Johan R. Valstar

A representer‐based inverse method for groundwater flow and transport applications

Groundwater inverse problems are concerned with the estimation of uncertain model parameters, such as hydraulic conductivity, from field or laboratory measurements. In practice, model and measurement errors compromise the ability of inverse procedures to provide accurate results. It is important to account for such errors in order to determine the proper weight to give to each source of information. Probabilistic descriptions of model and measurement errors can be incorporated into classical variational inverse procedures, but the computational demands are excessive if the model errors vary over time. An alternative approach based on representer expansions is able to efficiently accommodate time-dependent errors for large problems. In the representer approach, unknown variables are expanded in finite series which depend on unknown fonctions called representers. Each representer quantifies the influence of a given measurement on the estimate of a particular variable. This procedure replaces the original inverse problem by an equivalent problem where the number of independent unknowns is proportional to the number of measurements. The representer approach is especially advantageous in groundwater problems, where the total number of measurements is often small. This approach is illustrated with a synthetic flow and transport example that includes time-dependent model errors. The representer algorithm is able to provide good estimates of a spatially variable hydraulic conductivity field and good predictions of solute concentration. Its computational demands are reasonable, and it is relatively easy to implement. The example reveals that it is beneficial to account for model errors even when they are difficult to estimate.

Inverse modeling of groundwater flow using model reduction

Numerical groundwater flow models often have a very high number of model cells (greater than a million). Such models are computationally very demanding, which is disadvantageous for inverse modeling. This paper describes a low?dimensional formulation for groundwater flow that reduces the computational burden necessary for inverse modeling. The formulation is a projection of the original groundwater flow equation on a set of orthogonal patterns (i.e., a Galerkin projection). The patterns (empirical orthogonal functions) are computed by a decomposition of the covariance matrix over an ensemble of model solutions. Those solutions represent the behavior of the model as a result of model impulses and the influence of a chosen set of parameter values. For an interchangeable set of parameter values the patterns yield a low?dimensional model, as the number of patterns is often small. An advantage of this model is that the adjoint is easily available and most accurate for inverse modeling. For several synthetical cases the low?dimensional model was able to find the global minimum efficiently, and the result was comparable to that of the original model. For several cases our model even converged where the original model failed. Our results demonstrate that the proposed procedure results in a 60% time reduction to solve the groundwater flow inverse problem. Greater efficiencies can be expected in practice for large?scale models with a large number of grid cells that are used to compute transient simulations.

Constraining a compositional flow model with flow‐chemical data using an ensemble‐based Kalman filter

Isothermal compositional flow models require coupling transient compressible flows and advective transport systems of various chemical species in subsurface porous media. Building such numerical models is quite challenging and may be subject to many sources of uncertainties because of possible incomplete representation of some geological parameters that characterize the systems processes. Advanced data assimilation methods, such as the ensemble Kalman filter (EnKF), can be used to calibrate these models by incorporating available data. In this work, we consider the problem of estimating reservoir permeability using information about phase pressure as well as the chemical properties of fluid components. We carry out state-parameter estimation experiments using joint and dual updating schemes in the context of the EnKF with a two-dimensional single-phase compositional flow model (CFM). Quantitative and statistical analyses are performed to evaluate and compare the performance of the assimilation schemes. Our results indicate that including chemical composition data significantly enhances the accuracy of the permeability estimates. In addition, composition data provide more information to estimate system states and parameters than do standard pressure data. The dual state-parameter estimation scheme provides about 10% more accurate permeability estimates on average than the joint scheme when implemented with the same ensemble members, at the cost of twice more forward model integrations. At similar computational cost, the dual approach becomes only beneficial after using large enough ensembles.

Measurement network design including traveltime determinations to minimize model prediction uncertainty

Traveltime determinations have found increasing application in the characterization of groundwater systems. No algorithms are available, however, to optimally design sampling strategies including this information type. We propose a first-order methodology to include groundwater age or tracer arrival time determinations in measurement network design and apply the methodology in an illustrative example in which the network design is directed at contaminant breakthrough uncertainty minimization. We calculate linearized covariances between potential measurements and the goal variables of which we want to reduce the uncertainty: the groundwater age at the control plane and the breakthrough locations of the contaminant. We assume the traveltime to be lognormally distributed and therefore logtransform the age determinations in compliance with the adopted Bayesian framework. Accordingly, we derive expressions for the linearized covariances between the transformed age determinations and the parameters and states. In our synthetic numerical example, the derived expressions are shown to provide good first-order predictions of the variance of the natural logarithm of groundwater age if the variance of the natural logarithm of the conductivity is less than 3.0. The calculated covariances can be used to predict the posterior breakthrough variance belonging to a candidate network before samples are taken. A Genetic Algorithm is used to efficiently search, among all candidate networks, for a near-optimal one. We show that, in our numerical example, an age estimation network outperforms (in terms of breakthrough uncertainty reduction) equally sized head measurement networks and conductivity measurement networks even if the age estimations are highly uncertain. Copyright 2008 by the American Geophysical Union.

Measurement Network Design for Minimizing the Uncertainty in Breakthrough Predictions

We are developing a first-order measurement network optimization algorithm that can find the optimal combination and configuration of hydraulic conductivity, head and travel time measurements. The method is based on a representer-based inverse method for groundwater flow and transport applications (Valstar et al., Water Resources Research 2004). In the representer approach, unknown variables are linearly expanded in finite series which depend on unknown functions called representers. Each representer quantifies the influence of a given measurement on the estimate of a particular (state or static) variable. This procedure replaces the original inverse problem by an equivalent problem where the number of independent unknowns is reduced to the number of measurements. The representers can be shown to be equivalent to the linearized cross-covariance between the measurement and the (state or static) variable for which the representer is defined. As such, the representers can be used to estimate the prior covariances of certain goal variables, if a (pseudo) measurement for this variable is defined. Also, the representers can be used for a first-order approximation of the posterior covariances of the goal variables if a certain measurement set is assumed (because these covariances are functions of the prior variances and the cross-covariances between all measurements and the cross- covariances between the measurements and the goal variables). In our study, we are interested in minimizing the uncertainty in the prediction of contaminant breakthrough through confining layers. The uncertainty in breakthrough is a convolution between the contaminant arrival time and contaminant arrival location probability at the bottom of the confining layer. So the contaminant arrival time and the contaminant arrival location are our goal variables, and we derived representer definitions for advectively transported particles accordingly. The posterior covariance of the goal variables obtained by the representer method are based on a normal distribution. Using Monte Carlo calculation it turned out that arrival times are nearly log-normally distributed. Therefore we choose the logarithm of the arrival time rather than the arrival time itself as the goal variable. The derivation of the travel time representers now requires an additional linearization. However, the uncertainty estimates (both prior and posterior) of the arrival times as given by the thus derived travel time representers turned out to approximate Monte Carlo results very well , even for large variances of the hydraulic conductivity field. Once all representers that describe the relationships between a chosen set of potential measurements (with their locations) and the goal variables are known, the posterior covariances of the goal variables and therefore also the posterior breakthrough prediction uncertainty can in principle be calculated for every possible measurement set. Because the number of possible measurement combinations is usually excessively large, we use a Genetic Algorithm for an efficient search for a (close to) optimal measurement network within the available configuration space. The novelty of this work lies primarily in the incorporation of travel time measurements (for example Tritium/3He measurements) into a measurement network optimization algorithm, which to our knowledge has never been reported before. It requires an additional linearization in the derivation of the travel time representers, which is also new. Furthermore, our approach of evaluating the performance of a measurement network to the convolution of arrival time and arrival location probability is a new approach. Within the applied inverse method this requires the derivation of particle position representers, which has not been published before.