CN115371642A - Geological statistics inversion method for hydrogeological parameter dynamic change characteristics - Google Patents
Geological statistics inversion method for hydrogeological parameter dynamic change characteristics Download PDFInfo
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Abstract
The invention discloses a geologic statistics inversion method for hydrogeological parameter dynamic change characteristics, which comprises the following steps: establishing a numerical model of dynamic change of target hydrogeological parameters along with time, and acquiring time series observation data according to different model types; based on the principle that the dynamic change of the model parameters on the time domain is equivalent to the one-dimensional heterogeneity of the model parameters on the space domain, coupling the numerical model with a geostatistical inversion program; and (4) inverting the target hydrogeological parameters by using the time sequence observation data, and acquiring the dynamic change track of the target hydrogeological parameters at one time. The geological statistics inversion method for the dynamic change characteristics of the hydrogeological parameters, provided by the invention, has the advantages of rapidness and high precision, and is suitable for determining the hydrogeological parameters under the actual condition.
Description
Technical Field
The invention relates to a geologic statistics inversion method for dynamic change characteristics of hydrogeological parameters, and belongs to the technical field of hydrogeological parameter inversion.
Background
The traditional idea of obtaining the dynamic change characteristics of hydrogeological parameters is to divide the whole inversion window into a plurality of continuous or partially overlapped relatively small-scale inversion windows, consider that the parameters do not change along with time in each inversion window, and perform inversion identification on the parameters according to observation data in each inversion window. And moving to the next inversion window when the inversion of the current window is finished until the inversion of all windows is finished. Although the traditional 'moving window' type continuous inversion method is successful in inverting and identifying the dynamic change characteristics of the hydrogeological parameters, the traditional 'moving window' type continuous inversion method has the defects of time consumption in calculation, difficulty in identifying the sudden change characteristics of the parameters and strict requirements on setting of the window length and the advancing step length, and is difficult to apply to the determination of the hydrogeological parameters under the actual condition in a large scale.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, provides a geological statistics inversion method of the dynamic change characteristics of hydrogeological parameters, has the advantages of rapidness and high precision, and is suitable for determining hydrogeological parameters under the actual condition in a large scale.
In order to solve the above technical problems, the present invention proposes a principle that a dynamic change of a model parameter in a time domain is equivalent to a one-dimensional heterogeneity of the model parameter in the space domain, and the specific explanation is as follows:
the dynamic change of the model parameters in the time domain is equivalent to the one-dimensional heterogeneity of the model parameters in the space domain under the high-dimensional parameter inversion optimization scene. That is, when the continuous dynamic variation process of the parameters is discrete in the time domain (i.e. a discrete column vector with a sufficiently large size is used for approximation, each element in the vector represents the model parameter in each minute time period), the dynamic variation process of the model parameters in the time domain based on the observation data at different time points and the heterogeneity of the model parameters in the one-dimensional space domain based on the observation data at different positions are the same in the implementation path.
The technical scheme adopted by the invention is as follows:
a geologic statistics inversion method for hydrogeological parameter dynamic change characteristics comprises the following steps:
establishing a numerical model of dynamic change of target hydrogeological parameters along with time, and acquiring time series observation data according to different model types;
based on the principle that the dynamic change of the model parameters on the time domain is equivalent to the one-dimensional heterogeneity of the model parameters on the space domain, coupling the numerical model with a geostatistical inversion program;
and (4) inverting the target hydrogeological parameters by using the time sequence observation data, and acquiring the dynamic change track of the target hydrogeological parameters at one time.
The time domain geostatistical inversion method comprises the following steps:
establishing an objective function of an optimization problem:
L＝[F obs -F(s)] T R -1 [F obs -F(s)]+(s-Xβ) T Q -1 (s-Xβ) (8)
wherein L represents an objective function, F obs (nx1) is a column vector consisting of observed values at different moments, s (mx 1) is a column vector of dynamic parameters to be optimized, n is the total amount of observed data, m is the total number of discrete time steps, F(s) is a simulation value obtained by a positive algorithm,
for the covariance matrix of the observed errors, it is assumed that each observed error is white noise and independent of each other, so R is a diagonal matrix and the elements on the diagonal are equal to the variance of the white noise
I n×n Is an identity matrix; x is a matrix of one (2 m X2) expressed as
Wherein, X 1 And X 2 Is a column vector of (m × 1), the elements are all 1,
β (2 × 1) is the mean of the prior dynamic parameter s, Q (2 m × 2 m) is the correlation matrix of the prior dynamic parameter s in the time domain, and is described by the exponential type variation function widely used in groundwater research:
wherein the exponential-type variation function Q involvesIs the variance of two prior structural parameters
And
and correlation length λ lnK And λ lnSs ；
And obtaining the optimal solution of the dynamic parameter s in an iterative solution mode based on the objective function of the optimization problem.
The iterative solution specifically includes:
let the estimated value after k iterations be s k Then the provisional estimate after the k +1 th iteration
Comprises the following steps:
wherein, X eta k+1 And
respectively represent
The mean value and the disturbance term of (c),
is s is k Analog value of F(s) k ) To s k Is calculated by means of a numerical difference, in particular
Wherein the content of the first and second substances,
is a vector s k A slight variation of the ith element in (b),
η k+1 (2X 1) and ξ k+1 (nx1) solving by coordinating kriging equations as follows
In order to accelerate the process of parameter inversion, a linear search method is used for the temporary estimation value after k iterations
Further optimization, i.e.
In the formula s k+1 Representing the final estimated value after the k iteration, wherein x is a parameter used by a linear search algorithm and takes a value of 0-1, and after finite iterations, when two adjacent iterations s k Or the difference of the objective function L is within the allowable range, the iteration process is terminated, and the final dynamic parameter estimation value is s = s k 。
The dynamic parameters to be optimized are permeability coefficient and water storage rate, and logarithm is taken respectively.
The posterior covariance matrix of the parameter estimate s is
Wherein, COV pos The diagonal element of (a) is the posterior variance of s, and the 95% confidence interval of s can be calculated as [ -1.96 σ(s) according to the Gaussian normality assumption]Where σ is the standard deviation of s.
The numerical model includes but is not limited to a weak permeable layer water release deformation model, a river-groundwater hydrodynamic response model and a groundwater recharge aquifer blockage model.
The invention has the beneficial effects that: the geological statistics inversion method for the dynamic change characteristics of the hydrogeological parameters breaks through the conventional thought, provides the concept that the dynamic change of the model parameters in the time domain and the one-dimensional heterogeneity of the model parameters in the space domain are equivalent under the situation of high-dimensional parameter inversion optimization, and has extremely strong innovation; the dynamic trajectory of the hydrogeological parameters is obtained by utilizing time domain geological statistical inversion, the calculation result of the traditional method is closer to the reality, the theory is strict, and the parameter determination precision is high; the dynamic trajectory of the hydrogeological parameters in the observation time period can be obtained only by one inversion, and the time required by calculation is greatly reduced; the method can be used in different underground water systems to obtain the parameter dynamic track of each hydrogeological unit, and has wide application range.
Drawings
FIG. 1 is a logic diagram of equivalent time domain parameter dynamic variation and one-dimensional spatial domain heterogeneity according to the present invention;
FIG. 2 is a graph comparing a dynamic trajectory of a bed permeability coefficient and a river level obtained by inversion in embodiment 1 of the present invention;
FIG. 3 is a graph showing a comparison of the river level, the observed aquifer level, and the simulated aquifer level in example 1 of the present invention;
FIG. 4 is a trace diagram of the dynamic change of permeability coefficient of the aquifer inverted in embodiment 2 of the present invention;
FIG. 5 is a water storage capacity dynamic variation trace of the aquifer inverted in embodiment 2 of the present invention.
Detailed Description
The present invention is further described with reference to the accompanying drawings, and the following examples are only for clearly illustrating the technical solutions of the present invention, and should not be taken as limiting the scope of the present invention.
Detailed description of the preferred embodiment 1
As shown in fig. 1, the present invention provides a geostatistical inversion method for the dynamic variation characteristics of hydrogeological parameters, which breaks through the conventional thinking and provides a concept that the dynamic variation of model parameters in a time domain and the one-dimensional heterogeneity of model parameters in a space domain are equivalent under the situation of high-dimensional parametric inversion optimization, specifically, when a parameter continuous dynamic variation process is discretized in a time domain (i.e., a discrete column vector with a sufficiently large size is used for approximation, each element in the vector represents a model parameter in each tiny period), the dynamic variation process of the model parameters in the time domain is characterized based on the observation data of different time points, and the heterogeneity of the model parameters in the one-dimensional space domain is characterized by the same concept on the implementation path based on the observation data of different positions.
The time domain geology statistics inversion method is characterized in that the geology statistics inversion method is applied to solving the problem of model parameter dynamic change in the time domain, and the hydrogeology parameters are inverted by being coupled with numerical simulation and time sequence observation data, so that the dynamic change track of the target hydrogeology parameters is obtained at one time. The geostatistical inversion method comprises but is not limited to a quasi-linear geostatistical inversion method and all geostatistical inversion methods which can be applied to determining the dynamic change track of the time domain model parameter.
Taking a Quasi-linear geostatistical inversion algorithm (QLGA) proposed based on bayes inference theorem as an example, the hydrogeological parameter dynamic trajectory is identified by inversion. The specific implementation path of the time domain geological statistics inversion method is explained by the hydrogeological problem that the permeability coefficient and the water storage rate of the weakly permeable stratum dynamically change along with the water release deformation process.
Firstly, establishing an objective function of an optimization problem:
L＝[F obs -F(s)] T R -1 [F obs -F(s)]+(s-Xβ) T Q -1 (s-Xβ) (15)
wherein L represents the value of the objective function, F obs (nx1) is a column vector consisting of observed values at different moments, and s (mx 1) is a column vector of dynamic parameters to be optimized, wherein the dynamic parameters are permeability coefficients and water storage rates, and logarithm is taken, so that the stability of a sensitivity matrix is ensured during calculation; n is the total amount of observed data, and m is the total number of discrete time steps; f(s) correcting the analog value obtained by the algorithm;
for the covariance matrix of the observed errors, it is assumed that each observed error is white noise and independent of each other, so R is a diagonal matrix and the elements on the diagonal are equal to the variance of the white noise
I n×n Is an identity matrix; x is a matrix of one (2 m X2) expressed as
Wherein, X 1 And X 2 Is a column vector of (m × 1), the elements are all 1; β (2 × 1) is the mean of the a priori dynamic parameters s; q (2 m × 2 m) is a correlation matrix of the prior dynamic parameter s in the time domain, and can be described by an exponential type variation function widely used in groundwater research:
wherein, the two prior structure parameters related to the exponential type variation function Q are variance
And correlation length (λ) lnK 、
)。
Based on QLGA, the optimal solution of s can be obtained by means of iterative solution. Let the estimated value after k iterations be s k Then the temporary estimate after the k +1 th iteration is
Wherein, the two terms on the right side of the formula (4) respectively represent
The mean and disturbance terms of;
is s is k Analog value of F(s) k ) To s k Can be calculated by means of numerical differentiation, in particular
Wherein the content of the first and second substances,
is a vector s k Minor variations of the ith element in (1). Eta in formula (4) k+1 (2X 1) and ξ k+1 (nx 1) may be solved by the cooperative kriging equation as follows
In order to accelerate the process of parameter inversion, a linear search method is used for the temporary estimation value after k iterations
Further optimization, i.e.
In the formula s k+1 And (3) representing the final estimation value after the kth iteration, wherein x is a parameter used by a linear search algorithm and takes a value of 0-1. Subject to limitationAfter a sub-iteration, when two adjacent iterations s k Or the difference of the objective function L is within the allowable range, the iterative process is terminated. The final parameter estimate is s = s k And a posterior covariance matrix of
COV pos The diagonal element of (a) is the posterior variance of s. The 95% confidence interval for s can be calculated as [ -1.96 σ(s) according to the Gaussian normality hypothesis]Where σ is the standard deviation of s.
The numerical model in this embodiment is a river-groundwater dynamic response model:
(1) Introduction to the ground
Next, switzerland is used
And verifying the effectiveness and the practicability of the method by using aquifer water level data recorded by an observation well P2.2 of the river bank. Due to the influence of the activities of the human being,
the water level fluctuation of the river is about 1m on weekdays and about 0.4m on weekends. Permeability coefficient K of diving aquifer adjacent to river bed a Has a water storage coefficient S of 51.84m/day a Is 0.2; water storage coefficient of riverbed S sb Is set to be 2 x 10 -6 Coefficient of permeability K sb Changes occur over time.
(2) Parameter determination
Firstly, according to the field type, judging that the model is suitable for a river water-underground water dynamic response model. Considering horizontal semi-infinite river-aquifer system, supposing that the river water level change is within one tenth of the thickness of the aquifer, the equation for controlling the movement of the groundwater of the river-aquifer system is obtained by the Buxiesque formula and the Cubuyi formula
Wherein, K sb ＝K sb (t) is the riverbed permeability coefficient which changes with time, h sb ＝h sb (x, t) is the riverbed water level, h a ＝h a (x, t) is the aquifer water level; s sb And S a The water storage coefficients of the riverbed and the aquifer are respectively. K a Is the permeability coefficient of the aquifer, which does not change with time in the model; m is the thickness of the aquifer; b is the thickness of the riverbed.
Let the water level of the riverbed and the water level of the aquifer be the same and h under the initial condition 0 Is provided with
h a (x,0)＝h sb (x,0)＝h 0 (10)
Assuming that the dynamic change trajectory of the river level at the x =0 position is F (t), it can be obtained
h sb (x,t)| x＝0 ＝F(t) (11)
When x = ∞ time
And then, solving the problem by adopting a one-dimensional finite element method, and carrying out next step of inversion calculation after obtaining a numerical solution of the water head space-time distribution of the aquifer.
Then, lnK is set sb The mean and variance of (a) are 0.26 and 0.08 respectively, and the dynamic change trajectory of the riverbed permeability coefficient is determined by the time domain geostatistical inversion method, and the result is shown in fig. 2.
To quantitatively evaluate the fitting degree of the inversion simulation result and the measured data, the mean square error of the inversion simulation deformation is calculated:
wherein n is the total number of observed data,
is the measured value of the water level of the ith aquifer,
is an analog value of the aquifer water level. The smaller the mean square error MSE, the higher the degree of fit. As shown in FIG. 3, the MSE value based on QLGA is only 2.1 × 10 -3 m 2 The method fully embodies the characteristic that the parameter track obtained by inversion of the method has high accuracy.
Specific example 2
This example is the same as the specific example 1, except that the numerical model in this example is a model of deformation by water release from a permeable weakly-permeable layer:
(1) Introduction to the ground
Taking the aquifer system at the middle east part of the sea district hierarchical logo group F16 as an example, the hierarchical logo group F16 is built in 10 months in 1980, and the accumulated settlement deformation of the lower confined aquifer system can be calculated by comparing the monitoring data of the hierarchical logos. And an observation hole PD15-1 is arranged near the F16 and is used for monitoring the water level dynamic of the lower pressure-bearing aquifer system. In this aquifer system, a plurality of aquifer lenses are equated to a single aquifer, the equivalent thickness of which is less than the maximum thickness of a single lens. In addition, the water storage rate S of the confined aquifer s Is 2.64X 10 -6 m -1 。
(2) Parameter determination
And judging the water release deformation model of the permeable aquitard suitable for the place according to the place type. Considering a one-dimensional cross-flow aquifer system, assuming that the underground water motion meets Darcy's law, the control equation of the hydraulic response of the homogeneous aquifer is
Initial conditions and boundary conditions are
h(z,t)| t＝0 ＝h i (z) (16)
h(z,t)| z＝0 ＝h 0 (t) (17)
h(z,t)| z＝b ＝h b (t) (15)
Wherein h (z, t) is the head at position z at time t, K (t) and S s (t) is the dynamic permeability coefficient and water storage rate (average value) of the weakly permeable layer, h i (z) initial water level distribution in the weakly permeable layer, h 0 (t)、h b And (t) dynamic water heads of the water-permeable aquifer which is covered under and covered on the weakly permeable aquifer respectively.
And then, solving the initial value problem by adopting a one-dimensional finite element method, and further calculating the accumulated deformation of the aquitard after obtaining a numerical solution of the water head space-time distribution of the aquitard. According to the principle of water quantity equilibrium, the cumulative deformation F of the water-permeable thin layer is equal to the cumulative water release quantity per unit area, so that the water quantity
Wherein M is the total number of evenly divided cells, Δ z is the length of each cell,
is the average head value of the mth cell at time t.
Thereafter, K and S for each period of the initial state s Is uniformly set to 10 -3 m/year and 10-3m -1 (ii) a Dynamic lnK and lnS s Setting the variance and the correlation length of the variation function as 1.0 and 1.0year, and obtaining K and S once by a time domain geostatistical inversion method s The dynamic variation trajectory of (2). The results are shown in fig. 4 and 5, respectively. To quantitatively evaluate the fitting degree of the inversion simulation result and the measured data, the mean square error of the inversion simulation deformation is calculated:
wherein n is the total number of deformation observation data, F i obs Is the ith weakly permeable stratum accumulated deformation measured value, F i simu Is an analog value of the deformation. The smaller the mean square error MSE, the higher the degree of fit and vice versa. In this example, the MSE value based on QLGA is only 3.1 × 10 -3 mm 2 The effectiveness and accuracy of the method are proved again.
Through the two specific embodiments, the geological statistics inversion method for the dynamic change characteristics of the hydrogeological parameters is novel in theory, strict in logic, high in applicability, accurate in calculation result, short in calculation time and suitable for popularization and application, and has important application values in the weakly permeable stratum, the riverbed and related fields.
The above is only a preferred embodiment of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention, and such modifications and adaptations are intended to be within the scope of the invention.
Claims (7)
1. A geologic statistics inversion method of hydrogeological parameter dynamic change characteristics is characterized in that: the method comprises the following steps:
establishing a numerical model of dynamic change of the target hydrogeological parameters along with time, and acquiring time series observation data according to different numerical model types;
based on the principle that the dynamic change of the model parameters on the time domain is equivalent to the one-dimensional heterogeneity of the model parameters on the space domain, coupling the numerical model with a geostatistical inversion program;
and inputting the time series observation data into a coupled geological statistics inversion program to invert the target hydrogeological parameters, and acquiring the dynamic change track of the target hydrogeological parameters at one time.
2. A method of geostatistical inversion of a characteristic of dynamic variation of hydrogeological parameters according to claim 1, characterized in that: the inversion of the target hydrogeological parameters comprises the following steps:
establishing an objective function of an optimization problem:
L＝[F obs -F(s)] T R -1 [F obs -F(s)]+(s-Xβ) T Q -1 (s-Xβ) (1)
wherein L represents an objective function, F obs N x 1 column vectors composed of observed values at different moments, s is m x 1 column vector of dynamic parameter to be optimized, n is total observed data, m is total discrete time step, F(s) is analog value obtained by positive algorithm,
is an n × n covariance matrix of observation errors, where each observation error is assumed to be white noise and independent of each other, so R is a diagonal matrix and the elements on the diagonal are equal to the variance of the white noise
I n×n Is an identity matrix; x is a 2m X2 matrix expressed as
Wherein, X 1 And X 2 Is a column vector of (m × 1), elements are all 1, β (2 × 1) is a mean value of the prior dynamic parameter s, Q (2 m × 2 m) is a correlation matrix of the prior dynamic parameter s in a time domain, and is described by an exponential type variation function widely used in groundwater research:
wherein, the two prior structure parameters related to the exponential type variation function Q are variance
And
and correlation length λ lnK And
and obtaining the optimal solution of the dynamic parameter s in an iterative solution mode based on the objective function of the optimization problem.
3. A method of geostatistical inversion of a characteristic of dynamic variation of hydrogeological parameters according to claim 2, characterized in that: the iterative solution specifically includes:
let the estimated value after k iterations be s k Then the provisional estimate after the k +1 th iteration
Comprises the following steps:
wherein, X eta k+1 And
respectively represent
The mean value and the disturbance term of (c),
is s is k Analog value of F(s) k ) To s k Is calculated by means of a numerical difference, in particular
Wherein the content of the first and second substances,
is a vector s k A slight variation of the ith element in (b),
η k+1 (2X 1) and xi k+1 (n × 1) solving by coordinating kriging equations as follows
In order to accelerate the process of parameter inversion, a linear search method is used for the temporary estimation value after k iterations
Further optimization, i.e.
In the formula s k+1 Representing the final estimated value after the k iteration, wherein x is a parameter used by a linear search algorithm and takes a value of 0-1, and after finite iterations, when two adjacent iterations s k Or the difference of the objective function L is within the allowable range, the iteration process is terminated, and the final dynamic parameter estimation value is s = s k 。
4. A method of geostatistical inversion of a characteristic of dynamic variation of hydrogeological parameters according to claim 2, characterized in that: according to different numerical models, the dynamic parameters to be optimized include, but are not limited to, permeability coefficient and water storage rate.
5. A method of geostatistical inversion of a characteristic of dynamic variation of hydrogeological parameters according to claim 3, characterized in that: the posterior covariance matrix of the parameter estimate s is
Wherein, COV pos The diagonal element of (a) is the posterior variance of s, and the 95% confidence interval of s can be calculated as [ -1.96 σ(s) according to the Gaussian normality assumption]Wherein σ is the standard deviation of s.
6. The geostatistical inversion method for the dynamic variation characteristics of hydrogeological parameters according to claim 1, characterized in that: the numerical models include, but are not limited to, a aquifer water release deformation model, a river-groundwater hydrodynamic response model, and an aquifer plugging model for groundwater recharge.
7. The geostatistical inversion method for the dynamic variation characteristics of hydrogeological parameters according to claim 1, characterized in that: according to different model types, the time series observation data comprises but is not limited to surface water level observation data and underground aquifer water level observation data.
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| CN115935626A (en) * | 2022-11-25 | 2023-04-07 | 河南大学 | Inversion method of river water-underground water vertical transient interaction water flux based on VAE-PINN algorithm |
| CN117216455A (en) * | 2023-11-09 | 2023-12-12 | 中国地震局地质研究所 | Method and device for monitoring vertical flow velocity of underground water |
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| CN115935626A (en) * | 2022-11-25 | 2023-04-07 | 河南大学 | Inversion method of river water-underground water vertical transient interaction water flux based on VAE-PINN algorithm |
| CN115935626B (en) * | 2022-11-25 | 2023-09-08 | 河南大学 | Inversion method of river water-underground water vertical transient interaction water flow |
| CN117216455A (en) * | 2023-11-09 | 2023-12-12 | 中国地震局地质研究所 | Method and device for monitoring vertical flow velocity of underground water |
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