Haiyan Zhou

Modeling transient groundwater flow by coupling ensemble Kalman filtering and upscaling

[1] The ensemble Kalman filter (EnKF) is coupled with upscaling to build an aquifer model at a coarser scale than the scale at which the conditioning data (conductivity and piezometric head) had been taken for the purpose of inverse modeling. Building an aquifer model at the support scale of observations is most often impractical since this would imply numerical models with many millions of cells. If, in addition, an uncertainty analysis is required involving some kind of Monte Carlo approach, the task becomes impossible. For this reason, a methodology has been developed that will use the conductivity data at the scale at which they were collected to build a model at a (much) coarser scale suitable for the inverse modeling of groundwater flow and mass transport. It proceeds as follows: (1) Generate an ensemble of realizations of conductivities conditioned to the conductivity data at the same scale at which conductivities were collected. (2) Upscale each realization onto a coarse discretization; on these coarse realizations, conductivities will become tensorial in nature with arbitrary orientations of their principal components. (3) Apply the EnKF to the ensemble of coarse conductivity upscaled realizations in order to condition the realizations to the measured piezometric head data. The proposed approach addresses the problem of how to deal with tensorial parameters, at a coarse scale, in ensemble Kalman filtering while maintaining the conditioning to the fine-scale hydraulic conductivity measurements. We demonstrate our approach in the framework of a synthetic worth-of-data exercise, in which the relevance of conditioning to conductivities, piezometric heads, or both is analyzed.

A pattern‐search‐based inverse method

[1]xa0Uncertainty in model predictions is caused to a large extent by the uncertainty in model parameters, while the identification of model parameters is demanding because of the inherent heterogeneity of the aquifer. A variety of inverse methods has been proposed for parameter identification. In this paper we present a novel inverse method to constrain the model parameters (hydraulic conductivities) to the observed state data (hydraulic heads). In the method proposed we build a conditioning pattern consisting of simulated model parameters and observed flow data. The unknown parameter values are simulated by pattern searching through an ensemble of realizations rather than optimizing an objective function. The model parameters do not necessarily follow a multi-Gaussian distribution, and the nonlinear relationship between the parameter and the response is captured by the multipoint pattern matching. The algorithm is evaluated in two synthetic bimodal aquifers. The proposed method is able to reproduce the main structure of the reference fields, and the performance of the updated model in predicting flow and transport is improved compared with that of the prior model.

Pattern Recognition in a Bimodal Aquifer Using the Normal-Score Ensemble Kalman Filter

The ensemble Kalman filter (EnKF) is now widely used in diverse disciplines to estimate model parameters and update model states by integrating observed data. The EnKF is known to perform optimally only for multi-Gaussian distributed states and parameters. A new approach, the normal-score EnKF (NS-EnKF), has been recently proposed to handle complex aquifers with non-Gaussian distributed parameters. In this work, we aim at investigating the capacity of the NS-EnKF to identify patterns in the spatial distribution of the model parameters (hydraulic conductivities) by assimilating dynamic observations in the absence of direct measurements of the parameters themselves. In some situations, hydraulic conductivity measurements (hard data) may not be available, which requires the estimation of conductivities from indirect observations, such as piezometric heads. We show how the NS-EnKF is capable of retrieving the bimodal nature of a synthetic aquifer solely from piezometric head data. By comparison with a more standard implementation of the EnKF, the NS-EnKF gives better results with regard to histogram preservation, uncertainty assessment, and transport predictions.

Three-dimensional hydraulic conductivity upscaling in groundwater modeling

The main point of this paper is to propose a non-local three-dimensional hydraulic conductivity full tensor upscaling algorithm and code. The algorithm is capable of transforming very refined cell conductivity models into coarse block conductivity models for quick and accurate solution of the groundwater flow equation. Flow rate and hydraulic head gradient are the variables used to relate the outputs from the fine scale model to the outputs from the coarse scale model. The flows and gradients computed at the coarse scale blocks should match the average values of the corresponding quantities observed at the fine scale. The algorithm is geared towards its use in conjunction with finite-difference codes for the solution of the groundwater flow equation capable of handling full tensor hydraulic conductivities. The finite-difference formulation of the groundwater flow equation requires specifying the hydraulic conductivity at the block interface in order to compute the discharge crossing the interface. Most finite-difference codes take as input conductivities at the blocks and then perform some type of averaging to come up with the interblock conductivity. Typically the harmonic mean of the conductivities of neighboring blocks is used. We propose computing directly the interblock conductivity during the upscaling process thus avoiding the need to average already averaged (upscaled) values of adjacent blocks. This approach also circumvents the problem associated with averaging conductivity tensors when their principal directions are not aligned with the Cartesian axes. The proposed algorithm is successfully demonstrated in four synthetic examples with spatially isotropic, anisotropic and bimodal conductivities fields at the fine scale. The computer code is provided with explanation of the input parameters and output results.

Steady-state saturated groundwater flow modeling with full tensor conductivities using finite differences

We present a new 3D steady-state saturated groundwater-flow forward-simulator with full conductivity tensors using a 19-points block-centered finite-difference method. Hydraulic conductivity tensors are defined at the block interfaces eliminating the need to average conductivity tensors at adjacent blocks to approximate their values at the interfaces. The capabilities of the code are demonstrated in three heterogeneous formulations, two of the examples are 2D, and the third one is 3D and uses a nonuniform discretization grid. A benchmark, in the context of conductivity upscaling, is carried out with the MODFLOW LVDA module, which uses hydraulic conductivity tensors at block centers and then approximates their values at the interfaces. The results show that our code gives more accurate predictions for the interblock fluxes than the MODFLOW LVDA module when the block conductivity principal directions are not parallel to the Cartesian axis.

Simultaneous Estimation of Geologic and Reservoir State Variables Within an Ensemble-Based Multiple-Point Statistic Framework

Assessment of uncertainty due to inadequate data and imperfect geological knowledge is an essential aspect of the subsurface model building process. In this work, a novel methodology for characterizing complex geological structures is presented that integrates dynamic data. The procedure results in the assessment of uncertainty associated with the predictions of flow and transport. The methodology is an extension of a previously developed pattern search-based inverse method that models the spatial variation in flow parameters by searching for patterns in an ensemble of reservoir models. More specifically, the pattern-searching algorithm is extended in two directions: (1)xa0state values (such as piezometric head) and parameters (such as conductivities) are simultaneously and sequentially estimated, which implies that real-time assimilation of dynamic data is possible as in ensemble filtering approaches; and (2)xa0both the estimated parameter and state variables are considered when pattern searching is implemented. The new scheme results in two main advantages—better characterization of parameters, especially for delineating small scale features, and an ensemble of head states that can be used to update the parameter field using the dynamic data at the next instant, without running expensive flow simulations. An efficient algorithm for pattern search is developed, which works with a flexible search radius and can be optimized for the estimation of either large- or small-scale structures. Synthetic examples are employed to demonstrate the effectiveness and robustness of the proposed approach.

Parallelized ensemble Kalman filter for hydraulic conductivity characterization

The ensemble Kalman filter (EnKF) is nowadays recognized as an excellent inverse method for hydraulic conductivity characterization using transient piezometric head data. Its implementation is well suited for a parallel computing environment. A parallel code has been designed that uses parallelization both in the forecast step and in the analysis step. In the forecast step, each member of the ensemble is sent to a different processor, while in the analysis step, the computations of the covariances are distributed between the different processors. An important aspect of the parallelization is to limit as much as possible the communication between the processors in order to maximize execution time reduction. Four tests are carried out to evaluate the performance of the parallelization with different ensemble and model sizes. The results show the savings provided by the parallel EnKF, especially for a large number of ensemble realizations.

Characterizing Curvilinear Features Using the Localized Normal-Score Ensemble Kalman Filter

The localized normal-score ensemble Kalman filter is shown to work for the characterization of non-multi-Gaussian distributed hydraulic conductivities by assimilating state observation data. The influence of type of flow regime, number of observation piezometers, and the prior model structure are evaluated in a synthetic aquifer. Steady-state observation data are not sufficient to identify the conductivity channels. Transient-state data are necessary for a good characterization of the hydraulic conductivity curvilinear patterns. Such characterization is very good with a dense network of observation data, and it deteriorates as the number of observation piezometers decreases. It is also remarkable that, even when the prior model structure is wrong, the localized normal-score ensemble Kalman filter can produce acceptable results for a sufficiently dense observation network.

A local-global pattern matching method for subsurface stochastic inverse modeling

Inverse modeling is an essential step for reliable modeling of subsurface flow and transport, which is important for groundwater resource management and aquifer remediation. Multiple-point statistics (MPS) based reservoir modeling algorithms, beyond traditional two-point statistics-based methods, offer an alternative to simulate complex geological features and patterns, conditioning to observed conductivity data. Parameter estimation, within the framework of MPS, for the characterization of conductivity fields using measured dynamic data such as piezometric head data, remains one of the most challenging tasks in geologic modeling. We propose a new local-global pattern matching method to integrate dynamic data into geological models. The local pattern is composed of conductivity and head values that are sampled from joint training images comprising of geological models and the corresponding simulated piezometric heads. Subsequently, a global constraint is enforced on the simulated geologic models in order to match the measured head data. The method is sequential in time, and as new piezometric head become available, the training images are updated for the purpose of reducing the computational cost of pattern matching. As a result, the final suite of models preserve the geologic features as well as match the dynamic data. This local-global pattern matching method is demonstrated for simulating a two-dimensional, bimodally-distributed heterogeneous conductivity field. The results indicate that the characterization of conductivity as well as flow and transport predictions are improved when the piezometric head data are integrated into the geological modeling. A local-global pattern matching inverse method is proposed.The connectivity can be preserved through multiple point geostatistics.Static and dynamic data are integrated into the geological modeling.

When Steady-State Is Not Enough

Steady-state piezometric head data have always been regarded as containing only information about the major patterns of variability of hydraulic conductivity, but not about specific features, such as channels, or local scale heterogeneity. We have attempted to characterize a channeled aquifer using only steady-state piezometric head without success. However, in a companion paper, we demonstrate how transient piezometric head data can be used to characterize a bimodal aquifer for which no prior information on the aquifer spatial heterogeneity is available using a localized version of the normal-score ensemble Kalman filter.