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Multivariate calibration using measurements of multiple water balance components has emerged as a potential solution for improving the performance and realism of large scale hydrologic models. In this study we develop a novel multivariate calibration framework to rigorously test whether incorporation of multiple water balance components into calibration can result in sufficiently accurate (behavioral) solutions for all model responses. Unlike previous studies, we use Bayesian calibration to formally define limits of acceptability or error thresholds in order to distinguish behavioral solutions for each of the incorporated fluxes. We apply the framework in the Mississippi river basin for the calibration of a large scale distributed hydrologic model (Noah-MP) with different combinations of model responses - evapotranspiration (ET), soil moisture (SM), and streamflow (SF). The results of the study show that incorporation of additional fluxes and soil moisture (a storage variable) is not always valuable due to significant trade-offs in accuracy among the model responses. In our experiments, only ET and SF could be simulated simultaneously to a reasonable degree of accuracy. In addition, we quantify the trade-offs in accuracy between the model responses using the concept of Pareto optimality. We find that combining ET with other fluxes entails higher trade-offs in accuracy compared to either SM or SF. Unlike deterministic calibration, with the developed framework we are able to identify deficiencies in model parameterization that lead to significant trade-offs in accuracy, especially between ET and SM. We find that the parameters which are insensitive to individual model responses can influence the trade-off relationship between them.

Computer simulations of groundwater flow and solute transport are often burdened by long runtimes. The simulations are necessarily complex to capture the system dynamics and finely discretized spatial and temporal domains are often needed for solution accuracy and stability. Model reduction allows for the approximation of system state by solving equations in a reduced dimensional space. Proper orthogonal decomposition (POD) is an effective way to reduce the dimensionality of systems of differential equations that are discretized by finite difference or finite element methods. If the problems are nonlinear in nature, the discrete empirical interpolation method (DEIM) has been shown to supplement POD by further reducing the dimension of nonlinear calculations. Here, the combined POD-DEIM approach is shown to work on a problem of 2-dimensional groundwater flow with solute transport exhibiting nonlinear sorption. The application is restricted to largely dispersive problems (low Peclet number). Results show areas of high concentration are effectively identified with mean errors less than 2% of the full model.

Past research has demonstrated that a reduced-order model (ROM) can be two-to-three orders of magnitude smaller than the original model and run considerably faster with acceptable error. A standard method to construct the system matrices for a ROM is Proper Orthogonal Decomposition (POD), which projects the system matrices from the full model space onto a subspace whose range spans the full model space but has a much smaller dimension than the full model space. This projection can be prohibitively expensive to compute if it must be done repeatedly, as with a Monte Carlo simulation. We propose a Fast Algorithm to reduce the computational burden of constructing the system matrices for a parameterized, reduced-order groundwater model (i.e. one whose parameters are represented by zones or interpolation functions). The proposed algorithm decomposes the expensive system matrix projection into a set of simple scalar-matrix multiplications. This allows the algorithm to efficiently construct the system matrices of a POD reduced-order model at a significantly reduced computational cost compared with the standard projection-based method. The developed algorithm is applied to three test cases for demonstration purposes. The first test case is a small, two-dimensional, zoned-parameter, finite-difference model; the second test case is a small, two-dimensional, interpolated-parameter, finite-difference model; and the third test case is a realistically-scaled, two-dimensional, zoned-parameter, finite-element model. In each case, the algorithm is able to accurately and efficiently construct the system matrices of the reduced-order model.

Nonlinear groundwater flow models have the propensity to be overly complex leading to burdensome computational demands. Reduced modeling techniques are used to develop an approximation of the original model that has smaller dimensionality and faster run times. The reduced model proposed is a combination of proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). Solutions of the full model (snapshots) are collected to represent the physical dynamics of the system and Galerkin projection allows the formulation of a reduced model that lies in a subspace of the full model. Interpolation points are added through DEIM to eliminate the reduced model's dependence on the dimension of the full model. POD is shown to effectively reduce the dimension of the full model and DEIM is shown to speed up the solution by further reducing the dimension of the nonlinear calculations. To show the concept can work for unconfined groundwater flow model, with added nonlinear forcings, one-dimensional and two-dimensional test cases are constructed in MODFLOW-OWHM. POD and DEIM are added to MODFLOW as a modular package. Comparing the POD and the POD-DEIM reduced models, the experimental results indicate similar reduction in dimension size with additional computation speed up for the added interpolation. The hyper-reduction method presented is effective for models that have fine discretization in space and/or time as well as nonlinearities with respect to the state variable. The dual reduction approach ensures that, once constructed, the reduced model can be solved in an equation system that depends only on reduced dimensions.

We develop an experimental design algorithm to select locations for a network of observation wells that provide the maximum robust information about unknown hydraulic conductivity in a confined, anisotropic aquifer. Since the information that a design provides is dependent on an aquifer's hydraulic conductivity, a robust design is one that provides the maximum information in the worst-case scenario. The design can be formulated as a max–min optimization problem. The problem is generally non-convex, non-differentiable, and contains integer variables. We use a Genetic Algorithm (GA) to perform the combinatorial search. We employ proper orthogonal decomposition (POD) to reduce the dimension of the groundwater model, thereby reducing the computational burden posed by employing a GA. The GA algorithm exhaustively searches for the robust design across a set of hydraulic conductivities and finds an approximate design (called the High Frequency Observation Well Design) through a Monte Carlo-type search. The results from a small-scale 1-D test case validate the proposed methodology. We then apply the methodology to a realistically-scaled 2-D test case.

Projection-based model reduction techniques have been shown to be very effective for reducing the computational burden of high-dimensional groundwater simulations, but only applied to confined groundwater flow. A new methodology is proposed that reduces the dimension of a discretized, transient, unconfined groundwater-flow model. This unconfined model reduction technique is based on Galerkin projection and the Newton formulation of MODFLOW. The method is implemented following the standard package design and code structure that MODFLOW employs for all its features. When the package is invoked within MODFLOW it can collect snapshots, produce a basis, construct the reduced model and propagate the reduced model forward in time. The new formulation accurately represents the water-table surface under a variety of nonlinear settings, such as intraborehole flow from a Multi-Node Well. The unconfined model reduction is applied to four test cases to illustrate its flexibility in handling nonlinear features. Several test cases are discussed to demonstrate the unconfined model reduction applicability. The final test case applies the new model reduction methodology to a scoping MODFLOW model of Santa Barbara, CA composed of 113,578 cells, which requires solving 113,578 equations per time step, and reduces it to 127 equations.

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.

This paper develops an iterative procedure for capacity expansion studies for water distribution systems. We propose a methodology to analyze an existing water distribution system and identify the potential bottlenecks in the system. Based on the results, capacity expansion alternatives are proposed and evaluated for improving the efficiency of water supply. The methodology includes a network flow based optimization model, four evaluation indices, and a series of evaluation steps. We first use a directed graph to configure the water distribution system into a network. The network flow based model optimizes the water distribution in the system so that different expansion alternatives can be evaluated on a comparable basis. This model lends itself to linear programming (LP) and can be easily solved by a standard LP code. The results from the evaluation tool help to identify the bottlenecks in the water distribution system and provide capacity expansion alternatives. A useful complementary tool for decision making is composed of a series of evaluation steps with the bottleneck findings, capacity expansion alternatives, and the evaluation of results. We apply the proposed methodology to the Tou-Qian River Basin, located in the northern region of Taiwan, to demonstrate its applicability in optimization and capacity expansion studies.

The paper develops an efficient macro-evolutionary multiobjective genetic algorithm (MMGA) for optimizing the rule curves of a multi-purpose reservoir system in Taiwan. Macro-evolution is a new kind of high-level species evolution that can avoid premature convergence that may arise during the selection process of conventional GAs. MMGA enriches the capabilities of GA to handle multiobjective problems by diversifying the solution set. Simulation results using a benchmark test problem indicate that the proposed MMGA yields better-spread solutions and converges closer to the true Pareto frontier than the nondominated sorting genetic algorithm-II (NSGA-II). When applied to a real case study, MMGA is able to generate uniformly spread solutions for a two-objective problem involving water supply and hydropower generation. Results of this work indicate that the proposed MMGA is highly competitive and provides a viable alternative to solve multiobjective optimization problems for water resources planning and management.

A quasi-Newton algorithm allied with the adjoint state equation method is proposed to solve the inverse problem of parameter estimation. The proposed method utilizes the information from both groundwater flow and solute transport in parameter estimation. It is capable of identifying parameters which control both flow and transport simultaneously. It also avoids the calculation of sensitivity coefficients, which normally requires much computational effort. The BFGS (Broyden, 2 Fletcher, 10 Goldfarb, 13 and Shanno 21 ) formula is used to approximate the Hessian matrix required in the optimization procedure. The proposed inverse procedure is applied to a two-dimensional flow-transport problem. Two dimensionless numbers, the Peclet number and the Damkoehler number, are used to characterize the behavior of the transport processes. Numerical experiments which were conducted demonstrate the usefulness of the proposed methodology and show good results in practical application.