Implicit hydraulic fracturing simulation calculation method based on finite element-near field dynamics

Abstract

The invention belongs to the technical field of numerical simulation, and provides an implicit hydraulic fracturing simulation calculation method based on finite element-near field dynamics. According to the method, firstly, a hydraulic fracturing model is divided into three parts including solid deformation, crack propagation and flowing of fluid in a porous medium; a solid deformation part is simulated by a finite element, and near-field dynamic material points are arranged in a possible crack development area. In the crack propagation part, crack initiation is judged by a near-field dynamics method, and crack propagation is simulated. The flow of the fluid in the porous medium comprises the flow of the fluid in the crack and is simulated by a finite element. And finally, through fully implicit solution of the fluid-solid coupling equation set, an expansion path of hydraulic fracture, a fracture deformation mode and fluid pressure in the porous medium can be accurately predicted, and dynamic modeling and efficient simulation of propagation of the hydraulic fracture in the porous medium are realized.

Description

Technical Field

[0001] The present invention belongs to the technical field of numerical simulation, and more specifically, relates to an implicit hydraulic fracturing simulation calculation method based on finite element-peridynamics. Background Art

[0002] Hydraulic fracturing involves injecting a fluid containing a proppant into a wellbore to generate pressure, causing fractures to develop and propagate in the rock. It is widely used in the petroleum industry to develop oil and gas reservoirs, particularly in low-permeability, tight formations. Simulating hydraulic fracturing in porous media involves three physical processes: (i) deformation of the porous formation, (ii) fluid flow, and (iii) fracture propagation. These processes reveal a strong coupling between the solid porous formation and the fluid, making their numerical simulation a challenging task.

[0003] The finite element method has the universality and flexibility to deal with fracture propagation problems under various conditions, but it encounters challenges in meshing, selecting appropriate crack propagation criteria, and crack tip singularities. Peridynamics theory is a new non-local theory that uses spatial integral equations to describe the mechanical behavior of materials. When simulating crack propagation with the peridynamic method, compared with traditional numerical simulation methods, it does not require additional crack propagation criteria. In addition, by using spatial integral equations instead of differential equations to describe the mechanical behavior of materials, it avoids the singularity and complexity problems faced by the method based on differential equations when describing discontinuous behavior. However, although the peridynamic model eliminates the problems of traditional numerical methods such as the need for additional crack propagation criteria and mesh dependence, it has the disadvantages of large computational complexity and low model solution efficiency.

[0004] In summary, due to the complex fluid-structure coupling and large computational effort involved, near-field dynamics alone cannot adequately simulate the complex hydraulic fracturing process. Furthermore, given the limitations of conventional numerical simulation methods such as finite element methods in selecting crack propagation criteria and crack expansion, the development of more effective numerical simulation methods for hydraulic fracturing remains a challenge. Therefore, this paper proposes an implicit hydraulic fracturing simulation calculation method based on finite element and peri-field dynamics. Compared to explicit amplification, implicit methods generally have higher numerical stability and allow for larger time steps, making them more suitable for application in geological practice for oil and gas reservoir development.

[0005] By coupling peridynamics and the finite element method, a method for simulating hydraulic fracturing in porous media is proposed. The specific process includes: ① For hydraulic fractures, the control equations are given, including the solid deformation equation and the fluid flow equation in the fractures and pores; ② For fractures and pores in the reservoir, the fluid in the fractures and pores is considered to establish a solid-seepage-flow coupled fracture propagation model; ③ The fluid field is described using the finite element method, and the solid field is simulated using peridynamics, and a peridynamics and finite element coupling numerical method and high-performance computing program are developed to solve multi-field problems. Summary of the Invention

[0006] In response to the limitations of existing hydraulic fracturing simulation techniques, the present invention proposes an implicit hydraulic fracturing simulation calculation method based on finite element and peridynamics. This method can effectively and accurately predict the expansion path of hydraulic fracturing, the fracture deformation pattern, and the fluid pressure in porous media, thereby achieving dynamic modeling and efficient simulation of hydraulic fracture propagation in porous media. The implementation steps include:

[0007] Step 1: For hydraulic fractures, the governing equations are given, including the solid deformation equation and the fluid flow equations in the fractures and pores:

[0008] Step 1.1, construct the solid deformation equation;

[0009] Step 1.2, construct the fluid flow equation in the fracture;

[0010] Step 1.3, construct the fluid flow equation in the pore;

[0011] Step 2: Discretize the solid equations and fluid equations using the finite element method and peridynamics method respectively:

[0012] Step 2.1: Divide the computational domain into crack-free regions and regions where cracks may occur. The crack-free regions are meshed using finite element methods to construct a mesh and achieve spatial discretization. The regions where cracks may occur are spatially discretized using peridynamic material points.

[0013] Step 2.2: Replace the finite element and peridynamic material points. The specific method is to construct a grid and divide the calculation area into a finite element point area and a peridynamic point area. Since the calculation domain of some peridynamic points contains finite element points, these points are called coupling points. The bond connecting the peridynamic point and the coupling point only applies force to the peridynamic point.

[0014] The finite elements containing the peridynamic points act only on the finite element points. The final coupling is achieved by using the terms from the finite element theory (representing the forces acting on the finite element nodes) and the terms from the local discretization (representing the forces acting on the local nodes) in the final constructed global stiffness matrix on a per-region basis.

[0015] Step 2.3, discretize the fluid flow equation of the porous medium using the finite element method;

[0016] Step 3: Based on the reservoir conditions, the fracture criterion is given to determine whether the fracture is open and the size of the fracture using the peridynamic method;

[0017] Step 4. For fractures within the reservoir, a coupled solid-seepage-flow hydraulic fracture propagation model is established, taking into account the fluids within the fractures and the fluids within the voids. This achieves solid-fluid coupling by simultaneously solving the solid equations with the equations for the fluids within the pores and fractures. Solid deformation and fracture propagation are captured using peridynamics and the finite element method, while fluid flow in the reservoir and fractures is simulated using the finite element method. Because the equations are dynamic and nonlinear, the Newmark method is used for time discretization, and inertial forces and acceleration terms are considered. The entire process is solved using the implicit Newton-Raphson method.

[0018] The present invention provides an implicit hydraulic fracturing simulation calculation method based on finite element-peridynamics, which overcomes the limitations of the traditional finite element method in the selection of crack propagation criteria and crack propagation and the limitations of the peridynamics method in computational efficiency. It makes full use of the coupling method of peridynamics and finite element to establish a solid-seepage-flow coupled hydraulic fracture propagation model. This method can effectively and accurately predict the expansion path of hydraulic fracturing, the crack deformation mode and the fluid pressure in the porous medium, and realizes the dynamic modeling and efficient simulation of the propagation of hydraulic fractures in porous media. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] In order to more clearly illustrate the embodiments or implementation effects of the present invention, necessary drawings are provided in this specification. The following is a description of the drawings used:

[0020] Figure 1 This is a flow chart of an implicit hydraulic fracturing simulation calculation method based on finite element-peridynamics described in this specification; Figure 2 is a schematic diagram of the coupling of finite element and peridynamic material point replacement described in this specification;

[0021] Figure 3 is a schematic diagram of setting the initial crack as described in this specification;

[0022] Figure 4 This is a schematic diagram of hydraulic fracture fluid pressure characterized by the method of the present invention in an embodiment provided in this specification;

[0023] Figure 5 This is a schematic diagram of the fracture path and fluid pressure after the hydraulic fracture characterized by the method of the present invention intersects with the fracture in the reservoir in an embodiment provided in this specification. DETAILED DESCRIPTION

[0024] By coupling peridynamics and the finite element method and fully implicitly solving the fluid-solid coupling equations, the present invention accurately predicts the expansion path of hydraulic fracturing, the fracture deformation pattern, and the fluid pressure within the porous medium, thereby achieving dynamic modeling and efficient simulation of the propagation of hydraulic fractures in porous media. To better illustrate the technical solution of the present invention, the following, in conjunction with the accompanying drawings, uses an example of the present invention in hydraulic fracturing of a reservoir in a block in the Tarim Basin to explain in detail the specific implementation of the present invention. This example does not limit the present invention.

[0025] Figure 1 This is a flow chart of an implicit hydraulic fracturing simulation calculation method based on finite element-peridynamics described in this specification. A detailed description of the specific implementation steps is given below.

[0026] Step 1: For hydraulic fractures, the control equations are given, including the solid deformation equation and the fluid flow equation in the fractures and pores. In the fracture fluid equation, the permeability k of the fracture is represented by the fracture width.

[0027] Step 2: Divide the calculation area into the finite element area and the peridynamic area. The peridynamic area is where cracks may occur, and the finite element area is the area containing boundary conditions or no crack propagation. Both the finite element area and the peridynamic area are discretized, such as Figure 2 shown.

[0028] Step 2.1: Perform finite element meshing in the calculation area to obtain multiple finite element units, which are quadrilateral units.

[0029] In step 2.2, in the peridynamic region, the finite element nodes are replaced with peridynamic material points, so that the mesh in the finite element region and the material points in the peridynamic region are consistent in the discrete scale.

[0030] Step 2.3, couple near-field dynamics with the finite element method and set the coupling region. In the coupling region, the particle not only interacts with other particles in its neighborhood through bonds, but also interacts with the finite element nodes in its neighborhood. The coupling region is defined as:

[0031] Ω c ={x j |x j ∈(H xi ∩Ω F )andx i ∈Ω p}

[0032] x is a material point, i and j range from 1 to N, N is the total number of finite element nodes and peridynamic mass points. H is the computational domain of the peridynamic point, in which the material point x i and x j Interaction. According to the solid deformation equation, the stiffness matrix K is obtained by finite element method. FEM , the stiffness matrix K is obtained by peridynamics method PD Assumptions is the coupling matrix, and each material node is judged. If the point x i ∈Ω p , then using the peridynamic matrix, Otherwise, the finite element matrix is ​​used, Where j=2(i-1)+1.

[0033] Step 3: Set the initial crack and define the crack fracture conditions.

[0034] Step 3.1 Set the initial crack, distribute the peridynamic points on both sides of the crack, and break all the bonds passing through the initial crack, such as Figure 3 shown.

[0035] Step 3.2 defines the crack fracture condition, which refers to the condition where the bonds between the peridynamic material points break when the elongation of the bonds exceeds a certain critical value, i.e., s>s0, where s is the elongation of the bonds between the material points and s0 is the critical elongation of the bonds between the material points. When a series of bonds break, a discontinuous space is formed by these broken bonds. To describe the crack path, a scalar called the damage parameter is used. This scalar is the ratio of the number of broken bonds connected to the material point to the total number of bonds connected to the material point. Generally, it can be considered that when this scalar is greater than a certain value, such as 0.4, a crack surface has formed.

[0036] Step 4. For the fractures in the reservoir, considering the fluid in the fractures and the fluid in the voids, a solid-seepage-flow coupled hydraulic fracture propagation model is established, and the solid equations are solved simultaneously with the fluid equations in the pores and the fluid equations in the fractures.

[0037] Step 4.1: Consider the geological characteristics of the actual study area and set the geological parameters, time parameters, and boundary conditions. Discretize the model in time, set the time step, and the total calculation time.

[0038] Step 4.2, loop time step (initialize n=0), n=n+1;

[0039] Step 4.2.1, loop iteration step itn (initialize itn=0), itn=itn+1;

[0040] In step 4.2.2, the solid equations are solved together with the fluid equations in the pores and the fluid equations in the fractures to obtain the displacement field u and the pressure field p. Based on the results, the length, width, fluid pressure in the fracture, and fracture permeability of the fracture are calculated. Then, the velocity and acceleration of the material point are obtained using the Newmark method.

[0041] Step 4.2.3: Determine whether any peridynamic bond breaks at this time step. If so, find the broken bond, delete it, update the stiffness matrix, and repeat step 4.2.2. If no bond breaks, determine whether the convergence condition is met. If so, the convergence condition is met (the convergence tolerance in this example is 1.0×10 -5 ), then record the displacement field and pressure field at that moment, as well as the length of the crack, and proceed to the next step 4.3. If the convergence condition is not met, return to step 4.2.1.

[0042] Step 4.3: Determine whether the calculation time exceeds the set total calculation time. If it is equal to or exceeds, proceed to step 4.4; otherwise, proceed to step 4.2.

[0043] Step 4.4, end the calculation and output and save the calculation results, and use the computer to draw fluid pressure, crack path diagram, etc. Figure 4 FIG. 1 is a schematic diagram of hydraulic fracture fluid pressure characterized by the method of the present invention in an embodiment provided in this specification. Figure 5 This is a schematic diagram of the fracture path and fluid pressure after the hydraulic fracture characterized by the method of the present invention intersects with the fracture in the reservoir.

Claims

1. An implicit hydraulic fracturing simulation calculation method based on finite element-peridynamics, characterized in that: For hydraulic fractures, the governing equations are given, including the solid deformation equation and the fluid flow equation in the fractures and pores.

2. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 1, characterized in that: The solid equations and fluid equations are discretized using the finite element method and peridynamics method respectively.

3. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 2, characterized in that: The calculation area is divided into crack-free areas and areas where cracks may occur. The crack-free areas are divided by finite element meshes, and the meshes are constructed to achieve spatial discretization. The areas where cracks may occur are spatially discretized by peri-field dynamic material points.

4. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 3, characterized in that: To achieve the replacement of finite element and peridynamic material points, the computational domain is divided into regions of finite element points and regions of peridynamic points. Since some peridynamic points contain finite element points within their computational domain, these points are referred to as coupling points. The bond connecting a peridynamic point to a coupling point applies forces only to the peridynamic point, while the finite elements containing peridynamic points only apply forces to the finite element points. The final coupling is achieved by using terms from the finite element theory (representing forces acting on finite element nodes) and terms from the local discretization (representing forces acting on local nodes) in the final constructed global stiffness matrix, partitioned by region.

5. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 2, characterized in that: The fluid flow equations in porous media are discretized using the finite element method.

6. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 1, characterized in that: According to the reservoir conditions, the fracture criterion is given to judge whether the fracture is cracked and the size of the crack by using the peridynamic method.

7. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 1, characterized in that: For the fractures in the reservoir, considering the fluid in the fractures and the fluid in the voids, a solid-seepage-flow coupled hydraulic fracture propagation model is established to realize solid-fluid coupling.

8. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 1, characterized in that: The solid equations are solved simultaneously with the equations for the fluid in the pores and fractures. Solid deformation and fracture propagation are captured using peridynamics and the finite element method, while fluid flow in the reservoir and fractures is simulated using the finite element method. Because the equations are dynamic and nonlinear, the Newmark method is used for time discretization, accounting for inertial forces and acceleration terms. The entire process is solved using the implicit Newton-Raphson method.

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