A method for micro water test analysis and permeability coefficient inversion considering finite thickness negative thin wall layer and non-darcy effect in layer
By constructing a mathematical model for micro-water experiments that considers negative thin-walled layers and non-Darcy effects, and by using Forchheimer flow and Laplace transform to handle nonlinear terms, a semi-analytical solution was derived and the permeability coefficient was inverted. This solved the problem of calculation distortion in traditional models and achieved accurate inversion of the permeability coefficient.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-03-19
- Publication Date
- 2026-06-26
AI Technical Summary
Traditional micro-water test analytical models fail to effectively account for negative thin-wall effects and non-Darcy effects, resulting in distorted permeability coefficient calculations and affecting the accuracy of groundwater resource assessment and pollutant transport simulation.
A mathematical model for micro-water testing was constructed, considering the finite-thickness negative thin-walled layer and the non-Darcy effect within the layer. The radial dual-zone medium assumption was adopted, and Forchheimer-type seepage and Laplace transform were introduced. The Brady method was used to handle nonlinear terms, and a semi-analytical solution was derived. The permeability coefficient was then obtained by fitting data collected by a high-frequency water level gauge.
It significantly improves the accuracy of permeability coefficient calculation, enables simultaneous inversion of permeability coefficients of negative thin-walled layers and aquifers, and broadens the application scope of micro-water tests.
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Figure CN122287437A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to an analytical method for micro-water testing and permeability coefficient inversion calculation considering finite-thickness negative thin-walled layers and non-Darcy effects within the layers. Specifically, it relates to a method for solving analytical models and inverting permeability coefficients of confined aquifer systems at semi-infinite or finite domain scales under the influence of negative thin-walled effects and non-Darcy effects, and belongs to the technical field of aquifer hydrogeological parameter testing and groundwater resource assessment. Background Technology
[0002] Accurately obtaining the permeability coefficient K of an aquifer is crucial for groundwater resource assessment, pollutant transport simulation, and geological engineering evaluation. Under strict control of groundwater environmental disturbance, micro-water testing has become the mainstream method for inverting in-situ hydraulic parameters of confined and unconfined aquifers due to its ease of operation, low testing cost, and small disturbance range to the aquifer. However, traditional analytical models for micro-water testing are all limited by a stringent core assumption: that the well-perimeter medium is ideally homogeneous and isotropic, and that the seepage law strictly follows Darcy's law throughout the hydrodynamic response. When the formation is disturbed or the experimental conditions cause a significant increase in the local hydraulic gradient, the seepage state in this near-well region deviates from a linear relationship. The well-aquifer system is affected by both the near-well negative thin-wall effect and the non-Darcy effect. If the traditional linear analytical model is directly used for inversion calculations in this case, serious systemic biases will occur due to neglecting the negative thin-wall effect and the non-Darcy effect, leading to a distorted calculated permeability coefficient K value. Summary of the Invention
[0003] Objective: This invention aims to provide an analytical method for micro-water testing and permeability coefficient inversion calculation considering finite-thickness negative thin-walled layers and intra-layer non-Darcy effects. A mathematical model for micro-water testing describing finite-thickness negative thin-walled layers and intra-layer non-Darcy effects is constructed. Based on the radial dual-zone medium assumption, seepage within the negative thin-walled layer is generalized as Forchheimer-type seepage, while seepage within the aquifer is linear Darcy flow. The Laplace transform and Brady method are introduced to handle nonlinear terms, deriving semi-analytical solutions for well water level H(t) in semi-infinite and finite domains. During parameter inversion, high-frequency, high-precision water level gauges deployed in the test well are used to collect water level response data after excitation, generating a curve of H versus t. By comparing the measured values with a standard curve, the permeability coefficient K of both the negative thin-walled layer and the aquifer is simultaneously inverted and calculated. i The purpose is to avoid the systemic physical biases in describing the early and mid-stage decay lag of the overall response of the Ht curve using traditional analytical models, significantly improve the accuracy of measuring the permeability coefficient of negative thin-walled layers and aquifers using micro-water tests, and broaden the application scope of micro-water tests.
[0004] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:
[0005] A method for analytical analysis and permeability coefficient inversion calculation of micro-water experiments considering finite-thickness negative thin-walled layers and non-Darcy effects within the layer, characterized by the following steps:
[0006] (1) For a semi-infinite or finite domain scale well-aquifer system in an initial stable flow field state, construct the groundwater seepage control equation and boundary conditions considering the finite thickness negative thin-walled layer, the non-Darcy effect within the layer, and the constraint of the far-end constant water level boundary, form a mathematical model, and perform dimensionless processing on the mathematical model by defining a dimensionless factor to obtain a dimensionless model.
[0007] (2) Based on the dimensionless model, the frequency domain analytical solution of the water level in the test well is derived using the Laplace transform and the Brady method. By combining the Stehfest numerical inverse transform algorithm, a semi-analytical solution for the test well water level response in the time domain is obtained. ;
[0008] (3) Construct a physical test platform consisting of a large-scale sand box, a negative thin-walled layer, a test well and an observation system, and build an initial background seepage field with stable flow.
[0009] (4) Based on a stable background seepage field, pumping tests were carried out in well-aquifer systems with different thicknesses of negative thin-walled layers, and the theoretical values of the permeability coefficients of the aquifer and negative thin-walled layer media were calculated according to the Dupuit formula for single-pore and the Thiem formula for multi-pore, respectively.
[0010] (5) Instantaneous water injection excitation was used to obtain the full-process response data of micro-water tests under negative thin-walled layers of different thicknesses. Based on the well water level response data obtained from the micro-water tests, a curve showing the variation of well water level H with time t (measured curve) was generated. At the same time, the semi-analytical solution results were used. Get H D -t D The standard curve is obtained by comparing and fitting the measured curve with the standard curve. Based on the least squares method, the time scale corresponding point of the two curves at the best fit is determined, and then the permeability coefficient values of the negative thin-walled layer and the aquifer are calculated simultaneously.
[0011] The aforementioned analytical and permeability coefficient inversion calculation method for micro-water tests considering finite-thickness negative thin-walled layers and in-layer non-Darcy effects suffers from severe deviations in permeability coefficient K value inversion when micro-water tests are affected by negative thin-walled and non-Darcy effects. To address this, a mathematical model for micro-water tests considering finite-thickness negative thin-walled layers and in-layer non-Darcy effects is constructed. Based on the radial dual-zone medium assumption, seepage within the negative thin-walled layer is generalized as Forchheimer-type seepage, while seepage within the aquifer is linear Darcy flow. The Laplace transform and Brady method are introduced to handle nonlinear terms, deriving semi-analytical solutions for well water level H(t) in both semi-infinite and finite domains. During parameter inversion, high-frequency, high-precision water level gauges deployed in the test well are used to collect water level response data after excitation, generating a curve of H versus t. By comparing the measured data with a standard curve, the permeability coefficient K of both the negative thin-walled layer and the aquifer is simultaneously inverted and calculated. i This invention achieves, for the first time, the simultaneous inversion of the permeability coefficients of negative thin-walled layers and aquifers, and significantly improves the efficiency of obtaining the permeability coefficient K using micro-water tests. i The accuracy of the value.
[0012] In step (1) above, the theoretical model construction and dimensionless processing are as follows:
[0013] In a complete well-confined aquifer system with a negative thin-walled layer, the governing equation for radial groundwater seepage can be expressed as:
[0014]
[0015]
[0016] In the formula, (m) represents the water head within the negative thin-walled layer; (m) represents the water head within the aquifer; (m) is the distance from the origin; B(m) is the thickness of the aquifer; (m) represents the radius of the test wellbore; The water storage coefficient of the negative thin-walled layer (dimensionless); The aquifer storage coefficient (dimensionless); (m / s) is the theoretical permeability coefficient of the negative thin-walled layer medium; (m / s) is the theoretical permeability coefficient of the aquifer medium; (s) represents time; L(m) represents the thickness of the negative thin-walled layer.
[0017] The perforated pipe is the only hydraulic exchange interface between the wellbore and the aquifer. According to the principle of mass conservation, the instantaneous volume change caused by changes in well water level must be consistent with the seepage flow rate into the aquifer through the perforated pipe. Therefore, the mass balance condition between the well and the aquifer is satisfied at the perforated pipe.
[0018]
[0019] In the formula, H(m) is the well water level in the test well; (m / s) represents the groundwater seepage velocity within the negative thin-walled layer.
[0020] In radially dual-zone media, the seepage state within the negative thin-walled layer is dominated by nonlinear seepage, and the seepage law is described by the Forchheimer equation; while the seepage state within the aquifer is dominated by linear seepage, and the seepage law is described by Darcy's law.
[0021]
[0022]
[0023] In the formula, (s / m) is the Forchheimer coefficient, which reflects the intensity of the non-Darcy effect; (m / s) represents the groundwater seepage velocity within the aquifer medium.
[0024] The initial and boundary conditions can be expressed as:
[0025]
[0026]
[0027]
[0028]
[0029]
[0030] In the formula, (m) represents the disturbed water level. (m) represents the water level in the test well at the moment of activation.
[0031] To enhance the generality of the solution and simplify the derivation, the following dimensionless variables are introduced:
[0032]
[0033] In the formula, This refers to the dimensionless well water level. This refers to the water level within the dimensionless negative thin-walled layer. This refers to the water level within the dimensionless aquifer. The distance is dimensionless; The thickness of the dimensionless negative thin-walled layer; Time is dimensionless; The dimensionless flow velocity within the negative thin-walled layer; The dimensionless flow velocity within the aquifer; This is a dimensionless, non-Darcy effect parameter.
[0034] Substitute the dimensionless variable in equation (11) into equations (1)-(10) for dimensionless processing, and define the thin-wall factor. , This reflects the difference in permeability between the negative thin-walled layer and the aquifer medium, leading to a dimensionless boundary value problem:
[0035]
[0036]
[0037]
[0038]
[0039]
[0040]
[0041]
[0042]
[0043]
[0044]
[0045] Performing a Laplace transform on equations (12)-(21), we denot: p is the Laplace space parameter. For the Laplace operator, This represents the negative thin-walled water level in the Laplace domain. For the water level of the Laplace aquifer, The water level in the Laplace domain test well; The seepage velocity in the negative thin-walled layer of the Laplace domain; Let be the seepage velocity of the aquifer in the Laplace domain. Then, equation (12) in the Laplace domain is expressed as:
[0046]
[0047] Equation (22) is the zeroth-order imaginary argument Bessel equation, and its general solution can be expressed as:
[0048]
[0049] In the formula, and These are the first and second kind of zero-order imaginary argument Bessel functions, respectively. These are coefficients to be determined.
[0050] The function diverges as r increases. The function converges to 0 as r increases, considering the boundary conditions. Equation (13) and its general solution can be expressed in the Lagrange domain as:
[0051]
[0052]
[0053] In the formula, These are coefficients to be determined.
[0054] Differentiating equations (23) and (25) respectively, based on the derivative properties of the imaginary argument Bessel function: , We can obtain:
[0055]
[0056]
[0057] Equation (15) is a nonlinear equation, and its Laplace transform is calculated using the Brady (1969) method. Brady (1969) used the Laplace transform method and the Newton-Leibniz transform to calculate the nonlinear function f with an integer power of a. a In the Laplace transform of , its Laplace domain expression can be approximated as: Then equation (15) is expressed in the Laplace domain as:
[0058]
[0059] Combining equation (27), we can obtain:
[0060]
[0061] Combining equations (16) and (27), we can similarly obtain:
[0062]
[0063] Combining conditions ,and Therefore, in The following conditions must be met:
[0064]
[0065]
[0066] exist Place, ,satisfy:
[0067]
[0068]
[0069]
[0070]
[0071]
[0072] Combining equations (31), (32), (36), and (37), we obtain:
[0073]
[0074]
[0075]
[0076]
[0077]
[0078] Equations (38)-(42) are the semi-analytical solutions of the NNM model in the Laplace domain, which are in form highly nonlinear implicit equations. Therefore, this study uses the Newton-Raphson method to solve each discrete Laplace variable p required by the Stehfest numerical inversion algorithm. i The algorithm iteratively solves the problem for (i=1,2,3,…,N), defining the error function vector R(X):
[0079]
[0080] In the formula, It is a linear coefficient matrix; It is a nonlinear vector containing the square of the velocity term; Let be the source term vector. The goal is to find the solution vector. , making .
[0081] Introducing the Jacobian matrix J for iteration, its elements constitute the partial derivatives of the error vector with respect to X:
[0082]
[0083] The iterative process is updated using the following principles, and the convergence principle is defined as follows:
[0084]
[0085]
[0086] In the formula, This represents the number of iteration steps. The relative error between the two iterations is defined in this study as when... =10 -8 The iteration is determined to be converged at that time.
[0087] After achieving stability After obtaining the frequency domain solution, the Stehfest numerical inversion algorithm is used to obtain the time domain solutions corresponding to a series of p values:
[0088]
[0089]
[0090]
[0091] In the formula, It is the image primitive function in spacetime; Let be the image function of the Laplace space; N is the number of summation terms; These are the weighting coefficients.
[0092] In step (3) above, the micro-water test is carried out through the following device structure and procedures:
[0093] Based on the radial flow assumption and axisymmetry principle in groundwater well flow problems, a semi-circular model from the axisymmetric model was adopted as the physical experimental platform. A test well was placed at the center of the semi-cylindrical cross-section, and overflow outlets were uniformly arranged on the arc side of the semi-cylindrical section to ensure that the boundary remained a constant head boundary during the experiment. A high-frequency, high-precision water level gauge was used to ensure real-time observation and recording of water level changes within the system during the experiment. The test well was located at the center of the semi-cylindrical cross-section, and observation holes were distributed in the radial direction of the semi-circular bottom plate. The aquifer system simulated a pressurized environment, with impermeable layers on both the top and bottom plates. A layer of sandbags was laid on the upper part of the top plate to apply pressure and simulate pressurization. A gap was left at the arc boundary as a constant head recharge boundary. Simultaneously, two different thicknesses of negative thin-walled layers (0.1m and 0.2m) were used to study the influence of different thicknesses of thin-walled layers on the micro-water test results. The negative thin-walled layer is located within 20cm around the main borehole and is partially separated from the aquifer by a semi-cylindrical frame. The frame is divided into two semi-cylindrical areas: 0-0.1m and 0.1-0.2m. The negative thin-walled layer is filled with quartz sand gravel. In the absence of a thin-walled layer, the negative thin-walled layer is filled with the same medium-fine sand as the aquifer.
[0094] Based on the physical test platform, water was first injected into the aquifer from the arc side of the model to gradually saturate it, expelling air from the sand body. Water injection was stopped when the water level in the model completely submerged the aquifer sand body. The system was considered saturated when the difference in water level gauge readings did not exceed 1 cm and remained constant for 5 minutes. Water injection continued until the water level reached the overflow outlet of the model and water overflowed. Only when the water level in the model remained at the overflow outlet position and the difference in readings of all water level gauges did not exceed 1 cm and remained constant for 5 minutes could pumping tests and micro-water tests be conducted under thin-walled negative layers of different thicknesses. After obtaining the water level data in the test wells, the pumping test results were processed and calculated based on the Dupuit formula method (single-well) and the Thiem formula method (multi-well), respectively.
[0095] The Dupuit formula method is defined as follows:
[0096]
[0097] Thiem's formula method is defined as follows:
[0098]
[0099] In the formula, Q(m) 3 / d) is the pumping flow rate; s w (m) represents the drawdown in the test well; R(m) represents the distance from the boundary of the finite domain to the origin; H2(m) represents the water level in the far observation well; H1(m) represents the water level in the near observation well; r2(m) represents the distance from the far observation well to the origin; r1(m) represents the distance from the far observation well to the origin.
[0100] Furthermore, when a definite constant head boundary exists within the influence range of a micro-water experiment, or when conducting finite-scale physical sandbox model experiments indoors, the influence of the external boundary scale on the semi-analytical solution becomes significant. In this case, the mathematical model originally based on the assumption of a semi-infinite field needs to be transformed into a boundary value problem considering a finite field (radius R). A dimensionless external boundary radius is introduced. Then the general solutions of the governing equations (12) and (13) in the Laplace domain will be reconstructed as follows:
[0101]
[0102]
[0103] In the formula, and These are undetermined coefficients.
[0104] Differentiating equations (61) and (62), and combining this with the continuity condition at the interface between the thin-walled layer and the aquifer, we can obtain a system of two linear equations in two variables, E1 and E2:
[0105]
[0106]
[0107]
[0108] Based on the properties of the Lonski determinant, for the modified Bessel function, we have the equation... This holds true for all cases, therefore the determinant of the coefficient matrix is... Then, according to Cramer's method, we can obtain and They are respectively:
[0109]
[0110]
[0111]
[0112]
[0113] J1 and I1 are the derivatives of J0 and I0, respectively; combined with Given the conditions of head continuity and mass conservation, we can solve for... The frequency domain semi-analytical solution is shown as follows:
[0114]
[0115]
[0116] Equations (70) and (71) are mathematical models for micro-water experiments of finite-thickness negative thin-walled layers and non-Darcy effects within the finite field R. The semi-analytical solution, after obtaining the data on the change of water level H in the test well with time t, generates the curve of the Ht change law of the measured data. By comparing and fitting the measured curve with the standard curve calculated by the analytical model, the time scale point corresponding to the best fit of the two curves is determined based on the least squares method. Then, the permeability coefficient values of the negative thin-walled layer and the aquifer are calculated synchronously. The calculation formula of the permeability coefficient is determined in reverse by the dimensionless time transformation relationship in equation (11):
[0117]
[0118] Unless otherwise specified in this invention, all techniques are based on existing technologies.
[0119] Compared with the prior art, the present invention has the following advantages:
[0120] (1) A dual-zone medium seepage model was introduced to solve the problem of distortion in the inversion calculation of the permeability coefficient K value under the influence of the negative thin-wall effect of the traditional linear model.
[0121] (2) The introduction of the Forchheimer type nonlinear seepage equation solves the problem of distortion in the inversion calculation of the permeability coefficient K value of the traditional linear model when it is affected by the non-Darcy effect in the negative thin-walled layer.
[0122] (3) The Brady method was used to solve the problem that the traditional linear superposition and Laplace solution methods failed because the inertial term of the Forchheimer equation made the control equation nonlinear.
[0123] (4) High-precision solutions are obtained by using the Laplace transform and Stehfest numerical inverse transform algorithm, which ensures the stability of the evaluation under complex working conditions.
[0124] (5) The least squares algorithm is used to fit the data, which effectively avoids the disadvantages of low efficiency and large subjective error of the traditional manual wiring method.
[0125] (6) For the first time, the permeability coefficient values of negative thin-walled layers and aquifers were accurately inverted simultaneously. Attached Figure Description
[0126] Figure 1 This is a flowchart illustrating the operation of the method of the present invention;
[0127] Figure 2 This is a schematic diagram of a physical experimental platform;
[0128] Figure 3 The diagram shows the fitting of measured data and semi-analytical solutions.
[0129] In the diagram, 1-overflow outlet; 2-pump outlet; 3-observation system; 4-test well; 5-bottom plate of aquifer; 6-thin-walled layer; 7-top plate of aquifer; 8-injection outlet; 9-initial water level. Detailed Implementation
[0130] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading this invention, any modifications of the invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.
[0131] like Figure 1 As shown, a method for analytical analysis and permeability coefficient inversion calculation of micro-water experiments considering finite-thickness negative thin-walled layers and non-Darcy effects within the layer includes the following steps:
[0132] A mathematical model is constructed that considers a finite-thickness negative thin-walled layer, non-Darcy effect within the layer, and the constraint of a fixed water level boundary at the far end. The mathematical model is then dimensionless by defining a dimensionless factor.
[0133] In a complete well-confined aquifer system with a negative thin-walled layer, the governing equation for radial groundwater seepage can be expressed as:
[0134]
[0135]
[0136] In the formula, (m) represents the water head within the negative thin-walled layer; (m) represents the water head within the aquifer; (m) is the distance from the origin; B(m) is the thickness of the aquifer; (m) represents the radius of the test wellbore; The water storage coefficient of the negative thin-walled layer (dimensionless); The aquifer storage coefficient (dimensionless); (m / s) is the theoretical permeability coefficient of the negative thin-walled layer medium; (m / s) is the theoretical permeability coefficient of the aquifer medium; (s) represents time; L(m) represents the thickness of the negative thin-walled layer.
[0137] The perforated pipe is the only hydraulic exchange interface between the wellbore and the aquifer. According to the principle of mass conservation, the instantaneous volume change caused by changes in well water level must be consistent with the seepage flow rate into the aquifer through the perforated pipe. Therefore, the mass balance condition between the well and the aquifer is satisfied at the perforated pipe.
[0138]
[0139] In the formula, H(m) is the well water level in the test well; (m / s) represents the groundwater seepage velocity within the negative thin-walled layer.
[0140] In radially dual-zone media, the seepage state within the negative thin-walled layer is dominated by nonlinear seepage, and the seepage law is described by the Forchheimer equation; while the seepage state within the aquifer is dominated by linear seepage, and the seepage law is described by Darcy's law.
[0141]
[0142]
[0143] In the formula, (s / m) is the Forchheimer coefficient, which reflects the intensity of the non-Darcy effect; (m / s) represents the groundwater seepage velocity within the aquifer medium.
[0144] The initial and boundary conditions can be expressed as:
[0145]
[0146]
[0147]
[0148]
[0149]
[0150] In the formula, (m) represents the disturbed water level. (m) represents the water level in the test well at the moment of activation.
[0151] To enhance the generality of the solution and simplify the derivation, the following dimensionless variables are introduced:
[0152]
[0153] In the formula, This refers to the dimensionless well water level. This refers to the water level within the dimensionless negative thin-walled layer. This refers to the water level within the dimensionless aquifer. The distance is dimensionless; The thickness of the dimensionless negative thin-walled layer; Time is dimensionless; The dimensionless flow velocity within the negative thin-walled layer; The dimensionless flow velocity within the aquifer; This is a dimensionless, non-Darcy effect parameter.
[0154] Substitute the dimensionless variable in equation (11) into equations (1)-(10) for dimensionless processing, and define the thin-wall factor. , This reflects the difference in permeability between the negative thin-walled layer and the aquifer medium, leading to a dimensionless boundary value problem:
[0155]
[0156]
[0157]
[0158]
[0159]
[0160]
[0161]
[0162]
[0163]
[0164]
[0165] Performing a Laplace transform on equations (12)-(21), we denot: p is the Laplace space parameter. For the Laplace operator, This represents the negative thin-walled water level in the Laplace domain. For the water level of the Laplace aquifer, The water level in the Laplace domain test well; The seepage velocity in the negative thin-walled layer of the Laplace domain; Let be the seepage velocity of the aquifer in the Laplace domain. Then, equation (12) in the Laplace domain is expressed as:
[0166]
[0167] Equation (22) is the zeroth-order imaginary argument Bessel equation, and its general solution can be expressed as:
[0168]
[0169] In the formula, and These are the first and second kind of zero-order imaginary argument Bessel functions, respectively. These are coefficients to be determined.
[0170] The function diverges as r increases. The function converges to 0 as r increases, considering the boundary conditions. Equation (13) and its general solution can be expressed in the Lagrange domain as:
[0171]
[0172]
[0173] In the formula, These are coefficients to be determined.
[0174] Differentiating equations (23) and (25) respectively, based on the derivative properties of the imaginary argument Bessel function: , We can obtain:
[0175]
[0176]
[0177] Equation (15) is a nonlinear equation, and its Laplace transform is calculated using the Brady (1969) method. Brady (1969) used the Laplace transform method and the Newton-Leibniz transform to calculate the nonlinear function f with an integer power of a. a In the Laplace transform of , its Laplace domain expression can be approximated as: Then equation (15) is expressed in the Laplace domain as:
[0178]
[0179] Combining equation (27), we can obtain:
[0180]
[0181] Combining equations (16) and (27), we can similarly obtain:
[0182]
[0183] Combining conditions ,and Therefore, in The following conditions must be met:
[0184]
[0185]
[0186] exist Place, ,satisfy:
[0187]
[0188]
[0189]
[0190]
[0191]
[0192] Combining equations (31), (32), (36), and (37), we obtain:
[0193]
[0194]
[0195]
[0196]
[0197]
[0198] Equations (38)-(42) are the semi-analytical solutions of the NNM model in the Laplace domain, which are in form highly nonlinear implicit equations. Therefore, this study uses the Newton-Raphson method to solve each discrete Laplace variable p required by the Stehfest numerical inversion algorithm. i The algorithm iteratively solves the problem for (i=1,2,3,…,N), defining the error function vector R(X):
[0199]
[0200] In the formula, It is a linear coefficient matrix; It is a nonlinear vector containing the square of the velocity term; Let be the source term vector. The goal is to find the solution vector. , making .
[0201] Introducing the Jacobian matrix J for iteration, its elements constitute the partial derivatives of the error vector with respect to X:
[0202]
[0203] The iterative process is updated using the following principles, and the convergence principle is defined as follows:
[0204]
[0205]
[0206] In the formula, This represents the number of iteration steps. The relative error between the two iterations is defined in this study as when... =10 -8 The iteration is determined to be converged at that time.
[0207] After achieving stability After obtaining the frequency domain solution, the Stehfest numerical inversion algorithm is used to obtain the time domain solutions corresponding to a series of p values:
[0208]
[0209]
[0210]
[0211] In the formula, It is the image primitive function in spacetime; Let be the image function of the Laplace space; N is the number of summation terms; These are the weighting coefficients.
[0212] When a well-defined constant head boundary exists within the influence range of a micro-water experiment, or when conducting finite-scale physical sandbox model experiments indoors, the influence of the external boundary scale on the semi-analytical solution becomes significant. In this case, the mathematical model originally based on the assumption of a semi-infinite field needs to be transformed into a boundary value problem considering a finite field (radius R). A dimensionless external boundary radius is introduced. Then the general solutions of the governing equations (12) and (13) in the Laplace domain will be reconstructed as follows:
[0213]
[0214]
[0215] In the formula, and These are undetermined coefficients.
[0216] Differentiating equations (61) and (62), and combining this with the continuity condition at the interface between the thin-walled layer and the aquifer, we can obtain a system of two linear equations in two variables, E1 and E2:
[0217]
[0218]
[0219]
[0220] Based on the properties of the Lonski determinant, for the modified Bessel function, we have the equation... This holds true for all cases, therefore the determinant of the coefficient matrix is... Then, according to Cramer's method, we can obtain and They are respectively:
[0221]
[0222]
[0223]
[0224]
[0225] Combination Given the conditions of head continuity and mass conservation, we can solve for... The frequency domain semi-analytical solution is shown as follows:
[0226]
[0227]
[0228] Equations (70) and (71) are mathematical models for micro-water experiments of finite-thickness negative thin-walled layers and non-Darcy effects within the finite field R. semi-analytical solution
[0229] like Figure 2 As shown, based on the radial flow assumption and axisymmetry principle in groundwater well flow problems, and to save experimental space, a semi-cylindrical model from the axisymmetric model was used as the experimental platform. The overall model design is a semi-cylindrical sand trough with a height of 2.2m and a radius of 2m. The test well is located at the center of the semi-cylindrical section, with a radius of 0.05m. Overflow outlets are evenly distributed on the semi-cylindrical arc side 1.5m above the model base to ensure that the boundary remains a constant head boundary during the experiment. The observation system uses a high-frequency, high-precision water level gauge (Level Vent ventilated water level gauge, range 5m, accuracy...). The sampling frequency can reach up to 0.125 s / time, ensuring real-time observation and recording of water level changes within the system during the experiment. The test well is located at the center of the semi-cylindrical cross-section, and the observation holes are distributed along the radial direction of the semi-circular bottom plate.
[0230] The aquifer system simulates a pressurized environment with a medium layer thickness of 0.8m. The aquifer medium consists of medium-fine sand with a particle size of 0.25mm-0.50mm. Water-resistant layers are installed on both the top and bottom plates. A layer of sandbags is placed on the top plate to apply pressure and simulate pressurization. A gap is left at the arc-shaped boundary as a constant head recharge boundary. The setting of the negative thin-walled layer is crucial and is the focus of this micro-water test. This study uses two different thicknesses of negative thin-walled layers: no thin-walled layer (0m thickness) and layers of 0.1m and 0.2m, to investigate the impact of different thicknesses on the micro-water test results. The negative thin-walled layer is located within 20cm around the main borehole, partially separated from the aquifer by a semi-cylindrical frame. The frame is divided into two semi-cylindrical regions: 0-0.1m and 0.1-0.2m. The negative thin-walled layer is filled with quartz gravel; in the absence of a thin-walled layer, it is filled with the same medium-fine sand as the aquifer.
[0231] To verify the reliability and accuracy of the proposed analytical model for micro-water testing that considers a finite-thickness negative thin-walled layer and the non-Darcy effect within the layer, it is first necessary to determine the theoretical permeability coefficients of the negative thin-walled layer and the aquifer in the system. Pumping tests were conducted on the test platform with the same medium (considered as a scenario without a thin wall, used to measure the theoretical permeability coefficient of the aquifer medium) and different media (the negative thin-walled layer was filled with quartz gravel, and the aquifer was filled with medium-fine sand, considered as a negative thin-walled scenario, used to measure the theoretical permeability coefficient of the negative thin-walled layer).
[0232] Before the experiment began, water was injected into the aquifer from the side of the model's arc to gradually saturate it, expelling air from the sand mass. Water injection was stopped when the water level in the model completely submerged the aquifer sand mass. The system was considered saturated when the difference in water level gauge readings did not exceed 1 cm and remained constant for 5 minutes. Water injection continued until the water level reached the overflow outlet of the model and water overflowed. Only when the water level in the model remained at the overflow outlet position and the difference in readings of all water level gauges did not exceed 1 cm and remained constant for 5 minutes could the pumping test and micro-water test be conducted. The pumping test results were processed and calculated using the Dupuit formula method and the Thiem formula method, respectively.
[0233] The Dupuit formula method is defined as follows:
[0234]
[0235] Thiem's formula method is defined as follows:
[0236]
[0237] In the formula, Q(m) 3 / d) is the pumping flow rate; s w (m) represents the drawdown in the test well; R(m) represents the distance from the boundary of the finite domain to the origin; H2(m) represents the water level in the far observation well; H1(m) represents the water level in the near observation well; r2(m) represents the distance from the far observation well to the origin; r1(m) represents the distance from the far observation well to the origin.
[0238] After obtaining the theoretical permeability coefficients of the thin-walled layer and aquifer in the system, a micro-water test was conducted on a complete well in a finite-domain confined aquifer. In this test, the thickness of the medium was B = 0.8 m, and the radius of the test well was r. w =0.05m, finite domain radius R=2 m, negative thin-wall layer thickness L is 0.1 m and 0.2 m. The experiment used instantaneous water injection for excitation, doubling the excitation intensity, corresponding to injection volumes of 500 ml, 1000 ml and 2000 ml, with equivalent water level rises of 0.0637 m, 0.1273 m and 0.2546 m, respectively. For each set of experimental data, the permeability coefficient K of the medium was calculated based on the CBP model and the new model. i The value is inverted and calculated, β D It is then used as a dimensionless non-Darcy effect parameter to characterize the intensity of the near-wellbore non-Darcy effect.
[0239] To verify the actual effectiveness of the method of the present invention, this embodiment applies the above scheme to an example, and the specific experimental process is as follows:
[0240] 1. Aquifer Saturation: Before the test begins, water is injected into the aquifer from the side of the model arc to saturate it layer by layer, expelling the air from the sand body. When the water level in the model completely submerges the aquifer sand body, the water injection is stopped. When the readings of each water level gauge are basically consistent and remain unchanged for a long time, the system is considered to have reached saturation. Continue to inject water until the water level reaches the overflow port of the model and water overflows. When the water level in the model remains at the overflow port position and the readings of each water level gauge are stable and remain unchanged for a long time, the pumping test and micro-water test can be carried out.
[0241] 2. Obtaining theoretical values of permeability coefficients for the negative thin-walled layer and aquifer: Based on the water level change data of the main borehole and each observation borehole of the test well, the theoretical values of permeability coefficients for the negative thin-walled layer and aquifer were calculated according to the Thiem formula and the Dupuit formula, respectively, and were 3.5914 cm / s and 0.1143 cm / s.
[0242] 3. Transient Excitation of Micro-Water Test: Based on the established stable background flow field, the system was transiently excited using the instantaneous water injection method at different negative thin-wall layer thicknesses. A predetermined volume of water (500 ml, 1000 ml, and 2000 ml, respectively) was rapidly injected into the central test well, generating an initial transient disturbance H0 superimposed on the background seepage field (equivalent water level rise of 0.0637 m, 0.1273 m, and 0.2546 m, respectively).
[0243] 4. Data Acquisition of Water Level Changes in Test Wells: The data acquisition system synchronously records the entire process of water level response at the sensor in the test well at high frequency. The acquired measured data covers the complete evolution sequence from stable water level to sudden water level changes and then to complete recovery to stable water level.
[0244] 5. Permeability coefficient K i Value Inversion and Verification: The measured water level response data collected in the test wells were processed to be dimensionless. By comparing the fit between the measured curves and the standard curves calculated by the analytical model, the time scale point corresponding to the best fit between the two curves was determined based on the least squares method. The permeability coefficient K of the negative thin-walled layer and the aquifer was decoupled, calculated, and inverted. i Value. Simultaneously, a comparative calculation was performed using the traditional linear analytical model CBP model. The fitted curves of the measured data and the semi-analytical solution are shown below. Figure 3 As shown in Table 1, the calculated data of the micro-water test are summarized.
[0245] Table 1. Results of Permeability Coefficient K Value Inversion Calculation
[0246]
[0247] Comparing the data in Table 1, it can be seen that the average aquifer permeability coefficient calculated by the new model obtained in this patent is 0.1147 cm / s (0.1 m thick negative thin-walled layer) and 0.1183 cm / s (0.2 m thick negative thin-walled layer), which is relatively consistent with the theoretical value of 0.1143 cm / s for aquifer permeability coefficient; the average calculated negative thin-walled layer permeability coefficient is 3.4431 cm / s (0.1 m thick negative thin-walled layer) and 3.5761 cm / s (0.2 m thick negative thin-walled layer), which is relatively consistent with the theoretical value of 3.5914 cm / s for negative thin-walled layer permeability coefficient; and different excitation... The calculated permeability coefficient under the specified intensity is relatively stable; however, the traditional CBP model can only obtain the permeability coefficient of the aquifer. The average calculated permeability coefficient for a 0.1m thick negative thin-walled layer is 0.2209 cm / s, with an error of 93.26% compared to the theoretical value; the average calculated permeability coefficient for a 0.2m thick negative thin-walled layer is 0.2840 cm / s, with an error of 148.47% compared to the theoretical value. Furthermore, the calculated permeability coefficient increases with increasing negative thin-walled layer thickness and decreases with increasing excitation intensity, indicating that the traditional linear analytical model is strongly influenced by both the negative thin-walled effect and the non-Darcy effect. The new model proposed in this invention, by characterizing the negative thin-walled effect and the non-Darcy effect, can accurately obtain the permeability coefficient values of both the negative thin-walled layer and the aquifer during parameter inversion, avoiding the inversion distortion problem of the traditional micro-water test analytical model under the influence of the negative thin-walled effect and the non-Darcy effect, and significantly improving the accuracy of permeability coefficient K obtained from micro-water test inversion. i The accuracy of the values has broadened the application scope of micro-water tests.
Claims
1. A method for analytical analysis and permeability coefficient inversion calculation of micro-water tests considering finite-thickness negative thin-walled layers and non-Darcy effects within the layer, characterized in that, Includes the following steps: (1) For a semi-infinite or finite domain scale well-aquifer system in an initial stable flow field state, construct the groundwater seepage control equation and boundary conditions considering the finite thickness negative thin-walled layer, the non-Darcy effect in the layer and the constraint of the far-end constant water level boundary, form a mathematical model, and perform dimensionless processing on the mathematical model by defining a dimensionless factor to obtain a dimensionless model. (2) Based on the dimensionless model, the frequency domain analytical solution of the water level in the test well is derived using the Laplace transform and the Brady method. By combining the Stehfest numerical inverse transform algorithm, a semi-analytical solution for the test well water level response in the time domain is obtained. ; (3) Construct a physical test platform consisting of a large-scale sand box, a negative thin-walled layer, a test well and an observation system to build an initial background seepage field with stable flow; (4) Based on a stable background seepage field, pumping tests were carried out in well-aquifer systems with different thicknesses of negative thin-walled layers, and the theoretical values of the permeability coefficients of the aquifer and negative thin-walled layer media were calculated according to the Dupuit formula for single-pore and the Thiem formula for multi-pore, respectively. (5) The response data of the entire process of micro-water test under negative thin-walled layers of different thicknesses were obtained by instantaneous water injection excitation. Based on the well water level response data obtained by the micro-water test, the curve of the well water level H changing with time t was generated; at the same time, the semi-analytical solution results were used. Get H D -t D The standard curve is obtained by comparing and fitting the measured curve with the standard curve. Based on the least squares method, the time scale corresponding point of the two curves at the best fit is determined, and then the permeability coefficient values of the negative thin-walled layer and the aquifer are calculated simultaneously.
2. The method for analyzing micro-water tests and calculating permeability coefficients considering finite-thickness negative thin-walled layers and non-Darcy effects within the layer, as described in claim 1, is characterized in that... In step (1), the mathematical model is constructed as follows: In a complete well-confined aquifer system with a negative thin-walled layer, the governing equation for radial groundwater seepage can be expressed as: ； ； In the formula, (m) represents the water head within the negative thin-walled layer; (m) represents the water head within the aquifer; (m) is the distance from the origin; B(m) is the thickness of the aquifer; (m) represents the radius of the test wellbore; The water storage coefficient of the negative thin-walled layer (dimensionless); The aquifer storage coefficient (dimensionless); (m / s) is the theoretical permeability coefficient of the negative thin-walled layer medium; (m / s) is the theoretical permeability coefficient of the aquifer medium; (s) represents time; L(m) represents the thickness of the negative thin-walled layer; The perforated pipe is the only hydraulic exchange interface between the wellbore and the aquifer. According to the principle of mass conservation, the instantaneous volume change caused by the change in well water level must be consistent with the seepage flow rate into the aquifer through the perforated pipe. Therefore, the mass balance condition between the well and the aquifer is satisfied at the perforated pipe. ； In the formula, H(m) is the well water level in the test well; (m / s) represents the groundwater seepage velocity within the negative thin-walled layer; In radially dual-zone media, the seepage state within the negative thin-walled layer is dominated by nonlinear seepage, and the seepage law is described by the Forchheimer equation; while the seepage state within the aquifer is dominated by linear seepage, and the seepage law is described by Darcy's law. ； ； In the formula, (s / m) is the Forchheimer coefficient, which reflects the intensity of the non-Darcy effect; (m / s) represents the groundwater seepage velocity within the aquifer medium; The initial and boundary conditions can be expressed as: ； ； ； ； ； In the formula, (m) represents the disturbed water level. (m) represents the water level in the test well at the moment of activation.
3. The method for analyzing micro-water tests and calculating permeability coefficients considering finite-thickness negative thin-walled layers and non-Darcy effects within the layer, as described in claim 2, is characterized in that... In step (1), the method for making the mathematical model dimensionless by defining dimensionless factors is as follows: the following dimensionless variables are introduced: ； In the formula, This refers to the dimensionless well water level. This refers to the water level within the dimensionless negative thin-walled layer. This refers to the water level within the dimensionless aquifer. The distance is dimensionless; The thickness of the dimensionless negative thin-walled layer; Time is dimensionless; The dimensionless flow velocity within the negative thin-walled layer; The dimensionless flow velocity within the aquifer; The parameter is a dimensionless non-Darcy effect parameter. Substitute the dimensionless variable in equation (11) into equations (1)-(10) for dimensionless processing, and define the thin-wall factor. , This reflects the difference in permeability between the negative thin-walled layer and the aquifer medium, leading to a dimensionless boundary value problem: ； ； ； ； ； ； ； ； ； 。 4. The method for analyzing micro-water tests and calculating permeability coefficient inversion considering finite-thickness negative thin-walled layers and non-Darcy effects within the layer, as described in claim 3, is characterized in that... Step (2) specifically includes the following process: Performing a Laplace transform on equations (12)-(21), we denot: p is the Laplace space parameter. For the Laplace operator, This represents the negative thin-walled water level in the Laplace domain. For the water level of the Laplace aquifer, The water level in the Laplace domain test well; The seepage velocity in the negative thin-walled layer of the Laplace domain; Let be the seepage velocity of the aquifer in the Laplace domain. Then, equation (12) in the Laplace domain is expressed as: ； Equation (22) is the zeroth-order imaginary argument Bessel equation, and its general solution can be expressed as: ； In the formula, and These are the first and second kind of zero-order imaginary argument Bessel functions, respectively. These are coefficients to be determined; The function diverges as r increases. The function converges to 0 as r increases, considering the boundary conditions. Equation (13) and its general solution can be expressed in the Lagrange domain as: ； ； In the formula, These are coefficients to be determined; Differentiating equations (23) and (25) respectively, based on the derivative properties of the imaginary argument Bessel function: , We can obtain: ； ； Equation (15) is a nonlinear equation, and its Laplace transform is calculated using the Brady (1969) method. Brady (1969) used the Laplace transform method and the Newton-Leibniz transform to calculate the nonlinear function f with an integer power of a. a In the Laplace transform of , its Laplace domain expression can be approximated as: Then equation (15) is expressed in the Laplace domain as: ； Combining equation (27), we can obtain: ； Combining equations (16) and (27), we can similarly obtain: ； Combining conditions ,and Therefore, in The following conditions must be met: ； ； exist Place, ,satisfy: ； ； ； ； ； Combining equations (31), (32), (36), and (37), we obtain: ； ； ； ； ； Equations (38)-(42) are the semi-analytical solutions of the NNM model in the Laplace domain, which are in form highly nonlinear implicit equations. The Newton-Raphson method is used to solve each discrete Laplace variable p required by the Stehfest numerical inversion algorithm. i The algorithm iteratively solves the problem for (i=1,2,3,…,N), defining the error function vector R(X): ； In the formula, It is a linear coefficient matrix; It is a nonlinear vector containing the square of the velocity term; The source term vector is the solution vector; the goal is to find the solution vector. , making ; Introducing the Jacobian matrix J for iteration, its elements constitute the partial derivatives of the error vector with respect to X: ； The iterative process is updated using the following principles, and the convergence principle is defined as follows: ； ； In the formula, This represents the number of iteration steps. The relative error between the two iterations is set as when =10 -8 Determine if the iteration converges at that time; After achieving stability After obtaining the frequency domain solution, the Stehfest numerical inversion algorithm is used to obtain the time domain solutions corresponding to a series of p values: ； ； ； In the formula, It is the image primitive function in spacetime; Let be the image function of the Laplace space; N is the number of summation terms; These are the weighting coefficients.
5. A method for analyzing micro-water test results and calculating permeability coefficient inversion considering a finite-thickness negative thin-walled layer and non-Darcy effect within the layer, as described in any one of claims 1-4, characterized in that... In step (3), based on the radial flow assumption and axisymmetry principle in the groundwater well flow problem, the semi-cylindrical model in the axisymmetric model is used as the physical test platform: The test well is set at the center of the semi-cylinder, and overflow outlets are evenly set on the arc side of the semi-cylinder to ensure that the boundary is a constant water head boundary during the test. The observation holes are distributed in the radial direction of the semi-circular bottom plate, and together with the high-frequency and high-precision water level gauge, the real-time observation and recording of the system water level changes during the test can be realized. The top and bottom plates of the aquifer are sealed with waterproof layers. Sandbags are laid on the upper part of the top plate to apply pressure and simulate a confined water environment. The arc boundary is set as a constant head recharge boundary to ensure stable hydraulic recharge during the test.
6. The method for analyzing and calculating the permeability coefficient of micro-water tests considering finite-thickness negative thin-walled layers and non-Darcy effects within the layer, as described in claim 5, is characterized in that... In step (3), in order to study the effect of thin-walled layers of different thicknesses on the micro-water test results, two types of negative thin-walled layers with different thicknesses were set within a 20cm radius around the main borehole: 0-0.1m and 0.1-0.2m, which were separated from the main aquifer by a semi-cylindrical frame; the negative thin-walled layers were filled with quartz sand gravel; the control group without thin-walled layers was filled with medium and fine sand consistent with the aquifer in this area.
7. A method for analyzing micro-water test results and calculating permeability coefficient inversion considering a finite-thickness negative thin-walled layer and non-Darcy effect within the layer, as described in any one of claims 1-4, characterized in that... In step (4), the different thicknesses include: no thin wall, a negative thin wall layer with a thickness of 0.1m, and a negative thin wall layer with a thickness of 0.2m.
8. A method for analyzing micro-water tests and calculating permeability coefficients considering finite-thickness negative thin-walled layers and non-Darcy effects within the layer, as described in any one of claims 1-4, characterized in that... In step (4), based on the physical test platform, water is first injected into the aquifer from the side of the model arc to saturate it layer by layer, and the air in the sand body is vented outward. When the water level in the model completely submerges the aquifer sand body, the water injection is stopped. When the readings of the water level gauges do not differ by more than 1 cm and remain unchanged for 5 minutes, the system is considered to have reached saturation. Water injection continues until the water level reaches the overflow port of the model and water overflows. When the water level in the model is always at the overflow port position and the readings of each water level gauge do not differ by more than 1 cm and remain unchanged for 5 minutes, pumping tests and micro-water tests of negative thin-walled layers of different thicknesses can be carried out. After obtaining the water level data in the test well, the pumping test results are processed and calculated based on the Dupuit formula method and the Thiem formula method, respectively. The Dupuit formula method is defined as follows: ； Thiem's formula method is defined as follows: ； In the formula, Q(m) 3 / d) is the pumping flow rate; s w (m) represents the drawdown in the test well; R(m) represents the distance from the boundary of the finite domain to the origin; H2(m) represents the water level in the far observation well; H1(m) represents the water level in the near observation well; r2(m) represents the distance from the far observation well to the origin; r1(m) represents the distance from the far observation well to the origin.
9. A method for analyzing micro-water test results and calculating permeability coefficient inversion considering a finite-thickness negative thin-walled layer and non-Darcy effect within the layer, as described in any one of claims 1-4, characterized in that... In step (5), when there is a clear constant head boundary within the influence range of the micro-water test, or when a finite-scale physical sandbox model test is carried out indoors, the influence of the external boundary scale on the semi-analytical solution will not be ignored; in this case, the mathematical model originally based on the assumption of a semi-infinite field needs to be transformed into a boundary value problem considering a finite field condition with radius R; a dimensionless external boundary radius is introduced. Then the general solutions of the governing equations (12) and (13) in the Laplace domain will be reconstructed as follows: ； ； In the formula, and These are undetermined coefficients; Differentiating equations (61) and (62), and combining this with the continuity condition at the interface between the thin-walled layer and the aquifer, we can obtain a system of two linear equations in two variables, E1 and E2: ； ； ； Based on the properties of the Lonski determinant, for the modified Bessel function, we have the equation... This holds true for all cases, therefore the determinant of the coefficient matrix is... Then, according to Cramer's method, we can obtain and They are respectively: ； ； ； ； Combination Given the conditions of head continuity and mass conservation, we can solve for... The frequency domain semi-analytical solution is shown as follows: ； ； Equations (70) and (71) are mathematical models for micro-water experiments with finite-thickness negative thin-walled layers and non-Darcy effects within the finite field R. The semi-analytical solution, after obtaining the data on the change of water level H in the test well with time t, generates the curve of the Ht change law of the measured data. By comparing and fitting the measured curve with the standard curve calculated by the analytical model, the time scale point corresponding to the best fit of the two curves is determined based on the least squares method. Then, the permeability coefficient values of the negative thin-walled layer and the aquifer are calculated synchronously. The calculation formula of the permeability coefficient is determined in reverse by the dimensionless time transformation relationship in equation (11): 。