Optimal signal source frequency selection method and system based on oscillation hydraulic chromatography test

Abstract

The invention provides an optimal signal source frequency selection method and system based on an oscillation hydraulic chromatography test, and the method comprises the steps: building a sensitivity equation of water head observation to a water diversion coefficient under the condition of exporting oscillation water pumping; under the condition of visualized two-dimensional oscillation water pumping, the sensitive degree of a water head to the water guiding coefficients of different positions is observed; the sensitivity equation of the hydraulic conductivity coefficient is converted into a dimensionless form, a frequency selection scheme based on the observation position and the hydraulic diffusion coefficient is provided, and the optimal frequency is determined according to the sensitivity equation of the dimensionless form; and according to the hydraulic diffusion coefficient, forward modeling of the two-dimensional heterogeneous parameters of the detection area is carried out by using the established two-dimensional geological forward modeling model, and according to the determined optimal frequency, parameter inversion is carried out by using the established parameter inversion model. According to sensitivity analysis, the optimal frequency selection scheme is obtained according to the influence of parameters on observation data, the hydraulic diffusion coefficient of a simple geologic structure and the distance between an observation well and a pumping well.

Description

Technical Field

[0001] The present invention relates to the field of identification of hydrogeological and engineering geological parameters, and more specifically, to an optimal signal source frequency selection method and system based on oscillatory hydraulic tomography tests. Background Art

[0002] Accurately measuring the hydraulic conductivity (T) and storage coefficient (S) of an aquifer has important practical significance and far-reaching influence on the development, management, protection, and scientific research of groundwater resources. The hydraulic conductivity (T) represents the amount of water passing through a unit-width aquifer per unit time under a unit hydraulic gradient, reflecting the hydraulic conductivity of the aquifer. The storage coefficient (S) represents the amount of water released or stored per unit area of the aquifer when the unit head changes, reflecting the water storage capacity of the aquifer. The ratio of the hydraulic conductivity to the storage coefficient is the hydraulic diffusivity (D = T / S), which is an important parameter describing the propagation speed of hydraulic disturbances in the aquifer and reveals the relationship between the water-conducting ability and water storage ability of the aquifer. Precise hydraulic conductivity and storage coefficient are the basic data for calculating the groundwater resource reserves, exploitable amounts, and sustainable utilization, and are also the key input parameters in groundwater numerical simulation models.

[0003] Accurately measuring the hydraulic conductivity and storage coefficient helps to deeply understand the migration law of pollutants in groundwater, and thus provides a scientific basis for formulating effective pollution prevention and control measures. However, the current direct measurement of the hydraulic conductivity (T), storage coefficient (S), or hydraulic diffusivity (D) usually relies on aquifer core samples and conducts seepage and pressure disturbance experiments in laboratory seepage equipment. This method is not only costly but also requires destructive sampling, making it difficult to apply to large-scale measurements. In actual engineering, large-scale measurements often rely on indirect measurement methods such as pumping tests.

[0004] Oscillatory Hydraulic Tomography (OHT) is an advanced technology for determining the hydraulic conductivity and storage coefficient of an aquifer by observing the head response. In a test well, a periodic or oscillatory hydraulic disturbance is introduced by quickly injecting or extracting a certain volume of water, and the head response process is collected using a high-precision sensor, thereby more efficiently obtaining the spatial distribution information of the aquifer. Compared with traditional Hydraulic Tomography (HT), Oscillatory Hydraulic Tomography (OHT) has the following significant advantages: 1. When an oscillatory signal propagates in the groundwater system, it is less affected by background noise (such as natural water level fluctuations or other disturbances) compared to a single steady-state or transient pumping test, which is conducive to extracting effective periodic signal characteristics from the observed data; 2. Oscillatory hydraulic tomography can be tested without net water injection or pumping, effectively reducing the disturbance to the solute plume because the average flow velocity caused by this pumping is zero in all directions; 3. The oscillatory signal can propagate to a longer distance. Compared with the monitoring within a smaller disturbance range of traditional HT, oscillatory hydraulic tomography has a larger spatial coverage; 4. Oscillatory hydraulic tomography can be tested at multiple different frequencies between any pair of wells to obtain more characteristic information about aquifer heterogeneity. Therefore, oscillatory hydraulic tomography has significant advantages over traditional hydraulic tomography in terms of noise resistance, test efficiency, coverage, and data richness, especially in a highly heterogeneous aquifer environment.

[0005] Although oscillatory hydraulic tomography has significant technical advantages, the selection of the oscillation frequency of the signal source is usually based on the propagation distance of the signal in the aquifer, lacking a clear theoretical basis. And the cost of setting different signal source frequencies for observation is high, which limits the efficiency of practical applications. Therefore, to solve the problem of optimal frequency selection for oscillatory hydraulic tomography, it is necessary to optimize the frequency selection to obtain the maximum amount of information with the least number of frequency observations. Summary of the Invention

[0006] The present invention aims at the technical problems existing in the prior art, and provides an optimal signal source frequency selection method and system based on oscillatory hydraulic tomography tests, which helps to obtain a frequency selection scheme for the best estimation of parameter heterogeneity based on the least frequency observations. Through numerical tests, the frequency selection scheme shows excellent accuracy and reliability in the inversion model. In addition, the frequency selection scheme provided by the present invention only needs to know the approximate value of the hydraulic diffusion coefficient (D) of the test site and the distance between the pumping well and the observation well, and the frequency selection scheme is simple, making it more reliable in practical applications.

[0007] According to the first aspect of the present invention, there is provided an optimal signal source frequency selection method based on oscillatory hydraulic tomography tests, including the following steps: Under the condition of derived oscillatory pumping, establish a sensitivity equation of the head observation to the hydraulic conductivity; Based on the sensitivity equation, under the condition of visualizing two-dimensional oscillatory pumping, analyze the sensitivity degree of the observed head to the hydraulic conductivity at different positions when pumping at different frequencies; Convert the sensitivity equation of the hydraulic conductivity into a dimensionless form, propose a frequency selection scheme based on the dimensionless distance and the hydraulic diffusion coefficient, and determine the optimal frequency according to the dimensionless form of the sensitivity equation; A two-dimensional geological forward model and a parameter inversion model for the groundwater oscillation response under oscillatory pumping are established. According to the hydraulic diffusivity, the forward simulation of the two-dimensional heterogeneous parameters in the detection area is carried out using the established two-dimensional geological forward model. According to the determined optimal frequency, parameter inversion is performed using the established parameter inversion model.

[0008] Based on the above technical solution, the present invention can also be improved as follows.

[0009] Optionally, when deriving the oscillatory pumping, the expression form of the sensitivity equation of the head observation to the hydraulic conductivity is:

[0010] In the formula, represents the complex sensitivity function of the response function at the position Q with an amplitude of and an oscillation frequency of caused by the oscillatory pumping at the position to the hydraulic conductivity at the position , including the sensitivity of the amplitude and phase shift of the head to the hydraulic conductivity; T is the hydraulic conductivity in space, D is the hydraulic diffusivity in space, is the derivative of the modified Kelvin function of the second kind of order 0; represents the dimensionless distance between the pumping position and the parameter position ; represents the dimensionless distance between the observed head position and the parameter position ; Assuming the parameter position is , is expressed as the spatial gradient term, represents the distance between the pumping position and the parameter position ; represents the distance between the observed head position and the parameter position .

[0011] Optionally, based on the sensitivity equation, when visualizing two-dimensional oscillatory pumping, analyzing the sensitivity of the observed head to the hydraulic conductivity at different positions when pumping at different frequencies includes: Transform the sensitivity equation to extract the sensitivity equations of the head amplitude and head phase at the observation position to the hydraulic conductivity respectively.

[0012] Optionally, the dimensionless form of the sensitivity equation is expressed as:

[0013] In the formula: The dimensionless sensitivity function of the hydraulic conductivity T at the position where the representative amplitude is for the dimensionless distance is the dimensionless distance where r is the distance between the observation well and the pumping well, ω is the frequency, D is the hydraulic diffusivity; Ker 1 and Kei 1 are the real and imaginary parts of the modified Kelvin function of the second kind and first order.

[0014] Optionally, the conversion of the sensitivity equation of the hydraulic conductivity into a dimensionless form and the proposed frequency selection scheme based on the dimensionless distance and the hydraulic diffusivity include: Set the second derivative of the sensitivity equation in dimensionless form to zero, find the position of the dimensionless frequency peak. When the optimal frequency is combined with the optimal distance and the dimensionless distance, it is the selection scheme of the stimulation source frequency.

[0015] Optionally, the determination of the optimal frequency includes: Determine the pumping frequency of the pumping well according to the average hydraulic diffusivity of the site and the distance of the observation well; Based on the distance between the observation well and the pumping well and the average hydraulic parameters, select the frequency with the maximum information containing the heterogeneity parameters according to the sensitivity equation in dimensionless form, which is the optimal frequency.

[0016] Optionally, the calculation formula of the optimal frequency is expressed as follows:

[0017] In the formula, represents the optimal frequency, r is the distance between the observation well and the pumping well, and D represents the hydraulic diffusivity.

[0018] Optionally, the parameter inversion model calculates the parameter values using the ensemble iterative smoothing algorithm according to the head response data, and performs the inversion of the two-dimensional heterogeneity parameters in the detection area.

[0019] Optionally, the parameter inversion using the established parameter inversion model according to the determined optimal frequency includes: Determine the oscillation frequency of the stimulation source through the frequency selection scheme. According to the determined optimal frequency, use the amplitude data of the optimal frequency to perform tomography scanning of the permeability coefficient of the underground aquifer and the distribution of the underground hydraulic conductivity field.

[0020] According to the second aspect of the present invention, there is provided an optimal signal source frequency selection system based on an oscillatory hydraulic tomography test, including: A sensitivity equation construction module, which is used to establish a sensitivity equation of head observation to hydraulic conductivity under the condition of oscillatory pumping; An equation visualization module, which is used to analyze the sensitivity of the observed head to the hydraulic conductivity at different positions when pumping at different frequencies under the condition of visualizing two-dimensional oscillatory pumping; An optimal frequency selection module, which is used to transform the sensitivity equation of hydraulic conductivity into a dimensionless form, propose a frequency selection scheme based on the observation position and the hydraulic diffusivity coefficient, and determine the optimal frequency according to the dimensionless sensitivity equation; A model parameter inversion module, which is used to establish a two-dimensional geological forward model and a parameter inversion model for the groundwater oscillatory response under the condition of oscillatory pumping. According to the hydraulic diffusivity coefficient, the two-dimensional heterogeneous parameters of the detection area are forward modeled by using the established two-dimensional geological forward model, and parameter inversion is carried out by using the established parameter inversion model according to the determined optimal frequency.

[0021] The technical effects and advantages of the present invention: The present invention proposes an optimal signal source frequency selection method and system based on oscillatory hydraulic tomography for evaluating the influence of aquifer parameters on observed data (such as head fluctuations). By introducing an explicit form of the sensitivity equation of head observation to hydraulic conductivity (T) under oscillatory hydraulic tomography, the present invention quantifies the contribution degree of parameter perturbation (such as hydraulic conductivity T) to the head response at the observation position. On the basis of maximizing sensitivity information, combined with the hydraulic diffusivity coefficient and the observation distance, an optimal frequency selection scheme can be obtained according to the hydraulic diffusivity coefficient of a simple geological structure and the distance between the observation well and the pumping well. During the parameter inversion process, sensitivity analysis helps to determine which parameters have a significant impact on the observed data, thereby reducing the inversion of non-sensitive parameters and reducing the computational complexity. Description of the drawings

[0022] In order to more clearly illustrate the technical solutions in the present invention or the prior art, the following will briefly introduce the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained according to these drawings without creative efforts.

[0023] Figure 1 It is a flow chart of the optimal signal source frequency selection method based on oscillatory hydraulic tomography provided by an embodiment of the present invention; Figure 2 It is a sensitivity diagram of the fluctuation observation to the hydraulic conductivity (T) under the condition of oscillatory pumping provided by an embodiment of the present invention; wherein, Figure 2 The (a) in it shows the spatial distribution diagram of the normalized sensitivity of the amplitude A to the parameter T, Figure 2In (b), it shows the spatial distribution map of the normalized sensitivity of the amplitude P to the parameter T; Figure 3 It is a schematic diagram of the optimal observation frequency solved according to the graphical method provided by the embodiment of the present invention; Figure 4 It is an arrangement diagram of the pumping well and the observation well provided by the embodiment of the present invention; Figure 5 It is a distribution diagram of setting the mean and variance of the initial sample lnT and the reference field lnT provided by the embodiment of the present invention; among them, (a) in Figure 5 is the mean diagram of the initial sample lnT, (b) in Figure 5 is the variance diagram of the initial sample lnT, and (c) in Figure 5 is the distribution diagram of the reference field lnT; Figure 6 It is an inversion result diagram provided by the embodiment of the present invention; among them, Figure 6 In (a), it is the result diagram of the inverted lnT, Figure 6 In (b), it is the deviation diagram between the inverted lnT and the reference field lnT, Figure 6 In (c), it is the variance diagram of lnT of the sample after the inversion iteration is completed, Figure 6 In (d), it is the comparison diagram between the reference lnT and the inverted lnT. Specific embodiments

[0024] To make the purpose, technical solutions and advantages of the present invention clearer, the technical solutions in the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by those of ordinary skill in the art without making creative efforts belong to the scope of protection of the present invention.

[0025] It can be understood that, based on the defects in the background technology, the embodiment of the present invention proposes an optimal signal source frequency selection method based on the oscillatory hydraulic tomography test, specifically as Figure 1 shown, the method includes the following steps: Step S1, in the case of deriving oscillatory pumping, establish a sensitivity equation of the head observation to the hydraulic conductivity (T); It should be noted that the sensitivity equation describes the influence of the change of the hydraulic conductivity at a certain position on the periodic head observation. The sensitivity equation is one of the basic steps of groundwater parameter inversion, aiming to initially understand the influence of the heterogeneity of the aquifer on the head observation.

[0026] In the case of deriving oscillatory pumping, establishing a sensitivity equation of the head observation to the hydraulic conductivity (T) includes: According to the definitions of the groundwater flow equation and the sensitivity equation, the function of the periodic response head with respect to time t can be expressed as the periodic head , where is the complex frequency response function, is the frequency. The sensitivity of the head at location to the hydraulic conductivity T at is expressed as: , which is controlled by the following equation: (1) In the equation, represents the sensitivity function of the periodic head , is the periodic head at location with an amplitude of and an oscillation frequency of caused by the oscillatory pumping at location with the same frequency.

[0027] The boundary conditions of the sensitivity function φ are satisfied as: (2) Equation (1) can be expanded as: (3) In equation (3), the spatial derivative is involved as the source-sink term of the equation, and the governing equation and its solution can be understood more clearly through the following steps.

[0028] First, consider a governing equation where the source term appears in the spatial derivative term: (4) Equation (3) describes the oscillatory pumping test with the pump well located at , and its complex pumping rate can be expressed according to the groundwater response function as: (5) where is the complex frequency response function, which is closely related to the derivative : (6) In the equation, is the derivative of the modified Kelvin function of the second kind of order 0; represents the dimensionless distance, , is the actual distance, represents the pumping location and the parameter location The dimensionless distance. Consistent with the solution of the groundwater periodic equation, after reaching the steady state, can be expressed as , where can be obtained by the following formula: (7) In the formula, represents the position of the observed water head and the dimensionless distance from the parameter position . Assume there is another observation position, whose position is slightly offset relative to , denoted as , then its solution can be obtained by replacing in equation (6) with , where is and the radial distance between.

[0029] Consider the following equation: (8) The solution of equation (6) is obtained by taking the derivative with respect to : (9) Similarly, by replacing with , the solution of equation (9) is obtained, and then the solution of the sensitivity equation of the complex frequency response of formula (1) is derived. The result of the sensitivity equation includes the magnitudes of the sensitivity of the amplitude and phase of groundwater fluctuations to the parameter T. Here, the sensitivity equation is in complex form. It includes the sensitivity of the amplitude (A) and phase shift (P) of the water head to the hydraulic conductivity T. It represents the complex sensitivity function of the response function at the position with amplitude and oscillation frequency caused by oscillatory pumping at the position to the hydraulic conductivity T at the position . The function is expressed as: (10) In the formula, represents the complex sensitivity function of the periodic response , is the complex response form of the periodic water head, that is, ; T is the hydraulic conductivity in space, D is the hydraulic diffusivity in space, is the derivative of the modified Kelvin function of the second kind of order 0, represents the dimensionless distance, , r is the actual distance; represents the dimensionless distance between the pumping position and the parameter position ; represents the dimensionless distance between the observed head position and the parameter position ; Assuming the parameter position is , is expressed as the spatial gradient term and in coordinates as: , represents the distance between the pumping position and the parameter position ; represents the distance between the observed head position and the parameter position .

[0030] Step S2: Based on the sensitivity equation, analyze the sensitivity of the observed head to the hydraulic conductivity at different positions under the condition of visualizing two-dimensional oscillatory pumping; With the midpoint O between the observed head position and the pumping position as the origin, and the line between them as the x axis, establish a two-dimensional plane rectangular coordinate system x O y . Assuming the distances between the origin O and and are both c , then the observed position coordinates can be obtained as , and the coordinates of the pumping position are . Assuming the parameter position is , then there is: (11) Based on the complex frequency response function , the sensitivities of the amplitude (A) and phase shift (P) related to the hydraulic conductivity T can be derived. The complex frequency head response of oscillatory pumping , and the sensitivity complex function can be expressed as , where represents the real part of the complex frequency head response, represents the imaginary part of the complex frequency head response, represents the real part of the sensitivity complex function, represents the imaginary part of the sensitivity complex function. If the oscillatory pumping / injection rate is , then it can be obtained that:

[0031] Among them, , represents the real part of the complex-frequency head response, represents the amplitude of the complex-frequency head response under oscillatory pumping, represents the argument of the complex-frequency head response of oscillatory pumping. It should be noted that, in the context of the sensitivity map, the dimensionless distance between the head position and the pumping position is a fixed value and is related to the following three factors: the distance between the pumping well and the observation well, the frequency, and the average diffusivity.

[0032] According to the properties of complex numbers, Equation (10) can be written as: (12)

[0033] Assume , , according to the Cauchy-Riemann conditions, the sensitivities of the amplitude (A) and phase shift (P) related to the hydraulic conductivity T can be derived: the sensitivities of the amplitude (A) and phase (P) of groundwater fluctuations to the parameter T.

[0034] (13) (14) The analytical solution is a function of spatial coordinates, frequency and the average parameter D. Since and are the dimensionless radial distances between the pumping well and the parameter position, and between the observation well and the parameter position, and are normalized by multiplying by , the properties of the analytical solution are determined by the characteristics of the second-kind modified Kelvin function. It should be noted that the phase-shift sensitivity is not affected by the oscillation amplitude .

[0035] The coordinates of the pumping well and the observation well are set as and for the virtual example, where , which corresponds to , and , and the spatial coordinates x and y are normalized by multiplying by . Figure 2 is the sensitivity map of the fluctuation observation to the hydraulic conductivity (T) under the virtual example, Figure 2Visualization results showing the sensitivity of the amplitude (A) and phase (P) of groundwater fluctuations to parameter T. Figure 2 In (a) shows the normalized sensitivity of the amplitude (A) to the hydraulic conductivity T spatial distribution; Figure 2 In (b) shows the normalized sensitivity of the amplitude (P) to the hydraulic conductivity T spatial distribution, sensitivity and are normalized by and respectively. In this way, the normalized sensitivity is only determined by the combined modified Kelvin functions and where the parameters are related to , and Results show that in the sensitivity of the hydraulic conductivity T, the term is the cause of the sudden change in the sign of the sensitivity map inside and outside the circle, and this change occurs at the boundary of the circle .

[0036] Step S3, transform the sensitivity equation of the hydraulic conductivity into a dimensionless form, propose a frequency selection scheme based on the dimensionless distance and the hydraulic diffusion coefficient, and determine the optimal frequency according to the sensitivity equation in dimensionless form; Nondimensionalize the expression of the sensitivity equation of the amplitude (A) of groundwater fluctuations to the hydraulic conductivity T, transform the sensitivity equation of the hydraulic conductivity into a dimensionless form, and eliminate the influence of factors such as pumping flow rate on the sensitivity equation. The purpose is to make the relationship between the pumping frequency ( ω ), the distance of the observation well ( r ), and the average hydraulic parameter coefficient D more clearly shown, thus providing convenience for solving the problem.

[0037] Set when the observation point coincides with the oscillating pumping well ( ), the sensitivity of the amplitude related to the hydraulic conductivity T in step 1 becomes: (15) Where: is the sensitivity function of the amplitude of the water head to the hydraulic conductivity T at a position with a distance of when the observation point coincides with the oscillating pumping well. represents the distance between the parameter position and the observation point (pumping well).

[0038] Introduce the scale parameter and consider the dimensionless distance (frequency) , define the dimensionless amplitude sensitivity function , at this time, the expression of the sensitivity equation in dimensionless form is as follows: (16) In the formula: represents the dimensionless sensitivity function of the amplitude to the hydraulic conductivity T at the position where the dimensionless distance is , is the dimensionless distance , where r is the distance between the observation well and the pumping well, ω is the frequency, D is the hydraulic diffusivity, Ker 1 and Kei 1 are the real and imaginary parts of the modified Kelvin function of the second kind of the first order.

[0039] The proposed frequency selection scheme based on the observation position and the hydraulic diffusivity includes: Set the second derivative of the sensitivity equation in dimensionless form to zero, and find the peak position of the dimensionless frequency (distance), which is expressed as follows (17) The root of formula (17) can be given by the graphical method, and the characteristic root . When the optimal frequency is combined with the optimal distance and the dimensionless distance, it is the selection scheme of the stimulation source frequency.

[0040] Determining the optimal frequency according to the sensitivity equation in dimensionless form includes: Determine the pumping frequency of the pumping well according to the average hydraulic diffusivity of the site and the distance of the observation well ω ; Based on the distance r between the observation well and the pumping well and the average hydraulic parameter D, select the frequency with the maximum information containing the heterogeneity parameter according to the dimensionless sensitivity equation .

[0041] Among them, the optimal frequency can be combined with the optimal distance and the dimensionless distance (frequency) to give, and the calculation method is as follows: (18) In the formula, represents the optimal frequency, r is the distance between the observation well and the pumping well, and D represents the hydraulic diffusivity.

[0042] Equation (18) gives the optimal frequency selection method in oscillatory hydraulic tomography. The optimal frequency depends on the approximate value of the hydraulic diffusivity D and the distance between the pumping well and the observation wells. This result, combined with the characteristics of sensitivity and the explicit expression of sensitivity, analytically expresses the optimal selection scheme for designing the stimulation source frequency in oscillatory hydraulic tomography tests.

[0043] Figure 3 Shows a schematic diagram of the optimal observation frequency solved by the graphical method. The physical meaning of this optimal observation frequency ( ) is that for the two-dimensional radial flow situation of the observation point, the pumping well, and the confined aquifer, when pumping starts, a pressure fluctuation signal will be emitted in the well, and this fluctuation signal propagates outward radially along the way. The characteristic root corresponds to the characteristic frequency at which the fluctuation amplitude reaches the abnormal parameter position point and returns.

[0044] Step S4, establish a two-dimensional geological forward model and a parameter inversion model for the groundwater oscillation response under oscillatory pumping. According to the hydraulic diffusivity, use the established two-dimensional geological forward model to perform forward simulation of the two-dimensional heterogeneous parameters in the detection area. According to the determined optimal frequency, use the established parameter inversion model to perform parameter inversion.

[0045] In this embodiment, the simulation domain is a two-dimensional horizontal confined aquifer with a size of 866 [L] × 866 [L]. The numerical model discretizes the computational domain into square grids. To optimize the computational efficiency, the grid size of the pumping area (100 [L] × 100 [L]) near the center of the domain is 1 [L], while the grid size gradually increases to 32 [L] near the boundary. There are a total of 17,424 grid cells and 17,689 nodes. One pumping well and 13 observation wells are arranged in the aquifer, and one of the observation wells coincides with the pumping well. The arrangement is as Figure 4 shown, where the circles represent the observation wells and the squares represent the pumping well. The pumping well is located at the center of the area, and the peak pumping rate is 0.01 [L 3 / T]. The observation wells are distributed in a cross shape around the pumping well, with radial distances of 15 [L], 30 [L], and 45 [L] respectively. The forward method uses the basic numerical method of the groundwater movement equation. Perform numerical simulation to establish a two-dimensional geological forward model and a parameter inversion model for the groundwater oscillation response under oscillatory pumping conditions.

[0046] The parameter inversion model uses the Ensemble Iterative Smoother (IES) algorithm to calculate parameter values for the inversion of two-dimensional heterogeneous parameters in the detection area. Under the data assimilation framework of the Ensemble Iterative Smoother (IES), an initialization optimization scheme based on available observation data is proposed to improve the estimation accuracy of soil moisture and hydraulic parameters. The Ensemble Iterative Smoother (IES) is a global data assimilation method that can comprehensively consider the information of all observation data, thus enabling effective parameter estimation in a multi-observation environment.

[0047] In the Ensemble Iterative Smoother algorithm, the observation vector of each ensemble member is denoted as , where j = 1, 2, …, Ne represents the ensemble number. In this paper, it is assumed that the ensemble size Ne = 100, is the carrier of the true observation data; while has a dimension of Nd, representing the total number of all available observations.

[0048] In each iteration , the model parameter vector can be updated by combining the observation data with the prediction results, thereby continuously optimizing the parameter estimation.

[0049] (19) In the formula, represents the initial guess or estimated value of parameter j at iteration ; while the updated estimated value in iteration +1 is ; is the vector of predicted data at iteration , is the vector of observation data, is the Kalman gain, and its calculation formula is: (20) where represents the cross-covariance matrix between the parameter vector and the predicted data vector ; represents the self-covariance matrix of the predicted data vector with a dimension of Nd×Nd. represents the measurement error covariance matrix. is the dynamic stability multiplier (the prior value is 10), diag( ) is related to ​​A diagonal matrix with the same diagonal elements. Mathematically, the dynamic stabilizer is determined by the Levenberg-Marquardt method, which helps to switch the solution between the Gauss-Newton solution method and the steepest descent method.

[0050] Cross-covariance matrix It can be described by the statistics of the parameters and the model state variables: (21) In the formula, and represent the parameter at the position with the set number j during the simulation process, and the state variable at the frequency . The horizontal line above the letter represents the mean value of the set.

[0051] The oscillation frequency of the stimulus source can be determined through the frequency selection scheme. The optimal frequency of oscillatory hydraulic tomography is selected. According to formula (18) and different observation distances and the average value of the parameter D, the three optimal observation frequencies are: = 0.0062, 0.0016, and 0.0007 [T -1 . Using the observed inversion of the amplitudes of the three frequencies and 100 random lnT fields, the reference lnT field is inverted.

[0052] The example is set as follows: The geometric mean hydraulic conductivity (T) and the specific storage (S) are 1.0 × 10 -5 [L 2 / T] and 5.0 × 10 -6 [-] respectively. Through the spectral method, 100 random lnT fields are generated, and the variance of their logarithm (lnT) is 1, and the correlation length is 30 [L]. It should be noted that the random field is limited to the central area of the domain (100 [L] × 100 [L]), and the value of T in the boundary area is a constant value. Figure 5 For the embodiment of the present invention, the mean value and variance of the initial sample lnT and the distribution diagram of the reference field lnT are set. Among them, (a) in Figure 5 is the mean value diagram of the initial sample lnT, (b) in Figure 5 is the variance diagram of the initial sample lnT, and (c) in Figure 5 is the distribution diagram of the reference field lnT.

[0053] The inversion results are as Figure 6 shown, where Figure 6 (a) in Figure 6 is the result diagram of the inverted lnT, and Figure 6Among them, (c) is the variance diagram of lnT of the samples after the inversion iteration is completed. Figure 6 Among them, (d) is the comparison diagram of the reference lnT and the lnT obtained by inversion. The results show that the lnT obtained by inversion reflects that by using the amplitude data of the three optimal frequencies, efficient tomographic scanning of the permeability coefficient of the underground aquifer and the distribution of the underground hydraulic conductivity field can be carried out.

[0054] In summary, the embodiment of the present invention first uses the sensitivity equation method. From the perspective of sensitivity analysis, according to the influence of parameters on the observed data, the optimal observation frequency is selected. The proposed frequency selection scheme has significant physical significance and interpretability compared with the previous methods. The method can obtain the optimal frequency selection scheme according to the hydraulic diffusion coefficient of the simple geological structure and the distance between the observation well and the pumping well, and has better applicability in practical engineering.

[0055] According to the second aspect of the present invention, there is provided an optimal signal source frequency selection system based on an oscillating hydraulic tomography test, including: A sensitivity equation construction module, configured to establish a sensitivity equation of the head observation to the hydraulic conductivity coefficient in the case of deriving oscillating pumping. An equation visualization module, configured to analyze the sensitivity degree of the observed head to the hydraulic conductivity coefficient at different positions when pumping at different frequencies in the case of visualizing two-dimensional oscillating pumping. An optimal frequency selection module, configured to convert the sensitivity equation of the hydraulic conductivity coefficient into a dimensionless form, propose a frequency selection scheme based on the observation position and the hydraulic diffusion coefficient, and determine the optimal frequency according to the dimensionless form of the sensitivity equation. A model parameter inversion module, configured to establish a two-dimensional geological forward model and a parameter inversion model for the groundwater oscillation response in the case of oscillating pumping. According to the hydraulic diffusion coefficient, use the established two-dimensional geological forward model to perform forward inversion of the two-dimensional heterogeneous parameters of the detection area, and use the established parameter inversion model to perform parameter inversion according to the determined optimal frequency.

[0056] It can be understood that an optimal signal source frequency selection system based on an oscillating hydraulic tomography test provided by the present invention corresponds to an optimal signal source frequency selection method based on an oscillating hydraulic tomography test provided in the foregoing embodiment. The relevant technical features of an optimal signal source frequency selection system based on an oscillating hydraulic tomography test can refer to the relevant technical features of an optimal signal source frequency selection method based on an oscillating hydraulic tomography test, and will not be elaborated herein.

[0057] Although the preferred embodiments of the present invention have been described, additional changes and modifications can be made by those skilled in the art once they learn the basic inventive concept. Therefore, the appended claims are intended to be construed to include the preferred embodiments as well as all changes and modifications falling within the scope of the present invention.

[0058] Finally, it should be noted that the above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention. Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features. These modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the various embodiments of the present invention.

Claims

1. An optimal signal source frequency selection method based on oscillatory hydraulic tomography tests, characterized in that The method includes: Under the condition of induced oscillatory pumping, establishing a sensitivity equation of the head observation to the hydraulic conductivity; Based on the sensitivity equation, under the condition of visual two-dimensional oscillatory pumping, analyzing the sensitivity degree of the observed head to the hydraulic conductivity at different positions when pumping at different frequencies; Converting the sensitivity equation of the hydraulic conductivity into a dimensionless form, proposing a frequency selection scheme based on the dimensionless distance and the hydraulic diffusivity, and determining the optimal frequency according to the dimensionless form of the sensitivity equation; Establishing a two-dimensional geological forward model and a parameter inversion model for the groundwater oscillatory response under the condition of oscillatory pumping. According to the hydraulic diffusivity, using the established two-dimensional geological forward model to perform forward simulation of the two-dimensional heterogeneous parameters in the detection area, and according to the determined optimal frequency, using the established parameter inversion model to perform parameter inversion.

2. The optimal signal source frequency selection method based on the oscillatory hydraulic tomography test according to claim 1, wherein The expression form of establishing the sensitivity equation of the head observation to the hydraulic conductivity under the condition of induced oscillatory pumping is: In the formula, represents the response function Q at position caused by oscillatory pumping with amplitude and oscillation frequency at position to the complex sensitivity function of the hydraulic conductivity located at including the sensitivity of the amplitude and phase shift of the hydraulic head to the hydraulic conductivity; T is the hydraulic conductivity in space, and D is the hydraulic diffusivity in space. is the derivative of the modified Kelvin function of the second kind of order zero; represents the pumping location and the parameter location of the dimensionless distance; represents the observed head location and the parameter location of the dimensionless distance; assuming the parameter location is , is expressed as the spatial gradient term, represents the pumping location and the parameter location of the distance; represents the observed head location and the parameter location of the distance.

3. The optimal signal source frequency selection method based on the oscillating hydraulic tomography test according to claim 1, wherein Based on the sensitivity equation, under the condition of visual two-dimensional oscillatory pumping, analyzing the sensitivity degree of the observed head to the hydraulic conductivity at different positions when pumping at different frequencies includes: Transforming the sensitivity equation to extract the sensitivity equations of the head amplitude and the head phase at the observation position to the hydraulic conductivity respectively.

4. The optimal signal source frequency selection method based on the oscillatory hydraulic tomography test according to claim 1, characterized in that The dimensionless form of the sensitivity equation is expressed as: In the formula: represents the dimensionless sensitivity function of the hydraulic conductivity T at the position where the amplitude is for the dimensionless distance, is the dimensionless distance , where r is the distance between the observation well and the pumping well, ω is the frequency, D is the hydraulic diffusivity; Ker 1 and Kei 1 are the real and imaginary parts of the second-kind first-order modified Kelvin function.

5. The optimal signal source frequency selection method based on the oscillatory hydraulic tomography test according to claim 1, characterized in that, Converting the sensitivity equation of the hydraulic conductivity into a dimensionless form and proposing a frequency selection scheme based on the dimensionless distance and the hydraulic diffusivity includes: Setting the second derivative of the dimensionless form of the sensitivity equation to zero to find the position of the dimensionless frequency peak. When the optimal frequency is combined with the optimal distance and the dimensionless distance, it is the selection scheme of the stimulation source frequency.

6. The optimal signal source frequency selection method based on the oscillatory hydraulic tomography test according to claim 1, wherein Determining the optimal frequency according to the dimensionless form of the sensitivity equation includes: Determining the pumping frequency of the pumping well according to the average hydraulic diffusivity of the site and the distance of the observation well; Based on the distance between the observation well and the pumping well and the average hydraulic parameters, according to the dimensionless form of the sensitivity equation, selecting the frequency with the maximum information of the heterogeneous parameters as the optimal frequency.

7. The optimal signal source frequency selection method based on the oscillating hydraulic tomography test according to claim 6, characterized in that, The calculation formula of the optimal frequency is expressed as follows: In the formula, represents the optimal frequency, r is the distance from the observation well to the pumping well, and D represents the hydraulic diffusion coefficient.

8. The optimal signal source frequency selection method based on the oscillatory hydraulic tomography test according to claim 1, wherein The parameter inversion model calculates the parameter values by using the ensemble iterative smoothing algorithm according to the head response data, and performs the inversion of the two-dimensional heterogeneous parameters in the detection area.

9. The optimal signal source frequency selection method based on the oscillatory hydraulic tomography test according to claim 1, wherein Performing parameter inversion by using the established parameter inversion model according to the determined optimal frequency includes: Determining the oscillatory frequency of the stimulation source through the frequency selection scheme, and according to the selected optimal frequency, using the amplitude data of the optimal frequency to perform tomographic scanning of the permeability coefficient of the underground aquifer and the distribution of the underground hydraulic conductivity field.

10. An optimal signal source frequency selection system based on an oscillatory hydraulic tomography test, characterized in that, Including: A sensitivity equation construction module for establishing a sensitivity equation of the head observation to the hydraulic conductivity under the condition of induced oscillatory pumping; An equation visualization module for analyzing the sensitivity degree of the observed head to the hydraulic conductivity at different positions when pumping at different frequencies under the condition of visual two-dimensional oscillatory pumping; An optimal frequency selection module for converting the sensitivity equation of the hydraulic conductivity into a dimensionless form, proposing a frequency selection scheme based on the observation position and the hydraulic diffusivity, and determining the optimal frequency according to the dimensionless form of the sensitivity equation; The model parameter inversion module is used to establish a two-dimensional geological forward model and a parameter inversion model for the groundwater oscillation response under oscillatory pumping. According to the hydraulic diffusivity, the two-dimensional heterogeneous parameters of the detection area are forward modeled using the established two-dimensional geological forward model. According to the determined optimal frequency, parameter inversion is performed using the established parameter inversion model.

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