Application of Proper Orthogonal Decomposition (POD) to inverse problems in saturated groundwater flow

We develop a new Proper Orthogonal Decomposition (POD) reduced order model for saturated groundwater flow, and apply that model to an inverse problem for the hydraulic conductivity field. We use sensitivities as the POD basis. We compare the output when the optimizer uses the reduced order model against results obtained with a full PDE based model. The solutions generated using the POD reduced model are comparable in residual norm to the solutions formed using only the full-scale model. The material parameters are similarly comparable. The time to solution when using the reduced model is reduced by at least an order of magnitude, as are the number of calls to the full model.     

Pilot points method incorporating prior information for solving the groundwater flow inverse problem

The pilot points method is often used in nonlinear geostatistical calibration. The method consists of estimating the values of the hydraulic properties at a set of arbitrary (pilot) points so as to best fit the aquifer response as measured by available indirect observations (i.e., heads or drawdowns). Though this method remains general and appealing, no prior information of the hydraulic properties is usually included in the optimization process, which constrains the number of pilot points to ensure stability. In this paper, we present a modification of the pilot points method, including prior information in the optimization process by adding a plausibility term to the objective function to be minimized. This results from formulating the inverse problem in a maximum likelihood framework. The performance of the method is tested on a synthetic example. Results show that including the plausibility term improves the identification of heterogeneity. Furthermore, this term makes the inverse problem more stable and allows the use of larger number of pilot points, thus improving the identification of the heterogeneity as well. Therefore, the use of the plausibility term is recommended.     

Multi-level parameter structure identification for two-phase porous-media flow problems using flexible representations

We consider identification of absolute permeability (hydraulic conductivity) based on time series of pressure data in sparsely distributed wells for two-phase porous-media flow. For this problem, it is impossible to recover all details of the parameter function. On the other hand, a coarser, approximate recovery may be sufficient for many applications. We propose a novel solution approach, based on reparametrization, for such approximate identification of the parameter function. We use a nonlinear, composite representation, which is detached from the computational grid, allowing for a flexible representation of the parameter function at many resolution levels. This is utilized in a sequential multi-level estimation of the parameter function, starting at a coarse resolution, which is then gradually refined. The composite representation is designed to allow for smooth as well as sharp transitions between regions of nearly constant parameter value. Moreover, it facilitates the estimation also of the structure and smoothness of the parameter function itself. As a limiting case, the chosen representation is reduced to a zonation with implicit representation of the interior boundaries that is equivalent to a level-set representation. A motivation for the selected representation and the multi-level estimation is presented in terms of an analysis of sensitivity and nonlinearity. Numerical examples demonstrate identification of coarse-scale features of reference permeability distributions with varying degree of smoothness. Comparisons show how the multi-level strategy stabilize the identification and avoid local minima of the objective function compared to a single-level strategy.     

Knowledge,transparency, and refutability in groundwater models,an example from the Death Valley regional groundwater flow system

This work demonstrates how available knowledge can be used to build more transparent and refutable computer models of groundwater systems. The Death Valley regional groundwater flow system, which surrounds a proposed site for a high level nuclear waste repository of the United States of America, and the Nevada National Security Site (NNSS), where nuclear weapons were tested, is used to explore model adequacy, identify parameters important to (and informed by) observations, and identify existing old and potential new observations important to predictions. Model development is pursued using a set of fundamental questions addressed with carefully designed metrics. Critical methods include using a hydrogeologic model, managing model nonlinearity by designing models that are robust while maintaining realism, using error-based weighting to combine disparate types of data, and identifying important and unimportant parameters and observations and optimizing parameter values with computationally frugal schemes. The frugal schemes employed in this study require relatively few (10–1000 s), parallelizable model runs. This is beneficial because models able to approximate the complex site geology defensibly tend to have high computational cost. The issue of model defensibility is particularly important given the contentious political issues involved.     

Theoretical analysis and field evidence of reciprocity gaps during interference pumping tests

We address the question of the reciprocity of drawdowns observed during a sequence of interference pumping tests performed within a porous medium, in the context of different conceptual models of subsurface flow. We provide a generalization of the work by Bruggeman (1972) [5], extending his results obtained in 1972 for Darcian flows in an unbounded, heterogeneous porous medium. We then analyze reciprocity within a dual-continuum conceptualization, where the medium is viewed as being composed of two overlapping continua, representing the porous matrix and embedded fractures, respectively. We show theoretically, and demonstrate numerically, that only drawdowns associated with the fracture continuum display reciprocity under transient flow conditions. Conversely, non-flowing matrix and fracture continua display reciprocity behavior under steady-state conditions. We then provide field evidence of the insurgence of reciprocity gaps by analyzing interference test data from the karstic limestone aquifer of the Hydrogeological Experimental Site (HES) in Poitiers, France. On the basis of our theoretical results and experimental observations, we discuss different interpretations of the observed reciprocity gaps. These include (a) non-linear dependencies of local hydraulic parameters, (b) occurrence of internal boundaries within the domain, (c) inertial effects that develop through open conduits within the rock matrix, (d) modifications of the aquifer properties between subsequent pumping tests, and (e) significant contribution of the matrix pressure in monitored wells when the behavior of the aquifer is conceptualized by a dual-continuum approach.     

A quadratic element method for evaluating groundwater flow by the inversion of surface tilt with application to the Tono Area, Japan

By modifying a previous method with constant elements, we developed a quadratic element method for more accurately estimating groundwater flow by the inversion of tilt data. In this method: (1) a region of groundwater flow is divided into quadratic elements in which the change in groundwater volume per unit volume of rock (Δv) and the Skempton coefficient (B) vary in a quadratic manner with the coordinates, (2) the values of Δv are set to zero at the boundaries of the region of groundwater flow and (3) the sum of the squared second derivatives of Δv is adopted as a constraining condition that is weighted and added to the sum of the squared errors in tilt. First, analyses were performed for a flow model to determine the accuracy of this method for estimating groundwater flow and also to clarify the effect of the assumed size of a region of groundwater flow. These analyses showed that the quadratic element method proposed in this study gives a much better estimation of Δv than the constant element method and that a large region of groundwater flow should be assumed, rather than a small region, since the values of Δv at points outside of the actual region of groundwater flow are estimated to be nearly zero when a large region is assumed while these values are greatly overestimated when an excessively small region is assumed. Finally, the quadratic element method was applied to the site of the Mizunami Underground Research Laboratory in the Tono area, Japan. Inverse analyses were performed for tilt data measured by four tiltmeters with a resolution of 10⁻⁹ radians during the excavation of two shafts under the assumption that the rock mass is an isotropic and homogeneous half- space. The results showed that the method proposed in this study reproduced the tilt data very accurately. Thus, the distribution of Δv was estimated without sacrificing the reproducibility of the tilt data. The contour maps of B(1 + ν)Δv (ν: Poisson’s ratio) showed that the heterogeneous flow of groundwater occurred at the site and that groundwater volume decreased mainly in the area surrounded by two faults. The latter result is consistent with the finding obtained by previous investigations that these faults have low permeability in the direction perpendicular to the strike and may act as a flow barrier.     

Parameter-independent model reduction of transient groundwater flow models: Application to inverse problems

A new methodology is proposed for the development of parameter-independent reduced models for transient groundwater flow models. The model reduction technique is based on Galerkin projection of a highly discretized model onto a subspace spanned by a small number of optimally chosen basis functions. We propose two greedy algorithms that iteratively select optimal parameter sets and snapshot times between the parameter space and the time domain in order to generate snapshots. The snapshots are used to build the Galerkin projection matrix, which covers the entire parameter space in the full model. We then apply the reduced subspace model to solve two inverse problems: a deterministic inverse problem and a Bayesian inverse problem with a Markov Chain Monte Carlo (MCMC) method. The proposed methodology is validated with a conceptual one-dimensional groundwater flow model. We then apply the methodology to a basin-scale, conceptual aquifer in the Oristano plain of Sardinia, Italy. Using the methodology, the full model governed by 29,197 ordinary differential equations is reduced by two to three orders of magnitude, resulting in a drastic reduction in computational requirements.     

Multi-resolution adaptive modeling of groundwater flow and transport problems

Many groundwater flow and transport problems, especially those with sharp fronts, narrow transition zones, layers and fingers, require extensive computational resources. In this paper, we present a novel multi-resolution adaptive Fup approach to solve the above mentioned problems. Our numerical procedure is the Adaptive Fup Collocation Method (AFCM), based on Fup basis functions and designed through a method of lines (MOL). Fup basis functions are localized and infinitely differentiable functions with compact support and are related to more standard choices such as splines or wavelets. This method enables the adaptive multi-resolution approach to solve problems with different spatial and temporal scales with a desired level of accuracy using the entire family of Fup basis functions. In addition, the utilized collocation algorithm enables the mesh free approach with consistent velocity approximation and flux continuity due to properties of the Fup basis functions. The introduced numerical procedure was tested and verified by a few characteristic groundwater flow and transport problems, the Buckley–Leverett multiphase flow problem, the 1-D vertical density driven problem and the standard 2-D seawater intrusion benchmark–Henry problem. The results demonstrate that the method is robust and efficient particularly when describing sharp fronts and narrow transition zones changing in space and time.     

Stochastic analysis of spatial variability in unconfined groundwater flow

In this paper, spatial variability in steady one-dimensional unconfined groundwater flow in heterogeneous formations is investigated. An approach to deriving the variance of the hydraulic head is developed using the nonlinear filter theory. The nonlinear governing equation describing the one-dimensional unconfined groundwater flow is decomposed into three linear partial differential equations using the perturbation method. The linear and quadratic frequency response functions are obtained from the first- and second-order perturbation equations using the spectral method. Furthermore, under the assumption of the exponential covariance function of log hydraulic conductivity, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the linear system are derived. The results show that the variance derived herein is less than that of Gelhar (1977). The reason is that the log transmissivity is linearized in Gelhars work. In addition, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the quadratic system are derived as well. It is found that the correlation scale and the trend in mean of log hydraulic conductivity are important to the dimensionless variance ratio.     

Stochastic modelling of groundwater flow and solute transport in aquifers

This paper presents an introductory overview of recently developed stochastic theories for tackling spatial variability problems in predicting groundwater flow and solute transport. Advantages and limitations of the theories are discussed. Lastly, strategies based on the stochastic approaches to predict solute transport in aquifers are recommended.     

A BME solution of the inverse problem for saturated groundwater flow

In most real-world hydrogeologic situations, natural heterogeneity and measurement errors introduce major sources of uncertainty in the solution of the inverse problem. The Bayesian Maximum Entropy (BME) method of modern geostatistics offers an efficient solution to the inverse problem by first assimilating various physical knowledge bases (hydrologic laws, water table elevation data, uncertain hydraulic resistivity measurements, etc.) and then producing robust estimates of the subsurface variables across space. We present specific methods for implementing the BME conceptual framework to solve an inverse problem involving Darcys law for subsurface flow. We illustrate one of these methods in the case of a synthetic one-dimensional case study concerned with the estimation of hydraulic resistivity conditioned on soft data and hydraulic head measurements. The BME framework processes the physical knowledge contained in Darcys law and generates accurate estimates of hydraulic resistivity across space. The optimal distribution of hard and soft data needed to minimize the associated estimation error at a specified sampling cost is determined. This work was supported by grants from the National Institute of Environmental Health Sciences (Grant no. 5 P42 ES05948 and P30ES10126), the National Aeronautics and Space Administration (Grant no. 60-00RFQ041), the Army Research Office (Grant no. DAAG55-98-1-0289), and the National Science Foundation under Agreement No. DMS-0112069.     

Adaptive multi-scale parameterization for one-dimensional flow in unsaturated porous media

In the analysis of the unsaturated zone, one of the most challenging problems is to use inverse theory in the search for an optimal parameterization of the porous media. Adaptative multi-scale parameterization consists in solving the problem through successive approximations by refining the parameter at the next finer scale all over the domain and stopping the process when the refinement does not induce significant decrease of the objective function any more. In this context, the refinement indicators algorithm provides an adaptive parameterization technique that opens the degrees of freedom in an iterative way driven at first order by the model to locate the discontinuities of the sought parameters. We present a refinement indicators algorithm for adaptive multi-scale parameterization that is applicable to the estimation of multi-dimensional hydraulic parameters in unsaturated soil water flow. Numerical examples are presented which show the efficiency of the algorithm in case of noisy data and missing data.     

Sensitivity analysis of two-dimensional steady-state aquifer flow equations. Implications for groundwater flow model calibration and validation

This paper presents the analytical properties of the sensitivity of the two-dimensional, steady-state groundwater flow equation to the flow parameters and to the boundary conditions, based on the perturbation approach. These analytical properties are used to provide guidelines for model design, model calibration and monitoring network design. The sensitivity patterns are shown to depend on the nature of both the perturbed parameter and the variable investigated. Indeed, the sensitivity of the hydraulic head to the hydraulic conductivity extends mainly in the flow direction, while the sensitivity to the recharge spreads radially. Besides, the sensitivity of the flow longitudinal velocity to the hydraulic conductivity propagates in both the longitudinal and transverse directions, whereas the sensitivity of the flow transverse velocity propagates in the diagonal directions to the flow. The analytical results are confirmed by application examples on idealized and real-world simulations. These analytical findings allow some general rules to be established for model design, model calibration and monitoring network design. In particular, the optimal location of measurement points depends on the nature of the variable of interest. Measurement network design thus proves to be problem-dependent. Moreover, adequate monitoring well network design may allow to discriminate between the possible sources of error.     

Conditioning mean steady state flow on hydraulic head and conductivity through geostatistical inversion

Nonlocal moment equations allow one to render deterministically optimum predictions of flow in randomly heterogeneous media and to assess predictive uncertainty conditional on measured values of medium properties. We present a geostatistical inverse algorithm for steady-state flow that makes it possible to further condition such predictions and assessments on measured values of hydraulic head (and/or flux). Our algorithm is based on recursive finite-element approximations of exact first and second conditional moment equations. Hydraulic conductivity is parameterized via universal kriging based on unknown values at pilot points and (optionally) measured values at other discrete locations. Optimum unbiased inverse estimates of natural log hydraulic conductivity, head and flux are obtained by minimizing a residual criterion using the Levenberg-Marquardt algorithm. We illustrate the method for superimposed mean uniform and convergent flows in a bounded two-dimensional domain. Our examples illustrate how conductivity and head data act separately or jointly to reduce parameter estimation errors and model predictive uncertainty.This work is supported in part by NSF/ITR Grant EAR-0110289. The first author was additionally supported by scholarships from CONACYT and Instituto de Investigaciones Electricas of Mexico. Additional support was provided by the European Commission under Contract EVK1-CT-1999-00041 (W-SAHaRA-Stochastic Analysis of Well Head Protection and Risk Assessment).     

A study of the three-dimensional groundwater flow system in an upland area of Japan

A three-dimensional finite difference model was developed to study the groundwater flow system in an upland area bordering a lake. For a general perspective of the groundwater flow system, a steady state three-dimensional flow was employed. Having determined the flow net by using a three-dimensional model, the flow volumes under natural conditions have been used to establish the parameter values and for the analyses of flow patterns. Further, to study the effects of human impact and precipitation on groundwater flow conditions in a small area, a transient three-dimensional simulation was performed. Environmental tritium was used to trace the regional groundwater movement to verify the three-dimensional mathematical model. Results obtained using the three-dimensional mathematical model approach and tritium concentration analyses were in close agreement. The results demonstrated that the groundwater flow system should be analysed using a three-dimensional geometric concept of groundwater movement.     

The transition of flow patterns through critical stagnation points in two-dimensional groundwater flow

A flow pattern is characterized by aquifer features and the number, type, and distribution of stagnation points (locations where the discharge is zero). This article identifies a condition for transition of flow patterns in two-dimensional groundwater flow obeying Darcy’s law by examining changes in stagnation points, using the Taylor series expansion of the discharge vector. It is found that the three standard types of stagnation points (minimums, maximums, and saddle points) are completely characterized by the first-order term containing the discharge gradient tensor. However, when the determinant of the tensor becomes zero, stagnation points of other types characterized by higher-order terms come into existence. In this article, we call these zero-determinant stagnation points as critical stagnation points; they may emerge suddenly, split to a set of new stagnation points, or disappear from the flow, resulting in transitions of flow patterns. Examples of both transient and steady flows are used to illustrate the usefulness and significance of critical stagnation points.     

Conceptual and numerical models for groundwater flow in an arid inland river basin

The Heihe River Basin (HRB) is an inland watershed in northwest China with a total area of approximately 130,000 km², stretching from the Qilian Mountains in the south to the oases and agricultural fields in the middle and further to the Gobi desert in the north bordering Mongolia. As part of a major ecohydrological research initiative to provide a stronger scientific underpinning for sustainable water management in arid ecosystems, a regional‐scale integrated ecological and hydrological model is being developed, incorporating the knowledge based on the results of environmental isotope tracer analysis and the multiscale observation datasets. The first step in the model development effort is to construct and calibrate a groundwater flow model for the middle and lower HRB where the oases and vegetation along the Heihe river corridor are highly dependent on groundwater. In this study, the software tool ‘Arc Hydro Groundwater’ is used to build and visualize a hydrogeological data model for the HRB that links all relevant spatiotemporal hydrogeological data in a unified geodatabase within the ArcGIS environment. From the conceptual model, a regional‐scale groundwater flow model has been developed using MODFLOW‐2005. Critical considerations in developing the flow model include the representation of mountainous terrains and fluvial valleys by individual model layers, treatment of aquifer heterogeneities across multiple scales and selection of proper observation data and boundary conditions for model calibration. This paper discusses these issues in the context of the Heihe River Basin, but the results and insights from this study will have important implications for other large, regional groundwater modelling studies, especially in arid and semiarid inland river basins. Copyright © 2014 John Wiley & Sons, Ltd.     

A least-squares penalty method algorithm for inverse problems of steady-state aquifer models

Based on the generalized Gauss–Newton method, a new algorithm to minimize the objective function of the penalty method in (Bentley LR. Adv Wat Res 1993;14:137–48) for inverse problems of steady-state aquifer models is proposed. Through detailed analysis of the “built-in” but irregular weighting effects of the coefficient matrix on the residuals on the discrete governing equations, a so-called scaling matrix is introduced to improve the great irregular weighting effects of these residuals adaptively in every Gauss–Newton iteration. Numerical results demonstrate that if the scaling matrix equals the identity matrix (i.e., the irregular weighting effects of the coefficient matrix are not balanced), our algorithm does not perform well, e.g., the computation cost is higher than that of the traditional method, and what is worse is the calculations fail to converge for some initial values of the unknown parameters. This poor situation takes a favourable turn dramatically if the scaling matrix is slightly improved and a simple preconditioning technique is adopted: For naturally chosen simple diagonal forms of the scaling matrix and the preconditioner, the method performs well and gives accurate results with low computational cost just like the traditional methods, and improvements are obtained on: (1) widening the range of the initial values of the unknown parameters within which the minimizing iterations can converge, (2) reducing the computational cost in every Gauss–Newton iteration, (3) improving the irregular weighting effects of the coefficient matrix of the discrete governing equations. Consequently, the example inverse problem in Bentley (loc. cit.) is solved with the same accuracy, less computational effort and without the regularization term containing prior information on the unknown parameters. Moreover, numerical example shows that this method can solve the inverse problem of the quasilinear Boussinesq equation almost as fast as the linear one.In every Gauss–Newton iteration of our algorithm, one needs to solve a linear least-squares system about the corrections of both the parameters and the groundwater heads on all the discrete nodes only once. In comparison, every Gauss–Newton iteration of the traditional method has to solve the discrete governing equations as many times as one plus the number of unknown parameters or head observation wells (Yeh WW-G. Wat Resour Res 1986;22:95–108).All these facts demonstrate the potential of the algorithm to solve inverse problems of more complicated non-linear aquifer models naturally and quickly on the basis of finding suitable forms of the scaling matrix and the preconditioner.     

A multiscale method for subsurface inverse modeling: Single-phase transient flow

High-resolution geologic models that incorporate observed state data are expected to effectively enhance the reliability of reservoir performance prediction. One of the major challenges faced is how to solve the large-scale inverse modeling problem, i.e., to infer high-resolution models from the given observations of state variables that are related to the model parameters according to some known physical rules, e.g., the flow and transport partial differential equations. There are typically two difficulties, one is the high-dimensional problem and the other is the inverse problem. A multiscale inverse method is presented in this work to attack these problems with the aid of a gradient-based optimization algorithm. In this method, the model responses (i.e., the simulated state data) can be efficiently computed from the high-resolution model using the multiscale finite-volume method. The mismatch between the observations and the multiscale solutions is then used to define a proper objective function, and the fine-scale sensitivity coefficients (i.e., the derivatives of the objective function with respect to each node’s attribute) are computed by a multiscale adjoint method for subsequent optimization. The difficult high-dimensional optimization problem is reduced to a one-dimensional one using the gradient-based gradual deformation method. A synthetic single-phase transient flow example problem is employed to illustrate the proposed method. Results demonstrate that the multiscale framework presented is not only computationally efficient but also can generate geologically consistent models. By preserving spatial structure for inverse modeling, the method presented overcomes the artifacts introduced by the multiscale simulation and may enhance the prediction ability of the inverse-conditional realizations generated.     

Development and application of a coupled-process parameter inversion model based on the maximum likelihood estimation method

The coupled flow-mass transport inverse problem is formulated using the maximum likelihood estimation concept. An evolutionary computational algorithm, the genetic algorithm, is applied to search for a global or near-global solution. The resulting inverse model allows for flow and transport parameter estimation, based on inversion of spatial and temporal distributions of head and concentration measurements. Numerical experiments using a subset of the three-dimensional tracer tests conducted at the Columbus, Mississippi site are presented to test the model's ability to identify a wide range of parameters and parametrization schemes. The results indicate that the model can be applied to identify zoned parameters of hydraulic conductivity, geostatistical parameters of the hydraulic conductivity field, angle of hydraulic conductivity anisotropy, solute hydrodynamic dispersivity, and sorption parameters. The identification criterion, or objective function residual, is shown to decrease significantly as the complexity of the hydraulic conductivity parametrization is increased. Predictive modeling using the estimated parameters indicated that the geostatistical hydraulic conductivity distribution scheme produced good agreement between simulated and observed heads and concentrations. The genetic algorithm, while providing apparently robust solutions, is found to be considerably less efficient computationally than a quasi-Newton algorithm.