CN118114540B - Multi-scale coarse crack particle migration simulation method - Google Patents
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Abstract
The invention discloses a multi-scale coarse crack particle migration simulation method, which comprises the steps of constructing a coarse crack physical model; constructing a multi-scale grid, and endowing non-uniform seam width on the fine-scale grid; initializing parameters and boundary conditions in the seam; calculating a pressure field and a speed field on a coarse scale and a fine scale based on a multi-scale finite volume method; establishing and solving a particle transport model based on the fine-scale pressure field and the velocity field to obtain particle concentration distribution on the fine scale; calculating a time step, judging whether the current iteration step is converged or not, repeating the steps when the current iteration step is not converged, and calculating the next iteration step when the current iteration step is converged until the current iteration step is completed; repeating the steps until the simulation is completed, and finally determining the concentration distribution of the particles in the crack. The method solves the fluid flow and the grain migration behavior in the rough fracture by using a multi-scale method, realizes the fine and accurate simulation of the grain migration process while reducing the simulation cost, and has important significance for the accurate evaluation of the hydraulic fracturing grain migration behavior and the optimization of the sand conveying process.
Description
Technical Field
The invention relates to a multi-scale coarse crack particle migration simulation method, and belongs to the field of petroleum and natural gas development.
Background
The fractured hydrocarbon reservoirs are widely distributed in the global scope, have huge reserve potential, and play a vital role in global hydrocarbon exploration and development. These reservoirs are often found in typical geological environments such as carbonates, tight sandstones, and shale reservoirs. In the process of developing the oil and gas reservoirs, migration and deposition of particles in the cracks can directly influence the diversion capacity of the cracks, so that the final yield of oil and gas is obviously affected. The existing intra-slit particle migration numerical simulation technology has obvious limitations in treating complex situations of remarkable changes of the slit permeability and uneven slit width, and particularly has a defect in the capability of accurately simulating a complex slit network.
Therefore, the invention provides a multi-scale coarse crack particle migration simulation method. This method first calculates the pressure field and velocity field on a coarse scale and then maps these calculations precisely onto a fine scale grid. Compared with the traditional numerical simulation method, the method has the advantages that the calculation cost is remarkably reduced by reducing the need of direct solution on a fine grid. The method not only improves the calculation efficiency, but also ensures the accuracy and reliability of the model under various scales. Therefore, the method can realize the fine and accurate simulation of the grain migration process while reducing the simulation cost, and has important significance for the accurate evaluation of the hydraulic fracturing grain migration behavior and the optimization of the sand conveying process.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention aims to provide a multi-scale coarse crack particle migration simulation method.
The technical scheme provided by the invention for solving the technical problems is as follows: a multi-scale coarse crack particle migration simulation method comprises the following steps:
s10, constructing a rough crack physical model;
s20, constructing a coarse-scale grid and a fine-scale grid in a coarse crack physical model, and endowing uneven slit width on the fine-scale grid;
step S30, initializing parameters and boundary conditions in the seam;
Step S40, calculating a coarse-scale pressure field and a fine-scale pressure field and a speed field based on a multi-scale finite volume method;
s50, establishing and solving a particle transport model based on a fine-scale pressure field and a fine-scale velocity field to obtain particle concentration distribution on the fine scale;
step S60, carrying out iterative computation in one time step until iteration converges, repeating the steps S40-S50 when the iteration converges, and computing the next time step after the iteration converges;
and step S70, repeating the steps S40-S60 until the simulation is completed, and finally determining the concentration distribution of the particles in the crack.
The further technical scheme is that the specific process of step S20 includes:
s21, uniformly dividing M, N nodes in the x and y directions in the rough crack physical model to form a coarse scale grid;
s22, uniformly dividing m and n nodes along the x and y directions of the grid to form a fine-scale grid;
and S23, randomly generating non-uniform seam widths on the fine-scale grids through a probability distribution model.
The further technical scheme is that the probability distribution model is a normal distribution model.
The specific process of step S40 further includes:
S41, constructing a pressure equation of transient single-phase incompressible flow, and solving a coarse-scale pressure field;
S42, constructing a dual grid on a coarse scale, and solving a dual basis function according to the following equation;
wherein: Is a divergence operator; /(I) Pressure, pa; t is time, s; /(I)Is the viscosity of the fluid, pa.s; /(I)In order for the permeability to be a function of,；/>A kth dual basis function of the corresponding dual grid for any coarse grid; /(I)Is a dual grid;
the dual basis function satisfies the boundary condition:
wherein: ，k=1，2，3，4；
Respectively linearly combining the dual basis functions with corresponding coarse-scale pressures to obtain fine-scale pressures in the dual grid blocks;
wherein: is the fine scale pressure inside the dual grid block; /(I) The coarse grid of the horizontal ith row and the vertical jth column corresponds to the kth dual basis function of the dual grid; /(I)The coarse scale pressure corresponding to the kth dual basis function;
step S43, constructing a fine-scale grid on the fine scale Solving a fine-scale basis function according to the following equation;
wherein: Is a divergence operator; /(I) Pressure, pa; t is time, s; /(I)Is the viscosity of the fluid, pa.s; /(I)In order for the permeability to be a function of,；/>An s-th fine-scale basis function of any fine mesh;
the fine-scale basis function satisfies the boundary condition:
wherein: To pass the boundary/> Is a flux of (2); /(I)S th fine-scale basis function of fine grid of the i th row in the transverse direction and the j th column in the longitudinal direction/>A gradient along the boundary outward normal; /(I)To be in a fine scale region/>Fine grid/>, of the middle transverse i-th row, longitudinal j-th columnOn the boundary but not at boundary/>Is a flux of (2); n is the number of nodes of the fine grid in the y direction; /(I)The s-th fine scale basis function of the fine grid of the horizontal ith row and the vertical jth column;
The fine scale basis function is respectively and linearly combined with 8 coarse scale pressures around the grid block and the coarse scale pressure of the fine scale basis function, and the fine scale pressure of the grid block can be obtained:
wherein: fine scale pressure for the grid block; /(I) The s-th fine scale basis function of the fine grid of the horizontal ith row and the vertical jth column; /(I)The coarse scale pressure corresponding to the s-th fine scale basis function;
And S44, carrying out explicit calculation by using the pressure distribution of the fine-scale grids obtained by the multi-scale method through Darcy' S law to obtain a speed field on the fine-scale grids.
The further technical scheme is that the pressure equation of the transient single-phase incompressible flow is as follows:
wherein: Is a divergence operator; /(I) Pressure, pa; t is time, s; /(I)Is the viscosity of the fluid, pa.s; /(I)In order for the permeability to be a function of,；/>To solve for the region.
The further technical scheme is that the calculation formula in the step S44 is as follows:
wherein: is the pressure gradient, pa/m; /(I) Is the viscosity of the fluid, pa.s; /(I)For permeability,/>；/>Is the fluid velocity on a fine-scale grid, m/s.
The specific process of step S50 further includes:
Step S51, calculating the intra-slit particle migration speed according to the relation between the flow velocity of the fluid in the slit and the slit width and the particle diameter;
and step S52, solving a particle migration differential equation on a fine-scale grid based on a multi-scale finite volume method to obtain particle concentration distribution on the fine scale.
The further technical scheme is that the calculation formula of step S51 is:
wherein: Fluid velocity on a fine scale grid, m/s; /(I) The transport speed of the particles in the seam is m/s; a is the seam width, m; /(I)Is the particle diameter, m.
The further technical scheme is that the particle migration differential equation is as follows:
wherein: the volume fraction of particles in the crack is dimensionless; t is time, s; /(I) Is a divergence operator; /(I)The transport speed of the particles in the seam is m/s; a is the seam width, m; /(I)The transport speed of the particles in the seam is m/s.
According to a further technical scheme, in the step S60, the iteration convergence criterion is judged to be that the difference degree of the particle volume fractions in the cracks of the current iteration step and the previous iteration step is smaller than 0.1%.
The invention has the following beneficial effects:
1. The traditional numerical simulation method directly solves on the fine grid, and greatly increases the calculation cost when the grid is processed. The invention adopts a multi-scale finite volume method to effectively improve the calculation speed by calculating the pressure field and the speed field on a coarse scale and then mapping the pressure field and the speed field on a fine scale grid.
2. The multi-scale finite volume method can be used for accurately evaluating the migration behavior of the hydraulic fracturing particles and optimizing the sand conveying process, and has a great market prospect.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a partial grid partition diagram;
FIG. 3 is a seam width profile;
FIG. 4 is a cloud plot of the concentration profile of particles at time step 10;
FIG. 5 is a cloud plot of particle concentration distribution at time step 30;
FIG. 6 is a cloud plot of particle concentration distribution at time step 50;
FIG. 7 is a cloud plot of particle concentration distribution at time 60.
Detailed Description
The following description of the embodiments of the present invention will be made apparent and fully in view of the accompanying drawings, in which some, but not all embodiments of the invention are shown. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
As shown in FIG. 1, the multi-scale coarse crack particle migration simulation method disclosed by the invention specifically comprises the following steps of:
step S1, constructing a rough fracture physical model, and endowing required fracture simulation parameters as shown in a table 1:
TABLE 1
In the table: Is the particle diameter, m; /(I) Is liquid density, kg/m 3; /(I)The concentration of the particles is kg/m 3; s is the sand ratio; is the permeability of the granular sand pile,/> ；/>Is the viscosity of fracturing fluid, mPas; f is the roughness coefficient of the crack surface, and is dimensionless.
S2, constructing a coarse-scale grid and a fine-scale grid in the coarse crack physical model, and endowing the fine-scale grid with non-uniform slit width;
Step 21, consider a natural crack of 12m×1.0m, the single grid size of the crack in x, y direction is 0.02m,6×6 fine grids form a coarse grid, 100×10 coarse grids form the natural crack, see fig. 2 for details;
step 22, the non-uniform slit width can be randomly generated on a fine-scale grid according to a specific probability distribution model, and the probability distribution model can be a normal distribution model, as shown in fig. 3;
s3, initializing parameters and boundary conditions in the seam;
S4, calculating a coarse-scale pressure field and a fine-scale pressure field and a speed field based on a multi-scale finite volume method;
S41, constructing a pressure equation of transient single-phase incompressible flow, and solving a coarse-scale pressure field;
wherein: Is a divergence operator; /(I) Pressure, pa; t is time, s; /(I)Is the viscosity of the fluid, pa.s; /(I)In order for the permeability to be a function of,；/>To solve the area;
Wherein:
In the middle of Is a seam width, m; /(I)Is the particle diameter, m; /(I)The surface roughness coefficient of the crack is dimensionless; /(I)Is the permeability of the granular sand pile,/>；/>The volume fraction of particles in the crack is dimensionless;
the method adopts a finite volume method to carry out discrete solution to obtain coarse-scale pressure 。
S42, constructing a dual grid on a coarse scale, and solving a dual basis function according to the following equation;
wherein: Is a divergence operator; /(I) Pressure, pa; t is time, s; /(I)Is the viscosity of the fluid, pa.s; /(I)In order for the permeability to be a function of,；/>K (k=1, 2,3, 4) th dual basis functions of the corresponding dual grids for any coarse grid; /(I)Is a dual grid;
the dual basis function satisfies the boundary condition:
wherein: ，k=1，2，3，4；
the solving process uses the dual grids corresponding to the coarse grids of the horizontal ith row and the vertical jth column Coarse meshes of corresponding horizontal ith row and longitudinal jth column correspond to 1 st dual basis function/>, of dual meshesSolving according to the following equation and boundary conditions, the 2 nd dual basis function/>, corresponding to the dual grid, of the coarse grid of the transverse ith row and the longitudinal jth+1st column can be calculated similarlyCoarse grid of the (i+1) th row and the (j) th column in the horizontal direction corresponds to the 3 rd dual basis function/>, of the dual gridCoarse grid of the (i+1) th row and the (j+1) th column corresponds to the 4 th dual basis function/>, of the dual grid：
Wherein: the 1 st dual basis function of the dual grid is corresponding to the coarse grid of the horizontal ith row and the vertical jth column; Permeability for the first dual mesh;
the dual basis functions are respectively combined with corresponding coarse scale pressures in a linear manner, so that the fine scale pressure inside the dual grid block can be obtained, namely:
wherein: is the fine scale pressure inside the dual grid block; /(I) K (k=1, 2,3, 4) dual basis functions of the dual grids are corresponding to the coarse grids of the horizontal ith row and the vertical jth column; /(I)Coarse scale pressure for the kth (k=1, 2,3, 4) dual basis function;
step S43, constructing a fine-scale grid on the fine scale Solving a fine-scale basis function according to the following equation;
wherein: Is a divergence operator; /(I) Pressure, pa; t is time, s; /(I)Is the viscosity of the fluid, pa.s; /(I)In order for the permeability to be a function of,；/>S (s=1, 2, …, 9) th fine-scale basis function for any fine-mesh;
the fine-scale basis function satisfies the boundary condition:
wherein: To pass the boundary/> Is a flux of (2); /(I)S th fine-scale basis function of fine grid of the i th row in the transverse direction and the j th column in the longitudinal direction/>A gradient along the boundary outward normal; /(I)To be in a fine scale region/>Fine grid/>, of the middle transverse i-th row, longitudinal j-th columnOn the boundary but not at boundary/>Is a flux of (2); n is the number of nodes of the fine grid in the y direction;
S=1, 2, …,9 respectively (i-1, j-1), (i, j-1), (i+1, j-1), (i-1, j), (i, j), (i+1, j), (i-1, j+1), (i, j+1), (i+1, j+1) node-controlled boundaries; /(I) The s-th fine scale basis function of the fine grid of the horizontal ith row and the vertical jth column;
the solving process uses the fine grid of the transverse ith row and the longitudinal jth column Fine-scale basis function 1 of fine-mesh corresponding to the i-th row and the j-th column in the transverse direction/>Solving according to the following equation and boundary conditions, the 2 nd fine-scale basis function/>, of the fine grid of the transverse i-1 th row and the longitudinal j-1 th column, can be calculated similarlyFine-scale basis function 3 of fine-mesh of horizontal i-1 th row, vertical j-th column/>Fine-scale basis function 4 of fine-mesh of horizontal i-1 row, vertical j+1st column/>…, Fine-scale basis function 9 of fine-mesh of row i+1 in the transverse direction and column j+1 in the longitudinal direction/>；
The fine scale basis function is respectively and linearly combined with 8 coarse scale pressures around the grid block (i, j) and the coarse scale pressure of the fine scale basis function, and the fine scale pressure of the grid block (i, j) can be obtained:
wherein: Fine scale pressure for grid block (i, j); /(I) The s-th fine scale basis function of the fine grid of the horizontal ith row and the vertical jth column; /(I)The coarse scale pressure corresponding to the s-th fine scale basis function;
Step S44, carrying out explicit calculation by using the pressure distribution of the fine-scale grids obtained by the multi-scale method through Darcy' S law to obtain a speed field on the fine-scale grids;
wherein: is the pressure gradient, pa/m; /(I) Is the viscosity of the fluid, pa.s; /(I)For permeability,/>；/>Fluid velocity on a fine scale grid, m/s;
S5, establishing and solving a particle transport model based on a fine-scale pressure field and a fine-scale velocity field to obtain particle concentration distribution on the fine scale;
Step S51, calculating the intra-slit particle migration speed according to the relation between the flow velocity of the fluid in the slit and the slit width and the particle diameter;
wherein: Fluid velocity on a fine scale grid, m/s; /(I) The transport speed of the particles in the seam is m/s; a is the seam width, m; /(I)Is the particle diameter, m;
Step S52, solving a particle migration differential equation on a fine-scale grid based on a multi-scale finite volume method to obtain particle concentration distribution on the fine scale;
the particle migration differential equation is:
wherein: the volume fraction of particles in the crack is dimensionless; t is time, s; /(I) Is a divergence operator; /(I)The transport speed of the particles in the seam is m/s; a is the seam width, m; /(I)The transport speed of the particles in the seam is m/s;
Step S6, carrying out iterative computation in one time step until iteration converges, repeating the steps S4-S5 when the iteration converges, and computing the next time step after the iteration converges;
The iteration convergence judgment standard is that the difference degree of the volume fractions of particles in the cracks in the current iteration step and the previous iteration step is less than 0.1%;
And S7, repeating the steps S4-S6 until simulation is completed, and finally determining the concentration distribution of particles in the crack.
Wherein the cloud of the particle concentration distribution of the 10 th time step is shown in fig. 4; a cloud chart of the particle concentration distribution of the 30 th step is shown in fig. 5; the 50 th step particle concentration distribution cloud is shown in fig. 6, and the 60 th step particle concentration distribution cloud is shown in fig. 7.
From the particle concentration profile cloud, it can be found that: in the near-wellbore region, the flow rate of the working fluid is relatively fast, resulting in rapid diffusion of particles from the wellbore around the fracture. In addition, because of the non-uniformity of the slit width of the slit, when the slit width is less than three times the particle diameter, the throat of the slit is easily blocked; conversely, when the seam width exceeds three times the particle diameter, the crack exhibits better flow conductivity and the particle concentration increases accordingly.
The present invention is not limited to the above-mentioned embodiments, but is not limited to the above-mentioned embodiments, and any person skilled in the art can make some changes or modifications to the equivalent embodiments without departing from the scope of the technical solution of the present invention, but any simple modification, equivalent changes and modifications to the above-mentioned embodiments according to the technical substance of the present invention are within the scope of the technical solution of the present invention.
Claims (6)
1. The multi-scale coarse crack particle migration simulation method is characterized by comprising the following steps of:
s10, constructing a rough crack physical model;
s20, constructing a coarse-scale grid and a fine-scale grid in a coarse crack physical model, and endowing uneven slit width on the fine-scale grid;
step S30, initializing parameters and boundary conditions in the seam;
Step S40, calculating a coarse-scale pressure field and a fine-scale pressure field and a speed field based on a multi-scale finite volume method;
S41, constructing a pressure equation of transient single-phase incompressible flow, and solving a coarse-scale pressure field;
S42, constructing a dual grid on a coarse scale, and solving a dual basis function according to the following equation;
wherein: Is a divergence operator; /(I) Pressure, pa; t is time, s; /(I)Is the viscosity of the fluid, pa.s; /(I)For permeability,/>；A kth dual basis function of the corresponding dual grid for any coarse grid; /(I)Is a dual grid;
the dual basis function satisfies the boundary condition:
wherein: ，k=1，2，3，4；
Respectively linearly combining the dual basis functions with corresponding coarse-scale pressures to obtain fine-scale pressures in the dual grid blocks;
wherein: is the fine scale pressure inside the dual grid block; /(I) The coarse grid of the horizontal ith row and the vertical jth column corresponds to the kth dual basis function of the dual grid; /(I)The coarse scale pressure corresponding to the kth dual basis function;
step S43, constructing a fine-scale grid on the fine scale Solving a fine-scale basis function according to the following equation;
wherein: Is a divergence operator; /(I) Pressure, pa; t is time, s; /(I)Is the viscosity of the fluid, pa.s; /(I)For permeability,/>；An s-th fine-scale basis function of any fine mesh;
the fine-scale basis function satisfies the boundary condition:
wherein: To pass the boundary/> Is a flux of (2); /(I)S th fine-scale basis function of fine grid of the i th row in the transverse direction and the j th column in the longitudinal direction/>A gradient along the boundary outward normal; /(I)To be in a fine scale region/>Fine grid/>, of the middle transverse i-th row, longitudinal j-th columnOn the boundary but not at boundary/>Is a flux of (2); n is the number of nodes of the fine grid in the y direction; /(I)The s-th fine scale basis function of the fine grid of the horizontal ith row and the vertical jth column;
The fine scale basis function is respectively and linearly combined with 8 coarse scale pressures around the grid block and the coarse scale pressure of the fine scale basis function, and the fine scale pressure of the grid block can be obtained:
wherein: fine scale pressure for the grid block; /(I) The s-th fine scale basis function of the fine grid of the horizontal ith row and the vertical jth column; /(I)The coarse scale pressure corresponding to the s-th fine scale basis function;
Step S44, carrying out explicit calculation by using the pressure distribution of the fine-scale grids obtained by the multi-scale method through Darcy' S law to obtain a speed field on the fine-scale grids;
s50, establishing and solving a particle transport model based on a fine-scale pressure field and a fine-scale velocity field to obtain particle concentration distribution on the fine scale;
Step S51, calculating the intra-slit particle migration speed according to the relation between the flow velocity of the fluid in the slit and the slit width and the particle diameter;
wherein: Fluid velocity on a fine scale grid, m/s; /(I) The transport speed of the particles in the seam is m/s; a is the seam width, m; is the particle diameter, m;
Step S52, solving a particle migration differential equation on a fine-scale grid based on a multi-scale finite volume method to obtain particle concentration distribution on the fine scale;
wherein: the volume fraction of particles in the crack is dimensionless; t is time, s; /(I) Is a divergence operator; /(I)The transport speed of the particles in the seam is m/s; a is the seam width, m; /(I)The transport speed of the particles in the seam is m/s;
step S60, carrying out iterative computation in one time step until iteration converges, repeating the steps S40-S50 when the iteration converges, and computing the next time step after the iteration converges;
and step S70, repeating the steps S40-S60 until the simulation is completed, and finally determining the concentration distribution of the particles in the crack.
2. The multi-scale rough crack grain migration simulation method according to claim 1, wherein the specific process of step S20 comprises:
s21, uniformly dividing M, N nodes in the x and y directions in the rough crack physical model to form a coarse scale grid;
s22, uniformly dividing m and n nodes along the x and y directions of the grid to form a fine-scale grid;
and S23, randomly generating non-uniform seam widths on the fine-scale grids through a probability distribution model.
3. A multi-scale rough crack grain migration simulation method according to claim 2, wherein the probability distribution model is a normal distribution model.
4. The method of claim 1, wherein the pressure equation for the transient single-phase incompressible flow is:
wherein: Is a divergence operator; /(I) Pressure, pa; t is time, s; /(I)Is the viscosity of the fluid, pa.s; /(I)For permeability,/>；/>To solve for the region.
5. The multi-scale rough crack grain migration simulation method according to claim 1, wherein the calculation formula in the step S44 is:
wherein: is the pressure gradient, pa/m; /(I) Is the viscosity of the fluid, pa.s; /(I)For permeability,/>；/>Is the fluid velocity on a fine-scale grid, m/s.
6. The method according to claim 1, wherein the step S60 is characterized in that the iteration convergence criterion is determined to be that the intra-fracture particle volume fraction difference between the current iteration step and the previous iteration step is less than 0.1%.
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| CN102870087A (en) * | 2010-04-30 | 2013-01-09 | 埃克森美孚上游研究公司 | Method and system for finite volume simulation of flow |
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| Extended Finite Element Method for Predicting Productivity of Multifractured Horizontal Wells;Youshi Jiang等;Mathematical Problems in Engineering;20140723;第2014卷;第1-10页 * |
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