Two-phase delayed solute transport simulation method based on near-field dynamics

Abstract

The invention discloses a two-phase delayed solute transport simulation method based on near-field dynamics, and the method comprises the steps: dividing a region in a solid model, setting an initial substance point, and selecting a central substance point and an affiliated substance point in a neighborhood of the central substance point by taking a preset size as a radius; calculating mass flux transmission quantity among material points according to the local convection function and the local diffusion function, and constructing a diffusion-convection constitutive function; inputting a flux delay factor and a storage delay factor into the constitutive function to form a biphase delay migration integral-differential equation; and solving the equation to obtain a solute transport simulation result. According to the method, long-range interaction of solute particles is described in a non-local integral form, the method has remarkable advantages in depicting solute migration delay behaviors, interface reaction kinetics and the like in layered soil, anisotropic soil bodies and fissure media, and solute non-Fick migration behavior delay characteristics can be accurately predicted; and a novel theoretical tool is provided for solute migration analysis under complex geological conditions.

Description

Technical Field

[0001] The present invention relates to the field of hydrogeology, and in particular to a two-phase delayed solute transport simulation method based on peridynamics. Background Art

[0002] Quantitative simulation of solute transport in complex geological media is of great significance for explaining and analyzing the microscopic mechanisms that lead to non-Fickian behavior of solutes in actual formations. To date, many numerical models have been proposed to study the solute transport behavior in formations. These models usually regard advection and molecular diffusion as the main transport mechanisms of solutes, and these two mechanisms are significantly affected by groundwater flow conditions and the physical properties of porous media. At the same time, the heterogeneity of porous media also plays a key role in the solute transport process. Ultimately, the multi-scale heterogeneity and multiphase effects of porous media lead to the non-Fickian transport behavior of solutes in the medium.

[0003] However, traditional local continuity models, such as the conventional advection-diffusion equation (ADE) based on differential equations, have significant limitations in describing non-Fickian behavior. The linear assumption in the equation makes it difficult to accurately characterize the tailing phenomenon present in the solute concentration breakthrough curve, that is, the actual concentration decrease rate is slower than predicted by the ADE. Furthermore, the ADE has inherent limitations when dealing with discontinuous structures such as strong heterogeneity and cracks. Because it relies on the continuity assumption of spatial derivatives, it cannot describe key processes such as adsorption-desorption hysteresis at the solid-liquid interface and collision retention in tortuous pore paths. Summary of the Invention

[0004] In response to these challenges, this paper proposes a peridynamics-based two-phase delayed solute transport simulation method. Through numerical analysis, this method analyzes the solute migration patterns in different practical geological environments and the varying effects of two delay factors on migration behavior. The accuracy and applicability of the method are also verified. In structurally complex sites, this method can effectively identify the impact of actual environmental factors on solute transport in fluids, providing a theoretical basis for solute migration analysis under complex geological conditions.

[0005] To achieve the above objectives, this application provides a two-phase delayed solute transport simulation method based on peridynamics, which is applicable to solid models and includes:

[0006] Dividing the solid model into regions according to preset intervals, and setting a preset number of initial material points in each region;

[0007] With the preset size as the radius, select several initial material points as the center to divide the neighborhood according to the preset rules. The initial material point at the center is recorded as the central material point, and the other initial material points in the neighborhood are recorded as subsidiary material points.

[0008] The mass flux transfer between the attached material points and the central material point in the same neighborhood is calculated based on the local convection function and the local diffusion function, and the diffusion-convection constitutive function is constructed.

[0009] The flux delay factor and storage delay factor are input into the diffusion-convection constitutive function to obtain the two-phase delayed transport integral-differential equation;

[0010] The two-phase delayed transport integral-differential equation is solved to obtain the solute transport simulation results of the current neighborhood.

[0011] In some embodiments, the diffusion-convection constitutive function is expressed by formula (1), which is as follows:

[0012]

[0013] In formula (1), is the symbol for the derivative, J is the symbol for the mass flux, J d is the symbol of mass flux caused by diffusion behavior, J v is the symbol of the mass flux caused by convective behavior, C(x,t) is the solute concentration at the central material point x at time t, and t represents the transport time;

[0014] J d It is expressed by formula (2), which is as follows:

[0015]

[0016] In formula (2), is the first microscopic parameter corresponding to the traditional ADE in peridynamics, C(x ′ ,t) is the attached material point x ′ solute concentration at time t;

[0017] J v It is expressed by formula (3), which is as follows:

[0018]

[0019] In formula (3), It is the second microscopic parameter in peridynamics theory corresponding to traditional ADE.

[0020] In some embodiments, when the neighborhood range approaches 0 infinitely, and They converge to the local diffusion coefficient D and the convection velocity v, respectively, and are expressed by formula (4). Formula (4) is as follows:

[0021]

[0022] In formula (4), is the influence kernel function.

[0023] In some embodiments, the influence kernel function is configured to be expressed in a shape-constant form, a linear form;

[0024] In the case of a one-dimensional region, the influence kernel function is expressed by formula (5), which is as follows:

[0025]

[0026] In formula (5), δ is the peridynamic radius, ‖ξ‖ is the radius of the central material point x and the auxiliary material point x ′ the distance between them;

[0027] In the case of a two-dimensional region, the influence kernel function is expressed by formula (6), which is as follows:

[0028]

[0029] In formula (6), π is the ratio of the circumference of a circle to the circumference of a circle.

[0030] In some embodiments, the two-phase delayed transport integral-differential equation is expressed by formula (7), which is as follows:

[0031]

[0032] In formula (7), τ J is the flux delay factor, τ C is the storage delay factor, is the symbol for the time derivative, V x′ It refers to the morphological parameter of the attached material point x′, which is the length in one-dimensional state, the area in two-dimensional state, and the volume in three-dimensional state. x is the neighborhood of the central material point x.

[0033] In some embodiments, the two-phase delayed transport integral-differential equation is solved to obtain solute transport simulation results for the current neighborhood, including:

[0034] The Taylor expansion of the two-phase delayed transport integral-differential equation is expressed by formula (8), which is as follows:

[0035]

[0036] In formula (8), i is the derivative order;

[0037] Ignoring the higher-order terms in the Taylor series in formula (8), the non-local two-phase delayed transport equation is obtained, which is expressed by formula (9). Formula (9) is as follows:

[0038]

[0039] The non-local two-phase delayed transport equation finally obtained by series expansion is expressed by formula (10), which is as follows:

[0040]

[0041] In some embodiments, the method further comprises:

[0042] The non-local two-phase delayed transport equation is spatially discretized and expressed by formula (11), which is as follows:

[0043]

[0044] In formula (11), is the peridynamic convection coefficient in upwind difference form, ω′∈[0,1] is the mixing weight, represents the windward area within the neighborhood;

[0045] The initial boundary conditions, the first preset value of the flux delay factor, and the second preset value of the storage delay factor are set, and an explicit time forward difference calculation is performed on formula (11) to obtain the solute transport simulation results of the current neighborhood.

[0046] In some embodiments, an explicit time forward difference calculation is performed on formula (11), including:

[0047] Get the second derivative of solute concentration with respect to time at time k Spatial distribution, using explicit time forward difference to calculate the material point concentration C at time k+1 k+1 , expressed by formula (12), formula (12) is as follows:

[0048]

[0049] In formula (12), Δt is the time step, is the first-order derivative of the concentration at time k, is the first-order derivative of the concentration at time k+1, C k is the concentration at time k.

[0050] Different from the existing technology, the modeling method based on peridynamic theory of the above technical solution has high accuracy and good grid adaptability, and does not require special grid operations for special media in the computational domain; the present invention proposes to use the peridynamic method to simulate the non-Fickian behavior in solute transport, and establishes a solute transport model that considers both non-Fickian effects and non-local effects. The model is established under the framework of bond-based peridynamics (PD), and introduces the concept of dual-phase lag (DPL) effect, using the flux delay factor τ J Quantify the inertial collision delay caused by pore structure irregularities, the storage delay factor τ C It reflects the adsorption-desorption effect of the interface and overcomes the problems of numerical oscillation in discontinuous media caused by traditional local continuous methods, providing a new theoretical tool for analyzing and predicting solute migration in complex geological media.

[0051] The above-mentioned records related to the content of the invention are only an overview of the technical solution of this application. In order to enable ordinary technicians in this field to understand the technical solution of this application more clearly, and then implement it according to the text of the specification and the contents recorded in the drawings, and to make the above-mentioned purposes and other purposes, features and advantages of this application easier to understand, the following is an explanation in combination with the specific implementation methods and drawings of this application. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] The drawings are only used to illustrate the principles, implementation methods, applications, features and effects of the specific embodiments of the present invention and other related contents, and are not to be considered as limiting the present application.

[0053] In the drawings of the specification:

[0054] Figure 1 Flowchart of the two-phase delayed solute transport simulation method based on peridynamics described in the specific embodiment;

[0055] Figure 2 This is a schematic diagram of the distribution of heterogeneous soil layers and boundary conditions described in the specific implementation method;

[0056] Figure 3 This is a diagram of the relative solute concentration distribution of the entity model described in the specific implementation method under different delay factors. DETAILED DESCRIPTION

[0057] In order to explain in detail the possible application scenarios, technical principles, specific solutions that can be implemented, and the purpose and effects of this application, the following is a detailed description of the specific embodiments listed in conjunction with the accompanying drawings. The embodiments described herein are only used to more clearly illustrate the technical solutions of this application and are therefore only examples and are not intended to limit the scope of protection of this application.

[0058] References to "embodiments" herein mean that the specific features, structures, or characteristics described in conjunction with the embodiments may be included in at least one embodiment of the present application. The appearance of the word "embodiment" in various places in the specification does not necessarily refer to the same embodiment, nor does it particularly limit its independence or relevance to other embodiments. In principle, in this application, as long as there are no technical contradictions or conflicts, the various technical features mentioned in the embodiments can be combined in any manner to form a corresponding implementable technical solution.

[0059] Unless otherwise defined, the technical terms used herein have the same meanings as those generally understood by those skilled in the art to which this application belongs; the use of relevant terms herein is only for describing specific embodiments and is not intended to limit this application.

[0060] In the description of this application, the term "and / or" is used to describe a logical relationship between objects, indicating that three relationships can exist. For example, A and / or B means: A exists, B exists, and both A and B exist. In addition, the character " / " in this document generally indicates that the objects before and after are in a logical "or" relationship.

[0061] In this application, terms such as "first" and "second" are merely used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual quantity, priority or sequence relationship between these entities or operations.

[0062] Without further limitations, in this application, the words "include", "comprise", "have" or other similar open-ended expressions used in sentences are intended to cover non-exclusive inclusion. These expressions do not exclude the presence of additional elements in the process, method or product that includes the elements, so that the process, method or product that includes a series of elements may include not only those defined elements, but also other elements that are not explicitly listed, or also include elements inherent to such process, method or product.

[0063] Consistent with the understanding in the Examination Guidelines, in this application, expressions such as "greater than," "less than," and "exceed" are understood to exclude the number itself; expressions such as "above," "below," and "within" are understood to include the number itself. Furthermore, in the description of the embodiments of this application, "multiple" means more than two (including two), and similar expressions related to "multiple" are also understood in this manner, such as "multiple groups," "multiple times," etc., unless otherwise specifically defined.

[0064] In the description of the embodiments of the present application, the space-related expressions used, such as "center", "longitudinal", "lateral", "length", "width", "thickness", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "vertical", "top", "bottom", "inside", "outside", "clockwise", "counterclockwise", "axial", "radial", "circumferential", etc., indicate the orientation or position relationship based on the orientation or position relationship shown in the specific embodiments or drawings, and are only for the convenience of describing the specific embodiments of the present application or facilitating the reader's understanding, and do not indicate or imply that the device or component referred to must have a specific position, a specific orientation, or be constructed or operated in a specific orientation. Therefore, it should not be understood as a limitation on the embodiments of the present application.

[0065] The processor described in the embodiments of the present application can be implemented by hardware, firmware, software or a combination thereof, and can use a circuit, a single or multiple application-specific integrated circuits (ASIC), a digital signal processor (DSP), a digital signal processing device (DSPD), a programmable logic device (PLD), a field programmable gate array (FPGA), a central processing unit (CPU), a controller, a microcontroller, or at least one of a microprocessor, and also includes other physical, biological or chemical structures that can achieve similar or equivalent functions to the processors listed above, such as biological neurons, quantum computing units, DNA computing units, etc., so that the processor can execute some or all of the steps in the computer program or method involved in the various embodiments of the present application, or any combination of the steps mentioned therein.

[0066] The computer program involved in the embodiment can be stored in a computer device readable storage medium, which includes but is not limited to a disk, a tape, a magnetic card, a floppy disk, a flash memory, an optical disc, an optical card, a read-only memory (ROM), a random access memory (RAM), an erasable programmable ROM (EPROM) and an electrically erasable programmable ROM (EEPROM), etc., and also includes other biological, physical or chemical structures that can achieve similar or equivalent functions to the storage media listed above, such as DNA, RNA, protein and other units with information storage capabilities. In a specific embodiment, the storage medium involved can be one of the above-mentioned media types or a combination of the above-mentioned media types. In different embodiments, the computer program involved in the embodiment can be stored in a single medium in a centralized manner or in a distributed manner on multiple media. The memory containing the computer device readable storage medium can be a non-volatile memory or a random access memory. These computer device readable storage media can be built into the device or connected to the device involved in the embodiment as an external device or part of an external device. In some embodiments, a memory having a computer-readable storage medium is deployed locally; in other embodiments, a solution of deploying the memory away from the processor may also be adopted, such as a network-attached memory accessed via an RF circuit or an external port and a communication network, wherein the communication network may be the Internet, one or more intranets, a local area network (LAN), a wide area network (WLAN), a storage area network (SAN), etc., or a suitable combination thereof, as long as the computer device can access the memory. In addition, the computer program involved in the embodiment can be stored in plaintext / ciphertext form, or can be designed as training data, which can be integrated and reorganized through model training and implicitly stored in the parameter state of a deep neural network or other machine learning model.

[0067] See also Figures 1 to 3 This embodiment provides a two-phase delayed solute transport simulation method based on peridynamics, which is applicable to a solid model. The method includes:

[0068] Dividing the solid model into regions according to preset intervals, and setting a preset number of initial material points in each region;

[0069] With the preset size as the radius, select several initial material points as the center to divide the neighborhood according to the preset rules. The initial material point at the center is recorded as the central material point, and the other initial material points in the neighborhood are recorded as subsidiary material points.

[0070] The mass flux transfer between the attached material points and the central material point in the same neighborhood is calculated based on the local convection function and the local diffusion function, and the diffusion-convection constitutive function is constructed.

[0071] The flux delay factor and storage delay factor are input into the diffusion-convection constitutive function to obtain the two-phase delayed transport integral-differential equation;

[0072] The two-phase delayed transport integral-differential equation is solved to obtain the solute transport simulation results of the current neighborhood.

[0073] In this embodiment, the preset rules can be understood as follows: neighborhood search is a fundamental part of the peridynamics method and is a necessary modeling operation for all problems solved using the peridynamics method. If the neighborhood is divided with each initial material point as the center, there will be multiple overlapping neighborhoods. The purpose of the neighborhood search is to obtain the neighborhood of each material point in the area.

[0074] The method flow of this embodiment is as follows Figure 1 This example uses the method of this example to conduct peridynamic modeling analysis by studying the solute migration in a 0.2m×0.2m layered soil. x , Δy=D / N y , L is the length of the two-dimensional solid model, D is the width of the two-dimensional solid model, N x is the number of material points distributed along the x direction in the solid model, N x =N y =50. The neighborhood radius is generally set to δ = 3.015·Δx. When the distance between the auxiliary material point x′ and the central material point x is less than the neighborhood radius, the auxiliary material point x′ is considered to be in the neighborhood of the central material point x. The form of a central material point x in the field H x Other attached material points x′ are expressed as: P x ={x′≠x|‖xx′‖≤δ}, which is the preset rule.

[0075] This embodiment effectively constructs an interaction network between material points by using a preset interval to divide the region and set the discretization processing method of the initial material points, combined with a preset size neighborhood division strategy. This method introduces a flux delay factor and a storage delay factor to construct a two-phase delayed transport integral-differential equation that can accurately describe the non-local interaction characteristics in the solute migration process. In particular, the neighborhood of each material point in the region is obtained through a neighborhood search. This embodiment breaks through the limitations of the traditional continuity assumption through a modeling method based on near-field dynamics, and can more accurately characterize the non-Fickian migration behavior of solutes under complex geological conditions, providing an effective numerical simulation tool for studying the solute migration process in heterogeneous media such as layered soil layers.

[0076] In some embodiments, the diffusion-convection constitutive function is expressed by formula (1), which is as follows:

[0077]

[0078] In formula (1), is the symbol for the derivative, J is the symbol for the mass flux, J d is the symbol of mass flux caused by diffusion behavior, J v is the symbol of the mass flux caused by convective behavior, C(x,t) is the solute concentration at the central material point x at time t, and t represents the transport time;

[0079] J d It is expressed by formula (2), which is as follows:

[0080]

[0081] In formula (2), is the first microscopic parameter corresponding to the traditional ADE in peridynamics, C(x ′ ,t) is the attached material point x ′ solute concentration at time t;

[0082] J v It is expressed by formula (3), which is as follows:

[0083]

[0084] Formula (3), It is the second microscopic parameter in peridynamics theory corresponding to traditional ADE.

[0085] In this embodiment, the local convection and diffusion equations are used to calculate the distribution of other subsidiary material points x in the entire neighborhood of the central material point x. ′ The mass flux transmission between them is expressed by formula (2) and formula (3).

[0086] Then, the constitutive relation based on bond-based peridynamics for the diffusion-convection problem is constructed by performing integral calculations in the entire neighborhood, which is expressed by formula (1).

[0087] This embodiment integrates the local convection function and the local diffusion function into a unified integral form by constructing a diffusion-convection constitutive function, thereby achieving an accurate description of the solute transport process. and the second microscopic parameter The diffusion and convection effects are characterized separately, and the mass flux transmission between the central material point and the subsidiary material points is calculated by integration within the neighborhood. A constitutive relationship based on bond-based peridynamics is established, providing a theoretical basis for the simulation of solute transport under complex geological conditions.

[0088] In some embodiments, when the neighborhood range approaches 0 infinitely, and They converge to the local diffusion coefficient D and the convection velocity v, respectively, and are expressed by formula (4). Formula (4) is as follows:

[0089]

[0090] In formula (4), is the influence kernel function.

[0091] In this embodiment, To affect the kernel function, it has various forms, which are related to the shape of the integration region.

[0092] In this embodiment, the influence kernel function is introduced Realized peridynamic parameters and Compared to the traditional mathematical relationship between the local diffusion coefficient D and the convection velocity v, this method naturally degenerates into the classical convection-diffusion equation when the neighborhood range approaches zero, ensuring the compatibility and continuity of the model. The influence kernel function can be flexibly selected based on the shape of the integration region, enhancing the model's adaptability to diverse geological conditions.

[0093] In some embodiments, the influence kernel function is configured to be expressed in a shape-constant form, a linear form;

[0094] In the case of a one-dimensional region, the influence kernel function is expressed by formula (5), which is as follows:

[0095]

[0096] In formula (5), δ is the peridynamic radius, ‖ξ‖ is the radius of the central material point x and the auxiliary material point x ′ the distance between them;

[0097] In the case of a two-dimensional region, the influence kernel function is expressed by formula (6), which is as follows:

[0098]

[0099] In formula (6), π is the ratio of the circumference of a circle to the circumference of a circle.

[0100] In this embodiment, by configuring two influence kernel functions, shape-constant and linear, flexible mathematical expressions are provided for regions of different dimensions. Using the specific expressions of Equations (5) and (6) in one- and two-dimensional cases, respectively, the influence kernel function accurately describes the interaction between material points, enhancing the model's adaptability to different spatial dimensions and providing a more accurate computational basis for solute transport simulations under complex geological conditions.

[0101] In some embodiments, the two-phase delayed transport integral-differential equation is expressed by formula (7), which is as follows:

[0102]

[0103] In formula (1), τ J is the flux delay factor, τ C is the storage delay factor, is the symbol for the time derivative, V x′ Refers to the morphological parameters of the attached material point x′, which is the length in one-dimensional state, the area in two-dimensional state, and the volume in three-dimensional state. x is the neighborhood of the central material point x.

[0104] In this embodiment, a two-phase delayed transport integral-differential equation is constructed to introduce the flux delay factor τ J and storage delay factor τ C , achieving an accurate description of the nonlocal interactions and delay effects during solute transport. This equation unifies the diffusion and convection effects into an integral form, which can more accurately characterize the non-Fickian behavior of solute transport under complex geological conditions and provides an effective theoretical tool for studying the delay phenomenon of solute transport.

[0105] In some embodiments, the two-phase delayed transport integral-differential equation is solved to obtain solute transport simulation results for the current neighborhood, including:

[0106] The Taylor expansion of the two-phase delayed transport integral-differential equation is expressed by formula (8), which is as follows:

[0107]

[0108] In formula (8), i is the derivative order;

[0109] Ignoring the higher-order terms in the Taylor series in formula (8), the non-local two-phase delayed transport equation is obtained, which is expressed by formula (9). Formula (9) is as follows:

[0110]

[0111] In this embodiment, the flux delay factor τ J and the storage delay factor τ CThe constructed peridynamic convection-diffusion framework is introduced and expressed by formula (7). With the help of Taylor series expansion, the two-phase delayed transport integral-differential equation is derived and expressed by formula (8). After ignoring the second-order derivative and above derivatives, formula (9) is obtained. Substituting it into the constructed transport integral-differential equation formula (7), the simplified two-phase delayed transport integral-differential equation is obtained and expressed by formula (10). Formula (10) is as follows:

[0112]

[0113] This embodiment uses the Taylor expansion method to process the two-phase delayed transport integral-differential equation, reasonably simplifies the computational complexity while retaining the first-order derivative term, and accurately describes the flux delay factor τ J and storage delay factor τ C The delay effect of the method is reduced, and the calculation efficiency is guaranteed. This method realizes the simplified solution of the complex solute migration process through the mathematical transformation of formula (8) and formula (9), and provides a feasible numerical calculation scheme for practical engineering applications.

[0114] In some embodiments, the method further comprises:

[0115] The non-local two-phase delayed transport equation is spatially discretized and expressed by formula (11), which is as follows:

[0116]

[0117] In formula (11), is the peridynamic convection coefficient in upwind difference form, ω′∈[0,1] is the mixing weight, represents the windward area within the neighborhood;

[0118] The initial boundary conditions, the first preset value of the flux delay factor, and the second preset value of the storage delay factor are set, and an explicit time forward difference calculation is performed on formula (11) to obtain the solute transport simulation results of the current neighborhood.

[0119] In this embodiment, the nonlocal two-phase delayed transport equation is spatially discretized. Preferably, the calculation is performed using a spatial hybrid difference algorithm and a time-explicit forward difference numerical algorithm. The diffusion term is spatially calculated using a central difference method, while the convection term is calculated using an upwind kernel difference method. In formula (11), ω′∈[0,1] is the mixing weight, with an optimal value of 0.8.

[0120] This example achieves an efficient solution to the nonlocal two-phase delayed transport equation through spatial discretization and an explicit time forward differencing method. This method innovatively employs a hybrid differencing algorithm, using central differencing for the diffusion term and upwind kernel differencing for the convection term. The hybrid weight ω′ optimizes computational accuracy, and combined with preset values ​​for the flux delay factor and storage delay factor, accurately simulates the nonlocal interactions and delay effects during solute transport, providing a reliable numerical solution for solute transport analysis under complex geological conditions.

[0121] In some embodiments, an explicit time forward difference calculation is performed on formula (11), including:

[0122] Get the second derivative of solute concentration with respect to time at time k Spatial distribution, using explicit time forward difference to calculate the material point concentration C at time k+1 k+1 , expressed by formula (12), formula (12) is as follows:

[0123]

[0124] In formula (12), Δt is the time step, is the first-order derivative of the concentration at time k, is the first-order derivative of the concentration at time k+1, C k is the concentration at time k.

[0125] Furthermore, for the virtual material point at the boundary condition at time t0, C(x, t0)=C0 is set.

[0126] In this embodiment, the initial relative concentration of the solute at the left boundary is selected as 1, and the rest is 0%. The constructed solid model is segmented according to the distribution law of the set soil layers, and the actual width and location of each soil layer in the model are accurately determined, as shown in the attached figure. Figure 2 As shown. After setting the concentration boundary conditions of the solid model, directly substitute τ J With τ C Calculate the actual value of .

[0127] In this embodiment, the coefficients corresponding to various soil layers are: For high permeability soil layer, the diffusion coefficient D1 is set to 10 -7 m 2 / s, convection speed v1=10 -6 m / s;

[0128] For low permeability soil, set the diffusion coefficient D2 = 10 -8 m 2 / s, convection speed v2=10 -7 m / s;

[0129] The calculation time and delay factor are respectively taken as: t = 4 × 10 4 s,

[0130] (I)τ J =0s, τ C =0s;

[0131] (II)τ J =2×10 4 s, τ C =2×10 4 s;

[0132] (III)τ J =2×10 4 s, τ C =3×10 4 s;

[0133] (IV)τ J =3×10 4 s, τ C =2×10 4 s.

[0134] This example uses an explicit time-forward difference calculation method to achieve an efficient simulation of the evolution of solute concentration over time. This method is based on the second-order derivative at time k. Spatial distribution, the material point concentration C at time k+1 is calculated recursively by formula (12) k+1 , establishing a complete time-marching solution framework. Combined with the initial boundary condition settings and the actual value of the delay factor, this method accurately captures the dynamic characteristics of solute transport, providing a reliable numerical calculation method for analyzing solute transport behavior under different soil permeability conditions, and effectively solving the non-local delay effect problem that is difficult to handle with traditional methods.

[0135] In summary, through the establishment of non-local two-phase delayed migration model, such as Figure 3 As shown in Figure 2, the concentration distribution of the heterogeneous soil layer under the action of different delay factors at any time t can be described. From this, the effects of the flux delay factor and the storage delay factor in the high permeability area and the low permeability area and on the interface between the two can be observed.

[0136] By adopting the above technical solution, the present invention is different from the existing technology and has the following beneficial effects: the modeling method based on peridynamic theory of the present invention has high accuracy and good grid adaptability, and does not require special grid operations for special media in the computational domain; the above technical solution proposes to use the peridynamic method to simulate the non-Fickian behavior in solute transport, and establishes a solute transport model that considers both non-Fickian effects and non-local effects. The model is established under the framework of bond-based peridynamics (PD), and introduces the concept of dual-phase lag (DPL) effect, using the flux delay factor τ J Quantify the inertial collision delay caused by pore structure irregularities, the storage delay factor τ C It reflects the adsorption-desorption effect of the interface and overcomes the problems of numerical oscillation in discontinuous media caused by traditional local continuous methods, providing a new theoretical tool for analyzing and predicting solute migration in complex geological media.

[0137] Finally, it should be noted that although the above embodiments have been described in the specification and drawings of this application, this does not limit the scope of patent protection of this application. All technical solutions generated by replacing or modifying equivalent structures or equivalent processes based on the essential concepts of this application using the contents recorded in the specification and drawings of this application, as well as directly or indirectly implementing the technical solutions of the above embodiments in other related technical fields, are included in the scope of patent protection of this application.

Claims

1. A two-phase delayed solute transport simulation method based on peridynamics, characterized by: Applicable to a solid model, the method comprises: Dividing the solid model into regions according to preset intervals, and setting a preset number of initial material points in each region; Taking a preset size as a radius, a number of the initial material points are selected as the center to divide the neighborhood according to a preset rule, the initial material point at the center is recorded as the central material point, and the other initial material points in the neighborhood are recorded as subsidiary material points; The mass flux transfer between the attached material points and the central material point in the same neighborhood is calculated based on the local convection function and the local diffusion function, and the diffusion-convection constitutive function is constructed. Inputting the flux delay factor and the storage delay factor into the diffusion-convection constitutive function to obtain a two-phase delayed transport integral-differential equation; The two-phase delayed transport integral-differential equation is solved to obtain the solute transport simulation result of the current neighborhood.

2. The peridynamics-based two-phase delayed solute transport simulation method according to claim 1, characterized in that: The diffusion-convection constitutive function is expressed by formula (1), which is as follows: In formula (1), is the symbol for the derivative, J is the symbol for the mass flux, J d is the symbol of mass flux caused by diffusion behavior, J v is the symbol of the mass flux caused by convective behavior, C(x,t) is the solute concentration at the central material point x at time t, and t represents the transport time; J d It is expressed by formula (2), which is as follows: In formula (2), d(x′,x) is the first microscopic parameter of peridynamics corresponding to traditional ADE, and C(x′,t) is the solute concentration of the attached material point x′ at time t; J v It is expressed by formula (3), which is as follows: In formula (3), It is the second microscopic parameter in peridynamics theory corresponding to traditional ADE.

3. The peridynamics-based two-phase delayed solute transport simulation method according to claim 2, characterized in that: When the neighborhood range approaches 0 infinitely, and They converge to the local diffusion coefficient D and the convection velocity v, respectively, and are expressed by formula (4), which is as follows: In formula (4), is the influence kernel function.

4. The peridynamics-based two-phase delayed solute transport simulation method according to claim 3, characterized in that: The influence kernel function is configured to be expressed in shape-constant form, linear form; In the case of a one-dimensional region, the influence kernel function is expressed by formula (5), which is as follows: In formula (5), δ is the peridynamic field radius, ‖ξ‖ is the distance between the central material point x and the attached material point x′; In the case of a two-dimensional region, the influence kernel function is expressed by formula (6), which is as follows: In formula (6), π is the ratio of the circumference of a circle to the circumference of a circle.

5. The peridynamics-based two-phase delayed solute transport simulation method according to claim 2, characterized in that: The two-phase delayed transport integral-differential equation is expressed by formula (7), which is as follows: In formula (7), τ J is the flux delay factor, τ C is the storage delay factor, is the symbol for the time derivative, V x′ It refers to the morphological parameter of the attached material point x′, which is the length in one-dimensional state, the area in two-dimensional state, and the volume in three-dimensional state. x is the neighborhood of the central material point x.

6. The peridynamics-based two-phase delayed solute transport simulation method according to claim 5, characterized in that: Solve the two-phase delayed transport integral-differential equation to obtain the solute transport simulation results of the current neighborhood, including: Taylor expansion is performed on the two-phase delayed transport integral-differential equation, which is expressed by formula (8). The formula (8) is as follows: In formula (8), i is the derivative order; Ignoring the higher-order terms in the Taylor series in formula (8), the non-local two-phase delayed transport equation is obtained, which is expressed by formula (9), which is as follows: The non-local two-phase delayed transport equation finally obtained by series expansion is expressed by formula (10), which is as follows:

7. The peridynamics-based two-phase delayed solute transport simulation method according to claim 6, characterized in that: The method further comprises: The non-local two-phase delayed transport equation is spatially discretized and expressed by formula (11), which is as follows: In formula (11), is the peridynamic convection coefficient in upwind difference form, ω′∈[0,1] is the mixing weight, represents the windward area within the neighborhood; The initial boundary conditions, the first preset value of the flux delay factor, and the second preset value of the storage delay factor are set, and the explicit time forward difference calculation is performed on the formula (11) to obtain the solute transport simulation result of the current neighborhood.

8. The peridynamics-based two-phase delayed solute transport simulation method according to claim 7, characterized in that: The explicit time forward difference calculation of the formula (11) includes: Get the second derivative of solute concentration with respect to time at time k Spatial distribution, using explicit time forward difference to calculate the material point concentration C at time k+1 k+1 , expressed by formula (12), which is as follows: In formula (12), Δt is the time step, is the first-order derivative of the concentration at time k, is the first-order derivative of the concentration at time k+1, C k is the concentration at time k.

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