Analytical sensitivity analysis of transient groundwater flow in a bounded model domain using the adjoint method (original) (raw)
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Illustration and Verification of Adjoint Sensitivity Theory for Steady State Groundwater Flow
Water Resources Research, 1985
The application of adjoint sensitivity theory to steady state groundwater flow is illustrated with three one-dimensional flow problems. Adjoint states are analytically derived for four performance measures of these test problems: hydraulic head at a point, spatially average hydraulic head, Darcy velocity at a point, and flux from a prescribed head boundary. Sensitivity coefficients are analytically calculated for average head. The adjoint states are interpreted and their usefulness is discussed. The implementation of a numerical adjoint sensitivity flow code to solve for these adjoint states is described, and the computed adjoint states are used in the code to evaluate the sensitivities of model results to model input parameters. The one-dimensional flow problems provide a set of verification tests for the numerical code. The numerical code successfully reproduces both the analytically derived adjoint states, including those involving jump conditions, and the sensitivity coefficients for model output values that are nonlinear with respect to model parameters. INTRODUCTION Numerical models are used in many groundwater-related activities, including the design of subsurface waste disposal facilities, comparisons of methods for remedial action at existing disposal sites, and water resource evaluations. The sensitivity of the numerical solution to model parameters is often of interest in these analyses. Adjoint sensitivity theory provides a computationally efficient method for determining the sensitivities of user-defined scalar measures of system behavior or model performance, called performance measures, to various model parameters. In this method the sensitivity coefficients are exact derivatives (except for round-off error) of the performance measures with respect to the parameters, taken about the assumed parameter values. Sensitivity coefficients are used to indicate the relative sensitivity of performance measures to various parameters [e.g., Sykes et al., 1985], to guide a gradient search in nonlinear optimization [e.g., Neuman, 1980a, b; Townley and Wilson, 1983], or to perform first and second-order, second-moment uncertainty analyses [e.g., Dettinger and Wilson, 1981]. Adjoint states, which are computed as part of the process of determining sensitivity coefficients, may provide additional information regarding system performance. For examle, in groundwater flow models they indicate the impact that a unit injection of water, anywhere in the system, will have on the performance measure. Another and much more common approach to compute sensitivity coefficients for numerical models is to perform multiple computer runs of the model while slightly perturbing the parameters from run to run. This multiple-run approach to derivative ccomputation requires one base run and one or more additional runs for each parameter for which sensitivities are to be tested. In an application with many such parameters, the multiple-run approach to sensitivity coefficients can become prohibitively expensive. The adjoint method avoids the use of repetitive computer runs of the model; it employs
Sensitivity Analysis for Steady State Groundwater Flow Using Adjoint Operators
Water Resources Research, 1985
Adjoint sensitivity theory is currently being considered as a potential method for calculating the sensitivity of nuclear waste repository performance measures to the parameters of the system. For groundwater flow systems, performance measures of interest include piezometric heads in the vicinity of a waste site, velocities or travel time in aquifers, and mass discharge to biosphere points. The parameters include recharge‐discharge rates, prescribed boundary heads or fluxes, formation thicknesses, and hydraulic conductivities. The derivative of a performance measure with respect to the system parameters is usually taken as a measure of sensitivity. To calculate sensitivities, adjoint sensitivity equations are formulated from the equations describing the primary problem. The solution of the primary problem and the adjoint sensitivity problem enables the determination of all of the required derivatives and hence related sensitivity coefficients. In this study, adjoint sensitivity theo...
Advances in Water Resources, 2010
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 realworld 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.
Physically Based Groundwater Vulnerability Assessment Using Sensitivity Analysis Methods
Groundwater, 2013
A general physically based method is presented to assess the vulnerability of groundwater to external pressures by numerical simulation of groundwater flow. The concept of groundwater vulnerability assessment considered here is based on the calculation of sensitivity coefficients for a user-defined groundwater state for which we propose several physically based indicators. Two sensitivity analysis methods are presented: the sensitivity equation method and the adjoint operator method. We show how careful selection of a method can significantly minimize the computational effort. An illustration of the general methodology is presented for the Herten aquifer analog (Germany). This application to a simple, yet insightful, case demonstrates the potential use of this general and physically based vulnerability assessment method to complex aquifers.
Journal of Computational Physics, 2006
A multivariate Analysis of Variance (ANOVA) is used to measure the relative sensitivity of groundwater flow to two factors that indicate different dimensions of aquifer heterogeneity. An aquifer is modeled as the union of disjoint volumes, or blocks, composed of different materials with different hydraulic conductivities. The factors are correlation between the hydraulic conductivities of the different materials and the contrast between mean conductivities in the different materials. The precise values of aquifer properties are usually uncertain because they are only sparsely sampled, yet are highly heterogeneous. Hence, the spatial distribution of blocks and the distribution of materials in blocks are uncertain and are modeled as stochastic processes. The ANOVA is performed on a large sample of Monte Carlo simulations of a simple model flow system composed of two materials distributed within three disjoint blocks. Our key finding is that simulated flow is much more sensitive to the contrast between mean conductivities of the blocks than it is to the intensity of correlation, although both factors are statistically significant. The methodology of the experiment-ANOVA performed on Monte Carlo simulations of a multi-material flow system-constitutes the basis of additional studies of more complicated interactions between factors that define flow and transport in aquifers with uncertain properties.
Sensitivity Analysis for Hydraulic Models
Journal of Hydraulic Engineering, 2009
Sensitivity analysis is well recognized as being an important aspect of the responsible use of hydraulic models. This paper reviews a range of methods for sensitivity analysis. Two applications, one to a simple pipe bend example and the second to an advanced Shallow Water Equation solver, illustrate the deficiencies of standardized regression coefficients in the context of functionally nonlinear models. Derivatives and other local methods of sensitivity analysis are shown to give an incomplete picture of model response over the range of variability in the model inputs. The use of global variance-based sensitivity analysis is shown to be more general in its applicability and in its capacity to reflect nonlinear processes and the effects of interactions among variables.
Sensitivity of groundwater flow with respect to the drain–aquifer leakage coefficient
Journal of Hydroinformatics, 2017
Mitigation measures may be used to prevent soil and water pollution from waste disposal, landfill sites, septic or chemical storage tanks. Among them, drains and impervious barriers may be set up. The efficiency of this technique can be evaluated by means of groundwater modeling tools. The groundwater flow and the leakage drain-aquifer interactions are implemented in a conforming finite element method (FEM) and a mixed hybrid FEM (MHFEM) in a horizontal two-dimensional domain modeling regional aquifer below chemical storage tanks. Considering the influence of uncertainties in the drain-aquifer exchange rate parameter and using an automatic differentiation (AD) tool, the aim of this paper is to carry out a sensitivity analysis with respect to the leakage coefficient for the piezometric head, velocity field, and streamlines to provide a new insight into groundwater waterbody exchanges. Computations are performed with both an ideal homogeneous hydraulic conductivity and a realistic heterogeneous one. The tangent linear codes are validated using Taylor tests performed on the head and the velocity field. The streamlines computed using AD are well approximated in comparison with the nondifferentiated codes. Piezometric head computed by the MHFEM is the more sensitive, particularly near to the drain, than the FEM one.
MODFLOW-2005 ground water model—User guide to the adjoint state based sensitivity process (ADJ)
2007
This report describes the adjoint state based sensitivity process for MODFLOW-2005 that calculates the sensitivity of observations to parameters. The process is composed of three basic components, described here for one observation and one parameter; first is the solution of the groundwater flow problem, next this solution is used to calculate the adjoint state for the observation, and finally the sensitivity of the of the observation to the parameter is determined by summing the product of the adjoint state with the derivative of the groundwater flow equations with respect to the parameter for each time step of the flow simulation. The theoretical development presents the mathematical basis for the second two steps in the process.
Sensitivity analysis method of system identification and its potential in hydrologic research
Water Resources Research, 1969
Sensitivity analysis should be an integral part of nearly every hydrologic study. At an elementary level, sensitivity analysis is useful in studying the relative sensitivity of the result to the data input. Studying the sensitivity of a hydrologic system to changes in its parameters and initial conditions makes it possible not only to gain insight into a system's behavior but to derive simple computational algorithms for the identification of unknown parameters. The fact that sensitivity analysis leads to simple initial-value problems makes it ideal for mechanization on an analog computer. The computational steps involved in implementing identification algorithms based on sensitivity analysis are relatively simpler than those based on such other methods as quasilinearization. The applicability of this method to identify both lumped and distributed hydrologic systems with deterministic or statistical input-output data is demonstrated.
Water Resources Research, 1991
The net work and energy flux at the boundaries of an aquifer change its internal energy and overcome its resistance to flow. In saturated porous media, the change in internal (strain) energy is stored in the elastic soil matrix and in pore water compression. In unsaturated media, an additional term accounts for changes in gravitational potential energy. The energy approach complements conventional insight by allowing spatially distributed processes to be integrated into energy and work terms which characterize a system's response to a set of excitations. Specifically, a technique is developed in this paper to interpret the dynamic behavior of a one-dimensional leaky aquifer in terms of its composite energy functions. In particular, the work interaction at the leaky boundary is used as an index of the significance of the leakage: when the work parameter indicates a relatively small leakage, the flow components of the multiaquifer can be isolated and modeled separately with a controllable loss of accuracy.