Medium stable seepage field analysis method based on pore-karst cave structure

Abstract

The invention relates to the technical field of porous medium seepage mechanics, in particular to a pore-karst cave structure-based medium stable seepage field analysis method, which comprises the following steps of: selecting a pore seepage unit, establishing a corresponding stable seepage mathematical model, and solving a distribution function of a pressure field and a flow velocity field; and drawing a hole seepage space seepage field analysis flow line, flow velocity and other characteristics. The method comprises the following specific steps: establishing a hole seepage unit physical model; based on the hole seepage unit physical model, establishing a mathematical model and solving a pressure and flow velocity distribution function; and drawing a seepage field graph based on the analytic solution of the mathematical model. According to the scheme, full-field pressure and flow velocity explicit analytic solutions of medium stable seepage of the pore-karst cave structure are provided, numerical simulation dependence is broken through, the pore disturbance effect is directly quantified, and the problem of physical distortion of a traditional homogeneous model is solved. A strict mathematical basis is provided for karst cave type reservoir seepage, and a seepage mechanics theory is promoted to be coupled and spanned from a single pore to a hole.

Description

Technical Field

[0001] This invention relates to the field of porous media seepage mechanics, and in particular to an analytical method for stable seepage fields of media based on pore-cavity structures. Background Technology

[0002] For porous media materials such as soil and rock, single-pore porous media seepage models are often used to describe the flow characteristics of fluids in these materials. Porous porous media seepage models are the foundation and typical models for seepage research in this type of medium. However, in actual surface soil and rock, as well as underground reservoirs, large-sized pores and caverns are often developed, no longer constituting a single-pore medium. Taking marine carbonate oil and gas reservoirs as an example, these reservoirs have a typical pore-cavity structure, with numerous dissolution pores (millimeters to meters in size) developed in the rock matrix. This creates a very strong difference in flow space compared to conventional porous rock masses, mainly manifested in the change in flow method and velocity as fluid flows from the matrix pores to the cavern space.

[0003] Due to the significant differences in flow space and flow resistance, current Darcy flow analysis (which can only describe linear flow) cannot uniformly describe the flow patterns of matrix pores and the entire karst landscape. Seepage in porous media is difficult to calculate and analyze using a single, homogeneous porous media seepage analytical model; existing technologies and methods primarily employ numerical simulation. While this approach can accurately reflect the changing characteristics of the flow, its practicality faces two challenges. Firstly, discretizing porous structures using the finite element / finite volume method requires ultra-fine meshes, drastically increasing the model's degrees of freedom (especially for large-scale pore groups), and a single simulation can take several days, making it difficult to meet the real-time optimization needs of engineering projects. Secondly, commercial numerical modeling software (such as COMSOL and ANSYS) relies on simplified geometric assumptions (such as spherical / regular pores), still requiring a "one case, one model" approach, limiting its universality.

[0004] Therefore, considering the prevalence of porous media and the current inadequacy of the characterization of such seepage, this invention proposes an analytical method for stable seepage fields of porous media based on pore-cavity structures. Summary of the Invention

[0005] This invention provides an analytical method for stable seepage field of a medium based on a pore-cavity structure, which can effectively solve the problems in the background art.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] An analytical method for stable seepage field in a medium based on a pore-cavity structure includes the following steps:

[0008] Establish a physical model of the pore seepage unit;

[0009] Based on the physical model of the porous seepage unit, a mathematical model is established and the pressure and velocity distribution functions are solved.

[0010] The seepage field diagram is plotted based on the analytical solution of the mathematical model.

[0011] Furthermore, the establishment of the physical model of the pore seepage unit includes the following steps:

[0012] Define a square porous medium region with side length L, and set a cavity with radius R at the center position, satisfying 0.5L>3R;

[0013] Set the pore permeability k m and cave permeability k v , satisfying k v >>k m ;

[0014] The fluid viscosity μ is defined as constant.

[0015] Furthermore, the steady-state seepage boundary conditions of the physical model of the porous seepage unit are set as follows:

[0016] The left side of the model represents the inflow end of flow velocity u0, the right side represents the outflow end of flow velocity u0, and the remaining boundaries are closed boundaries.

[0017] Furthermore, based on the physical model of the porous seepage unit, a rotationally symmetric steady-state seepage control equation is established, and the analytical expressions of the pressure field and velocity field are solved by combining the method of separation of variables with boundary conditions.

[0018] Furthermore, the governing equations are expressed as Laplace equations in spherical coordinates, specifically as follows:

[0019]

[0020] The general solution is simplified by utilizing rotational symmetry, and is specifically expressed as follows:

[0021]

[0022] Where p represents pressure, r, θ, The three independent variables are Q in spherical coordinates. n (cosθ) is a spherical harmonic function, A n B n represents the undetermined coefficients of the radial equation after separation of variables, and n represents the order.

[0023] Furthermore, the radial coefficient is locked by the boundary conditions, which include:

[0024] Given a finite center point, at the center point of the cave, r = 0, and the pressure is a definite value, expressed as:

[0025] The continuity of flow velocity on the cave wall, where r = R at the interface between the cave and the pores, indicates the same seepage velocity, expressed as:

[0026] For the far-field asymptotic Darcy flow, at the element boundary rcosθ=±0.5L, the flow reverts to Darcy seepage flow with a seepage velocity of u0, expressed as:

[0027] Furthermore, the analytical solutions for the pressure field and the velocity field are specifically expressed as follows:

[0028] The particular solution of the Laplace equation for pressure distribution in steady flow of a porous medium is:

[0029]

[0030] The corresponding Cartesian coordinate representation is:

[0031]

[0032] The flow velocity within the porous medium is expressed as:

[0033]

[0034] The corresponding Cartesian coordinate representation is:

[0035]

[0036] Where, p v For the pressure in the cave region where r ≤ R, p m For the pressure in the pore region where r > R, u v Let u be the flow velocity in the cave region where r ≤ R. m Let r be the flow velocity in the pore region where r > R, and x and y be the Cartesian coordinate positions.

[0037] Furthermore, drawing the seepage field diagram includes the following steps:

[0038] Based on the pressure and velocity distribution function, pressure and velocity values ​​are sampled and calculated along a preset path;

[0039] Isobars and streamlines are generated by interpolation;

[0040] Output the full-field seepage field diagram.

[0041] Furthermore, the preset path includes: horizontal axis y = 0; 45° direction y = x; vertical axis x = 0 and 135° direction y = -x.

[0042] Furthermore, the interpolation method is based on 8-direction numerical interpolation, which extends the discrete path data into full-field isobars and streamlines.

[0043] The beneficial effects of this invention are as follows:

[0044] This invention proposes an explicit analytical solution for the full-field pressure and velocity of stable seepage in a pore-cavity structure, overcoming the dependence on numerical simulation: directly quantifying the pore disturbance effect and solving the physical distortion problem of traditional homogeneous models. It provides a rigorous mathematical foundation for seepage in cavitary reservoirs, promoting a leap in seepage mechanics theory from single-pore to pore-cavity coupled systems. Attached Figure Description

[0045] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0046] Figure 1 This is a schematic diagram of the perforated seepage unit model in this invention;

[0047] Figure 2 This is a schematic diagram of simulated pressure distribution in an embodiment of the present invention;

[0048] Figure 3 This is a schematic diagram of the pressure and flow velocity distribution along the X-axis in an embodiment of the present invention;

[0049] Figure 4 This is a schematic diagram of the pressure and flow velocity distribution along the Y=X axis in an embodiment of the present invention;

[0050] Figure 5 This is a schematic diagram of the pressure and flow velocity distribution along the Y-axis in an embodiment of the present invention;

[0051] Figure 6 This is a schematic diagram of the pressure and flow velocity distribution along the Y = -X axis in an embodiment of the present invention;

[0052] Figure 7 A seepage field diagram is drawn for an embodiment of the present invention. Detailed Implementation

[0053] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0054] It should be noted that when an element is referred to as being "fixed to" another element, it can be directly attached to the other element or there may be an intervening element. When an element is referred to as being "connected to" another element, it can be directly connected to the other element or there may be an intervening element. The terms "vertical," "horizontal," "left," "right," and similar expressions used herein are for illustrative purposes only and do not represent the only possible implementation.

[0055] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

[0056] This invention provides an analytical method for stable seepage field analysis based on a porous-cavity structure. By selecting pore seepage elements, establishing a corresponding stable seepage mathematical model, solving for the distribution functions of the pressure and velocity fields, and plotting the spatial seepage field of the pore seepage to analyze streamlines and velocity characteristics, the method includes the following steps:

[0057] A physical model of a porous seepage unit is established; based on the physical model of the porous seepage unit, a mathematical model is established and the pressure and velocity distribution functions are solved; based on the analytical solution of the mathematical model, a seepage field diagram is plotted.

[0058] The establishment of a physical model for the pore seepage unit includes the following steps:

[0059] Define a square porous medium region with side length L, and set a karst cave with radius R at the center, satisfying 0.5L>3R to ensure that the boundary does not affect the flow field of the karst cave; set the pore permeability k. m and cave permeability k v , satisfying k v >>k m The fluid viscosity μ is defined as constant.

[0060] Furthermore, the steady-state seepage boundary conditions for the physical model of the porous seepage element are set as follows:

[0061] The left side of the model represents the inflow end of flow velocity u0, the right side represents the outflow end of flow velocity u0, and the remaining boundaries are closed boundaries.

[0062] In the specific implementation process, a repeating unit of porous media seepage was selected as the research object. Specifically, a square porous medium with a side length of L was selected, with a cavity space of radius R set at the center; where 0.5L > 3R. To describe the difference between pore seepage and permeability, the pore permeability was set to k. mThe permeability of the karst cave is k v Considering the permeability k of the karst cave. v Much greater than the pore permeability k m In this case, the fluid viscosity μ remains constant, and the seepage unit is a steady-state seepage, meaning that the flow parameters, including velocity and pressure, do not change with time but are only functions of space; for example... Figure 1 As shown, the left end of the seepage element is the inflow end, and the rightmost end is the outflow end. Because it is steady seepage, the flow velocity on the left is equal to the flow velocity on the right, denoted as u0; the other boundaries are closed boundaries.

[0063] This invention simplifies complex porous media into repetitive units that can be analytically solved, avoiding the high computational cost of full-scale numerical simulation; it ignores boundary interference through geometric constraints and focuses on the interaction between pores and voids; and it establishes a correspondence between physics and mathematics, laying the foundation for analytical solutions.

[0064] In this embodiment, a steady-state seepage mathematical model for the porous seepage unit is further established. The pressure distribution in the steady-state flow process of any porous medium satisfies the Laplace equation. Based on the boundary conditions of two different differential equations, the solutions of the two equations in spherical coordinates are obtained, yielding the pressure p at any location in the porous medium. m and flow velocity u m The equation, and the pressure p at any location in the cavern medium. v and flow velocity u v equation.

[0065] Specifically, based on the physical model of the porous seepage unit, a rotationally symmetric stable seepage control equation is established, and the analytical expressions of the pressure field and velocity field are solved by combining the method of separation of variables with boundary conditions.

[0066] The governing equations are expressed as Laplace equations in spherical coordinates, as follows:

[0067]

[0068] Assume the pressure function in spherical coordinates is:

[0069]

[0070] Solving the above equation by separating variables, the general solution for pressure p is:

[0071]

[0072] In the formula, m < n;

[0073] Because the seepage element has rotational symmetry about the polar axis, seepage and parameters Regardless of the specific solution, we can simplify it using rotational symmetry, as shown below:

[0074]

[0075] Therefore, the pressure distribution for steady flow in porous media is as follows:

[0076]

[0077] Determining coefficients based on boundary conditions

[0078] Where p represents pressure, r, θ, The three independent variables are Q in spherical coordinates. n (cosθ) is a spherical harmonic function used to describe the pressure distribution in spherical coordinates related to angle θ. n B n The coefficients of the radial equation after separation of variables are represented, respectively corresponding to r. n and r -(n+1) The amplitude of the term, n represents the order, and θ is the polar angle.

[0079] A n r n This indicates that the pressure contribution increases with radius, and is dominant when moving away from the cavern; B n / r n+1 This indicates that the pressure contribution decreases with the radius and is dominant near the center of the cave.

[0080] By using the method of separation of variables, the pressure p(r,θ) is decomposed into a radial function R(r) and an angular function Q. n The product of (cosθ) transforms the Laplace equation into a solvable system of ordinary differential equations.

[0081] Furthermore, the radial coefficient is locked by the boundary conditions, which include:

[0082] Given a finite center point, at the center point of the cave, r = 0, and the pressure is a definite value, expressed as:

[0083] The continuity of flow velocity on the cave wall, where r = R at the interface between the cave and the pores, indicates the same seepage velocity, expressed as:

[0084] For the far-field asymptotic Darcy flow, at the element boundary rcosθ=±0.5L, the flow reverts to Darcy seepage flow with a seepage velocity of u0, expressed as:

[0085] The analytical solutions for the pressure field and the velocity field are expressed as follows:

[0086] The particular solution of the Laplace equation for pressure distribution in steady flow of a porous medium is:

[0087]

[0088] The corresponding Cartesian coordinate representation is:

[0089]

[0090] The flow velocity within the porous medium is expressed as:

[0091]

[0092] The corresponding Cartesian coordinate representation is:

[0093]

[0094] Where, p v For the pressure in the cave region where r ≤ R, p m For the pressure in the pore region where r > R, u v Let u be the flow velocity in the cave region where r ≤ R. m Let r be the flow velocity in the pore region where r > R, and x and y be the Cartesian coordinate positions.

[0095] In this embodiment, drawing the seepage field diagram includes the following steps:

[0096] Based on the pressure and velocity distribution function, pressure and velocity values ​​are sampled and calculated along a preset path; isobars and streamlines are generated through interpolation; and a full-field seepage field diagram is output.

[0097] The preset path includes the horizontal axis: y = 0, the 45° direction: y = x, the vertical axis: x = 0, and the 135° direction: y = -x. The interpolation method is based on 8-direction numerical interpolation, which expands the discrete path data into full-field isobars and streamlines to form a flow field diagram, such as... Figure 7 As shown, the analytical solution is transformed into a visualized flow field, which intuitively demonstrates the disturbance effect of the holes on the flow.

[0098] This invention proposes an analytical solution for pressure and velocity distribution under steady-state seepage conditions in porous media, and then a method for plotting hydrodynamic field diagrams based on the solution functions. Compared with existing analytical methods for simple seepage models and numerical simulation methods for complex seepage models, this invention provides a method for calculating analytical solutions for pressure and velocity in steady-state seepage in porous media, and for plotting field diagrams of steady-state seepage in porous media.

[0099] In existing technologies, numerical simulation is required to obtain pressure and velocity distributions for complex porous media. This invention fills the gap in the lack of analytical solutions for stable seepage in dual-porous media such as pore-cavity systems. Traditional numerical simulations using a single model take hours to days (requiring mesh generation and iterative solutions), while this invention directly outputs the full-field pressure / velocity distribution through analytical solutions, reducing computation time to the second level (an efficiency improvement of over 100 times), meeting real-time optimization needs (such as dynamic adjustments in oilfield development).

[0100] This invention quantifies the impact of porosity on development effectiveness: the analytical solution directly reveals the permeability ratio (k v / k m The control of the flow field by the pore size (R) and the porosity (R), such as how high permeability pores increase the flow velocity by 2k. v / (k v +k m () times, guiding the well network to avoid high-conductivity channels or to target and transform inefficient pores.

[0101] This invention is applicable to the analysis of stable seepage fields in pore-cavity dual-medium structures, such as in oil and gas field development (e.g., pressure distribution calculation and flow field analysis during injection and production in karst reservoirs); surface hydrology (e.g., groundwater flow analysis in the presence of large karst caves, including head pressure analysis along the flow path); rock engineering (e.g., early warning of tunnel water inrush disasters, analyzing the water pressure distribution in pores ahead of the tunnel face in karst tunnels, and predicting the risk of water inrush (e.g., subway projects in karst landform areas); and dam foundation seepage control (e.g., optimizing the design of anti-seepage curtains for dam foundations containing karst caves, reducing uplift pressure through flow field analysis (e.g., limestone dam foundations in the Three Gorges Reservoir area)).

[0102] In a specific embodiment, numerical simulation software is used to simulate a typical stable seepage process in a porous medium, outputting the flow velocity parameters along the central axis. These parameters are then compared with the flow velocity parameters calculated by this invention to verify the applicability and accuracy of the invention. First, the typical model size is set. The pore medium size is 100cm * 100cm, and the pore medium permeability k... m =100D; The cave is located at the center of the porous medium, with a radius of 30cm, and the cave's permeability k v =1000*k m The fluid viscosity was 1 cp, and the reservoir temperature was 25°C. Next, a steady-state seepage process was simulated with constant flow as the boundary condition: the injection rate was 1.0 × 10⁻⁶. -7 m / s, simulated pressure results are as follows Figure 2 As shown.

[0103] Further calculate the flow velocity and pressure values, and plot the field diagram; calculate the flow velocity and pressure values ​​along the X-axis, such as... Figure 3 As shown; calculate the flow velocity and pressure values ​​along the Y=X axis, as follows. Figure 4 As shown; calculate the flow velocity and pressure values ​​along the Y-axis, as follows. Figure 5 As shown; calculate the flow velocity and pressure values ​​along the Y = -X axis, as follows. Figure 6 As shown; the final flow field diagram is drawn as follows. Figure 7 As shown.

[0104] Those skilled in the art should understand that this invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to this invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A method for analytically determining a steady flow field in a porous-cavernous medium based on a pore-cavern structure, characterized in that, The method comprises the following steps: establishing a physical model of a porous medium with a cavity; based on the physical model of the porous medium with a cavity, establishing a mathematical model and solving a pressure and flow rate distribution function; based on the analytical solution of the mathematical model, drawing a seepage field diagram.

2. The pore-cavern structure based medium stabilized seepage field analytical method according to claim 1, wherein, The establishment of the physical model of the porous medium with a cavity comprises the following steps: defining a square porous medium region with a side length L and setting a cavity with a radius R at the center position, satisfying 0.5L>3R; Set the pore permeability k m and cavern permeability k v , meet k v > > k m ; defining a constant fluid viscosity μ.

3. The analytical method for steady seepage field of a pore-cave structure based medium according to claim 2, wherein, The stable seepage boundary condition of the physical model of the porous medium with a cavity is set as: The left side of the model is an inflow end with a flow rate u0, the right side is an outflow end with a flow rate u0, and the remaining boundaries are closed boundaries.

4. The pore-cavern structure based medium stabilized seepage field analytical method according to claim 1, wherein, Based on the physical model of the porous medium with a cavity, a rotationally symmetric stable seepage control equation is established, and an analytical expression of the pressure field and the flow rate field is solved by the separation of variables method combined with the boundary condition.

5. The pore-cave structure based medium stabilized seepage field analytical method according to claim 4, wherein, The control equation is expressed as a Laplace equation in a spherical coordinate system, specifically as follows: The general solution is simplified by using the rotational symmetry, specifically as follows: where p represents pressure, r, θ, are three independent variables in spherical coordinates, Q n (cos θ) are spherical harmonics, A n , B n represent undetermined coefficients of the radial equation after separation of variables, and n represents order.

6. The pore-cavern structure based medium stabilized seepage field analytical method according to claim 5, wherein, The radial coefficient is locked by the boundary condition, which comprises: The center point is limited, and the pressure at the center point of the cave r = 0 is a certain value, which is expressed as: The flow velocity continuity of the cave wall is that the seepage velocity is the same at the interface of the cave and the pore, which is expressed as: Far field asymptotic Darcy flow, at the cell boundaries rcosθ = ±0.5L, flow reverts to Darcy seepage, with seepage velocity u0, expressed as:

7. The pore-cavern structure based medium stabilized seepage field analytical method according to claim 6, wherein, The analytical solution of the pressure field and the flow rate field is specifically expressed as follows: The particular solution of the Laplace equation of the pressure distribution of the stable flow of the porous medium with a cavity is: Corresponding to the expression in the Cartesian coordinate system is: The flow rate in the porous medium with a cavity is expressed as: Corresponding to the expression in the Cartesian coordinate system is: where p v is the pressure in the cavern region r≤R, p m is the pressure in the pore region r>R, u v is the flow rate in the cavern region r≤R, u m is the flow rate in the pore region r>R, x, y are Cartesian coordinate positions.

8. The pore-cavern structure based medium stabilized seepage field analytical method of claim 1, wherein, Drawing a seepage field diagram comprises the following steps: According to the pressure and flow rate distribution function, sampling and calculating the pressure and flow rate values along a preset path; Generating isobaric lines and flow lines by interpolation; Outputting a full-field seepage field diagram.

9. The pore-cave structure based medium stabilized seepage field analytical method according to claim 8, wherein, The preset path comprises a horizontal axis y=0, a 45° direction y=x, a vertical axis x=0, and a 135° direction y=-x.

10. The pore-cave structure based medium stabilized seepage field analytical method of claim 8, wherein, The interpolation method is based on 8-direction numerical interpolation, which expands the discrete path data into full-field isobaric lines and flow lines.

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