock property changes; including: Establish a group of acidification models based on the DBF framework in two-dimensional polar coordinates, as follows: ; (1) ; (2)

[0028] ; (3) ; (4) Among them, equations (1) and (2) represent the fluid flow process based on the DBF framework, that is, the acidification model for describing fluid flow; equation (1) is a vector equation representing the momentum conservation equation. The right side of equation (1): represents the Darcy term, which is used to characterize the Darcy seepage phenomenon in porous media; the second term on the left side of equation (1): represents the Brinkman term, which is used to characterize the transitional flow between boundaries; the last term on the left side of equation (1): represents the Forchheimer term, also known as the inertial term, which is used to describe the significant inertial effect of the fluid when the flow velocity is high; equation (2) is the mass conservation equation; equation (3) is the acid concentration reaction transport equation, that is, the acidification model for describing solute reaction transport; equation (4) reflects the change of rock porosity with time evolution, that is, the acidification model for describing the change of rock properties; Among them, , is the flow radius, with the unit of ; is an annular region; is time, is the final moment, with the unit of ; is the fluid velocity vector, are the velocities of the fluid in the direction of the polar radius and the direction of the polar angle respectively, with the unit of ; , represent the boundaries in the direction of the polar radius ; is the pressure of the fluid, with the unit of ; is the concentration of the acid, with the unit of ; is the porosity of the rock, a dimensionless quantity; the velocity , pressure , concentration , porosity these four variables are all unknowns; is the density of the fluid, with the unit of ; is the viscosity of the fluid, with the unit of ; is a pseudo-compressibility coefficient, with the unit of , causing a slight change in the fluid density during the solute dissolution process, is a very small positive number to ensure the invertibility of the coefficient matrix; is the local mass transfer coefficient, with the unit of ; is the injection concentration, with the unit of ; is the acid dissolution capacity, with the unit of ; is the density of the rock, with the unit of ; is the Forchheimer coefficient, a dimensionless quantity; is the permeability of the rock, with the unit of ; , where is the injection rate, is the production rate, with the unit of ; positive definite matrix is the diffusion coefficient of the acid in the porous medium, with the unit of , and are in the and directions; for simplicity, assume is a diagonal matrix, where, is the molecular diffusion rate, is the identity matrix, represents a diagonal matrix with as the diagonal elements; are all given functions, while is a function of the rock porosity; is the concentration of the acid at the interface between the fluid (acid) and the solid (rock), with the unit of , according to the first-order kinetic reaction, the concentration of the acid at the fluid-solid interface and the concentration of the acid in the fluid have the following relationship: ; (5) where, is the surface reaction rate constant, with the unit of ; The pore-scale model depicting the change in rock properties is as follows: ; (6) ; (7) Among them, formula (6) was established by Carman–Kozeny to describe the relationship between rock porosity and permeability. represents the permeability, and are the initial porosity and initial permeability of the rock, respectively; the porosity and permeability are calculated as , where is the interfacial area (specific surface area) used for reaction per unit volume of the medium, with the unit of , is the initial interfacial area; The boundary and initial conditions are as follows: (8) Among them, the two conditions in the first line: are boundary conditions, indicating that at the inlet boundary and outlet boundary, the gradients of velocity and concentration are both 0; the following four are all initial conditions, , , , represent the initial distribution functions of velocity, pressure, concentration, and porosity in the region respectively; By combining the acidification model formulas (1)-(4) based on the DBF framework in two-dimensional polar coordinates with the pore-scale model formulas (6)-(7) and the initial and boundary condition formula (8), the acidification model of the present invention is constructed; considering the flow properties of non-Darcy seepage in porous media during the carbonate matrix acidification process, compared with the prior art, the acidification model established by the present invention can more accurately describe the acid-etched wormhole process; To simplify the equation and the writing of the following discrete format, an auxiliary variable is defined, where . Then, combined with formulas (2), (5)-(7), the acid concentration reaction transport formula (3) is simplified and transformed into the following: (9).

[0029] For the simulation time and the annular solution region are meshed; then different variables of the acidification model are defined at different positions of the grid cells; including: The total simulation time (final time) is evenly divided into parts, then the time step and the th moment ; To better simulate the actual situation, for the given simulation region That is, the circular solution region is meshed non-uniformly. The length of the region in the direction is divided into parts, and the length of the region in the direction is divided into parts, forming grid cells. The grid point coordinates are: and ; Denote the midpoints of the edges of the grid cells in the , directions, as well as the step lengths , , , as:

[0030] The characteristic of the staggered grid is that different variables can be stored at different positions in the grid cells. For the strongly coupled non-linear acidification model of pressure , velocity , concentration , porosity established in Step 1, i.e., Equations (1)-(4), define the numerical solutions of pressure , concentration , porosity at the centers of the grid cells; define the numerical solution of velocity at the midpoints of the four edges of the grid cells. Among them, the component of velocity in the direction, i.e., velocity, is defined at the midpoint of the tangential edge of the grid cell, and the component of velocity in the direction is defined at the midpoint of the radial edge of the grid cell; Numerically discretize the acidification model constructed in Step 1 to form a large linear equation system; finally, solve it in combination with the set parameters to achieve numerical simulation of acid-etched wormholes, including: To simplify the writing of subsequent discretization formats, first define the variable simplification rules, including: The independent variable is and the discrete function with function values at appropriate discrete nodes is denoted as , represents at time the value of the function at the grid point . Among them, are the coordinates of the grid point in the directions respectively, and takes , Take ; Denote , where the superscript represents the th moment , and the subscript represents the spatial position . In the case of no ambiguity, the superscript is usually omitted ; If is a vector function, then on this basis, the superscript is also needed to distinguish which component in which direction; the abbreviations of other variables are similar; Give the symbol definition of replacing the derivative with the difference quotient. Denote , , , , as the derivative of the function with respect to , , at a certain grid point, and its definition is as follows:

[0031] where , represent approximating the derivative at the midpoint of the grid cell edge with the difference quotient of the function value at the center of the grid cell; , represent approximating the derivative at the center of the grid cell with the difference quotient of the function value at the midpoint of the grid cell edge; Definitions of the interpolation operator and the root mean square operator, including: For the point , assume . It should be noted that the direction is the periodic boundary, . Define the following bilinear interpolation operator with the value of : When , there is the following two-point extrapolation formula:

[0032] where ; For the discrete function , define a piecewise constant function on that satisfies: ; Then, for the vector function , its component is a pair of discrete functions . Define the interpolation operator As follows: ; Among them, respectively represent the components of the interpolation operator in the direction and direction, represents at the point the function value at, represents at the point the function value at; Let be the norm function of the vector , and there is ; Define the square root mean operator and , as shown in the following formula:

[0033] Specifically expressed as:

[0034] Use to represent the corresponding numerical solution, that is, P is the numerical solution of the pressure p, W is the numerical solution of the fluid velocity u, is the numerical solution of the acid concentration , Q is the numerical solution of the auxiliary variable q, T is the numerical solution of the rock porosity ; among them, the superscript represents the time layer; the velocity and the auxiliary variable are vectors, so the superscript is used to distinguish the direction; Use backward Euler in time and staggered grid finite difference method in space to numerically discretize the acidification model constructed in step 1, and obtain the staggered grid finite difference format in polar coordinates, namely formulas (10)-(16), including: A. Given and , representing the th moment , the concentration , the porosity at the grid point , according to the equation formula (4) of the porosity changing with time, obtain , as shown below: ; (10) Among them, , ; Denote the initial moment, the rock porosity at the grid point . B. Given , where denotes the th moment , the component of the velocity in the direction at the grid point , denotes the th moment , the component of the velocity in the direction at the grid point , denotes the th moment , the pressure of the fluid at the grid point ; According to the momentum conservation equation formula (1) and mass conservation equation formula (2) describing fluid flow, obtain as follows: (11) (12) ; (13) where denotes the interpolation operator, denotes the permeability of the rock, denotes the Forchheimer coefficient; C. Given , according to the acid concentration reaction transport equation formula (9) and the definition of the auxiliary variable , obtain as follows: (14) ; (15) ; (16) where , , are the numerical approximations of respectively; D. Initial and boundary conditions: (17) Combining the above numerical format, using the initial and boundary conditions given in step D, i.e., formula (17), start the time loop from ​ Solve the explicit equation, namely formula (10), and the large linear equations, namely formulas (11)-(13) and (14)-(16), in sequence. Repeat the three steps A, B, and C until the numerical solutions of the four unknown variables at the final time are obtained, namely the porosity , pressure , velocity , and concentration . By solving the acidizing model using the staggered grid finite difference method to obtain the numerical solutions, the distribution of the rock porosity at any discrete time point can be simulated, and then the evolution of the rock porosity over time can be plotted; The numerical simulation of acid-etched wormholes in carbonate rock matrix acidizing is realized through Step 1, Step 2, and Step 3; based on the flow characteristics of non-Darcy seepage in porous media, a non-linear strongly coupled acidizing model of Darcy-Brinkman-Forchheimer in a two-dimensional polar coordinate system is established, and it is numerically discretized and efficiently solved using the staggered grid finite difference method to achieve a more practical and accurate numerical simulation of the acid-etched wormhole process.

[0035] To verify the accuracy of the acidizing model established by the present invention, the feasibility of the numerical solution algorithm, and the numerical simulation effect, four numerical experiments are listed, and a program is written using computer software for numerical calculation and simulation. First, Table 1 shows some physical parameters required for the numerical simulation and their values. Then, different initial porosities, permeability distributions, and different acid injection ranges are given in the four numerical examples, and the numerical simulation effects all verify the effectiveness of the technical solution of the present invention.

[0036] Table 1

[0037] In the numerical simulation of acid-etched wormhole reservoirs, let the simulation time interval be , about 46 days in total, the time step be , the radius of the simulation area be , the radius of the wellbore be , that is , and a non-uniform grid of cells be adopted, as shown in Figure 2 . Each small quadrilateral represents a grid cell, and all the vertices of the grid cells are called grid points. Let the initial pressure be , the initial concentration be 0. The initial permeability follows a uniform distribution in the range , and the initial porosity follows a uniform distribution in the range , and the injection and production rates of the acid are as follows:

[0038]

[0039] When the initial porosity and initial permeability of the carbonate rock matrix follow a uniform distribution, acid is continuously injected into the carbonate rock matrix through a circular wellbore. The acid reacts chemically with the rock, changing the rock structure. As the matrix acidification process progresses, the porosity and permeability of the rock increase. Since the resistance formed by the high-porosity region to the fluid is smaller, the acid preferentially flows into this region, forming randomly distributed high-conductivity channels (dominant wormholes with protruding branches). The numerical simulation effect of acid-etched wormholes is as Figure 3 shown.

[0040] Example 3 In this example, the simulated time interval is set as , and about 24 days are simulated in total. The time step is . The radius of the simulation area is , and the wellbore radius is , that is, , and a non-uniform grid of cells is adopted, as shown in Figure 2. The initial pressure is set as , and the initial concentration is 0. The initial porosity, permeability, and injection and production rates are as follows:

[0041]

[0042]

[0043] When the initial porosity and permeability of the carbonate rock matrix are only large at individual points, acid is continuously injected into the carbonate rock matrix through a circular wellbore. As the acidification time evolves, the acid extends radially outward faster at several points with larger initial porosity and permeability, and then forms long and narrow high-conductivity channels. This trend conforms to the expected effect. The numerical simulation effect of acid-etched wormholes is as Figure 4 shown.

[0044] Example 4 In this example, the simulated time interval is set as , and about 58 days are simulated in total. The time step is . The radius of the simulation area is , and the wellbore radius is , that is, , and a non-uniform grid of cells is adopted, as shown in Figure 2. The initial pressure is set as , and the initial concentration is 0. The initial permeability follows a range of Uniform distribution, the initial porosity follows a uniform distribution within the range Uniform distribution, and the injection and production rates of the acid are as follows:

[0045]

[0046] When the initial porosity and permeability of the carbonate rock matrix follow a uniform distribution, if only through A circular wellbore continuously injects acid into the carbonate rock matrix. Then, as the acidification time evolves, the acid diffuses continuously towards the surroundings and preferentially flows into the high-porosity regions. Since the acid extends radially outward faster in the high-porosity regions, randomly distributed high-conductivity channels (dominant wormholes with protruding branches) are gradually formed. The numerical simulation effect of acid-etched wormholes is as Figure 5 shown.

[0047] Example 5 In this example, let the simulation time interval be , and a total of about 92 days are simulated. The time step is . The radius of the simulation area is , and the radius of the wellbore is , that is , and a non-uniform grid of cells is adopted, as shown in Figure 2. Set the initial pressure to be , the initial concentration is 0. The initial porosity, permeability, and injection and production rates are as follows:

[0048]

[0049]

[0050] When the initial porosity and permeability of the carbonate rock matrix are relatively large in only two continuous regions along the radial direction, similar to the existence of two small fractures in the carbonate rock matrix. At this time, if acid is continuously injected into the matrix with the above characteristics only through a small-range wellbore near the two, since the porosity and permeability at the fractures are very large, equivalent to pipelines, they will play a certain role in guiding the flow. Then, the acid preferentially flows into this region. As the acidification time evolves, a large amount of acid flows into these two high-porosity channels and forms prominent branches. The final numerical simulation effect of acid-etched wormholes is as Figure 6 shown.

[0051] The numerical simulation results of the above examples effectively illustrate the accuracy and feasibility of the acidification model and numerical solution method of the present invention. Therefore, the technical solution of the present invention can not only accurately perform numerical simulation on the acid-etched wormhole process of carbonate rock matrix acidification, but also play an important guiding role in oil and gas exploitation.

[0052] Example 6 A computer device, comprising a memory and a processor, where the memory stores a computer program, and when the processor executes the computer program, it implements the steps of the method for numerically simulating the radial flow of porous medium fluid based on the acidification model of the DBF framework described in any one of Embodiments 1-5.

[0053] Embodiment 7 A computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, it implements the steps of the method for numerically simulating the radial flow of porous medium fluid based on the acidification model of the DBF framework described in any one of Embodiments 1-5.

[0054] Embodiment 8 A numerical simulation system for the radial flow of porous medium fluid based on the acidification model of the DBF framework, comprising: An acidification model establishment module, configured to: construct an acidification model for describing fluid flow, solute reaction transport, and rock property changes; A grid meshing module, configured to: perform grid meshing on the simulation time and the annular solution region; and then define different variables of the acidification model at different positions of the grid cells; A numerical solution module, configured to: numerically discretize the acidification model constructed by the acidification model establishment module to form a large linear equation set; and finally solve it in combination with the set parameters to achieve numerical simulation of acid-etched wormholes.

Claims

1. A numerical simulation method for the radial flow of fluid in porous media by implementing an acidification model based on the DBF framework, characterized in that, include: Step 1: Construct an acidification model to describe fluid flow, solute reaction transport, and rock property changes; Step 2: Divide the simulation time and the annular solution area into grids; then define different variables of the acidification model at different locations of the grid cells; Step 3: Numerically discretize the acidification model constructed in step 1 to form a large linear equation system; finally, solve it in combination with the set parameters to realize the numerical simulation of acid wormholes.

2. A numerical simulation method for realizing the radial flow of fluid in porous media based on an acidification model of the DBF framework according to claim 1, characterized in that, Construct acidizing models to describe fluid flow, solute reaction transport, and rock property changes; include: A set of acidification models based on DBF framework in two-dimensional polar coordinates is established as follows: ； (1) ； (2) ； (3) ； (4) Among them, Equations (1) and (2) represent the fluid flow process based on the DBF framework, that is, the acidification model for describing fluid flow; Equation (1) is a vector equation representing the momentum conservation equation, and the right end of Equation (1): represents the Darcy term, which is used to characterize the Darcy seepage phenomenon in porous media; the second term on the left end of Equation (1): represents the Brinkman term, which is used to characterize the transitional flow between boundaries; the last term on the left end of Equation (1): represents the Forchheimer term, which is used to describe the significant inertial effect of the fluid when the flow rate is high; Equation (2) is the mass conservation equation; Equation (3) is the acid concentration reaction transport equation, that is, the acidification model for describing solute reaction transport; Equation (4) reflects the change of rock porosity with time evolution, that is, the acidification model for describing the change of rock properties; Among them, , is the flow radius, with the unit of ; is the flow angle; is an annular region; is the time, is the final moment, with the unit of ; is the fluid velocity vector, are respectively the velocity of the fluid in the direction of the polar radius and the polar angle direction, with the unit of ; , represent the boundaries in the direction of the polar radius ; is the fluid pressure, with the unit of ; is the concentration of the acid, with the unit of ; is the porosity of the rock, a dimensionless quantity; the velocity , pressure , concentration , porosity these four variables are all unknowns; is the density of the fluid, with the unit of ; is the viscosity of the fluid, with the unit of ; is a pseudo - compressibility coefficient, with the unit of , represents a positive number, is the local mass transfer coefficient, with the unit of ; is the injection concentration, with the unit of ; is the corrosion ability of the acid, with the unit of ; is the density of the rock, with the unit of ; is the Forchheimer coefficient, a dimensionless quantity; is the permeability of the rock, with the unit of ; , where is the injection rate, is the production rate, with the unit of ; positive definite matrix is the diffusion coefficient of the acid in the porous medium, with the unit of , and are in​​​​​​​​​​​​​​​​​​​​​​​​​ and the component in the is a diagonal matrix, where is the molecular diffusion rate, is the identity matrix, denotes the diagonal matrix with as the diagonal elements; is the concentration of acid at the fluid-solid interface, with the unit of , according to the first-order kinetic reaction, the concentration of acid at the fluid-solid interface and the concentration of acid in the fluid have the following relationship: ； (5) Among them, is the surface reaction rate constant, with the unit of ; The pore-scale model that describes the changes in rock properties is as follows: ； (6) ； (7) Among them, formula (6) is used to describe the relationship between rock porosity and permeability. represents the permeability, and are the initial porosity and initial permeability of the rock, respectively; the porosity and permeability are calculated as , where is the interfacial area used for reaction per unit volume of the medium, with the unit of , is the initial interfacial area. The boundary and initial conditions are as follows: (8) Among them, the two conditions in the first row: are boundary conditions, indicating that at the inlet boundary and the outlet boundary, the gradients of velocity and concentration are both 0; the following four are all initial conditions, , , , respectively represent the initial distribution functions of velocity, pressure, concentration, and porosity in the region ; The acidification model formulas (1)-(4) based on the DBF framework in two-dimensional polar coordinates are combined with the pore-scale model formulas (6)-(7) and the initial boundary condition formula (8) to form the acidification model; Define an auxiliary variable , where , combined with formulas (2), (5)-(7), the acid concentration reaction transport formula (3) is simplified and transformed as follows: (9)。 3. A numerical simulation method for realizing the radial flow of porous medium fluid based on an acidification model of a DBF framework according to claim 2, characterized in that, For the simulation time and the annular solution region perform mesh generation; Then define different variables of the acidification model at different locations of the grid cells; including: Divide the total simulation time evenly into parts, then the time step and the th moment ; For the given simulation region i.e., perform non-uniform grid division on the toroidal solution region, divide the region length in the direction into parts, divide the region length in the direction into grid cells, and the grid points are: and there are ; Denote the midpoints of the grid cell edges in the direction and the direction , as well as the step sizes of the division , , , are respectively: For the non-linear strongly coupled acidification model of pressure , velocity , concentration , porosity established in step 1, i.e., formulas (1)-(4), the numerical solutions of pressure , concentration , porosity are defined at the centers of grid cells; the numerical solutions of velocity are defined at the midpoints of the four sides of grid cells. Among them, the component of velocity in the direction, i.e., -direction velocity, is defined at the midpoint of the tangential side of the grid cell, and the component of velocity in the direction is defined at the midpoint of the radial side of the grid cell.

4. A numerical simulation method for realizing the radial flow of fluid in porous media based on an acidification model of a DBF framework according to claim 3, characterized in that The acidification model constructed in step 1 is numerically discretized to form a large linear equation system; finally, the solution is combined with the set parameters to realize the numerical simulation of acid wormholes; including: Define the variable simplification rule, and represent the discrete function with independent variables as and having function values at appropriate discrete nodes as , denote as the moment, and the function at the grid point ; where are respectively the coordinates of the grid point in the direction and the direction, takes , takes ; denote , where the superscript represents the th moment , and the subscript represents the spatial position ; Give the symbolic definition of replacing the derivative with the difference quotient, denoted as 、 、 、 、 are the derivatives of the function with respect to 、 、 at a certain grid point, and their definitions are as follows: Among them, , represent using the difference quotient of the function value at the center of the grid cell to approximate the derivative at the midpoint of the grid cell edge; , represent using the difference quotient of the function value at the midpoint of the grid cell edge to approximate the derivative at the center of the grid cell; Definitions of interpolation operators and square root mean operators, including: For a point assume , the direction is a periodic boundary , and use to define the following bilinear interpolation operator as follows: When , there are the following two extrapolation formulas: Among them, ; For a discrete function , define a piecewise constant function on that satisfies: ; For a vector function , its components are a pair of discrete functions , and define an interpolation operator as follows: ； Among them, respectively represent the interpolation operator in the direction and the component in the direction, represents the function value at the point ; represents the function value at the point ; Let be the norm function of the vector , and there is ; Define the square root mean operator and , as shown in the following formula: Specifically expressed as: Use to represent the corresponding numerical solutions, i.e., P is the numerical solution of pressure p, W is the numerical solution of fluid velocity u, is the numerical solution of the concentration of acid , is the numerical solution of the auxiliary variable , T is the numerical solution of rock porosity ; among them, the superscript represents the time level; the velocity and the auxiliary variable are vectors, so the superscript is used to distinguish directions.

5. A numerical simulation method for realizing the radial flow of fluid in porous media based on an acidification model of the DBF framework according to claim 4, characterized in that, The acidification model constructed in step 1 is numerically discretized using the backward Euler method in time and the staggered grid finite difference method in space, and the staggered grid finite difference format in polar coordinates is obtained, including: A. Known and , representing the th moment , the concentration , porosity at the grid point . According to the equation formula (4) for the change of porosity with time, is obtained as follows: ； (10) Among them, , ; represents the initial moment, and the rock porosity at the grid point value at the location; B. Known , where represents the th moment , the component of the velocity with respect to the direction at the grid point ; represents the th moment , the component of the velocity with respect to the direction at the grid point ; represents the th moment , the pressure of the fluid at the grid point ; According to the momentum conservation equation formula (1) and the mass conservation equation formula (2) that describe fluid flow, we obtain , as follows: (11) (12) ； (13) Among them, represents an interpolation operator, represents the permeability of the rock, represents the Forchheimer coefficient; C. Known , according to the acid concentration reaction transport equation formula (9) and the definition of the auxiliary variable , we obtain as follows: (14) ； (15) ； (16) Among them, , , are respectively numerical approximations of . D. Initial boundary conditions: (17) Combined with the above numerical format, using the initial and boundary conditions given in step D, namely formula (17), starting from the time loop is started, successively solve the explicit equation, namely formula (10), and the large linear equations, namely formulas (11)-(13) and (14)-(16), and repeat the three steps A, B, and C until the numerical solutions of the four unknown variables at the final moment are obtained, namely the porosity , pressure , velocity , and concentration .

6. A computer device, comprising a memory and a processor, the memory storing a computer program, characterized in that, When the processor executes the computer program, the steps of the method for realizing numerical simulation of radial flow of porous media fluid by using the acidification model based on the DBF framework as described in any one of claims 1 to 5 are implemented.

7. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, the steps of the method for realizing numerical simulation of radial flow of porous media fluid by using an acidification model based on a DBF framework as described in any one of claims 1 to 5 are implemented.

8. A numerical simulation system for radial flow of fluid in porous media based on an acidification model of the DBF framework, characterized in that, include: The acidizing model building module is configured to: build an acidizing model for describing fluid flow, solute reaction transport and rock property changes; The gridding module is configured to: grid the simulation time and the annular solution area; and then define different variables of the acidification model at different locations of the grid cells; The numerical solution module is configured to: numerically discretize the acidification model constructed by the acidification model establishment module to form a large linear equation group; and finally solve the set parameters to realize the numerical simulation of acid corrosion wormholes.

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