Fine integration method based on two-dimensional fine fracture seepage heat transfer solution

Abstract

The application discloses a fine integration method based on two-dimensional fine fissure seepage heat transfer solution, and comprises the following steps: S1, establishing a control equation in a two-dimensional fine fissure lower domain, and performing grid division on a rectangular domain through discrete grids; S2, performing unit analysis on grid points in the rectangular domain and obtaining control equations of different unit grids; S3, converting partial differential equations into ordinary differential equations according to a finite difference method; S4, establishing a time domain recursion formula for solving ordinary differential equation groups; S5, solving a high-precision fine integration of a general solution and a special solution; and S6, establishing a discrete iteration recursion formula to solve the temperature of any point in the rectangular domain at any time. The application can realize the fine integration method based on two-dimensional fine fissure seepage heat transfer solution.

Description

Technical Field

[0001] This invention relates to the field of fine integration, and in particular to a fine integration method based on heat transfer infiltration in two-dimensional fine fractures. Background Technology

[0002] In recent years, with the rapid development of geothermal energy use, nuclear waste landfill, groundwater pollution, groundwater extraction, and oil extraction, the impact of temperature changes on deep fractured rock masses has received increasing attention. Geothermal energy is a renewable energy source with enormous potential, and my country ranks among the world's leading countries in the total amount of direct geothermal energy utilization. Compared with field measurement and analytical methods, numerical methods are less expensive and can consider complex underground structures and physical processes. Therefore, there is an urgent need for a numerical analysis method with high accuracy, good stability, and fast convergence speed.

[0003] Since its introduction by Zhong Wanxie in 1994, the precise integration method has been increasingly widely used in structural dynamic analysis. On the one hand, it possesses extremely high accuracy, essentially representing an exact solution on a computer; on the other hand, its solution process exhibits good stability, thus demonstrating high efficiency. However, for calculating particular solutions to non-homogeneous equations, matrix inversion is frequently involved. This matrix inversion operation is not only computationally intensive but can sometimes be unstable or even impossible, which limits the application of the precise integration method to some extent. To address this issue, many scholars have proposed various methods for finding particular solutions using the precise integration method, including homogeneous expansion and dimensionality enhancement methods, direct numerical integration methods, and response matrix methods. Currently, the precise integration method is relatively well-developed, and more and more scholars are applying it to various fields involving parabolic equations, elliptic equations, and hyperbolic equations. In the field of heat transfer in fractured rock masses, compared to the finite element method, the precise integration method can obtain accurate numerical solutions with high precision, good stability, and fast convergence speed. Summary of the Invention

[0004] The purpose of this invention is to provide a refined integration method based on the heat transfer solution of two-dimensional fine crack seepage, which aims to solve the refined integration problem of heat transfer solution of two-dimensional fine crack seepage.

[0005] This invention provides a refined integral method for solving heat transfer in two-dimensional fine fracture seepage, comprising:

[0006] The refined integral method based on the heat transfer solution of two-dimensional fine fracture seepage includes the following steps:

[0007] S1: Establish the governing equations within the two-dimensional fine crack domain, and perform meshing of the rectangular domain using a discrete mesh;

[0008] S2: Perform element analysis on the grid points within the rectangular domain and obtain the control equations for different element grids;

[0009] S3: Convert the partial differential equation into an ordinary differential equation using the finite difference method;

[0010] S4: Establish the time-domain recursive formula for the solution of the system of ordinary differential equations;

[0011] S5: High-precision fine integration of general and particular solutions;

[0012] S6: Establish a discrete iterative recursive formula to solve for the temperature at any point within the rectangular domain at any time.

[0013] Using the embodiments of the present invention, the present invention can realize precise integration based on heat transfer solution of two-dimensional fine crack seepage.

[0014] The above description is merely an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention and to implement it in accordance with the contents of the specification, and in order to make the above and other objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention are described below. Attached Figure Description

[0015] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0016] Figure 1 This is a flowchart of a refined integral method for solving heat transfer in two-dimensional fine fractures according to an embodiment of the present invention.

[0017] Figure 2 This is a schematic diagram of a two-dimensional fine crack conceptual model based on a refined integral method for solving heat transfer infiltration in two-dimensional fine cracks according to an embodiment of the present invention.

[0018] Figure 3 This is a schematic diagram of a discrete mesh of a rectangular domain in a fine crack, based on a fine integration method for solving heat transfer infiltration in two-dimensional fine cracks, according to an embodiment of the present invention.

[0019] Figure 4 This is a schematic diagram of material parameters according to an embodiment of the present invention;

[0020] Figure 5 This is a schematic diagram illustrating the calculation results of the numerical method, finite element method, and analytical solution of the present invention on the change of fracture water temperature at x = 10m over 200 days with two different time steps;

[0021] Figure 6This is a schematic diagram of the two-dimensional distribution of the rectangular domain temperature over 200 days according to an embodiment of the present invention;

[0022] Figure 7 This is a schematic diagram of the temperature distribution of the rectangular domain at 20, 100, and 180 days according to an embodiment of the present invention.

[0023] Figure 8 This is a schematic diagram comparing the results of the numerical method and the finite element method for calculating the temperature at x=10m over 100 days in an embodiment of the present invention, using different time steps. Detailed Implementation

[0024] The technical solution of the present invention will be clearly and completely described below with reference to the embodiments. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0025] Method Implementation Examples

[0026] According to embodiments of the present invention, a refined integral method for solving heat transfer through seepage in two-dimensional fine cracks is provided. Figure 1 This is a flowchart of a refined integral method for solving heat transfer in two-dimensional fine fractures according to an embodiment of the present invention, as shown below. Figure 1 As shown, it specifically includes:

[0027] like Figure 1 As shown, this invention proposes a refined integration method for solving heat transfer infiltration in two-dimensional fine cracks. This method, based on finite difference and refined integration techniques, first establishes the governing equations within the two-dimensional fine crack domain and then meshes the rectangular domain using a discrete mesh. Next, element analysis is performed on the mesh points within the rectangular domain to obtain the governing equations for different element meshes. Then, the partial differential equations are converted into ordinary differential equations using the finite difference method, and a time-domain recursive formula for the solution of the ordinary differential equation system is established. Next, high-precision refined integration is performed on the general solution and particular solutions. Finally, a discrete iterative recursive formula is established to solve for the temperature at any point within the rectangular domain at any time. Compared to the finite element method, this invention exhibits better stability, significantly reduces runtime and memory requirements, increases the time step of discrete iteration, and improves both computational efficiency and accuracy.

[0028] To achieve the above objectives, this invention proposes a refined integral method for solving heat transfer in two-dimensional fine fracture seepage, comprising the following steps:

[0029] Step 1: Establish the governing equations within the two-dimensional fine-crack domain, and mesh the rectangular domain using a discretized grid, including:

[0030] Figure 2 This is a schematic diagram of a two-dimensional fine crack conceptual model based on a refined integral method for solving heat transfer infiltration in two-dimensional fine cracks according to an embodiment of the present invention.

[0031] like Figure 2 As shown, a two-dimensional fine fracture is analyzed using a discrete fracture network model. It is assumed that the porosity within the fracture is 1, while the porosity within the bedrock is 0. Due to the small fracture width and its relative size compared to the bedrock, the fine fracture is analyzed using a one-dimensional element form throughout the temperature field analysis. This means that seepage and heat conduction along the direction perpendicular to the fracture extension are not considered, and the compressibility of the fluid within the fracture is ignored. For the heat exchange between the bedrock and the fracture, the assumption that the bedrock surface temperature is equal to the fracture fluid temperature is adopted, assuming that the heat transfer between the fracture and the bedrock is mainly related to the temperature gradient of the bedrock. The governing equation for the bedrock region is a heat source-free two-dimensional transient heat conduction equation. From this, the governing equations for the entire rectangular domain can be obtained.

[0032] Based on the principle of energy conservation, the heat transfer control equation for seepage within fine cracks is established as follows:

[0033]

[0034] The first term in the formula is the change in temperature over time, the second term is the energy carried by heat transfer through seepage, the third term is the heat conduction of the fluid within the fissure, and the fourth term is the heat exchange between the fissure and the rock.

[0035] Similarly, the heat conduction equation within bedrock is obtained:

[0036]

[0037] The width of the fine crack is 2b, the initial temperature within the entire rectangular domain is T0, and the temperature of the injected fluid is T. in The subscripts "r", "f", "x", and "y" represent bedrock, fluid within the fracture, the x-axis direction, and the y-axis direction, respectively. Here, r takes the value 1 or 2, ρc is the volumetric heat capacity, T is the temperature, t is the time, and λ is the time. f , λ rx , λ ry These represent the thermal conductivity coefficients of the fracture fluid, bedrock along the x-axis, and bedrock along the y-axis, respectively, with the fracture extending along the x-axis. f The constant flow velocity of the fluid within the fissure.

[0038] Step 2: Perform element analysis on the grid points within the rectangular domain and obtain the governing equations for different element grids, including:

[0039] Figure 3 This is a schematic diagram of a discrete mesh of a rectangular domain in a fine crack, based on a fine integration method for solving heat transfer infiltration in two-dimensional fine cracks, according to an embodiment of the present invention.

[0040] like Figure 3As shown, the governing equations for the bedrock region unit grid are based on the bedrock internal heat conduction equations, while the governing equations for the fracture region unit grid are based on the fine fracture seepage heat transfer governing equations. For the interface between the bedrock and the fracture, the interface is assumed to be the fine fracture itself, and the fine fracture seepage heat transfer governing equations are also used.

[0041] Step 3: Use the finite difference method to transform the partial differential equation into an ordinary differential equation, including:

[0042] For any bivariate function f(x,y), let f ij =f(x) i ,y j ),

[0043] Finite difference finite difference method is used for the governing equations within the rectangular domain;

[0044] The heat transfer control equation for seepage within the fracture, after finite difference, is expressed as:

[0045]

[0046] Applying boundary condition treatment to the fractured region yields:

[0047]

[0048] The equation for heat conduction within bedrock, after finite difference, is expressed as:

[0049]

[0050] After applying boundary condition treatment to the bedrock, we get:

[0051]

[0052] Where Ω represents the interior of the rectangular region (lx×ly), and ly=ly1+ly2+2b, ly1 is the length in the y-direction of bedrock region 1, and ly2 is the length in the y-direction of bedrock region 2; for ease of calculation, based on the assumption of fine fractures, we can consider ly=ly1+ly2. The rectangular domain is defined by its boundary, divided into m rows and n columns. T represents the temperature distribution within the rectangular domain. Subscripts i and j represent the row and column, respectively. Rows 1 to m1-1 belong to bedrock 1, row m1 belongs to fractures, and rows m1+1 to m belong to bedrock 2. The step sizes of bedrock 1 and 2 along the y-axis are respectively... The step size in the x-axis direction is...

[0053] In the formula δ ij The symbol is Kronecker_δ;

[0054] Substituting equations (3) and (5), the governing equations within the rectangular domain can be written in the following form:

[0055]

[0056] The matrices in the above formula are as follows:

[0057]

[0058]

[0059]

[0060]

[0061]

[0062]

[0063]

[0064]

[0065]

[0066]

[0067]

[0068] I n×n Since is an n-order identity matrix, equation (7) is an ordinary differential equation. It depends only on time t, denoted as

[0069] Step 4: Establish the time-domain recursive formula for the solution of the system of ordinary differential equations, including:

[0070] Transform the initial conditions into:

[0071]

[0072] The solution to equation (7) can be expressed as:

[0073]

[0074] in For a particular solution of the equation, if the time domain t is uniformly discretized with step size dt, it is not difficult to verify that at t = t k+1 The solution for time is:

[0075]

[0076] in t k =kdt, T(dt) = eHdt , This is a particular solution to the equation; from this, we obtain the time-domain recursive formula for the solution of the system of ordinary differential equations.

[0077] Step 5: Perform high-precision integration to find the general and particular solutions, including:

[0078] T(dt)=e Hdt It is the transfer matrix, which can be calculated precisely in the following way:

[0079] Divide the time step dt into M=2 N Divide into equal parts, let Δt = dt / M, then:

[0080] T(dt)=e Hdt =(e HΔt ) M ; (twenty two)

[0081] Since dt is a small time step, when N = 20, M = 1048576, Δt = dt / M will be a very small time interval for e. HΔt Performing a Taylor expansion and ignoring higher-order terms, we have:

[0082] e HΔt ≈I+HΔt+(HΔt) 2 / 2+(HΔt) 3 / 6; (23)

[0083] remember The above formula can then be written as:

[0084]

[0085] It can be calculated using the following recursive formula:

[0086]

[0087] For the particular solution part, let:

[0088]

[0089] Here we only list the first four terms on the right-hand side of the above equation for analysis. Substitute the above equation into the particular solution. The integral expression can be obtained as follows:

[0090]

[0091] in:

[0092]

[0093] In the above formula, q = 0, 1, 2, 3 are used as variables to replace s = t k+1-t, and note t k+1 -t k =dt, then the above expression can be simplified to:

[0094]

[0095] Therefore, S q Regardless of the segment it belongs to, it only needs to be calculated once in the entire calculation process; similar to the previous method, the interval [0, dt] is divided into M equal parts with an interval of Δt, and denoted as Y = e HΔt By numerically integrating the terms of the above equation using the rectangular integral formula, we obtain:

[0096]

[0097] use The first 2 in the above equation a The sum of the terms can be easily verified by the following recursive formula:

[0098]

[0099]

[0100]

[0101]

[0102] in:

[0103]

[0104]

[0105] This completes the high-precision fine integration calculations for the general and particular solutions.

[0106] Step Six: Establish a discrete iterative recursive formula to solve for the temperature at any point within the rectangular domain at any time, including:

[0107] The discrete iterative recurrence formula can be easily derived from the results obtained using the above steps:

[0108]

[0109] Where the transfer matrix T(dt) = e Hdt It can be obtained from the following formula:

[0110]

[0111] The part S0r0+S1r1+S2r2+S3r3 can also be obtained by the following formula:

[0112]

[0113]

[0114]

[0115]

[0116] Where Y 2a It can be obtained from the following formula:

[0117]

[0118] Therefore, the precise temperature value of any grid point at any time within the rectangular domain can be calculated iteratively.

[0119] Figure 4 This is a schematic diagram of material parameters according to an embodiment of the present invention;

[0120] Example 1: Material parameters are as follows Figure 4 As shown, only the heat exchange between the upper surface of the fracture and the bedrock is considered, while the lower surface of the fracture is insulated. The fracture width is taken as 1 mm, T0 = 90°, T in =20°, the temperature of the bedrock at the infinite boundary along the x-axis and the infinite boundary along the y-axis is T0, the temperature of the fissure at the infinite boundary along the x-axis is also T0, the flow velocity of the water in the fissure is taken as 0.01m / s, and the change of fissure water temperature at x=10m with time is observed.

[0121] Figure 5 This is a schematic diagram illustrating the calculation results of the numerical method, finite element method, and analytical solution of the present invention on the change of fracture water temperature at x = 10m over 200 days with two different time steps.

[0122] In implementing example 1, such as Figure 5 As shown, the calculation results of the change of fracture water temperature at x = 10m over 200 days using the numerical method and finite element method proposed in this invention are compared with the analytical solution.

[0123] Figure 6 This is a schematic diagram of the two-dimensional distribution of the rectangular domain temperature over 200 days according to an embodiment of the present invention;

[0124] In implementing example 1, such as Figure 6 As shown, the two-dimensional distribution of temperature in a rectangular domain at 200 days was calculated using the numerical method proposed in this invention.

[0125] Figure 7 This is a schematic diagram of the temperature distribution of the rectangular domain at 20, 100, and 180 days according to an embodiment of the present invention.

[0126] In implementing example 1, such as Figure 7As shown, the temperature distribution of the rectangular domain at 20, 100, and 180 days was calculated using the numerical method proposed in this invention.

[0127] Figure 8 This is a schematic diagram comparing the results of the numerical method and the finite element method for calculating the temperature at x=10m over 100 days in an embodiment of the present invention, using different time steps;

[0128] In implementing example 1, such as Figure 8 As shown, the numerical method proposed in this invention and the finite element method are compared to calculate the temperature at x=10m over 100 days. Stability analysis is performed using different time steps. It is easy to see that the numerical algorithm proposed in this invention has higher calculation accuracy and stability than the finite element method. Its numerical error is almost independent of the selection of time step. In other words, under the premise of ensuring calculation accuracy, a larger time step can be used for calculation to improve calculation efficiency and reduce memory consumption.

[0129] The beneficial effects of this invention are as follows: by employing the refined integral method of this invention based on the heat transfer solution of two-dimensional fine crack seepage, the following benefits are obtained.

[0130] For the governing equations within a rectangular domain, this invention uses the finite difference method to discretize the spatial partial differential operators and the fine integration technique to process the time partial differential operators, which greatly improves the stability of the algorithm, increases the time step, reduces the number of iterations, reduces memory usage, reduces running time, and improves computational efficiency.

[0131] In the calculation of the general solution and the particular solution, the fine integration technique is also used for the particular solution. The problem of matrix inversion is solved in the iteration process, while the particular solution maintains the same accuracy as the general solution.

[0132] The numerical error of this invention is almost independent of the time step. Under the premise of meeting the calculation conditions, the iteration time step can be selected as large as possible, thereby improving the calculation efficiency.

[0133] The basic idea and solution steps of this invention are universal and can be applied not only to solving heat transfer problems in two-dimensional fine crack seepage, but also to solving other heat transfer problems.

[0134] The shortcomings of this invention are: the method of this invention is essentially a semi-analytical method, which uses finite difference discretization in space; in other words, when the numerical solution calculated by the method of this invention is compared with the analytical solution, there is still an error in spatial discretization.

[0135] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions to the technical solutions of the embodiments of the present invention do not cause the essence of the corresponding technical solutions to deviate from the scope of the present solution.

Claims

1. A refined integral method for solving heat transfer through seepage in two-dimensional fine cracks, characterized in that, Includes the following steps: S1: Establish the governing equations within the two-dimensional fine crack domain, and perform meshing of the rectangular domain using a discrete mesh; S2: Perform element analysis on the grid points within the rectangular domain and obtain the control equations for different element grids; S3: Convert the partial differential equation into an ordinary differential equation using the finite difference method; S4: Establish the time-domain recursive formula for the solution of the system of ordinary differential equations; S5: High-precision fine integration of general and particular solutions; S6: Establish a discrete iterative recursive formula to solve for the temperature at any point within the rectangular domain at any time; Specifically, S3 includes: For any bivariate function ,remember , ; Finite difference finite difference method is used for the governing equations within the rectangular domain; The differential expression of the heat transfer control equations for seepage within fractures is as follows: （3） Applying boundary condition treatment to the fractured region yields: （4） The equation for heat conduction within bedrock, after difference, is expressed as: （5） After applying boundary condition treatment to the bedrock, we get: （6） in Inside the rectangular region ,in , Bedrock 1 area directional length, Bedrock 2 area directional length; fine crack is , To define the boundary of the rectangular domain, divide the rectangular domain into... OK List, Temperature distribution within a rectangular region, subscript and They represent rows and columns respectively. Line 1 belongs to bedrock. The line belongs to the fissure. The line belongs to bedrock 2, and bedrock 1 and 2 are along The step sizes in the axial direction are respectively , , The step size in the axial direction is ; In the formula For Kronecker_ symbol; Substituting equations (3) and (5), the governing equations within the rectangular domain can be written in the following form: （7） The matrices in the above formula are as follows: （8） ；（9） （10） （11） （12） （13） （14） （15） （16） （17） （18） for The identity matrix of order 1, from which equation (7) becomes an ordinary differential equation, Only with time Related, record .

2. The method according to claim 1, characterized in that, S1 specifically includes: Two-dimensional fine fractures are analyzed using a one-dimensional element model with a discrete fracture network. The porosity within the fractures is 1, while the porosity within the bedrock is 0. Seepage and heat conduction along the direction perpendicular to the fracture extension are neglected, and the compressibility of the fluid within the fractures is ignored. The bedrock surface temperature is equal to the fracture fluid temperature. The heat transfer between the fractures and the bedrock is mainly related to the bedrock temperature gradient. The governing equation for the bedrock region is a heat source-free two-dimensional transient heat conduction equation. Based on the principle of energy conservation, the governing equation for seepage and heat transfer within the fine fractures in the entire rectangular domain is obtained, as follows: （1） In the formula, the first term is the change in temperature over time, the second term is the energy carried by heat transfer through seepage, the third term is the heat conduction of the fluid within the fissure, and the fourth term is the heat exchange between the fissure and the rock. Similarly, the heat conduction equation within bedrock is obtained: （2） The width of the fine crack is The initial temperature within the entire rectangular region is The temperature of the injected fluid is ; Subscript , , and Representing bedrock, fluid within fractures, and Axial direction and axial direction, here Choose 1 or 2. For volumetric heat capacity, For temperature, For time, , , These are respectively fissure fluid and bedrock along Axis and bedrock along Thermal conductivity of the shaft, along the crack Extending along the axial direction, The constant flow velocity of the fluid within the fissure.

3. The method according to claim 1, characterized in that, S2 specifically includes: The governing equations for the bedrock region unit grid are based on the bedrock internal heat conduction equations, while the governing equations for the fracture region unit grid are based on the fine fracture seepage heat transfer governing equations. For the interface between the bedrock and the fracture, which is considered a fine fracture, the fine fracture seepage heat transfer governing equations are used.

4. The method described in step four of claim 1, characterized in that, S4 specifically includes: Transform the initial conditions into: （19） The solution to equation (7) can be expressed as: （20） in For a particular solution of the equation, the time domain With step size Uniformly discrete, it is not difficult to verify that in The solution for time is: （21） in , , , This is a particular solution to the equation; from this, we obtain the time-domain recursive formula for the solution of the system of ordinary differential equations.

5. The method according to claim 1, characterized in that, S5 specifically includes: It is a transfer matrix, calculated as follows: Time step Divided into Divide into equal parts, let ,but: （22） because It is a small time step, when hour, , It will be a very small time segment, for Performing a Taylor expansion and ignoring higher-order terms, we have: （23） remember Then (22) can be written as: （24） Calculated using recursive formula (25): （25） For the particular solution part, let: （26） Substitute (26) into the particular solution The integral expression is obtained as follows: （27） in: （28） In the above formula (28) Perform variable substitution And noted The above expression then becomes: （29） Regardless of the segment it is located in, the interval will be... Divided into Divide into equal parts, with intervals of . ,remember By numerically integrating the terms of (29) using the rectangular integral formula, we obtain: （30） use Indicate the first part of the above equation The sum of terms verifies the recursive formulas (31)-(34): （31） （32） （33） （34） in: （35） （36） High-precision fine integration calculations were completed for the general solution and particular solution.

6. The method according to claim 1, characterized in that, S6 specifically includes: The discrete iterative recurrence formula can be easily derived from the results obtained using the above steps: （37） Where the transfer matrix It can be obtained from the following formula: （38） Partially, it can also be obtained from the following formula: （39） （40） （41） （42） in It can be obtained from the following formula: （43） The precise temperature value of any grid point at any time within the rectangular domain is calculated iteratively.

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