Nonlinear Inversion of an Unconfined Aquifer: Simultaneous Estimation of Heterogeneous Hydraulic Conductivities, Recharge Rates, and Boundary Conditions
Abstract
A new inverse method is developed to simultaneously estimate heterogeneous hydraulic conductivities, source/sink rates, and unknown boundary conditions for steady-state flow in an unconfined aquifer. Unlike objective function-based techniques, the new method does not optimize any data-model misfits. Instead, its formulation is developed by honoring physical flow principles as well as observation data at sampled locations. Under the Dupuit–Forchheimer assumption of negligible vertical flow, accuracy and stability of the new method are demonstrated using synthetic heterogeneous aquifer problems with increasingly complex flow: (1) aquifer domains without source/sink effects; (2) aquifer domains with a point sink (a pumping well operating under a constant discharge rate); (3) aquifer domains with constant or spatially variable recharge; (4) aquifer domains with constant or spatially variable recharge undergoing single-well pumping. For all problems, inversion yields stable solutions under increasing head measurement errors (up to \(\pm \)10 % of the total head variation in a problem), although accuracy of the estimated parameters degrades with the increasing errors. The inverse method is successfully tested on problems with high hydraulic conductivity contrasts—up to 10,000 times between the maximum and minimum values. In inverting several heterogeneous problems, if the aquifer is assumed homogeneous with a constant recharge rate, physically meaningful parameter estimates (i.e., equivalent conductivities and mean recharge rates) can be determined. Alternatively, if the inverse parameterization contains spurious parameters, inversion can identify such parameters, while the simultaneous estimation of non-spurious parameters is not affected. The method obviates the well-known issues associated with model “structure errors”, when inverse parameterization either simplifies or complexifies the true parameter field.
Keywords
Unconfined aquifer Inversion Hydraulic conductivity Recharge rate Boundary conditions1 Introduction
Unconfined aquifers, or water table aquifers, underlie most areas of the earth and are important freshwater resources. Water table variation of unconfined aquifers is subject to direct infiltration of rainfall recharge, or losses due to evapotranspiration, while pumping (and less frequently, injection) modifies the water table locally near wells. In unconfined aquifers, groundwater flow is not only influenced by the intrinsic hydraulic parameters of the aquifer, e.g., hydraulic conductivity (\(K\)), transmissibility (\(T\)), storage coefficients, but also the source/sink effects as a result of recharge, evapotranspiration, and well operations. For scientific and management purposes, there exists a need to estimate not only the hydraulic parameters of an unconfined aquifer, but also the parameters characterizing its source/sink strengths. For an unconfined aquifer, this study focuses on the simultaneous estimation of hydraulic conductivities, recharge rates, and the unknown (steady-state) aquifer boundary conditions, based on observation data such as hydraulic heads, groundwater fluxes, or pumping rates.
A variety of techniques exist for estimating conductivities and recharge rates for unconfined aquifers. Traditional aquifer test methods develop analytical solutions or type curves based on the assumption that aquifer conductivity is homogeneous (Dagan 1967; Neuman 1972), or aquifer exhibits simple layering (Hantush and Jacob 1955), although conductivity estimated with such assumptions may exhibit “scale effect” due to aquifer heterogeneity (Neuman 1994; Sanchez-Vila et al. 1996; Schulze-Makuch et al. 1999). Other techniques, including slug tests, borehole flowmeters, and geophysical measurements, can estimate conductivities of small aquifer volumes near wellbores (Cooper et al. 1967; Bouwer and Rice 1976; Dagan 1978; Zlotnik and Zurbuchen 2003; Darnet et al. 2003; Crisman et al. 2007). Methods have also been developed that combine geostatistics with the inverse theory to directly infer heterogeneous aquifer conductivity, while quantifying its estimation uncertainty (Li et al. 2008; Liu et al. 2008; Cardiff et al. 2009). In many studies, source/sink effects are either not accounted for, assumed known, or eliminated using specialized formulations. As a result, conductivity estimation tends to be the focus of the investigations, while recharge (or evapotranspiration) rate is rarely estimated. Moreover, many techniques exist for estimating the recharge rate (\(N\)) of an unconfined aquifer (Simmers 1998): some are based on water or chemical mass balances (Dettinger 1989; Scalon et al. 2002; Healy and Cook 2002; Tan et al. 2007; Lin et al. 2009), others infer recharge rates from physically-based, highly detailed, vadose-zone or rainfall-runoff models (Pan et al. 1997; Russo et al. 2001; Jyrkama et al. 2002), while still others use model calibration to infer the recharge rate(s) as one or more unknown model parameters (Portniaguine and Solomon 1998; Moench et al. 2001; Hill and Tiedeman 2007). With the exception of model calibration, with which aquifer conductivity and recharge rate can be simultaneously estimated, many methods assume conductivity to be known, homogeneous, or piecewise homogeneous.
In natural systems, both aquifer conductivity and aquifer recharge rate are typically variable in space. To simultaneously estimate these parameters, inverse techniques offer the most flexibility. On the one hand, if large amounts of aquifer characterization data are collected, these techniques can act as a data integration tool to help develop highly detailed flow models (McKenna and Poeter 1995; Harvey and Gorelick 1995; Day-Lewis et al. 2006; Reynolds and Marimuthu 2007; Fienen et al. 2009; Sakaki et al. 2009; Keating et al. 2010; Liu and Kitanidis 2011). On the other hand, in data-poor environments, such techniques can help develop insights into the types of observation data to collect that are important for estimating different parameters (Carrera and Neuman 1986; Tiedeman et al. 2003; Saiers et al. 2004; Hill and Tiedeman 2007). However, most of the existing inverse methods are developed by minimizing an objective function, which is typically defined as a form of mismatch between measurement data and the corresponding model simulated values (Hill and Tiedeman 2007; Oliver et al. 2008). During inversion, parameters are updated iteratively using a forward model that provides the link between the parameters and the data. Because a forward model is needed, boundary conditions (BC) of the model are either assumed known, or are calibrated as part of the inversion process. However, objective function-based techniques may lead to non-uniqueness in the estimated parameters and model BC, even for relatively simple problems. To address this issue, a new steady-state aquifer inverse theory was developed, adopting a set of approximating functions of hydraulic head and groundwater fluxes as the fundamental solutions of inversion (Irsa and Zhang 2012). By enforcing the continuity of these functions at a set of collocation points on the interfaces of the inversion grid cells, physics of flow was preserved locally within each cell and globally extending to the boundary. To incorporate noisy observed data, the method penalized the mismatch between the measurements and the values predicted by the fundamental solutions. After enforcing the continuity constraint and the data fit requirement, inversion led to a system of linear equations that can be solved with least-squares techniques. Parameters were estimated from the solution of these equations, based on which heads and flow fields were reconstructed directly from the approximating functions. Boundary conditions were then inferred from these fields.
The method described in Irsa and Zhang (2012), however, has limitations. First of all, the governing flow equation must be linear for which the approximating functions can be obtained via integration, which leads to a set of polynomial functions. This limits the applicability of the method to confined aquifer problems without source/sink effects. Second, because source/sink effects (e.g., pumping well) cannot be accommodated, groundwater flux or flow rate measurements must be sampled from the subsurface. In theory, such measurements can be obtained using borehole logging or baseflow separation techniques, but both the cost and the level of measurement uncertainty are expected to be high. Third, for problems with multiple hydrofacies, only a single hydrofacies conductivity can be estimated: conductivities of the other hydrofacies are estimated using known ratios that are specified as a set of prior information equations. For unconfined aquifers for which the flow equation is nonlinear and source/sink effects are important, the method proposed by Irsa and Zhang (2012) is not applicable. This study improves the previous work by developing a new unconfined aquifer inverse method, where the aquifer is subject to areal recharge, well discharge, or both. Multiple can be hydraulic conductivities (\(Ks\)), recharge rates (\(Ns\)), as well as the unknown model BC can be simultaneously estimated, i.e., known ratios between parameters are no longer needed. The unconfined flow equation is linearized, allowing its solution by analytical techniques. Specifically, depending on the type of aquifer forcing, the fundamental solutions of inversion are obtained by superposing appropriate analytical flow solutions for homogeneous sub-domains. Compared to those developed in Irsa and Zhang (2012), these solutions are, considered physically-based. Due to linearization and superposition, inversion gives rise to a system of nonlinear equations, for which nonlinear equation solvers are used.
To condition the inverse method, observation data include hydraulic heads and a minimum of one measurement related to its gradient, e.g., Darcy flux or flow rate. Flux or flow rate data are needed because unconfined aquifer parameter estimation suffers a well-known issue of parameter identifiability. For example, if a homogeneous unconfined aquifer is receiving uniform recharge, the flow equation is: \(\nabla ^2 h^2 = - 2N/K\), where \(\nabla ^2\) is the Laplace operator. Clearly, as long as the ratio of recharge versus conductivity remains the same, infinite combinations of these two parameters can yield identical hydraulic head distribution in the aquifer. Fitting or inverting only the hydraulic head data, therefore, cannot lead to unique and simultaneous estimation of both parameters. Measurements related to the hydraulic head gradient must be provided, and this limitation cannot be overcome with any inverse methods. However, in view of the difficulty in subsurface sampling, for aquifers with nonuniform distributions of \(K\) and \(N\), we investigate problems whereby a single pumping rate, in addition to head measurements, is provided to inversion. Because well rates can be easily measured at the surface, data requirement for this problem is not much greater than that needed for interpreting pumping tests. Clearly, the method would be less useful if individual hydrofacies heterogeneity required a separate well rate (i.e., head gradient) measurement.
Moreover, with traditional inversion techniques, if small-scale parameter heterogeneities are ignored during inversion, model “structure errors” con arise that conuld lead to biased parameter estimates (Cooley and Christensen 2005; Gaganis and Smith 2008; Doherty and Welter 2010). In this work, the jointly estimated \(Ks\) and \(Ns\) will be tested for biases. For example, at sites where detailed measurements are not available, the inverse method should provide physically meaningful bulk parameter estimates that represent the effect of underlying parameter variations on flow. On the other hand, due to incomplete understanding of the flow process, spurious parameters can be introduced into inversion. Several problems are tested by developing inverse parameterizations that (1) do not explicitly account for small-scale heterogeneities (“simplifying” model structure error), and (2) contain spurious parameters (“complexifying” model structure error). For (1), the inverse solution is considered unbiased if an equivalent conductivity and average recharge rate can be simultaneously estimated along with the unknown model BC. For (2), an unbiased solution is defined as one where the inverse solution reveals the existence of spurious parameters, while the simultaneous estimation of non-spurious parameters is not affected.
To test the accuracy of the new method, a suite of one-dimensional (1D) inversions is conducted for synthetic heterogeneous aquifers with increasingly complex flow: (1) aquifer domains without source/sink effects; (2) aquifer domains with a point sink (a pumping well operating under a constant discharge rate); (3) aquifer domains with constant or spatially variable recharge rates; (4) aquifer domains with constant or spatially variable recharge rates undergoing single-well pumping. All test problems employ a hydrofacies parameterization in the distributions of conductivity and recharge rate, data requirement is thus low, e.g., up to 20 observed hydraulic heads and a few flux or flow rate measurements are used to condition the inversion. For problems (2) and (4), only a single pumping rate (in addition to hydraulic heads) are provided. For select problems, stability of inversion is investigated by adding increasingly larger measurement noise to the observed heads. The inverse solution is considered stable if small measurement errors do not lead to large estimation errors. In the remainder of this article, the inverse theory is introduced first, followed by results testing the above set of problems, whereby the inverse solution is compared to the “true” solution of a forward model. Issues related to stability, accuracy, and structure errors are addressed. Limitations and future research are indicated, before results are summarized in the Conclusion section.
2 Theory
2.1 The Forward Problem
2.2 The Inverse Problem
2.3 Fundamental Solutions
2.4 Test Problems
The development of the fundamental solutions of inversion is illustrated with a set of test problems with increasingly complex flow. For each problem, a forward (true) model is first created to generate a set of observation data under a set of (true) model BC. Inverse analysis is then carried out according to the steps of Fig. 1. Inversion accuracy is evaluated by comparing the estimated parameters and the recovered BC against those of the forward model. For all test problems, aquifers overlie an impervious base, which is set as the head datum. The inverse solution is developed in one dimension that is aligned with the flow direction.
Flow equations and the corresponding approximating functions for the test problems
Flow equation | \(\widetilde{h}\) | \(\widetilde{q}\) | Notes |
---|---|---|---|
Test problem 1: unconfined flow without source/sink | |||
\(\frac{\partial ^2 (h^2)}{\partial x^2} +\frac{\partial ^2 (h^2)}{\partial y^2} =0\) | \(\widetilde{h}^2(x)= a_0 + a_1 x\)\(^{1}\) | \(\widetilde{q}(x)=-\frac{K a_1}{2 \sqrt{a_0 + a_1 x}}\) | Forward model is 2D for which a 1D analytical solution along the \(x\) axis can be found, i.e., flow along both \(y\) and \(z\) axes is ignored |
Test problem 2: unconfined flow with a pumping well (no areal recharge) | |||
\(\frac{1}{r}\frac{\partial }{\partial r} \left( r \frac{\partial h^2}{\partial r} \right) + \frac{1}{r^2} \frac{\partial ^2 h^2}{\partial \theta ^2}=\quad \frac{2Q_\mathrm{w}}{K} \delta (\mathbf x -\mathbf x _0)\) | \(\widetilde{h}^2(r)=a_0+a_1 r + \frac{Q_\mathrm{w}}{K \pi } \ln r\) | \(\widetilde{q}(r)=- \frac{K a_1+Q_\mathrm{w}/\pi /r}{2 \sqrt{a_0 + a_1 r + \left( Q_\mathrm{w}/K/\pi \right) \ln r}} \) | Forward model is 3D and is radially symmetric around the pumping well, where the origin of \(r\) is |
Test problem 3: unconfined flow with areal recharge (no pumping) | |||
\(\frac{\partial ^2 h}{\partial x^2} +\frac{\partial ^2 h}{\partial y^2}+\frac{N(x,y)}{T} =0\) | \(\widetilde{h}(x)=-\frac{N}{2T} x^2+a_0 x + a_1 \) | \(\widetilde{q}(x)=\frac{N x-T a_0}{-\frac{N}{2T} x^2+a_0 x + a_1}\) | Forward model is 2D for which a 1D analytical solution along the \(x\) axis can be found, i.e., flow along both \(y\) and \(z\) axes is ignored |
Test problem 4: unconfined flow with areal recharge and a pumping well | |||
\(\frac{\partial ^2 (h^2)}{\partial x^2} +\frac{\partial ^2 (h^2)}{\partial y^2} + \frac{2N(x,y)}{K}=\quad \frac{2Q_\mathrm{w}}{K} \delta (\mathbf x -\mathbf x _0)\) | \(\widetilde{h}^2(r)=-\frac{N}{2K} r^2+ \frac{Q_\mathrm{w}}{K \pi } \ln r + a_0\) | \(\widetilde{q}(r)=\frac{Nr-\frac{Q_\mathrm{w}}{\pi r}}{2 \sqrt{-\frac{N}{2K} r^2+ \frac{Q_\mathrm{w}}{K \pi } \ln r + a_0}}\) | Forward model is 3D and is radially symmetric around the pumping well, where the origin of \(r\) is |
In test problem 3, the forward model is similar to that of test problem 1 (Fig. 2), except that aquifer is receiving recharge at the water table (Table 1). For this problem, Eq. (6) is rewritten to be transmissibility based, and the flow equation becomes linear with respect to hydraulic head. By inverting for \(T\), we aim to understand if the reduced nonlinearity in the flow equation will lead to a different inversion performance, i.e., stability, speed of solver convergence, and accuracy. The flow rate approximating function is again: \(\widetilde{Q(x)}=\widetilde{q_x(x)} \widetilde{h(x)}\). The inverse solution is: \(x^\mathrm{T}=[a_0^l, a_1^l, T_1, \ldots , T_{L}, N_1, \ldots , N_{L}]\). Hydraulic conductivity of a hydrofacies is estimated from the transmissibilities: \(K=T/\overline{h}^\mathrm{o}\), where \(\overline{h}^\mathrm{o}\) is an average of the observed heads from the hydrofacies.
In test problem 4, a single well pumps water at a constant rate similar to that of test problem 2 (Fig. 3), except aquifer is receiving recharge. Again, the well rate is considered an observation and subsurface fluxes are not sampled. The head at the pumping well is not sampled either. The inverse solution is: \(x^\mathrm{T}=[a_0^l, K_1, \ldots , K_{L}, N_1, \ldots , N_{L}]\).
For all the test problems, though in theory \(K\) and \(N\) can be discretized at every grid cell, without a large number of measurements, this will result in an underdetermined inversion system of equations. Although nonlinear solvers for underdetermined problems do exist, and similar to many classic inverse methods, one or more regularization constraints can be imposed on the parameters (thus increasing the number of equations), such solutions are not explored here. Instead, all the test problems adopt a deterministic zoned parameterization for both the conductivity and the recharge rate. In addition, the decision to carry out 1D inversion is not a limitation of the inverse method, as analytical solutions exist for 2D and 3D flows for which new problems have been successfully inverted. By presenting a set of 1D analyses, issues of data worth, parameter identifiability, and inversion stability can be clearly illustrated. As long as the Dupuit–Forchheimer assumption (and in the pumping cases, radial symmetry) is satisfied, the 1D analysis reveals how seemingly complex problems can be inverted easily in 1D with small grids.
2.5 Inverse Solution Techniques
With lsqnonlin, the equation system can be exact or overdetermined. Constraints can also be placed on \(x\), e.g., enforcing positive conductivities or recharge rates. fsolve solves a similar minimization problem, although no constraints can be placed on the solution. fsolve’s Levenberg–Marquardt algorithm can also minimize underdetermined problems. Both fsolve and lsqnonlin require that functions \(f_1(x), f_2(x), \ldots , f_M(x)\) be continuous over the solution domain. This does not pose a problem because the approximating functions are created by superposing continuous flow solutions.
To use the nonlinear solvers, an initial guess of the solution (\(x_0\)) must be provided. \(x_0\) can be estimated in two ways: (1) minimize Eq. (5) with an one-cell inverse grid, excluding the continuity equations, (2) solve an analytical problem assuming an infinite aquifer with homogeneous parameters. Neither approaches require the knowledge of aquifer BC. As an example, for test problem 1, \(x_0\) can be obtained by creating a single-cell inverse grid for which the solution is \(x_0^\mathrm{T}=[a_0, a_1, K]\). Alternatively, \(x_0\) can be obtained by fitting the analytical solutions (\(h^2(x)=a_0+a_1x,\; q(x)=-\frac{K a_1}{2 \sqrt{a_0 + a_1 x}}\), or \(Q(x)=-\frac{K a_1}{2}\)) to the observed heads, fluxes, or flow rate (the above solutions also illustrate that unique estimation of K requires at least one flux or flow rate measurement). Moreover, the parameterization adopted for generating \(x_0\) needs not be identical to that of the full inversion, e.g., \(x_0\) can contain a single \(K\) even if full inversion estimates a number of \(Ks\). These initial estimates, whether they are obtained with numerical or analytical means, provide a set of physically reasonable parameter values with which the full inversion can be carried out. Because of the homogeneity assumption implicit in the above approaches, data requirement for obtaining \(x_0\) is small.
2.6 Model Structure Errors
Due to the limited subsurface access, inverse parameterization often does not correspond to the true parameter fields, which can result in two types of model structure errors: simplifying and complexifying. For an inverse method to be useful, it is necessary to understand how it performs when its parameterization contains structure errors. For select problems in this study, inverse parameterization is modified from that of the forward model. For example, in test problem 4 the, observed data will be sampled from a forward model with multiple \(Ks\) and multiple \(Ns\), but inversion only estimates two parameters, assuming uniform \(K\) and \(N\) over the solution domain. This corresponds to data-poor situations where the underlying parameter variability is unknown. The estimated parameters will be tested for bias by comparing them to an equivalent conductivity (\(K_\mathrm{eq}\)) and an average recharge rate (\(\overline{N}\)), which are determined from the forward model using mass balance and upscaling techniques (Zhang et al. 2006). Other types of structure error also exist. For the above problem, the inverted parameters can be multiple \(Ks\) and a single \(N\) (\(K\) zonation is known to inversion, but \(N\) is assumed homogeneous over the solution domain), or multiple \(Ns\) and a single \(K\) (\(N\) zonation is known, but \(K\) is assumed homogeneous). In all these cases, model structure error arises from simplifying the true parameter fields. In addition, structure error can arise from complexifying the true parameter fields. For example, the forward model does not receive recharge (e.g., test problem 2), but, without data on moisture content, such information may be difficult to infer from limited field data. A modeler may choose to estimate a number of recharge rates by conceptualizing the inversion as that of test problem 4. For this envisioned scenario, several \(Ns\) are estimated along with several \(Ks\). The estimated \(Ns\) are spurious and how this structure error affects joint estimation of the parameters is also of interest.
3 Results
In the following sub sections, the forward model of each test problem is described, followed by results of the inverse analysis. Some problems are sufficiently simple for which the forward models are analytical solutions; others are more complex, for which the forward models are detailed finite difference solutions by MODFLOW2000. To accurately model well flow with a cartesian grid, the forward model employs local grid refinement at the well. MODFLOW2000 is implemented in the software package Groundwater Vista, which adopts the English unit. Results of this study are presented in: heads and distances (ft), fluxes, conductivities, recharge rates (ft/day or ft/d), and flow rates (ft\(^3\)/day or ft\(^3\)/d). Alternatively, all dimensional information can be removed, assuming that a consistent set of units is used (Neuman et al. 2007).
To obtain the inverse solutions, the observation data (hydraulic heads, fluxes, or flow rates) are sampled from the forward model and are provided to inversion, which results in the estimation of hydraulic conductivities, recharge rates, and the flow field including the model BC. Compared to test problems 1 and 2, test problems 3 and 4 are more complex with a greater number of parameters, thus stability tests are presented for these problems only. For problems 1 and 2, measurements sampled from the forward models are considered error-free (these data are strictly error-free if the forward model is analytical; they are approximately error-free if the forward model is a finite difference solution). For problems 3 and 4, hydraulic heads sampled from the FDM are corrupted by increasing measurement errors: \(h^\mathrm{measure}=h^\mathrm{FDM}\pm {\Delta } h\), where \({\Delta } h\) is a noise, assigned as 1, 5, and 10 % of the total head variation in the forward model. For example, \({\Delta } h\) of 10 % results in a set of head measurements that fluctuate over an interval that is 20 % of the total head change. The larger errors are imposed mainly to test the stability of inversion. Only the measured heads are subject to errors; flow rates or fluxes, when sampled from the forward model, are not affected by errors.
3.1 Test Problem 1: Flow Without Source/Sink Effect
The forward models are a suite of analytical and numerical solutions of unconfined flow between two constant head reservoirs (Fig. 2): \(h_0=15\) ft and \(h_{L}=10\) ft. Because there are no source/sink effects, uniform flow from the high head reservoir toward the low head reservoir prevails.
Estimated conductivities (in ft/d) for test problem 1 with three conductivity zones
Parameters: | \(K_1\) | \(K_2\) | \(K_3\) | \(h_0\) | \(h_{L}\) |
---|---|---|---|---|---|
True solution: | 10.00 | 1.00 | 5.00 | 15.00 | 10.00 |
Low K contrast | |||||
Inverse grid (6 cells) | |||||
Two heads (\(x=40,\; x=80\)) | 13.34 | 12.90 | 12.84 | 17.60 | 4.80 |
Two heads \(+\) 1\(Q_x\)\(^\mathrm{a}\) | 1.04 | 1.00 | 1.00 | 17.55 | 4.80 |
Two heads \(+\) 1\(q_x\)\(^\mathrm{b}\) | 1.09 | 1.05 | 1.05 | 17.54 | 4.81 |
Two heads \(+\) 1\(Q_x\)\(^\mathrm{a}+\) 1\(q_x\)\(^\mathrm{b}\) | 10.00 | 1.00 | 1.01 | 15.00 | 4.93 |
Eight heads (equal spacing) | 27.95 | 2.79 | 13.97 | 15.00 | 10.00 |
Eight heads \(+\) 1\(Q_x\)\(^\mathrm{a}\) | 10.00 | 1.00 | 5.00 | 15.00 | 10.00 |
Eight heads \(+\) 1\(q_x\)\(^\mathrm{b}\) | 10.00 | 1.00 | 5.00 | 15.00 | 10.00 |
Eight heads \(+\) 1\(Q_x\)\(^\mathrm{a}\)\(+\) 1\(q_x\)\(^\mathrm{b}\) | 10.00 | 1.00 | 5.00 | 15.00 | 10.00 |
Eight heads \(+\) 3\(Q_x\)\(^\mathrm{c}\)\(+\) 5\(q_x\)\(^\mathrm{c}\) | 10.00 | 1.00 | 5.00 | 15.00 | 10.00 |
hline True solution: | 100.00 | 10.00 | 1.00 | 15.00 | 10.00 |
Medium K contrast | |||||
Inverse grid (24 cells) | |||||
Two heads \(+\) 1\(Q_x\)\(^\mathrm{a}\) | 149.38 | 9.89 | 5.60 | 14.99 | 13.87 |
Eight heads (equal spacing) | 148.27 | 14.83 | 1.48 | 15.00 | 10.00 |
Eight heads \(+\) 1\(Q_x\)\(^\mathrm{a}\) | 100.00 | 10.00 | 1.00 | 15.00 | 10.00 |
Eight heads \(+\) 3\(Q_x\)\(^\mathrm{c}\)\(+\) 5\(q_x\)\(^\mathrm{c}\) | 100.00 | 10.00 | 1.00 | 15.00 | 10.00 |
hline True solution | 0.10 | 10.00 | 1000.00 | 15.00 | 10.00 |
High K contrast | |||||
Inverse grid (24 cells) | |||||
Two heads \(+\) 1\(Q_x\)\(^\mathrm{a}\) | 0.21 | 10.00 | 2499.8 | 12.64 | 10.00 |
Eight heads (equal spacing) | 0.25 | 25.00 | 2499.5 | 15.00 | 10.00 |
Eight heads \(+\) 1\(Q_x\)\(^\mathrm{a}\) | 0.13 | 13.14 | 1297.4 | 15.00 | 10.00 |
Eight heads \(+\) 3\(Q_x\)\(^\mathrm{c}\)\(+\) 5\(q_x\)\(^\mathrm{c}\) | 0.10 | 10.00 | 1000.00 | 15.00 | 10.00 |
Similar to the homogeneous problems, underdetermined systems, despite being supplied with both heads and fluxes (or flow rates), lead to less accurate results. In overdetermined problems, if the 8 observed heads are the only measurements used, the estimated conductivities suffer large errors, but the head recovery is accurate regardless of the conductivity contrast. If one or more \(q_x\) (or \(Q_x\)) are additionally sampled, conductivity estimations then become accurate. This is expected: unique estimation of \(K\) requires head gradient information as embodied in the flux or flow rate data, in addition to hydraulic head measurements. Furthermore, when the conductivity contrast increases, given the same observed data (e.g., 8 heads \(+\) 1\(Q_x\)), conductivities of the medium- to high-\(K\)-contrast problems suffer greater errors. In this case, adding more data (e.g., 8 heads \(+\) 3\(Q_x\)\(+\) 5\(q_x\)) leads to more accurate \(K\) estimates.
Estimated conductivities for test problem 1 with a triangular aquifer bottom
Homogeneous | Low K contrast | ||||||
---|---|---|---|---|---|---|---|
Inverse grid (24 cells) | Inverse grid (6 cells) | ||||||
Parameters: | \(K\) | \(K_1\) | \(K_2\) | \(K_3\) | |||
True solution: | 5.00 | 10.00 | 1.00 | 5.00 | |||
Nine heads \(+\) 3 \(q_x\) | 4.71 | Nine heads \(+\) 3\(q_x\) | 12.20 | 1.06 | 3.42 | ||
Nine heads \(+\) 3 \(q_x+\) 2 \(Q_x\) | 4.60 | Nine heads \(+\) 3\(q_x +\) 3\(Q_x\) | 9.99 | 0.73 | 2.72 |
Medium K contrast | High K contrast | ||||||
---|---|---|---|---|---|---|---|
Inverse grid (24 cells) | Inverse grid (24 cells) | ||||||
Parameters: | \(K_1\) | \(K_2\) | \(K_3\) | \(K_1\) | \(K_2\) | \(K_3\) | |
True solution: | 100.00 | 10.00 | 1.00 | 0.10 | 10.00 | 1,000.00 | |
Nine heads \(+\) 3\(q_x\) | 292.57 | 9.28 | 0.91 | Twelve heads \(+\) 3\(q_x +\) 3 \(Q_x\) | 0.36 | 6.56 | 921.5 |
Eleven heads \(+\) 3\(q_x\)\(^\mathrm{a}\) | 99.80 | 9.34 | 0.91 | Twelve heads \(+\) 4\(q_x\)\(^\mathrm{b}+\) 3\(Q_x\) | 0.36 | 6.56 | 989.3 |
3.2 Test Problem 2: Flow With a Pumping Well (No Areal Recharge)
For a set of four \(K\) distributions with increasing contrasts, the forward model (\(1{,}800 \,\times 1{,}800 \,\times 30\) ft\(^3\)) is simulated with a no-flow boundary at the bottom and a specified head of 20 ft along the sides (Fig. 3). The top boundary (water table) is solved iteratively by MODFLOW2000 using convertible layers. Areal recharge is not applied. A single pumping well, with a constant discharge rate, induces radial flow toward the model center. For each \(K\) distribution, \(Q_\mathrm{w}\) is adjusted to ensure that hydraulic head at the well will not drop below the top of the well screen (screen length is adjusted as well). The forward model initially employs a coarse grid (\(N_x=60\), \(N_y=60\), \(N_z=60\)), before it is refined at the pumping well and at the interfaces between hydrofacies. With the refined grid, global mass balance errors are less than 0.1 % for all \(K\) distributions. From the forward models, hydraulic heads are sampled quasi-regularly along the radial axis at an elevation of 3.5 ft. Darcy fluxes are not sampled, nor is the head at the pumping well. The well rate is considered known and is incorporated into the inverse formulation. Inversion is carried out along the same radial axis.
For a case with low \(K\) contrast (\(K_1=5\), \(K_2=1\), and \(K_3=10\)), a preliminary inversion is carried out with ten observed heads using a 24-cell, grid (\({\Delta } x=37.5\) ft). Result is inaccurate: (1) the estimated conductivities contain up to 100 % errors; (2) the recovered head at the pumping well, exhibits the largest deviation from the true head (smallest head deviation occurs at the constant head boundary). The inverse grid is then refined: the previous 24th cell, located closest to the pumping well, is split into: cell 24 (\({\Delta } x=18.75\) ft), cell 25 (\({\Delta } x=9.375\) ft), and cell 26 (\({\Delta } x=9.375\) ft). Given the same observed data, both the estimated conductivities and the recovered head profile improve immediately.
Estimated conductivities for test problem 2 with three conductivity zones and a pumping well which lies in the center of the \(K_3\) zone (see Fig. 3)
Low K contrast | Medium K contrast | ||||||||
---|---|---|---|---|---|---|---|---|---|
\(Q_\mathrm{w}=250\)\(\mathrm{ft}^3/\mathrm{d}\) | \(Q_\mathrm{w}=250\)\(\mathrm{ft}^3/\mathrm{d}\) | ||||||||
Parameters: | \(K_1\) | \(K_2\) | \(K_3\) | \(h_\mathrm{well}\) | \(K_1\) | \(K_2\) | \(K_3\) | \(h_\mathrm{well}\) | |
True solution: | 5.00 | 1.00 | 10.00 | 17.65 | 1.0 | 10.00 | 100.00 | 19.10 | |
Eleven heads \(+ Q_\mathrm{w}\) | 5.14 | 0.97 | 11.50 | 17.67 | Eleven heads \(+ Q_\mathrm{w}\) | 1.11 | 9.98 | 113.53 | 19.08 |
Eleven heads \(+ Q_\mathrm{w} +\) 3\(q_x{^\mathrm{a}}\) | 0.96 | 14.57 | 126.44 | 19.08 |
Medium K contrast | High K contrast | ||||||||
---|---|---|---|---|---|---|---|---|---|
\(Q_\mathrm{w}=125\)\(\mathrm{ft}^3/\mathrm{d}\) | \(Q_\mathrm{w}=125\)\(\mathrm{ft}^3/\mathrm{d}\) | ||||||||
Parameters: | \(K_1\) | \(K_2\) | \(K_3\) | \(h_\mathrm{well}\) | \(K_1\) | \(K_2\) | \(K_3\) | \(h_\mathrm{well}\) | |
True solution: | 10.00 | 100.0 | 1.00 | 15.20 | 1,000.00 | 0.10 | 10.00 | 11.30 | |
Eleven heads \(+ Q_\mathrm{w}\) | 11.14 | 88.10 | 1.15 | 15.23 | Eleven heads \(+ Q_\mathrm{w}\) | 1204.3 | 0.10 | 11.48 | 11.26 |
Twenty heads\({^\mathrm{b}}\)\(+ Q_\mathrm{w}\) | 11.17 | 103.62 | 1.19 | 15.23 | Twenty heads \(+ Q_\mathrm{w}\) | 698.64 | 0.12 | 10.70 | 11.26 |
3.3 Test Problem 3: Flow With Areal Recharge (No Pumping)
The forward model is similar to test problem 1 (Fig. 2; \(L\) is now 900 ft). Areal recharge is applied to the water table throughout the model. Initially, a uniform recharge rate is assigned (\(N=0.005\) ft/d), along with these parameters and BC: \(h_0= 15\) ft, \(h_{l}=10\) ft, \(K_1=30\) ft/d, \(K_2=100\) ft/d, \(K_3=10\) ft/d. The same finite difference grid of test problem 1 is used, except here the aquifer bottom is flat. Because of the recharge, water table builds up near the model center, where a hydrological divide occurs at \(x\simeq \) 300 ft, and diminishes toward the boundaries. From the forward model, 8 hydraulic heads, 2 \(Q_x\), and 4 \(q_x\) are sampled semi-regularly at a fixed elevation.
Measurement errors (\({\pm }1\) %) are then added to the observed heads, leading to these results: (1) nearly perfect head recovery (Fig. 5); (2) \(K_1=35.69\) ft/d; (3) \(K_2=93.38\) ft/d; (4) \(K_3=10.82\) ft/d; and (5) \(N=0.0068\) ft/d. When \({\pm }5\) % errors are used, inversion yields: (1) fairly good head recovery with the leftmost heads slightly overestimated (Fig. 5); (2) \(K_1=105.32\) ft/d; (3) \(K_2=173.98\) ft/d; (4) \(K_3=10.08\) ft/d; (5) \(N=0.0056\) ft/d. When \({\pm }10\) % errors are used, inversion yields: (1) reasonably accurate head recovery (Fig. 5); (2) \(K_1=1457.30\) ft/d; (3) \(K_2=114.31\) ft/d; (4) \(K_3=9.73\) ft/d; (5) \(N=0.0056\) ft/d. Clearly, parameter estimation error increases with increasing measurement error. Compared to the estimated recharge rate and head recovery, \(K\) estimation is more sensitive to the measurement errors. Moreover, \(K_1\) is consistently overestimated as a result of the specific measurements used and their specific errors. For the case with \({\pm }10\) % errors, if error signs are switched among the observed heads, inversion yields: (1) \(K_1=12.59\) ft/d; (2) \(K_2=85.12\) ft/d; (3) \(K_3=11.57\) ft/d; (4) \(N=0.0056\) ft/d. \(K_1\) is now underestimated, while the recovered head near the left boundary (not shown) slightly underestimates the true water table.
Next, the forward model is rerun with spatially variable recharge: \(N_1=0.005\) ft/d, \(N_2=0.0167\) ft/d, \(N_3=0.00167\) ft/d. The recharge zones are identical to the conductivity zones. From this model, 17 heads, 3\(Q_x\), and 5\(q_x\) are sampled in a semi-regular pattern. With error-free data, inversion yields: (1) perfect head recovery (not shown); (2) \(K_1= 31.02\) ft/d; (3) \(K_2=94.15\) ft/d; (4) \(K_3=10.64\) ft/d; (5) \(N_1=0.007\) ft/d; (6) \(N_2=0.0164\) ft/d; and (7) \(N_3=0.0065\) ft/d. With the exception of \(N_3\), all parameters are accurately estimated. When measurement errors are increased, inverse solutions behave similarly to those observed previously for the problem with uniform recharge: inversion is stable under increasing errors, but accuracy of the estimated parameters degrades.
Finally, from the same forward model (i.e., three \(K\)s and three \(N\)s), a new set of observations—8 heads, 2 \(Q_x\), and 4 \(q_x\), are sampled. Inversion adopts a simplified parameterization, whereby three \(Ks\) and a single \(N\) are estimated, yielding: (1) perfect head recovery (not shown); (2) \(K_1= 30.48\) ft/d; (3) \(K_2=107.82\) ft/d; (4) \(K_3=9.53\) ft/d; and (5) \(N=0.0072\) ft/d. The estimated \(N\) is close to the mean aquifer recharge rate: 0.0078 ft/d. Despite this structure error, accuracy of the simultaneously estimated conductivities is not affected. In addition, stability tests under increasing measurement errors are conducted, yielding similar behaviors as those observed for problems without structure errors.
3.4 Test Problem 4: Flow With Areal Recharge and A Pumping Well
The forward model is similar to test problem 2 (Fig. 3), except areal recharge is additionally applied to the water table. The same finite difference grid of test problem 2 is used, containing grid refinement at the pumping well and at the hydrofacies interfaces. Except for the recharge boundary, BC are identical to those of test problem 2. The true conductivities are: \(K_1=10\) ft/d, \(K_2=100\) ft/d, and \(K_3=30\) ft/d. Initially, uniform recharge is applied at the top boundary, and then, nonuniform recharge. In both cases, a steady-state pumping rate of 3,000 ft\(^3\)/d can be maintained. Because of the recharge, flow pattern is complex in both cases though it is still radially symmetric: near the pumping well, converging flow is observed; away from the well, water table builds up in a circular mound, where recharge water on the inner side of the divide flows toward the well, and that on the outer side flows toward the boundary. From the forward model, 19 heads are sampled semi-regularly at a fixed elevation, from the well toward the boundary. In inversion, conductivity-based formulations are adopted (Table 1), leading to the following results:
When a uniform recharge is applied to the forward model, inversion recovers the hydraulic heads, the three conductivities, and the single \(N\) (not shown), with a similar level of accuracy as that of test problem 3. Stability analysis is also conducted, with similar error characteristics as those of test problem 3.
3.5 Grid Refinement
Previous experimentations with the inverse grid suggest that, for a given set of measurements, local grid refinement in the high head gradient region (i.e., near the well) can improve the inversion accuracy. However, this is true only to a certain extent. For example, for test problem 4 with three \(Ks\) and three \(Ns\), given the same error-free data (19 heads and \(Q_\mathrm{w}\)), inversion with a 52-cell grid (i.e., the 26-cell grid is uniformly refined) did not improve the results. The estimated parameters are: (1) nearly perfect head recovery; (2) \(K_1=11.00\) ft/d; (3) \(K_2=93.54\) ft/d; (4) \(K_3=20.99\) ft/d; (5) \(N_1=0.021\) ft/d; (6) \(N_2=0.019\) ft/d; and (7) \(N_3=0.014\) ft/d, which do not differ significantly from those obtained using the coarser grid. On the other hand, for a fixed inverse grid, increasing measurements generally leads to more accurate results, as discussed above. Clearly, both inverse grid discretization and data density affect the inversion accuracy.
3.6 Model Structure Errors
A single \(K\) and a single \(N\) are estimated. In this case, head recovery is reasonably accurate (Fig. 6). The estimated parameters are: \(K=12.88\) ft/d, \(N=0.016\) ft/d. \(K\) underestimates \(K_\mathrm{eq}\), but \(N\) is close to the average recharge rate.
Three conductivities and a single \(N\) are estimated. Head recovery is accurate (Fig. 6). The estimated parameters are: \(K_{1}=7.84\) ft/d, \(K_2=67.66\) ft/d, \(K_3=20.92\) ft/d, and \(N=0.015\) ft/d. The estimated recharge rate is close to the average value, while the estimated conductivities are similar to those of test problem 4 without model structure error (see Sect. 3.4). The accuracy of conductivity estimation is not affected by the incorrect recharge parameterization.
Three recharge rates and a single \(K\) are estimated. Head recovery is again accurate (Fig. 6). The estimated parameters are: \(K=18.58\) ft/d, \(N_1=0.033\) ft/d, \(N_2=0.0089\) ft/d, and \(N_3=0.014\) ft/d. The estimated conductivity is not far from \(K_\mathrm{eq}\). The estimated recharge rates are similar to those of test problem 4 without model structure error. The accuracy of recharge estimation is not affected by the incorrect conductivity parameterization.
Next, complexifying structure error is investigated. For example, when the inverse formulation contains the recharge term, what will happen to its solution if the actual recharge is zero? Test problem 4 is inverted again, but the forward model is assigned zero recharge, while maintaining the same pumping rate. From this model, the same 19 error-free heads are sampled. Inversion with the 26-cell grid yields excellent head recovery (not shown), with these estimated parameters: (1) \(K_1=14.08\) ft/d; (2) \(K_2=89.70\) ft/d; (3) \(K_3=23.88\) ft/d; (4) \(N_1=-1.8\times 10^{-4}\) ft/d; (5) \(N_2=5.8\times 10^{-5}\) ft/d; and (6) \(N_1=8.2\times 10^{-4}\) ft/d. The three estimated recharge rates are extremely small, pointing to the error in recharge parameterization. This result can also lead to a more parsimonious future model, e.g., inverse parameterization can be simplified after new data are collected to confirm the structure error. The error in parameterizing recharge similarly does not affect the simultaneous estimation of the conductivities, i.e., all \(Ks\) are comparable to those found for test problem 4 without model structure error (see Sect. 3.4).
4 Discussion
By providing as few as one pumping rate in addition to hydraulic heads, multiple \(Ks\) and \(Ns\) can be uniquely and simultaneously estimated, along with the model flow field and BC. This demonstrates that inversion can succeed for heterogeneous problems as long as a single head-gradient-based measurement is provided. Data requirement of the method is low.
Inversion result is stable with increasing measurement errors (up to \({\pm }10\) % of the total head variation). As measurement error increases, parameter estimation becomes less accurate, as expected;
Inversion is computationally efficient. For well-posed problems, convergence time on a PC workstation is typically a few seconds. For problems that are not well-posed (e.g., lacking flux or flow rate measurement, underdetermined equations due to insufficient data), the method typically does not converge, no matter how stringent the solver convergence criteria are (in these cases, parameter estimates are those obtained at the maximum solver iteration);
Inversion accuracy is relatively insensitive to parameter variability, e.g., \(K_{\max }/K_{\min }\) is successfully tested up to 10,000. However, for a given problem, when \(K_{\max }/K_{\min }\) increases, more observation data are needed to achieve the same accuracy. Inversion accuracy is also insensitive to the transmissibility versus conductivity-based formulations. In the latter formulation, the approximating functions are more strongly nonlinear;
Inversion accuracy is insensitive to flow patterns. The flow path, as long as it is largely horizontal, can be unidirectional (test problem 1), convergent (test problem 2), divergent (test problem 3), and quite complex with converging and diverging flows (test problem 4). This suggests that by developing approximating functions that superpose individual well solutions, the method can address problems with multiple injection and production wells. This has been confirmed by successfully inverting a 2D problem with four conductivities, two recharge rates, and a pair of well dipole.
Despite the Dupuit–Forchheimer assumption with which the inverse formulation is developed, inversion accuracy is not significantly affected by minor violation of this assumption, suggesting a broader applicability of these solutions to aquifers with irregular shapes.
In this work, parameter zonation is known to inversion, leading to a set of solutions without uncertainty measures. The zonation pattern, in effect, enforces a deterministic constraint on the parameters. To remove this constraint and to account for static data uncertainty, the inverse method can be integrated with geostatistics, whereby uncertainty in both the estimated parameters and the estimated state variables (including model BC) can be quantified (Wang et al. 2013). Because analytical solutions are used to generate the approximating functions, future work can explore inverting such solutions within a stochastic framework, i.e., conductivity and recharge become spatial random functions. Moreover, though the current method adopts a zoned parameterization, whereby the number of parameters is small compared to the number of observations, future work can explore highly parameterized inversion for which regularization (e.g., smoothness constraint, spatial covariances, cross correlation between hydrological parameters and geophysical measurements) can be incorporated into the inversion equations. The additional constraints will allow us to solve for a larger number of parameters, e.g., one \(K\) at each inverse grid cell. Finally, the inverse method does not explicitly account for unsaturated flow process which is treated as an instantaneous recharge to the water table. There is a growing body of work that couples and inverts flow in the unsaturated and saturated zones for which parameters specific to each zone are estimated (Moench 2004; Mishra and Neuman 2010, 2011; Mao et al. 2013). Whether or not the continuity concept of this study can be extended to unsaturated flow and coupled processes will require further investigations. For such problems, data requirement to obtain well-posed inversion is likely higher (Mao et al. 2013).
5 Conclusion
A physically-based inverse method is developed to analyze steady-state flow in an unconfined aquifer with heterogeneous hydraulic conductivities and significant source/sink effects. The method extends the confined aquifer inversion of Irsa and Zhang (2012), where a single conductivity and model boundary conditions were estimated for problems without source/sink effects. In this work, to address nonlinearity in the unconfined flow equation, the hydraulic head approximating function is created by superposing analytical flow solutions for homogeneous sub-domains. Given appropriate measurements, the inversion system of equations becomes well-posed and can be solved with nonlinear optimization, which allows the simultaneous estimation of multiple conductivities and recharge rates. From the inverse solution, the flow field including the unknown BC can be recovered. Because the inverse method does not optimize any objective functions (i.e., data-model mismatch) which require forward flow simulations, the method is computationally efficient.
Under the Dupuit–Forchheimer assumption, accuracy and stability of the method are demonstrated by inverting heterogeneous synthetic aquifer problems with increasingly complex flow: (1) aquifer domains without source/sink effects; (2) aquifer domains with a point sink (a pumping well operating under a constant discharge rate); (3) aquifer domains with constant or spatially variable recharge; (4) aquifer domains with constant or spatially variable recharge undergoing single-well pumping. In the problems without the pumping well, observation data are hydraulic heads, Darcy fluxes, and/or flow rates; in the problems with the pumping well, the observation data are hydraulic heads and the pumping rate. For all problems, inverse solutions are stable under increasing measurement errors, although accuracy of the estimated parameters degrades with increasing errors. The method is successfully tested on strongly heterogeneous problems with conductivity contrast up to 10,000.
The inverse problem must be well-posed to obtain accurate solutions, i.e., the observation data must be of sufficient quantity, adequate quality, and of the necessary types. For example, given only head measurements, inversion can yield accurate head profiles extending to the model boundaries. Accurate estimation of the conductivities and the recharge rates, however, requires both hydraulic head and Darcy flux or flow rate measurements. Interestingly, a single flux or flow rate measurement suffices to enable the unique and simultaneous estimation of multiple conductivity and recharge parameters. Inversion accuracy is also affected by the resolution of the inverse grid: at locations where hydraulic head gradient is large, local grid refinement can improve the solution. Moreover, the inverse method obviates the well-known issue associated with model “structure errors,” whereas inverse parameterization simplifies or complexifies the true parameter field. For several heterogeneous problems, when aquifer is assumed homogeneous with a constant recharge, physically meaningful parameter estimates (i.e., equivalent conductivities and mean recharge rates) can be obtained. Alternatively, if the inverse parameterization contains spurious parameters, inversion can identify such parameters, while the simultaneous estimation of non-spurious parameters is not affected.
Notes
Acknowledgments
This research is supported by the University of Wyoming Center for Fundamentals of Subsurface Flow (WYDEQ49811ZH). The author acknowledges helpful comments of two anonymous reviewers who helped to improve the content and organization of this paper.
References
- Bear, J.: Dynamics of Fluids in Porous Media, vol. 764, 1st edn. Elsevier, New York (1972)Google Scholar
- Bouwer, H., Rice, R.C.: A slug test for determining hydraulic conductivity of unconfined aquifers with completely or partially penetrating wells. Water Resourc. Res. 12(3), 423–428 (1976)CrossRefGoogle Scholar
- Cardiff, M.: W, B., Kitanidis, P., Malama, B., Revil, A., Straface, S., Rizzo, E.: A potential-based inversion of unconfined steady-state hydraulic tomography. Ground Water 47(2), 259–270 (2009)CrossRefGoogle Scholar
- Carrera, J., Neuman, S.P.: Estimation of aquifer parameters under transient and steady state conditions: III. Application to synthetic and field data. Water Resourc. Res. 22(2), 228–242 (1986)Google Scholar
- Cooley, R.L., Christensen, S.: Bias and uncertainty in regression-calibrated models of groundwater flow in heterogeneous media. Adv. Water Resourc. 29(5), 639–656 (2005)CrossRefGoogle Scholar
- Cooper, H.H., Bredehoeft, J.D., Papadopulos, I.S.: Response of a finite-diameter well to an instantaneous charge of water. Water Resourc. Res. 3, 263–269 (1967)CrossRefGoogle Scholar
- Crisman, S.A., Molz, F.J., Dunn, D.L., Sappington, F.C.: Application procedures for the electromagnetic borehole flowmeter in shallow unconfined aquifers. Groundw. Monit. Remediat. 12(3), 96–100 (2007)Google Scholar
- Dagan, G.: A method determining the permeability and effective porosity of unconfined anisotropic aquifers. Water Resourc. Res. 3(4), 1059–1071 (1967)CrossRefGoogle Scholar
- Dagan, G.: A note on packer, slug, and recovery tests in unconfined aquifers. Water Resourc. Res. 14, 929–934 (1978)CrossRefGoogle Scholar
- Darnet, M., Marquis, G., Sailhac, P.: Estimating aquifer hydraulic properties from the inversion of surface streaming potential (SP) anomalies. Geophys. Res. Lett. 30(13), 1679 (2003). doi:10.1029/2003GL017631 CrossRefGoogle Scholar
- Day-Lewis, F., Lane, J.W., Gorelick, S.M.: Combined interpretation of radar, hydraulic, and tracer data from a fractured-rock aquifer. Hydrogeol. J. 14(1–2), 1–14 (2006)CrossRefGoogle Scholar
- Dettinger, M.D.: Reconnaissance estimates of natural recharge to desert basins in Nevada, U.S.A., by using chloride-balance calculations. J. Hydrol. 106, 55–78 (1989)CrossRefGoogle Scholar
- Doherty, J., Welter, D.: A short exploration of structure noise. Water Resourc. Res. 46, W05525 (2010). doi:10.1029/2009WR008377 CrossRefGoogle Scholar
- Fienen, M., Hunt, R., Krabbenhoft, D., Clemo, T.: Obtaining parsimonious hydraulic conductivity fields using head and transport observations: A Bayesian geostatistical parameter estimation approach. Water Resourc. Res. 45, W08405 (2009). doi:10.1029/2008WR007431 CrossRefGoogle Scholar
- Gaganis, P., Smith, L.: Accounting for model error in risk assessments: Alternative to adopting a bias towards conservative risk estimates in decision models. Adv. Water Resourc. 31(8), 1074–1086 (2008)CrossRefGoogle Scholar
- Haitjema, H.M., Mitchell-Bruker, S.: Are water tables a subdued replica of the topography? Ground Water 43(6), 781–786 (2005)Google Scholar
- Hantush, M.S., Jacob, C.E.: Non-steady radial flow in an infinite leaky aquifer. Trans. Am. Geophys. Union 36(1), 95–100 (1955)CrossRefGoogle Scholar
- Harvey, C.F., Gorelick, S.M.: Mapping hydraulic conductivity: Sequential conditioning with measurements of solute arrival time, hydraulic head, and local conductivity. Water Resourc. Res. 31(7), 1615–1626 (1995)CrossRefGoogle Scholar
- Healy, R.W., Cook, P.: Using groundwater levels to estimate recharge. Hydrogeol. J. 10(10), 91–109 (2002)CrossRefGoogle Scholar
- Hill, M.C., Tiedeman, C.R.: Effective Groundwater Model Calibration: With Analysis of Data, Sensitivities, Predictions, and Uncertainty, vol. 480, 1st edn. Wiley-Interscience, Berlin (2007)Google Scholar
- Irsa, J., Zhang, Y.: A new direct method of parameter estimation for steady state flow in heterogeneous aquifers with unknown boundary conditions. Water Resourc. Res. 48, W09526 (2012). doi:10.1029/2011WR011756 CrossRefGoogle Scholar
- Jyrkama, M.I., Sykes, J.F., Norman, S.D.: Recharge estimation for transient ground water modeling. Ground Water 40(6), 638–648 (2002)CrossRefGoogle Scholar
- Keating, E.H., Doherty, J., Vrugt, J.A., Kang, Q.: Optimization and uncertainty assessment of strongly nonlinear groundwater models with high parameter dimensionality. Water Resourc. Res. 46, W10517 (2010). doi:10.1029/2009WR008584 CrossRefGoogle Scholar
- Li, W., Englert, A., Cirpka, O.A., Vereecken, H.: Three dimensional geostatistical inversion of flowmeter and pumping test data. Ground Water 46(2), 193–201 (2008)CrossRefGoogle Scholar
- Lin, Y.F., Wang, J., Valocchi, A.: PRO-GRADE: GIS toolkits for ground water recharge and discharge estimation. Ground Water 47(1), 122–128 (2009)CrossRefGoogle Scholar
- Liu, G., Chen, Y., Zhang, D.: Investigation of flow and transport processes at the MADE site using ensemble Kalman filter. Adv. Water Resourc. 31, 975–986 (2008)CrossRefGoogle Scholar
- Liu, X., Kitanidis, P.: Large-scale inverse modeling with an application in hydraulic tomography. Water Resourc. Res. 47, W02501 (2011). doi:10.1029/2010WR009144 Google Scholar
- Mao, D., Yeh, T.C.J., Wan, L., Wen, J.C., Lu, W., Lee, C.H., Hsu, K.C.: Joint interpretation of sequential pumping tests in unconfined aquifers. Water Resourc. Res. 49, 1782–1796 (2013)CrossRefGoogle Scholar
- Mao, D., Yeh, T.C.J., Wan, L., Hsu, K.C., Lee, C.H., Wen, J.C.: Necessary conditions for inverse modeling of flow through variably saturated porous media. Adv. Water Resourc. 52, 50–61 (2013)CrossRefGoogle Scholar
- McKenna, S., Poeter, E.: Field example of data fusion for site characterization. Water Resourc. Res. 31(12), 3229–3240 (1995)CrossRefGoogle Scholar
- Mishra, P.K., Neuman, S.P.: Improved forward and inverse analyses of saturated-unsaturated flow toward a well in a compressible unconfined aquifer. Water Resourc. Res. 46, W07508 (2010). doi:10.1029/2009WR008899 CrossRefGoogle Scholar
- Mishra, P.K., Neuman, S.P.: Saturated-unsaturated flow to a well with storage in a compressible unconfined aquifer. Water Resourc. Res. 47(5), W05553 (2011). doi:10.1029/2010WR010177 CrossRefGoogle Scholar
- Moench, A., Garabedian, S., LeBlanc, D.: Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, Cape Cod, Massachusetts. In: US Geological Survey Professional Paper, vol. 1629, pp. 1–51. US Geological Survey, Reston (2001)Google Scholar
- Moench, A.F.: Importance of the vadose zone in analyses of unconfined aquifer tests. Ground Water 42(2), 223–233 (2004). doi:10.1111/j.1745 CrossRefGoogle Scholar
- Neuman, S.P.: Theory of flow in unconfined aquifers considering delayed response of the water table. Water Resourc. Res. 8(4), 1031–1045 (1972)CrossRefGoogle Scholar
- Neuman, S.: Generalized scaling of permeabilities; validation and effect of support scale. Geophys. Res. Lett. 21(5), 349–352 (1994)CrossRefGoogle Scholar
- Neuman, S.P., Blattstein, A., Riva, M., Tartakovsky, D.M., Guadagnini, A., Ptak, T.: Type curve interpretation of late-time pumping test data in randomly heterogeneous aquifers. Water Resourc. Res. 43, W10421 (2007). doi:10.1029/2007WR005871 CrossRefGoogle Scholar
- Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching, vol. 380, 1st edn. Cambridge Unviersity Press, Cambridge (2008)Google Scholar
- Pan, L., Warrick, A.W., Wierenga, P.: Downward water flow through sloping layers in the vadose zone: time-dependence and effect of slope length. J. Hydrol. 199, 36–52 (1997)CrossRefGoogle Scholar
- Portniaguine, O., Solomon, D.: Parmater estimation using groundwater age and head data, Cap Cod. Massachusetts. Water Resourc. Res. 34(4), 637–645 (1998)CrossRefGoogle Scholar
- Reynolds, D.A., Marimuthu, S.: Deuterium composition and flow path analysis as additional calibration targets to calibrate groundwater flow simulation in a coastal wetlands system. Hydrogeol. J. 15, 515–535 (2007)CrossRefGoogle Scholar
- Russo, D., Zaidel, J., Lauter, A.: Numerical analysis of flow and transport in a combined heterogeneous vadose zone-groundwater system. Adv. Water Resourc. 24, 49–62 (2001)CrossRefGoogle Scholar
- Saiers, J.E., Genereux, D.P., Bolster, C.H.: Influence of calibration methodology on ground water flow predictions. Ground Water 42(1), 32–44 (2004)CrossRefGoogle Scholar
- Sakaki, T., Frippiat, C.C., Komatsu, M., Illangasekare, T.H.: On the value of lithofacies data for improving groundwater flow model accuracy in a three-dimensional laboratory-scale synthetic aquifer. Water Resourc. Res. 45, W11404 (2009). doi:10.1029/2008WR007229 Google Scholar
- Sanchez-Vila, X., Carrera, J., Girardi, J.: Scale effects in transmissivity. J. Hydrol. 183, 1–22 (1996)CrossRefGoogle Scholar
- Scalon, B.R., Healy, R.W., Cook, P.G.: Choosing appropriate techniques for quantifying groundwater recharge. Hydrogeol. J. 10, 18–37 (2002)CrossRefGoogle Scholar
- Schulze-Makuch, D., Carlson, D., Cherkauer, D., Malik, P.: Scale dependency of hydraulic conductivity in heterogeneous media. Ground Water 37(6), 904–919 (1999)CrossRefGoogle Scholar
- Simmers, I.: Groundwater Recharge: An Overview Of Estimation “Problems” and Recent Developments, pp. 107–115. Geological Society of London, London (1998)Google Scholar
- Tan, S., Shuy, E., Chua, L.: Regression method for estimating rainfall recharge at unconfined sandy aquifers with an equatorial climate. Hydrogeol. Process. 21, 3514–3526 (2007)CrossRefGoogle Scholar
- The Mathworks Inc.: Optimization Toolbox\(^{\rm TM}\) Users Guide. Mathworks, Natick (2012)Google Scholar
- Tiedeman, C.R., Hill, M.C., D’Agnese, F.A., Faunt, C.C.: Methods for using groundwater model predictions to guide hydrogeologic data collection, with application to the Death Valley regional groundwater flow system. Water Resourc. Res. 39(1), 1010 (2003). doi:10.1029/2001WR001255 CrossRefGoogle Scholar
- Wang, D., Zhang, Y., Irsa, J.: Proceeding of the 2013 AGU Hydrology Days (2013). http://hydrologydays.colostate.edu/Papers_13/Dongdong_paper.pdf. Accessed 16 August 2013
- Zhang, Y., Gable, C.W., Person, M.: Equivalent hydraulic conductivity of an experimental stratigraphy–implications for basin-scale flow simulations. Water Resourc. Res. 42(7), W05404 (2006). doi:10.1029/2005WR004720 Google Scholar
- Zlotnik, V.A., Zurbuchen, B.R.: Estimation of hydraulic conductivity from borehole flowmeter tests considering head losses. J. Hydrol. 281, 115–128 (2003)CrossRefGoogle Scholar