Using data assimilation method to calibrate a heterogeneous conductivity field conditioning on transient flow test data

A data assimilation method is developed to calibrate a heterogeneous hydraulic conductivity field conditioning on transient pumping test data. The ensemble Kalman filter (EnKF) approach is used to update model parameters such as hydraulic conductivity and model variables such as hydraulic head using available data. A synthetical two-dimensional flow case is used to assess the capability of the EnKF method to calibrate a heterogeneous conductivity field by assimilating transient flow data from observation wells under different hydraulic boundary conditions. The study results indicate that the EnKF method will significantly improve the estimation of the hydraulic conductivity field by assimilating continuous hydraulic head measurements and the hydraulic boundary condition will significantly affect the simulation results. For our cases, after a few data assimilation steps, the assimilated conductivity field with four Neumann boundaries matches the real field well while the assimilated conductivity field with mixed Dirichlet and Neumann boundaries does not. We found in our cases that the ensemble size should be 300 or larger for the numerical simulation. The number and the locations of the observation wells will significantly affect the hydraulic conductivity field calibration.     

The power of transient piezometric head data in inverse modeling: An application of the localized normal-score EnKF with covariance inflation in a heterogenous bimodal hydraulic conductivity field

The localized normal-score ensemble Kalman filter (NS-EnKF) coupled with covariance inflation is used to characterize the spatial variability of a channelized bimodal hydraulic conductivity field, for which the only existing prior information about conductivity is its univariate marginal distribution. We demonstrate that we can retrieve the main patterns of the reference field by assimilating a sufficient number of piezometric observations using the NS-EnKF. The possibility of characterizing the conductivity spatial variability using only piezometric head data shows the importance of accounting for these data in inverse modeling.     

Worth of secondary data compared to piezometric data for the probabilistic assessment of radionuclide migration

A common approach for the performance assessment of radionuclide migration from a nuclear waste repository is by means of Monte-Carlo techniques. Multiple realizations of the parameters controlling radionuclide transport are generated and each one of these realizations is used in a numerical model to provide a transport prediction. The statistical analysis of all transport predictions is then used in performance assessment. In order to reduce the uncertainty on the predictions is necessary to incorporate as much information as possible in the generation of the parameter fields. In this regard, this paper focuses in the impact that conditioning the transmissivity fields to geophysical data and/or piezometric head data has on convective transport predictions in a two-dimensional heterogeneous formation. The Walker Lake data based is used to produce a heterogeneous log-transmissivity field with distinct non-Gaussian characteristics and a secondary variable that represents some geophysical attribute. In addition, the piezometric head field resulting from the steady-state solution of the groundwater flow equation is computed. These three reference fields are sampled to mimic a sampling campaign. Then, a series of Monte-Carlo exercises using different combinations of sampled data shows the relative worth of secondary data with respect to piezometric head data for transport predictions. The analysis shows that secondary data allows to reproduce the main spatial patterns of the reference transmissivity field and improves the mass transport predictions with respect to the case in which only transmissivity data is used. However, a few piezometric head measurements could be equally effective for the characterization of transport predictions.     

Worth of secondary data compared to piezometric data for the probabilistic assessment of radionuclide migration

A common approach for the performance assessment of radionuclide migration from a nuclear waste repository is by means of Monte-Carlo techniques. Multiple realizations of the parameters controlling radionuclide transport are generated and each one of these realizations is used in a numerical model to provide a transport prediction. The statistical analysis of all transport predictions is then used in performance assessment. In order to reduce the uncertainty on the predictions is necessary to incorporate as much information as possible in the generation of the parameter fields. In this regard, this paper focuses in the impact that conditioning the transmissivity fields to geophysical data and/or piezometric head data has on convective transport predictions in a two-dimensional heterogeneous formation. The Walker Lake data based is used to produce a heterogeneous log-transmissivity field with distinct non-Gaussian characteristics and a secondary variable that represents some geophysical attribute. In addition, the piezometric head field resulting from the steady-state solution of the groundwater flow equation is computed. These three reference fields are sampled to mimic a sampling campaign. Then, a series of Monte-Carlo exercises using different combinations of sampled data shows the relative worth of secondary data with respect to piezometric head data for transport predictions. The analysis shows that secondary data allows to reproduce the main spatial patterns of the reference transmissivity field and improves the mass transport predictions with respect to the case in which only transmissivity data is used. However, a few piezometric head measurements could be equally effective for the characterization of transport predictions.     

Inverse sequential simulation: Performance and implementation details

For good groundwater flow and solute transport numerical modeling, it is important to characterize the formation properties. In this paper, we analyze the performance and important implementation details of a new approach for stochastic inverse modeling called inverse sequential simulation (iSS). This approach is capable of characterizing conductivity fields with heterogeneity patterns difficult to capture by standard multiGaussian-based inverse approaches. The method is based on the multivariate sequential simulation principle, but the covariances and cross-covariances used to compute the local conditional probability distributions are computed by simple co-kriging which are derived from an ensemble of conductivity and piezometric head fields, in a similar manner as the experimental covariances are computed in an ensemble Kalman filtering. A sensitivity analysis is performed on a synthetic aquifer regarding the number of members of the ensemble of realizations, the number of conditioning data, the number of piezometers at which piezometric heads are observed, and the number of nodes retained within the search neighborhood at the moment of computing the local conditional probabilities. The results show the importance of having a sufficiently large number of all of the mentioned parameters for the algorithm to characterize properly hydraulic conductivity fields with clear non-multiGaussian features.     

Using data assimilation method to calibrate a heterogeneous conductivity field and improve solute transport prediction with an unknown contamination source

Hydraulic conductivity distribution and plume initial source condition are two important factors affecting solute transport in heterogeneous media. Since hydraulic conductivity can only be measured at limited locations in a field, its spatial distribution in a complex heterogeneous medium is generally uncertain. In many groundwater contamination sites, transport initial conditions are generally unknown, as plume distributions are available only after the contaminations occurred. In this study, a data assimilation method is developed for calibrating a hydraulic conductivity field and improving solute transport prediction with unknown initial solute source condition. Ensemble Kalman filter (EnKF) is used to update the model parameter (i.e., hydraulic conductivity) and state variables (hydraulic head and solute concentration), when data are available. Two-dimensional numerical experiments are designed to assess the performance of the EnKF method on data assimilation for solute transport prediction. The study results indicate that the EnKF method can significantly improve the estimation of the hydraulic conductivity distribution and solute transport prediction by assimilating hydraulic head measurements with a known solute initial condition. When solute source is unknown, solute prediction by assimilating continuous measurements of solute concentration at a few points in the plume well captures the plume evolution downstream of the measurement points.     

Numerical method of moments for solute transport in a nonstationary flow field conditioned on hydraulic conductivity and head measurements

The unconditional stochastic studies on groundwater flow and solute transport in a nonstationary conductivity field show that the standard deviations of the hydraulic head and solute flux are very large in comparison with their mean values (Zhang et al. in Water Resour Res 36:2107–2120, 2000; Wu et al. in J Hydrol 275:208–228, 2003; Hu et al. in Adv Water Resour 26:513–531, 2003). In this study, we develop a numerical method of moments conditioning on measurements of hydraulic conductivity and head to reduce the variances of the head and the solute flux. A Lagrangian perturbation method is applied to develop the framework for solute transport in a nonstationary flow field. Since analytically derived moments equations are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. Instead of using an unconditional conductivity field as an input to calculate groundwater velocity, we combine a geostatistical method and a method of moment for flow to conditionally simulate the distributions of head and velocity based on the measurements of hydraulic conductivity and head at some points. The developed theory is applied in several case studies to investigate the influences of the measurements of hydraulic conductivity and/or the hydraulic head on the variances of the predictive head and the solute flux in nonstationary flow fields. The study results show that the conditional calculation will significantly reduce the head variance. Since the hydraulic head measurement points are treated as the interior boundary (Dirichlet boundary) conditions, conditioning on both the hydraulic conductivity and the head measurements is much better than conditioning only on conductivity measurements for reduction of head variance. However, for solute flux, variance reduction by the conditional study is not so significant.     

Discrete stability of the Differential System Method evaluated with geostatistical techniques

The Differential System Method (DSM) permits identification of the physical parameters of finite-difference groundwater flow models in a confined aquifer when piezometric head and source terms are known at each point of the finite-difference lattice for at least two independent flow situations for which the hydraulic gradients are not parallel. Since piezometric head data are usually few and sparse, interpolation of the measured data onto a regular grid can be performed with geostatistical techniques. We apply kriging to the sparse data of a synthetic aquifer to evaluate the stability of the DSM with respect to uncorrelated measurement errors and interpolation errors. The numerical results show that the DSM is stable.     

Discrete stability of the Differential System Method evaluated with geostatistical techniques

The Differential System Method (DSM) permits identification of the physical parameters of finite-difference groundwater flow models in a confined aquifer when piezometric head and source terms are known at each point of the finite-difference lattice for at least two independent flow situations for which the hydraulic gradients are not parallel. Since piezometric head data are usually few and sparse, interpolation of the measured data onto a regular grid can be performed with geostatistical techniques. We apply kriging to the sparse data of a synthetic aquifer to evaluate the stability of the DSM with respect to uncorrelated measurement errors and interpolation errors. The numerical results show that the DSM is stable.     

Assimilating transient groundwater flow data via a localized ensemble Kalman filter to calibrate a heterogeneous conductivity field

A localized ensemble Kalman filter (EnKF) method is developed to assimilate transient flow data to calibrate a heterogeneous conductivity field. To update conductivity value at a point in a study domain, instead of assimilating all the measurements in the study domain, only limited measurement data in an area around the point are used for the conductivity updating in the localized EnKF method. The localized EnKF is proposed to solve the problems of the filter divergence usually existing in a data assimilation method without localization. The developed method is applied, in a synthetical two dimensional case, to calibrate a heterogeneous conductivity field by assimilating transient hydraulic head data. The simulations by the data assimilation with and without localized EnKF are compared. The study results indicate that the hydraulic conductivity field can be updated efficiently by the localized EnKF, while it cannot be by the EnKF. The covariance inflation and localization are found to solve the problem of the filter divergence efficiently. In comparison with the EnKF method without localization, the localized EnKF method needs smaller ensemble size to achieve stabilized results. The simulation results by the localized EnKF method are much more sensitive to conductivity correlation length than to the localization radius. The developed localized EnKF method provides an approach to improve EnKF method in conductivity calibration.     

Including geophysical data in ground water model inverse calibration

A nonlinear regression method is developed that can be used to estimate parameters of a ground waterflow model from a combination of observations of hydrological variables and observations of geophysical properties that are functionally related with the hydraulic conductivity. The procedure estimates: parameters characterizing the hydraulic conductivity field (e.g., zonal or pilot point values); geophysical properties that have been observed and that are functionally related with the hydraulic conductivity parameters; and a few parameters of the function that relates the hydraulic conductivity parameters with the geophysical properties (the type of function is assumed known). A fidelity factor, sigma(r)2, of a term of the minimized objective function reflects the faith one has in the validity of this functional relationship. The estimation methodology has been tested by means of synthetic models. The experimental results demonstrate that the number of estimated hydraulic conductivity parameters can be increased by adding geophysical observations to the set of hydrological observations that are traditionally used for model calibration. The improvement of the estimated hydraulic conductivity field and the simulated hydraulic head field can be significant but is dependent on the number, the locations, and the uncertainty of geophysical observations. The sensitivity of the estimation results to the value of sigma(r) is small for the studied problems except when the uncertainty of geophysical observations is high. In the latter case, a large sigma(r) value was found to be optimal to avoid that hydraulic conductivity estimates are closely tied to corresponding but highly uncertain geophysical observations.     

Steady state flow with hydraulic conductivity change around large diameter wells

Darcian flow law in aquifers assumes that the aquifer hydraulic conductivity is constant and the groundwater movement is due only to the piezometric level changes through hydraulic gradient. In practice, after the well development the aquifer just around the well has comparatively larger hydraulic conductivity and gradient. Patchy aquifer solutions in the literature consider sudden hydraulic conductivity changes with distance for the steady state flow. The change of transmissivity is demonstrated by the application of slope‐matching procedure to actual field data. It is the main purpose of this paper to derive simple analytical expressions for aquifer parameter evaluations with steadily decreasing hydraulic conductivity around the well. Spatial nonlinear hydraulic conductivity changes around a large‐diameter well within the depression cone of a confined aquifer are considered as exponentially decreasing functions of the radial distance. Copyright © 2010 John Wiley & Sons, Ltd.     

AN EXERCISE IN GROUND-WATER MODEL CALIBRATION AND PREDICTION

Abstract. For a classroom exercise, nine groups of graduate students calibrated a numerical ground-water flow model to a set of perfectly observed hydraulic head data for a hypothetical phreatic aquifer. All groups used exactly the same numerical model and identical sets of observed data. After calibration, the students predicted the hydraulic head distribution in the aquifer resulting from a modification in one boundary condition. A quantitative analysis of the results of this calibration-prediction exercise vividly demonstrates some of the difficulties in parameter identification for ground-water flow models. Group predictions differed significantly. Successful prediction was strongly correlated with successful estimation of conductivity values, and was essentially unrelated to successful estimation of aquifer bottom elevations or with the number of trial-and-error simulations required for calibration. Most importantly, success in prediction was unrelated to success in matching observed heads under premodification conditions. In this sense, good calibration did not lead to good prediction.     

Pore size distributions and hydraulic conductivities of rocks derived from Magnetic Resonance Sounding relaxation data using multi-exponential decay time inversion

In hydrogeology there is a variety of empirical formulae available for determination of hydraulic conductivity of porous media, all based on the analysis of grain size distributions of aquifer materials. Sensitivity of NMR measurements to pore sizes makes it a good indicator of hydraulic conductivity. Analogous to laboratory NMR, Magnetic Resonance Sounding (MRS) relaxation data are of a multi-exponential (ME) nature due to the distribution of different pore sizes in an investigated rock layer. ME relaxation behaviour will also arise due to the superposition of NMR signals which originate from different layers. It has been shown, that both kinds of ME behaviour coexist in MRS and can principally be separated by ME inversion of the field data. Only a few publications exist that have proposed approaches to qualitatively and quantitatively estimate petrophysical parameters such as the hydraulic conductivity from MRS measurements, i.e. MRS porosity and decay times. The so far used relations for the estimation of hydraulic conductivity in hydrogeology and NMR experiments are compared and discussed with respect to their applicability in MRS. Taking into account results from a variety of laboratory NMR and MRS experiments mean rock specific calibration factors are introduced for a data-base-calibrated estimation of hydraulic conductivity when no on-site calibration of MRS is available. Field data have been analysed using conventional and ME inversion using such mean calibration values. The results for conventional and ME inversion agree with estimates obtained from well core analysis for shallow depths but are significantly improved using a ME inversion approach for greater depths.     

Three-dimensional geostatistical inversion of synthetic tomographic pumping and heat-tracer tests in a nested-cell setup

A main purpose of groundwater inverse modeling lies in estimating the hydraulic conductivity field of an aquifer. Traditionally, hydraulic head measurements, possibly obtained in tomographic setups, are used as data. Because the groundwater flow equation is diffusive, many pumping and observation wells would be necessary to obtain a high resolution of hydraulic conductivity, which is typically not possible. We suggest performing heat tracer tests using the same already installed pumping wells and thermometers in observation planes to amend the hydraulic head data set by the arrival times of the heat signals. For each tomographic combinations of wells, we recommend installing an outer pair of pumping wells, generating artificial ambient flow, and an inner well pair in which the tests are performed. We jointly invert heads and thermal arrival times in 3-D by the quasi-linear geostatistical approach using an efficiently parallelized code running on a mid-range cluster. In the present study, we evaluate the value of heat tracer versus head data in a synthetic test case, where the estimated fields can be compared to the synthetic truth. Because the sensitivity patterns of the thermal arrival times differ from those of head measurements, the resolved variance in the estimated field is 6 to 10 times higher in the joint inversion in comparison to inverting head data only. Also, in contrast to head measurements, reversing the flow field and repeating the heat-tracer test improves the estimate in terms of reducing the estimation variance of the estimate. Based on the synthetic test case, we recommend performing the tests in four principal directions, requiring in total eight pumping wells and four intersecting observation planes for heads and temperature in each direction.     

Detecting a Defective Casing Seal at the Top of a Bedrock Aquifer

An improperly sealed casing can produce a direct hydraulic connection between two or more originally isolated aquifers with important consequences regarding groundwater quantity and quality. A recent study by Richard et al. (2014) investigated a monitoring well installed in a fractured rock aquifer with a defective casing seal at the soil–bedrock interface. A hydraulic short circuit was detected that produced some leakage between the rock and the overlying deposits. A falling‐head permeability test performed in this well showed that the usual method of data interpretation is not valid in this particular case due to the presence of a piezometric error. This error is the direct result of the preferential flow originating from the hydraulic short circuit and the subsequent re‐equilibration of the piezometric levels of both aquifers in the vicinity of the inlet and the outlet of the defective seal. Numerical simulations of groundwater circulation around the well support the observed impact of the hydraulic short circuit on the results of the falling‐head permeability test. These observations demonstrate that a properly designed falling‐head permeability test may be useful in the detection of defective casing seals.     

Hydraulic Gradient Comparison Method to Estimate Aquifer Hydraulic Parameters Under Steady-State Conditions

The hydraulic gradient comparison method is an inverse method for estimation of aquifer hydraulic conductivity (or trans-missivity) and boundary conductance for a ground water flow model under steady-state conditions. This method, following formal optimization techniques, defines its objective function to minimize differences between interpreted (observed) and simulated hydraulic gradients, which results in minimization of differences between observed and simulated hydraulic heads. The key features of this method are that (1) the derived optimality conditions have an explicit form with a clear hydrology concept that is con-sistent with Darcy's law, and (2) the derived optimality conditions are spatially independent as they are a function of only local hydraulic conductivity and local hydraulic gradient. This second feature allows a multidimensional optimization problem to be solved by many one-dimensional optimization procedures simultaneously, which results in a substantial reduction in computation time. The results of the numerical performance testing on a heterogeneous hypothetical case confirm that minimizing gradient residuals in the entire model domain leads to minimizing head residuals. Application of the method in real-world projects requires rigorous conceptual model development, use of a global calibration target, and an iterative calibration proess. The conceptual model development includes interpretation of a potentiometric surface and estimation of other hydrologic parameters. This method has been applied to a wide range of real-world modeling projects, including the Rocky Mountain Arsenal and Rocky Flats sites in Colorado, which demonstrates that the method is efficient and practical.     

Identification of hydraulic conductivity distributions in density dependent flow fields of submarine groundwater discharge modeling using adjoint-state sensitivities

A mathematical optimal control method is developed to identify a hydraulic conductivity distribution in a density dependent flow field. Using a variational method, the adjoint partial differential equations are obtained for the density- dependent state equations used for the saline aquifer water flow. The adjoint equations are numerically solved in through a finite difference method. The developed method is applied to identify the hydraulic conductivity distribution through the numerical solution of an optimal control problem. To demonstrate the effectiveness of the optimal control method, three numerical experiments are conducted with artificial observation data. The results indicate that the developed method has the potential to accurately identify the hydraulic conductivity distribution in a saline water aquifer flow system.     

Comparison between gradient based UCODE_2005 and the ensemble Kalman Filter for transient groundwater flow inverse modeling

Gradient based UCODE_2005 and data assimilation based on the Ensemble Kalman Filter(EnKF) are two different inverse methods. A synthetic two-dimensional flow case with four no-flow boundaries is used to compare the UCODE_2005 with the Ensemble Kalman Filter(EnKF) for their efficiency to inversely calculate and calibrate a hydraulic conductivity field based on hydraulic head data. A zonal, random heterogeneous conductivity field is calibrated by assimilating the time series of heads observed in monitoring wells. The study results indicate that the two inverse methods, UCODE_2005 and EnKF, could be used to calibrate the hydraulic conductivity field to a certain degree. More available observations and information about the conductivity field, more accurate inverse results will be obtained for the UCODE_2005. On the other hand, for a realistic zonal heterogeneous hydraulic conductivity field, EnKF can only efficiently determine the hydraulic conductivity field at the first several assimilated time steps. The results obtained by the UCODE_2005 look better than those by the EnKF. This is possibly due to the fact that the UCODE_2005 uses observed head data at every time step, while EnKF can only use observed heads at first several steps due to the filter divergence problem.     

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

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