CN120409129A - A numerical simulation method and system for radial flow of porous media fluid using an acidification model based on DBF framework - Google Patents
A numerical simulation method and system for radial flow of porous media fluid using an acidification model based on DBF frameworkInfo
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Abstract
本发明涉及一种基于DBF框架的酸化模型实现多孔介质流体径向流动数值模拟方法及系统，包括：步骤1：构建用于描述流体流动、溶质反应运输和岩石性质变化的酸化模型；步骤2：对模拟时间和圆环型求解区域进行网格剖分；再将酸化模型的不同变量定义在网格单元的不同位置；步骤3：对步骤1所构建的酸化模型进行数值离散，形成大型线性方程组；最终结合设定参数求解，实现酸蚀蚓孔数值模拟。本发明提供了一种基于DBF框架的酸化模型实现多孔介质流体径向流动数值模拟的交错网格有限差分法，对酸化模型进行数值离散及高效求解，实现对酸蚀蚓孔过程较为精准的数值模拟，进而对油气开采发挥重要的指导作用。
The present invention relates to a method and system for numerical simulation of radial flow of porous media fluids using an acidification model based on a DBF framework, comprising: step 1: constructing an acidification model for describing fluid flow, solute reaction transport, and changes in rock properties; step 2: gridding the simulation time and the annular solution region; and then defining different variables of the acidification model at different positions of the grid units; step 3: numerically discretizing the acidification model constructed in step 1 to form a large linear equation group; and finally solving the equations in combination with the set parameters to realize numerical simulation of acid wormholes. The present invention provides a staggered grid finite difference method for numerical simulation of radial flow of porous media fluids using an acidification model based on a DBF framework, numerically discretizing and efficiently solving the acidification model, and realizing a relatively accurate numerical simulation of the acid wormhole process, thereby playing an important guiding role in oil and gas production.
Description
Technical Field
The invention relates to a method and a system for realizing radial flow numerical simulation of porous medium fluid based on an acidification model of a DBF frame, and belongs to the technical field of numerical simulation of acidizing acid-etched wormholes of carbonate matrixes.
Background
Acidizing is a widely used stimulation technique in the petroleum industry that aims to increase the effective connectivity of a wellbore and a carbonate reservoir, thereby efficiently extracting the oil of the reservoir to the surface. Matrix acidizing is considered to be an effective and efficient technique for acidizing carbonate formations, which increases the permeability and porosity of the carbonate, improves the acidizing efficiency and at the same time increases the production of hydrocarbon production.
The acid etching earthworm holes are formed by injecting acid liquid into a shaft, allowing the acid to react with mineral substances in stratum after flowing into a carbonate reservoir, changing rock structure, increasing permeability and porosity of carbonate, and further generating a series of irregular holes like earthworms. The acid etched earthworm holes are used as high-porosity diversion channels, so that the flow characteristics of porous medium fluid can be changed, the acidification efficiency is improved, and meanwhile, the oil gas yield is improved. In addition, the growth track and trend of wormholes are closely related to the acidification effect of carbonate matrixes, so that numerical simulation of the process of pickling wormholes is important in order to achieve a good acidification effect, and oil gas exploitation work is guided to a certain extent.
At present, numerical simulation of acid-etched wormholes is mostly realized by solving an acidification model based on a Darcy frame under a Cartesian coordinate system. Kou and Sun et al in [ comput. Methods appl. Mech. Engrg., 2016, 298:279-302 ] propose a global conservation mixed finite element method for solving an acidification model based on the Darcy-Forchheimer framework in a cartesian coordinate system. Li and Rui propose in [ J.Sci.Comput., 2018, 74:1115-1145 ] a finite difference method for solving the block center of a Darcy frame-based compressible acidification model in a Cartesian coordinate system and in [ J.fluid Mech., 2019, 872:438-471 ] a global conservation finite difference method for modeling a Darcy-Brinkman-Forchheimer acidification model in a Cartesian coordinate system. Yang et al in J.Comput.Phys., 2023, 473 propose a nonlinear complementary simulator with CPR pre-processor for simulating the Darcy-Brinkman frame acid etching wormhole process in a Cartesian coordinate system. Furthermore, guo et al in [ j.sci.comp., 2021, 89 (1) ] propose a high-order preservation finite difference method for solving an incompressible acidification model based on a Darcy frame in a cartesian coordinate system. Xia Ning et al, in [ science: mathematics, 2024, 1-30], show algorithmic studies of a Darcy frame-based compressible acidification model in two dimensions of polar coordinates.
The acidizing model based on the Darcy frame can well describe the process of acid etching wormholes when the porosity is not changed significantly, however, once the porosity is changed significantly with time, the flow speed of fluid in a high-porosity area is increased, and then the acidizing model under the Darcy frame is insufficient to describe the process of acid etching wormholes. In order to be able to better realize the simulation of the earthworms Kong Shuzhi, it is further necessary to build an acidification model based on the Darcy-Brinkman-Forchheimer framework. Meanwhile, the actual matrix acidification is to inject acid into a circular shaft, and the flow of the acid in the process to a carbonate reservoir through the shaft is radial, so that a Darcy-Brinkman-Forchheimer acidification model under a polar coordinate system is further established to more accurately simulate the acid etching earthworm pore process in actual industrial application.
The staggered grid is characterized in that different variables can be stored at different positions of the grid, namely, the pressure, the concentration and the porosity are defined in the center of the grid unit aiming at the established acidification model related to the pressure, the speed, the concentration and the porosity, and the speed is defined at the midpoints of four sides of the grid unit. The acidizing model under polar coordinates is subjected to numerical discrete by using the staggered grid finite difference method, so that the original physical properties of the model, such as conservation of mass and conservation of momentum, can be well maintained, and the numerical solving process is simple and efficient.
The matrix acidification effect can be effectively described in the acid etching earthworm pore process, a practical acidification model is built aiming at radial flow of porous medium fluid in a carbonate reservoir, numerical dispersion and solution are carried out by adopting a simple and efficient staggered grid finite difference method, the numerical simulation precision can be effectively improved, and the method has a certain guiding significance for oil and gas exploitation.
Disclosure of Invention
The invention provides a method for realizing radial flow numerical simulation of porous medium fluid by an acidification model based on a DBF framework, which comprises the following steps:
Step 1, constructing an acidification model for describing fluid flow, solute reaction transportation and rock property change;
Step 2, meshing the simulation time and the annular solving area, and defining different variables of the acidification model at different positions of the mesh unit;
And 3, performing numerical discrete on the acidizing model constructed in the step 1 to form a large linear equation set, and finally solving by combining with set parameters to realize numerical simulation of the acid etching wormholes.
According to a preferred embodiment of the invention, the construction of an acidification model for describing fluid flow, solute reaction transport and rock property changes comprises:
an acidification model based on a DBF frame under a set of two-dimensional polar coordinates is established as follows:
; (1)
; (2)
; (3)
; (4)
Wherein, the formulas (1) and (2) represent the fluid flow process based on the DBF framework, namely an acidification model for describing the fluid flow, the formula (1) is a vector equation representing a momentum conservation equation, and the right end of the formula (1): representing Darcy term for describing Darcy seepage phenomenon of porous medium, and a second term at left end of formula (1): The term Brinkman is used for describing the transition flow between boundaries, and the last term at the left end of the formula (1): The term Forchheimer, which may also be referred to as an inertia term, is used to describe the significant inertial effects of the fluid when the flow rate is high, equation (2) is a mass conservation equation, equation (3) is an acid concentration reaction transfer equation, i.e., an acidification model used to describe the transport of solute reactions, equation (4) reflects the evolution of rock porosity over time, i.e., an acidification model used to describe the change in rock properties;
Wherein, the ,Is the flow radius in units of;Is a circular ring area; It is the time that is required for the device to be in contact with the substrate, The final time is given in units of;Is a vector of the velocity of the fluid,Velocity of the fluid respectivelyAt the polar diameterDirection and polar angleThe component of the direction in units of;、Representing the polar diameterA boundary of the direction; Is the pressure of the fluid in units of ;Is the acid concentration in unit of;Is the porosity of rock, dimensionless quantity and speedPressure and forceConcentration ofPorosity of the porous bodyAll four variables are unknown; Is the density of the fluid in units of ;Is the viscosity of the fluid in units of;Is a pseudo-compression coefficient in units of,Resulting in a slight change in fluid density during dissolution of the solute,Representing a positive number to ensure that the coefficient matrix is invertible; is a local mass transmission coefficient, and has the unit of ;For the injection concentration, the unit is;Is the acid corrosion capacity in units of;Is the density of rock, the unit is;Is Forchheimer coefficient, dimensionless; is the permeability of rock in units of ;WhereinIn order to achieve the injection rate,For throughput rate, unit isPositive definite matrixIs the diffusion coefficient of acid in porous medium, and has the unit of,AndIs thatAt the position ofAndComponents in the direction, for simplicity, assumeIs a diagonal matrix, wherein,In order to achieve a molecular diffusion rate,Is an identity matrix of the unit cell,Expressed in terms ofA diagonal matrix that is a diagonal element; are all given functions of Is a function of the porosity of the rock;
is the concentration of acid at the interface of fluid (acid) and solid (rock), and has the unit of According to the primary kinetic reaction, the concentration of acid at the fluid-solid interfaceWith the concentration of acid in the fluidThe following relationship exists:
; (5)
Wherein, the Is the surface reaction rate constant, and has the unit of;
The pore scale model characterizing rock property changes is as follows:
; (6)
; (7)
wherein, the formula (6) is established by Carman-Kozeny to characterize the relationship between the rock porosity and permeability, The permeability is indicated as a function of the permeability,AndThe initial porosity and the initial permeability of the rock are calculated respectively, wherein,Is the interfacial area (specific surface area) per unit volume of medium for reaction, in units of,Is the initial interface area;
the boundary and initial conditions are as follows:
(8)
Wherein, two conditions in the first row: Is a boundary condition, indicating that at the entrance boundary and the exit boundary, the gradients of velocity and concentration are 0, the latter four are initial conditions, 、、、Respectively the speed, pressure, concentration and porosity in the regionAn initial distribution function within;
The acidification model formulas (1) - (4) based on the DBF frame in two-dimensional polar coordinates, the pore scale model formulas (6) - (7) and the initial boundary condition formula (8) are combined to form the acidification model, and the flow property of the porous medium non-Darcy seepage in the process of acidifying the carbonate matrix is comprehensively considered;
To simplify the equation and writing in discrete format below, an auxiliary variable is defined , wherein,In combination with formulas (2), (5) - (7), the concentration reaction of acid is simplified to transfer formula (3) and then converted into the following formula:
(9)。
according to the invention, the simulation time is preferably And an annular solution areaPerforming mesh dissection, defining different variables of the acidification model at different positions of the mesh unit, and comprising:
total simulation time Average division into (final moment)Time stepAnd the firstAt a moment ofFor better simulation of real world conditions, for a given simulation areaNamely, the annular solving area is subjected to non-uniform mesh dissection, andThe area length of the direction is divided intoThe weight of the components is calculated by the weight,The area length of the direction is divided intoPart(s) to formThe grid points are: And has Record(s)Direction and directionGrid cell edge midpoint of direction、Step size of subdivision、、、The method comprises the following steps of:
The interlaced grid is characterized in that different variables can be stored in different locations of the grid cells, the pressure is established for step 1 Speed and velocity ofConcentration ofPorosity of the porous bodyThe nonlinear strong coupling acidification model of (1) is expressed by formulas (1) - (4), and the pressure is calculatedConcentration ofPorosity of the porous bodyThe numerical solution of (2) is defined in the center of the grid cell, the speed is determinedIs defined at the midpoint of the four sides of the grid cell, where velocity is related toThe component of direction, i.e.The speed of the direction is defined at the midpoint of the tangential edge of the grid cell, the speed being related toThe component of the direction is defined at the midpoint of the radial edge of the grid cell.
According to the invention, the acidizing model constructed in the step 1 is subjected to numerical discrete to form a large linear equation set, and finally, the acidizing earthworm Kong Shuzhi is simulated by combining with the set parameter solution, and the method comprises the following steps:
to simplify writing in a subsequent discrete format, variable simplification rules are first defined, and the independent variables are defined as And the discrete function with function values at the appropriate discrete nodes is expressed as,Representation ofTime of day, functionAt grid pointsThe value at which the value is to be found, wherein,Respectively grid pointsAt the position ofDirection and directionThe coordinates in the direction of the light are,Taking out,Taking outRecord(s)Wherein the superscript indicates the firstAt a moment ofSubscript indicates spatial positionIn the absence of ambiguity, superscripts can generally be omittedIf (1)Is a vector function, and on the basis of the vector function, a superscript is also neededTo distinguish which direction component is in particular;
Giving a symbolic definition of the derivative replaced by the difference quotient, note 、、、、As a function ofWith respect to、、The derivative at a certain grid point is defined as follows:
Wherein, the 、Representing 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;、 representing approximating the derivative at the center of the grid cell with the difference quotient of the function values at the midpoints of the edges of the grid cell;
Definition of interpolation operator and square root mean operator, including:
For points Assume thatIt is noted that the number of the elements,The direction is the boundary of the period of time,By usingThe values of (2) define the bilinear interpolation operator as follows:
When (when)When there are two extrapolation formulas:
Wherein, the ;
For discrete functionsDefine aThe above piecewise constant function satisfies:;
For vector functions Its componentIs a pair of discrete functionsDefining interpolation operatorsThe method comprises the following steps:
;
Wherein, the Representing interpolation operators, respectivelyAt the position ofDirection and directionThe component in the direction of the light is,Representation ofAt the pointThe function value of the position,Representation ofAt the pointA function value at the location;
Is provided with Is a vectorIs a norm function of (1), and has;
Defining square root mean operatorAndThe following is shown:
the concrete steps are as follows:
By using To representThe corresponding numerical solution, i.e., P is the numerical solution of pressure P, W is the numerical solution of fluid velocity u,Is the acid concentrationIs the numerical solution of the auxiliary variable Q, T is the rock porosityWherein, superscriptRepresenting temporal layers, speed and auxiliary variables are vectors, so superscriptTo distinguish between directions.
According to the present invention, the acidizing model constructed in the step 1 is numerically discretized by using a backward Euler in time and using a staggered grid finite difference method in space to obtain a staggered grid finite difference format in polar coordinates, namely formulas (10) - (16), which comprises:
A. Is known to be AndRepresents the firstAt a moment ofConcentration ofPorosity of the porous bodyAt grid pointsThe value at the point is obtained according to equation (4) of the change of the porosity with timeThe following is shown:
; (10)
Wherein, the ,;Indicating the initial moment, the rock porosityAt grid pointsA value at;
B. is known to be WhereinRepresent the firstAt a moment ofSpeed is related toComponent of directionAt grid pointsThe value at which the value is to be calculated,Represent the firstAt a moment ofSpeed is related toComponent of directionAt grid pointsThe value at which the value is to be calculated,Represent the firstAt a moment ofPressure of fluidAt grid pointsA value at;
obtaining according to a momentum conservation equation (1) and a mass conservation equation (2) describing fluid flow The following is shown:
(11)
(12)
; (13)
Wherein, the The interpolation operator is represented as a function of the interpolation,Indicating the permeability of the rock,Representing Forchheimer coefficients;
C. Is known to be Equation (9) and auxiliary variables are transmitted according to the acid concentration reactionDefinition of (1) to obtainThe following is shown:
(14)
; (15)
; (16)
Wherein, the 、、Are respectively toIs a numerical approximation of (a);
D. Initial boundary conditions:
(17)
combining the numerical formats, using the initial value conditions given in step D, equation (17), from The time cycle is turned on and,Solving the explicit equation (10) and the large linear equation set (equations (11) - (13) and (14) - (16)) in turn, repeating the steps A, B, C untilObtaining four unknown variables at the final moment, namely the porosityPressure and forceSpeed and velocity ofConcentration ofThe acidizing model is solved through a staggered grid finite difference method to obtain a numerical solution, so that the distribution condition of the rock porosity at any discrete time point can be simulated, and further the evolution of the rock porosity along with time is drawn.
A computer device comprising a memory storing a computer program and a processor implementing steps of a method for implementing a radial flow numerical simulation of a porous medium fluid based on an acidification model of a DBF framework when executing the computer program.
A computer readable storage medium having stored thereon a computer program which when executed by a processor implements the steps of a method for implementing a radial flow numerical simulation of a porous media fluid based on an acidification model of a DBF framework.
In a second aspect, the present invention provides a system for implementing radial flow numerical simulation of porous medium fluid based on an acidification model of a DBF framework, comprising:
an acidification model creation module configured to construct an acidification model for describing fluid flow, solute reaction transport and rock property changes;
The grid subdivision module is configured to perform grid subdivision on the simulation time and the annular ring type solving area, and then define different variables of the acidification model at different positions of the grid unit;
the numerical solution module is configured to carry out numerical dispersion on the acidification model constructed by the acidification model construction module to form a large linear equation set, and finally, the numerical simulation of the acid etching wormholes is realized by combining with the set parameter solution.
The invention has the beneficial effects that:
1. The existing polyacid model is a mathematical model based on a Darcy frame under a Cartesian coordinate system. However, in consideration of the fact that the porosity of the rock is continuously changed along with the time evolution in the acidification process of the carbonate matrix and the flow direction of acid liquor flowing into the carbonate matrix through a shaft is radial, the acidification model based on the Darcy-Brinkman-Forchheimer framework in the polar coordinate system is established, the flow phenomenon of a porous medium which is not Darcy seepage in the acidification process of the carbonate matrix can be more accurately described, more accurate acid etching earthworm Kong Shuzhi simulation is further realized, and an important guiding effect is exerted on oil and gas exploitation.
2. The invention provides a discrete solving method for an acidification model under polar coordinates by using a staggered grid finite difference method, which is characterized in that different variables can be stored at different positions of grids, the method can well keep the original physical properties of the model, such as conservation of mass, conservation of momentum, and the like, can also keep high precision of four unknown quantities of porosity, pressure, speed and concentration under non-uniform grids, and the numerical solving process is simple and efficient.
Drawings
FIG. 1 is a diagram of a framework of a method for simulating radial flow values of porous medium fluid by using an acidification model based on a DBF framework;
FIG. 2 shows the present invention A non-uniform grid of time;
FIG. 3 shows the rock porosity of the invention when the initial porosity and permeability of the matrix is subject to uniform distribution and the acid is continuously injected into the matrix through a circular well bore A profile of time of day;
FIG. 4 shows the rock porosity at the point of the initial porosity and permeability of the matrix of the invention when the matrix is only larger at individual points and the acid injection into the matrix is continued through the circular well bore A profile of time of day;
FIG. 5 shows the initial porosity and permeability of the matrix according to the present invention when the matrix is uniformly distributed and passed only The porosity of the rock is that when the circular shaft continuously injects acid into the matrixA profile of time of day;
FIG. 6 shows the rock porosity in the case of a matrix having only two small continuous sections in the radial direction of the matrix with very high initial porosity and permeability and continuous acid injection into the matrix through only a small range of the well bore in the vicinity of the matrix Distribution map of time of day.
Detailed Description
The invention will now be further illustrated by way of example, but not by way of limitation, with reference to the accompanying drawings.
Example 1
An acidification model based on a DBF framework realizes a porous medium fluid radial flow numerical simulation method, as shown in fig. 1, comprising the following steps:
Step 1, constructing an acidification model for describing fluid flow, solute reaction transportation and rock property change;
Step 2, meshing the simulation time and the annular solving area, and defining different variables of the acidification model at different positions of the mesh unit;
And 3, performing numerical discrete on the acidizing model constructed in the step 1 to form a large linear equation set, and finally solving by combining with set parameters to realize numerical simulation of the acid etching wormholes.
Example 2
The method for realizing radial flow numerical simulation of porous medium fluid by using the acidification model based on the DBF framework according to the embodiment 1 is characterized in that:
establishing an acidification model for describing fluid flow, solute reaction transport and rock property change, wherein the method comprises the following steps:
an acidification model based on a DBF frame under a set of two-dimensional polar coordinates is established as follows:
; (1)
; (2)
; (3)
; (4)
Wherein, the formulas (1) and (2) represent the fluid flow process based on the DBF framework, namely an acidification model for describing the fluid flow, the formula (1) is a vector equation representing a momentum conservation equation, and the right end of the formula (1): representing Darcy term for describing Darcy seepage phenomenon of porous medium, and a second term at left end of formula (1): The term Brinkman is used for describing the transition flow between boundaries, and the last term at the left end of the formula (1): The term Forchheimer, which may also be referred to as an inertia term, is used to describe the significant inertial effects of the fluid when the flow rate is high, equation (2) is a mass conservation equation, equation (3) is an acid concentration reaction transfer equation, i.e., an acidification model used to describe the transport of solute reactions, equation (4) reflects the evolution of rock porosity over time, i.e., an acidification model used to describe the change in rock properties;
Wherein, the ,Is the flow radius in units of;Is a circular ring area; It is the time that is required for the device to be in contact with the substrate, The final time is given in units of;Is a vector of the velocity of the fluid,Velocity of the fluid respectivelyAt the polar diameterDirection and polar angleThe component of the direction in units of;、Representing the polar diameterA boundary of the direction; Is the pressure of the fluid in units of ;Is the acid concentration in unit of;Is the porosity of rock, dimensionless quantity and speedPressure and forceConcentration ofPorosity of the porous bodyAll four variables are unknown; Is the density of the fluid in units of ;Is the viscosity of the fluid in units of;Is a pseudo-compression coefficient in units of,Resulting in a slight change in fluid density during dissolution of the solute,Is a very small positive number to ensure that the coefficient matrix is reversible; is a local mass transmission coefficient, and has the unit of ;For the injection concentration, the unit is;Is the acid corrosion capacity in units of;Is the density of rock, the unit is;Is Forchheimer coefficient, dimensionless; is the permeability of rock in units of ;WhereinIn order to achieve the injection rate,For throughput rate, unit isPositive definite matrixIs the diffusion coefficient of acid in porous medium, and has the unit of,AndIs thatAt the position ofAndComponents in the direction, for simplicity, assumeIs a diagonal matrix, wherein,In order to achieve a molecular diffusion rate,Is an identity matrix of the unit cell,Expressed in terms ofA diagonal matrix that is a diagonal element; are all given functions of Is a function of the porosity of the rock;
is the concentration of acid at the interface of fluid (acid) and solid (rock), and has the unit of According to the primary kinetic reaction, the concentration of acid at the fluid-solid interfaceWith the concentration of acid in the fluidThe following relationship exists:
; (5)
Wherein, the Is the surface reaction rate constant, and has the unit of;
The pore scale model characterizing rock property changes is as follows:
; (6)
; (7)
wherein, the formula (6) is established by Carman-Kozeny to characterize the relationship between the rock porosity and permeability, The permeability is indicated as a function of the permeability,AndThe initial porosity and the initial permeability of the rock are calculated respectively, wherein,Is the interfacial area (specific surface area) per unit volume of medium for reaction, in units of,Is the initial interface area;
the boundary and initial conditions are as follows:
(8)
Wherein, two conditions in the first row: Is a boundary condition, indicating that at the entrance boundary and the exit boundary, the gradients of velocity and concentration are 0, the latter four are initial conditions, 、、、Respectively the speed, pressure, concentration and porosity in the regionAn initial distribution function within;
The acidification model formulas (1) - (4) based on the DBF frame in two-dimensional polar coordinates, the pore scale model formulas (6) - (7) and the initial boundary condition formula (8) are combined to form the acidification model, and the flow property of the porous medium non-Darcy seepage in the process of acidifying the carbonate matrix is comprehensively considered;
To simplify the equation and writing in discrete format below, an auxiliary variable is defined , wherein,In combination with formulas (2), (5) - (7), the concentration reaction of acid is simplified to transfer formula (3) and then converted into the following formula:
(9)。
Simulation time And an annular solution areaPerforming mesh dissection, defining different variables of the acidification model at different positions of the mesh unit, and comprising:
total simulation time Average division into (final moment)Time stepAnd the firstAt a moment ofFor better simulation of real world conditions, for a given simulation areaNamely, the annular solving area is subjected to non-uniform mesh dissection, andThe area length of the direction is divided intoThe weight of the components is calculated by the weight,The area length of the direction is divided intoPart(s) to formGrid point coordinates are: And has Record(s)Direction and directionGrid cell edge midpoint of direction、Step size of subdivision、、、The method comprises the following steps of:
The interlaced grid is characterized in that different variables can be stored in different locations of the grid cells, the pressure is established for step 1 Speed and velocity ofConcentration ofPorosity of the porous bodyThe nonlinear strong coupling acidification model of (1) is expressed by formulas (1) - (4), and the pressure is calculatedConcentration ofPorosity of the porous bodyThe numerical solution of (2) is defined in the center of the grid cell, the speed is determinedIs defined at the midpoint of the four sides of the grid cell, where velocity is related toThe component of direction, i.e.The speed of the direction is defined at the midpoint of the tangential edge of the grid cell, the speed being related toThe component of the direction is defined at the midpoint of the radial edge of the grid cell;
Performing numerical discrete on the acidizing model constructed in the step 1 to form a large linear equation set, finally solving by combining set parameters to realize the simulation of the acid-etched earthworm Kong Shuzhi, wherein the method comprises the following steps:
To simplify writing in a subsequent discrete format, variable simplification rules are first defined, including:
The independent variable is And the discrete function with function values at the appropriate discrete nodes is expressed as,Representation ofTime of day, functionAt grid pointsThe value at which the value is to be found, wherein,Respectively grid pointsAt the position ofDirection and directionThe coordinates in the direction of the light are,Taking out,Taking outRecord(s)Wherein the superscript indicates the firstAt a moment ofSubscript indicates spatial positionIn the absence of ambiguity, superscripts can generally be omittedIf (1)Is a vector function, and on the basis of the vector function, a superscript is also neededTo distinguish which direction component is in particular;
Giving a symbolic definition of the derivative replaced by the difference quotient, note 、、、、As a function ofWith respect to、、The derivative at a certain grid point is defined as follows:
Wherein, the 、Representing 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;、 representing approximating the derivative at the center of the grid cell with the difference quotient of the function values at the midpoints of the edges of the grid cell;
Definition of interpolation operator and square root mean operator, including:
For points Assume thatIt is noted that the number of the elements,The direction is the boundary of the period of time,By usingThe values of (2) define the bilinear interpolation operator as follows:
When (when)When there are two extrapolation formulas:
Wherein, the ;
For discrete functionsDefine aThe above piecewise constant function satisfies:;
Then, for vector functions Its componentIs a pair of discrete functionsDefining interpolation operatorsThe following are provided:
;
Wherein, the Representing interpolation operators, respectivelyAt the position ofDirection and directionThe component in the direction of the light is,Representation ofAt the pointThe function value of the position,Representation ofAt the pointA function value at the location;
Is provided with Is a vectorIs a norm function of (1), and has;
Defining square root mean operatorAndThe following formula is shown:
the concrete steps are as follows:
By using To representThe corresponding numerical solution, i.e., P is the numerical solution of pressure P, W is the numerical solution of fluid velocity u,Is the acid concentrationIs the numerical solution of the auxiliary variable Q, T is the rock porosityWherein, superscriptRepresenting temporal layers, speed and auxiliary variables are vectors, so superscriptTo distinguish directions;
performing numerical discrete on the acidizing model constructed in the step 1 by using a backward Euler in time and using a staggered grid finite difference method in space to obtain a staggered grid finite difference format under polar coordinates, namely formulas (10) - (16), wherein the numerical discrete acidizing model comprises the following steps:
A. Is known to be AndRepresents the firstAt a moment ofConcentration ofPorosity of the porous bodyAt grid pointsThe value at the point is obtained according to equation (4) of the change of the porosity with timeThe following is shown:
; (10)
Wherein, the ,;Indicating the initial moment, the rock porosityAt grid pointsA value at;
B. is known to be WhereinRepresent the firstAt a moment ofSpeed is related toComponent of directionAt grid pointsThe value at which the value is to be calculated,Represent the firstAt a moment ofSpeed is related toComponent of directionAt grid pointsThe value at which the value is to be calculated,Represent the firstAt a moment ofPressure of fluidAt grid pointsA value at;
obtaining according to a momentum conservation equation (1) and a mass conservation equation (2) describing fluid flow The following is shown:
(11)
(12)
; (13)
Wherein, the The interpolation operator is represented as a function of the interpolation,Indicating the permeability of the rock,Representing Forchheimer coefficients;
C. Is known to be Equation (9) and auxiliary variables are transmitted according to the acid concentration reactionDefinition of (1) to obtainThe following is shown:
(14)
; (15)
; (16)
Wherein, the 、、Are respectively toIs a numerical approximation of (a);
D. Initial boundary conditions:
(17)
combining the numerical formats, using the initial value conditions given in step D, equation (17), from The time cycle is turned on and,Solving the explicit equation (10) and the large linear equation set (equations (11) - (13) and (14) - (16)) in turn, repeating the steps A, B, C untilObtaining four unknown variables at the final moment, namely the porosityPressure and forceSpeed and velocity ofConcentration ofSolving an acidification model through a staggered grid finite difference method to obtain a numerical solution, and simulating the distribution condition of rock porosity at any discrete time point so as to draw the evolution of the rock porosity along with time;
the method comprises the steps of (1) realizing the simulation of acidizing acid-etched earthworm Kong Shuzhi by a carbonate matrix through the steps of (2) and (3), establishing a non-linear strong coupling acidizing model of Darcy-Brinkman-Forchheimer in a two-dimensional polar coordinate system based on the flow characteristic of non-Darcy seepage of a porous medium, and carrying out numerical discrete and efficient solution on the model by using a staggered grid finite difference method to realize the more practical and more accurate numerical simulation on the acid-etched earthworm pore process.
In order to verify the accuracy of the acidification model, the feasibility of a numerical solution algorithm and the numerical simulation effect, four numerical experiments are listed, and numerical calculation and simulation are performed by utilizing computer software programming. First, table 1 shows some physical parameters and their values required for numerical simulation. Then, the four numerical examples are respectively provided with initial porosity, permeability distribution and different acid injection ranges with different characteristics, and the effectiveness of the technical scheme of the invention is verified by numerical simulation effects.
TABLE 1
In numerical simulation of the acid etched earthworm Kong Youcang, a simulation time interval is set asSimulating for 46 days, the time step isThe radius of the simulation area isA wellbore radius ofI.e.And adoptsA non-uniform grid of cells, as shown in fig. 2, in which each small quadrilateral represents a grid cell, the vertices of all grid cells are referred to as grid points. Will initiate pressureIs set asInitial concentrationIs 0. Initial permeability obeys in the rangeIs uniformly distributed and has initial porosity obeying in a rangeAnd the injection and production rates of the acid are as follows:
when the initial porosity and the initial permeability of the carbonate matrix are subjected to uniform distribution, acid is continuously injected into the carbonate matrix through the circular shaft, the acid and the rock react chemically, the structure of the rock is changed, and the porosity and the permeability of the rock are increased along with the advancement of the matrix acidification process. Because the resistance formed by the high porosity region to the fluid is smaller, the acid preferentially flows into the region, high diversion channels (dominant wormholes with protruding branches) which are distributed randomly are formed, and the numerical simulation effect of the acid etching wormholes is shown in figure 3.
Example 3
In this example, the analog time interval is set asSimulating for about 24 days, the time step isRadius of simulation areaA wellbore radius ofI.e.And adoptsA non-uniform grid of individual cells is shown in fig. 2. Will initiate pressureIs set asInitial concentrationInitial porosity and permeability and injection and production rates were as follows:
When the initial porosity and permeability of the carbonate matrix are only larger at individual points, acid is continuously injected into the carbonate matrix through the circular shaft, and the acid radially extends outwards and is faster at the points with larger initial porosity and permeability along with the evolution of the acidification time, so that a long and narrow high-diversion channel is formed, the trend accords with the expected effect, and the numerical simulation effect of the acid etching wormholes is shown in fig. 4.
Example 4
In this example, the analog time interval is set asSimulating for about 58 days, the time step isRadius of simulation areaA wellbore radius ofI.e.And adoptsA non-uniform grid of individual cells is shown in fig. 2. Will initiate pressureIs set asInitial concentrationIs 0. Initial permeability obeys in the rangeIs uniformly distributed and has initial porosity obeying in a rangeAnd the injection and production rates of the acid are as follows:
When the initial porosity and permeability of the carbonate matrix are subject to uniform distribution, if only by The circular shaft continuously injects acid into the carbonate matrix, so that the acid continuously diffuses to the periphery and preferentially flows into a high-porosity area along with the evolution of the acidification time, and as the acid radially extends out of the high-porosity area more quickly, high-diversion channels (branched protruding dominant wormholes) which are distributed randomly are gradually formed, and the numerical simulation effect of the acid etching wormholes is shown in figure 5.
Example 5
In this example, the analog time interval is set asSimulating for 92 days, the time step isRadius of simulation areaA wellbore radius ofI.e.And adoptsA non-uniform grid of individual cells is shown in fig. 2. Will initiate pressureIs set asInitial concentrationInitial porosity, permeability, and injection and production rates were as follows:
when the carbonate matrix has two sections of continuous areas along the radial direction, the initial porosity and the permeability are larger, similar to two small cracks in the carbonate matrix, and at the moment, if acid is continuously injected into the matrix with the characteristics through a small-range shaft near the two small-range cracks, a certain diversion effect can be achieved due to the fact that the porosity and the permeability of the cracks are large and correspond to a pipeline, then the acid preferentially flows into the areas, and along with the evolution of the acidification time, a large amount of acid flows into the two high-porosity channels and forms protruding branches, and the numerical simulation effect of the final acid etching wormholes is shown in fig. 6.
The numerical simulation results of the above calculation example effectively illustrate the accuracy and feasibility of the acidification model and the numerical solving method of the invention. Therefore, the technical scheme of the invention not only can carry out numerical simulation on the acidizing and acid etching earthworm pore process of the carbonate matrix more accurately, but also can play an important guiding role in oil and gas exploitation.
Example 6
A computer device comprising a memory storing a computer program and a processor implementing the steps of any one of embodiments 1-5 of the DBF framework-based acidification model implementing a porous media fluid radial flow numerical simulation method when the computer program is executed.
Example 7
A computer readable storage medium having stored thereon a computer program which when executed by a processor performs the steps of any of embodiments 1-5 of a DBF framework based acidification model implementing a porous media fluid radial flow numerical simulation method.
Example 8
An acidification model based on a DBF framework realizes a porous medium fluid radial flow numerical simulation system, comprising:
an acidification model creation module configured to construct an acidification model for describing fluid flow, solute reaction transport and rock property changes;
The grid subdivision module is configured to perform grid subdivision on the simulation time and the annular ring type solving area, and then define different variables of the acidification model at different positions of the grid unit;
the numerical solution module is configured to carry out numerical dispersion on the acidification model constructed by the acidification model construction module to form a large linear equation set, and finally, the numerical simulation of the acid etching wormholes is realized by combining with the set parameter solution.
Claims (8)
1. The method for realizing radial flow numerical simulation of porous medium fluid based on an acidification model of a DBF framework is characterized by comprising the following steps of:
Step 1, constructing an acidification model for describing fluid flow, solute reaction transportation and rock property change;
Step 2, meshing the simulation time and the annular solving area, and defining different variables of the acidification model at different positions of the mesh unit;
And 3, performing numerical discrete on the acidizing model constructed in the step 1 to form a large linear equation set, and finally solving by combining with set parameters to realize numerical simulation of the acid etching wormholes.
2. The method for simulating radial flow values of a porous media fluid by using an acidification model based on a DBF framework according to claim 1, wherein the method for simulating radial flow values of the porous media fluid by using the acidification model comprises the following steps of:
an acidification model based on a DBF frame under a set of two-dimensional polar coordinates is established as follows:
; (1)
; (2)
; (3)
; (4)
Wherein, the formulas (1) and (2) represent the fluid flow process based on the DBF framework, namely an acidification model for describing the fluid flow, the formula (1) is a vector equation representing a momentum conservation equation, and the right end of the formula (1): representing Darcy term for describing Darcy seepage phenomenon of porous medium, and a second term at left end of formula (1): The term Brinkman is used for describing the transition flow between boundaries, and the last term at the left end of the formula (1): the expression Forchheimer is used for describing the significant inertia effect of the fluid when the flow rate is high, the formula (2) is a mass conservation equation, the formula (3) is an acid concentration reaction transmission equation, namely an acidification model used for describing solute reaction transportation, and the formula (4) is used for describing an acidification model used for describing rock property change, namely the change of rock porosity evolution with time;
Wherein, the ,Is the flow radius in units of; Is the flow angle; Is a circular ring area; It is the time that is required for the device to be in contact with the substrate, The final time is given in units of;Is a vector of the velocity of the fluid,Velocity of the fluid respectivelyAt the polar diameterDirection and polar angleThe component of the direction in units of;、Representing the polar diameterA boundary of the direction; Is the pressure of the fluid in units of ;Is the acid concentration in unit of;Is the porosity of rock, dimensionless quantity and speedPressure and forceConcentration ofPorosity of the porous bodyAll four variables are unknown; Is the density of the fluid in units of ;Is the viscosity of the fluid in units of;Is a pseudo-compression coefficient in units of,A positive number is indicated and the number of the positive numbers,Is a local mass transmission coefficient, and has the unit of; For the injection concentration, the unit is;Is the acid corrosion capacity in units of;Is the density of rock, the unit is;Is Forchheimer coefficient, dimensionless; is the permeability of rock in units of ;WhereinIn order to achieve the injection rate,For throughput rate, unit isPositive definite matrixIs the diffusion coefficient of acid in porous medium, and has the unit of,AndIs thatAt the position ofAndComponent in the direction, assumingIs a diagonal matrix, wherein,In order to achieve a molecular diffusion rate,Is an identity matrix of the unit cell,Expressed in terms ofA diagonal matrix that is a diagonal element;
Is the concentration of acid at the interface of fluid and solid, and has the unit of According to the primary kinetic reaction, the concentration of acid at the fluid-solid interfaceWith the concentration of acid in the fluidThe following relationship exists:
; (5)
Wherein, the Is the surface reaction rate constant, and has the unit of;
The pore scale model characterizing rock property changes is as follows:
; (6)
; (7)
wherein, the formula (6) is used for describing the relation between the porosity and the permeability of the rock, The permeability is indicated as a function of the permeability,AndThe initial porosity and the initial permeability of the rock are calculated respectively, wherein,Is the interfacial area for reaction per unit volume of medium, in units of,Is the initial interface area;
the boundary and initial conditions are as follows:
(8)
Wherein, two conditions in the first row: Is a boundary condition, indicating that at the entrance boundary and the exit boundary, the gradients of velocity and concentration are 0, the latter four are initial conditions, 、、、Respectively the speed, pressure, concentration and porosity in the regionAn initial distribution function within;
the acidification model formulas (1) - (4) based on the DBF frame in two-dimensional polar coordinates are combined with the pore scale model formulas (6) - (7) and the primary boundary condition formula (8) to form an acidification model;
Defining an auxiliary variable , wherein,In combination with formulas (2), (5) - (7), the concentration reaction of acid is simplified to transfer formula (3) and then converted into the following formula:
(9)。
3. the method for simulating radial flow of porous media fluid by using acidification model based on DBF framework according to claim 2, wherein simulation time is as follows And an annular solution areaPerforming mesh dissection, defining different variables of the acidification model at different positions of the mesh unit, and comprising:
total simulation time Average division intoTime stepAnd the firstAt a moment ofFor a given simulation areaNamely, the annular solving area is subjected to non-uniform mesh dissection, andThe area length of the direction is divided intoThe weight of the components is calculated by the weight,The area length of the direction is divided intoPart(s) to formThe grid points are: And has Record(s)Direction and directionGrid cell edge midpoint of direction、Step size of subdivision、、、The method comprises the following steps of:
the pressure established for step 1 Speed and velocity ofConcentration ofPorosity of the porous bodyThe nonlinear strong coupling acidification model of (1) is expressed by formulas (1) - (4), and the pressure is calculatedConcentration ofPorosity of the porous bodyThe numerical solution of (2) is defined in the center of the grid cell, the speed is determinedIs defined at the midpoint of the four sides of the grid cell, where velocity is related toThe component of direction, i.e.The speed of the direction is defined at the midpoint of the tangential edge of the grid cell, the speed being related toThe component of the direction is defined at the midpoint of the radial edge of the grid cell.
4. The method for realizing radial flow numerical simulation of porous medium fluid by using an acidification model based on a DBF framework according to claim 3, wherein the method is characterized by performing numerical dispersion on the acidification model constructed in the step 1 to form a large linear equation set, finally solving by combining set parameters to realize the simulation of acid etching earthworm Kong Shuzhi, and comprises the following steps:
defining variable simplification rules and taking independent variables as And the discrete function with function values at the appropriate discrete nodes is expressed as,Representation ofTime of day, functionAt grid pointsThe value at which the value is to be found, wherein,Respectively grid pointsAt the position ofDirection and directionThe coordinates in the direction of the light are,Taking out,Taking outRecord(s)Wherein the superscript indicates the firstAt a moment ofSubscript indicates spatial position;
Giving a symbolic definition of the derivative replaced by the difference quotient, note、、、、As a function ofWith respect to、、The derivative at a certain grid point is defined as follows:
Wherein, the 、Representing 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;、 representing approximating the derivative at the center of the grid cell with the difference quotient of the function values at the midpoints of the edges of the grid cell;
Definition of interpolation operator and square root mean operator, including:
For points Assume that,The direction is the boundary of the period of time,By usingThe values of (2) define the bilinear interpolation operator as follows:
When (when)When there are two extrapolation formulas:
Wherein, the ;
For discrete functionsDefine aThe above piecewise constant function satisfies:;
For vector functions Its componentIs a pair of discrete functionsDefining interpolation operatorsThe following are provided:
;
Wherein, the Representing interpolation operators, respectivelyAt the position ofDirection and directionThe component in the direction of the light is,Representation ofAt the pointThe function value of the position,Representation ofAt the pointA function value at the location;
Is provided with Is a vectorIs a norm function of (1), and has;
Defining square root mean operatorAndThe following formula is shown:
the concrete steps are as follows:
By using To representThe corresponding numerical solution, i.e., P is the numerical solution of pressure P, W is the numerical solution of fluid velocity u,Is the acid concentrationIs a solution to the numerical value of (a),Is an auxiliary variableIs the numerical solution of T is the rock porosityWherein, superscriptRepresenting temporal layers, speed and auxiliary variables are vectors, so superscriptTo distinguish between directions.
5. The method for realizing radial flow numerical simulation of porous medium fluid by using an acidification model based on a DBF framework according to claim 4, wherein the method for realizing numerical dispersion of the acidification model constructed in the step1 by using a backward Euler in time and using an interleaving grid finite difference method in space to obtain an interleaving grid finite difference format under polar coordinates comprises the following steps:
A. Is known to be AndRepresents the firstAt a moment ofConcentration ofPorosity of the porous bodyAt grid pointsThe value at the point is obtained according to equation (4) of the change of the porosity with timeThe following is shown:
; (10)
Wherein, the ,;Indicating the initial moment, the rock porosityAt grid pointsA value at;
B. is known to be WhereinRepresent the firstAt a moment ofSpeed is related toComponent of directionAt grid pointsThe value at which the value is to be calculated,Represent the firstAt a moment ofSpeed is related toComponent of directionAt grid pointsThe value at which the value is to be calculated,Represent the firstAt a moment ofPressure of fluidAt grid pointsA value at;
obtaining according to a momentum conservation equation (1) and a mass conservation equation (2) describing fluid flow The following is shown:
(11)
(12)
; (13)
Wherein, the The interpolation operator is represented as a function of the interpolation,Indicating the permeability of the rock,Representing Forchheimer coefficients;
C. Is known to be Equation (9) and auxiliary variables are transmitted according to the acid concentration reactionDefinition of (1) to obtainThe following is shown:
(14)
; (15)
; (16)
Wherein, the 、、Are respectively toIs a numerical approximation of (a);
D. Initial boundary conditions:
(17)
combining the numerical formats, using the initial value conditions given in step D, equation (17), from The time cycle is turned on and,Solving the explicit equation (10) and the large linear equation set (equations (11) - (13) and (14) - (16)) in turn, repeating the steps A, B, C untilObtaining four unknown variables at the final moment, namely the porosityPressure and forceSpeed and velocity ofConcentration ofIs a numerical solution to (a).
6. A computer device comprising a memory and a processor, the memory storing a computer program, characterized in that the processor, when executing the computer program, performs the steps of the DBF framework-based acidification model realization porous media fluid radial flow numerical simulation method of any one of claims 1-5.
7. A computer readable storage medium having stored thereon a computer program, wherein the computer program when executed by a processor performs the steps of the DBF framework based acidification model of any one of claims 1-5 for performing a numerical simulation method of radial flow of a porous medium fluid.
8. An acidification model based on a DBF framework realizes a porous medium fluid radial flow numerical simulation system, which is characterized by comprising:
an acidification model creation module configured to construct an acidification model for describing fluid flow, solute reaction transport and rock property changes;
The grid subdivision module is configured to perform grid subdivision on the simulation time and the annular ring type solving area, and then define different variables of the acidification model at different positions of the grid unit;
the numerical solution module is configured to carry out numerical dispersion on the acidification model constructed by the acidification model construction module to form a large linear equation set, and finally, the numerical simulation of the acid etching wormholes is realized by combining with the set parameter solution.
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