High-resolution truncated plurigaussian simulations for the characterization of heterogeneous formations

Integrating geological concepts, such as relative positions and proportions of the different lithofacies, is of highest importance in order to render realistic geological patterns. The truncated plurigaussian simulation method provides a way of using both local and conceptual geological information to infer the distributions of the facies and then those of hydraulic parameters. The method (Le Loc'h and Galli 1994) is based on the idea of truncating at least two underlying multi-Gaussian simulations in order to create maps of categorical variable. In this manuscript we show how this technique can be used to assess contaminant migration in highly heterogeneous media. We illustrate its application on the biggest contaminated site of Switzerland. It consists of a contaminant plume located in the lower fresh water Molasse on the western Swiss Plateau. The highly heterogeneous character of this formation calls for efficient stochastic methods in order to characterize transport processes.Comment: 12 pages, 9 figure

Probability fields revisited in the context of ensemble Kalman filtering

Hu et al. (2013) proposed an approach to update complex geological fades models generated by multiple-point geostatistical simulation while keeping geological and statistical consistency. Their approach is based on mapping the fades realization onto the spatially uncorrelated uniform random numbers used by the sequential multiple-point simulation to generate the facies realization itself. The ensemble Kalman filter was then used to update the uniform random number realizations, which were then used to generate a new fades realization by multiple-point simulation. This approach has not a good performance that we attribute to the fact that, being the probabilities random and spatially uncorrelated, their correlation with the state variable (piezometric heads) is very weak, and the Kalman gain is always small. The approach is reminiscent of the probability field simulation, which also maps the conductivity realizations onto a field of uniform random numbers; although the mapping now is done using the local conditional distribution functions built based on a prior statistical model and the conditioning data. Contrary to Hu et al. (2013) approach, this field of uniform random numbers, termed a probability field, displays spatial patterns related to the conductivity spatial patterns, and, therefore, the correlation between probabilities and state variable is as strong as the correlation between conductivities and state variable could be. Similarly to Hu et al. (2013), we propose to use the ensemble Kalman filter to update the probability fields, and show that the existence of this correlation between probability values and state variables provides better results.The first author acknowledges the financial support from the China Scholarship Council (CSC). Part of this work was done while the second author was on sabbatical with the Kansas Geological Survey, Kansas University, Lawrence, KS, USA, which was funded by the Spanish Ministry of Education, Culture and Sports through grant PRX14/00501. Financial support to carry out this work was also received from the Spanish Ministry of Economy and Competitiveness through project CGL2014-59841-P.Xu, T.; Gómez-Hernández, JJ. (2015). Probability fields revisited in the context of ensemble Kalman filtering. Journal of Hydrology. 531(1):40-52. https://doi.org/10.1016/j.jhydrol.2015.06.062S4052531

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. (C) 2015 Elsevier Ltd. All rights reserved.The authors gratefully acknowledge the financial support by DOE through projects DE-FE0004962 and DE-SC0001114. The last author acknowledges the support of the Spanish Ministry of Economy and Competitiveness through project CGL2011-23295. We greatly thank the three anonymous reviewers for their comments, which substantially improved the manuscript.Li ., L.; Srinivasan, S.; Zhou, H.; Gómez Hernández, JJ. (2015). A local global pattern matching method for subsurface stochastic inverse modeling. Environmental Modelling and Software. 70:55-64. https://doi.org/10.1016/j.envsoft.2015.04.008S55647

Characterization of non-Gaussian conductivities and porosities with hydraulic heads, solute concentrations, and water temperatures

Reliable characterization of hydraulic parameters is important for the understanding of groundwater flow and solute transport. The normal-score ensemble Kalman filter (NS-EnKF) has proven to be an effective inverse method for the characterization of non-Gaussian hydraulic conductivities by assimilating transient piezometric head data, or solute concentration data. Groundwater temperature, an easily captured state variable, has not drawn much attention as an additional state variable useful for the characterization of aquifer parameters. In this work, we jointly estimate non-Gaussian aquifer parameters (hydraulic conductivities and porosities) by assimilating three kinds of state variables (piezometric head, solute concentration, and groundwater temperature) using the NS-EnKF. A synthetic example including seven tests is designed, and used to evaluate the ability to characterize hydraulic conductivity and porosity in a non-Gaussian setting by assimilating different numbers and types of state variables. The results show that characterization of aquifer parameters can be improved by assimilating groundwater temperature data and that the main patters of the non-Gaussian reference fields can be retrieved with more accuracy and higher precision if multiple state variables are assimilated.Financial support to carry out this work was provided by the Spanish Ministry of Economy and Competitiveness through project CGL2014-59841-P. All data used in this analysis are available from the authors.Xu, T.; Gómez-Hernández, JJ. (2016). Characterization of non-Gaussian conductivities and porosities with hydraulic heads, solute concentrations, and water temperatures. Water Resources Research. 52(8):6111-6136. https://doi.org/10.1002/2016WR019011S61116136528Alcolea, A., Carrera, J., & Medina, A. (2006). Pilot points method incorporating prior information for solving the groundwater flow inverse problem. Advances in Water Resources, 29(11), 1678-1689. doi:10.1016/j.advwatres.2005.12.009Anderson, M. P. (2005). Heat as a Ground Water Tracer. Ground Water, 43(6), 951-968. doi:10.1111/j.1745-6584.2005.00052.xBravo, H. R., Jiang, F., & Hunt, R. J. (2002). Using groundwater temperature data to constrain parameter estimation in a groundwater flow model of a wetland system. Water Resources Research, 38(8), 28-1-28-14. doi:10.1029/2000wr000172Capilla, J. E., & Llopis-Albert, C. (2009). Gradual conditioning of non-Gaussian transmissivity fields to flow and mass transport data: 1. Theory. Journal of Hydrology, 371(1-4), 66-74. doi:10.1016/j.jhydrol.2009.03.015Chang, H., Zhang, D., & Lu, Z. (2010). History matching of facies distribution with the EnKF and level set parameterization. Journal of Computational Physics, 229(20), 8011-8030. doi:10.1016/j.jcp.2010.07.005Chen , Y. D. S. Oliver 2010Chen, Y., Oliver, D. S., & Zhang, D. (2009). Data assimilation for nonlinear problems by ensemble Kalman filter with reparameterization. Journal of Petroleum Science and Engineering, 66(1-2), 1-14. doi:10.1016/j.petrol.2008.12.002Doussan, C., Toma, A., Paris, B., Poitevin, G., Ledoux, E., & Detay, M. (1994). Coupled use of thermal and hydraulic head data to characterize river-groundwater exchanges. Journal of Hydrology, 153(1-4), 215-229. doi:10.1016/0022-1694(94)90192-9Dovera, L., & Della Rossa, E. (2010). Multimodal ensemble Kalman filtering using Gaussian mixture models. Computational Geosciences, 15(2), 307-323. doi:10.1007/s10596-010-9205-3Evensen, G. (2003). The Ensemble Kalman Filter: theoretical formulation and practical implementation. 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Stochastic conditional inverse modeling of subsurface mass transport: A brief review and the self-calibrating method. Stochastic Environmental Research and Risk Assessment (SERRA), 17(5), 319-328. doi:10.1007/s00477-003-0153-5Gordon , N. D. Salmond A. Smith 1993 Novel approach to nonlinear/non-Gaussian Bayesian state estimation Proc. Inst. Electr. Eng. 140 107 113Gu, Y., & Oliver, D. S. (2005). The Ensemble Kalman Filter for Continuous Updating of Reservoir Simulation Models. Journal of Energy Resources Technology, 128(1), 79-87. doi:10.1115/1.2134735Gu, Y., & Oliver, D. S. (2007). An Iterative Ensemble Kalman Filter for Multiphase Fluid Flow Data Assimilation. SPE Journal, 12(04), 438-446. doi:10.2118/108438-paHu, L. Y. (2000). Mathematical Geology, 32(1), 87-108. doi:10.1023/a:1007506918588Kalman, R. E. (1960). A New Approach to Linear Filtering and Prediction Problems. 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Oliver 2005 Critical evaluation of the ensemble Kalman filter on history matching of geologic facies SPE Reservoir Eval. Eng. 8 6 470 477Losa, S. N., Kivman, G. A., Schröter, J., & Wenzel, M. (2003). Sequential weak constraint parameter estimation in an ecosystem model. Journal of Marine Systems, 43(1-2), 31-49. doi:10.1016/j.jmarsys.2003.06.001Ma , R. C. Zheng 2010 Effects of density and viscosity in modeling heat as a groundwater tracer, Groundwater 48 3 380 389Ma, R., Zheng, C., Zachara, J. M., & Tonkin, M. (2012). Utility of bromide and heat tracers for aquifer characterization affected by highly transient flow conditions. Water Resources Research, 48(8). doi:10.1029/2011wr011281McDonald , M. A. Harbaugh 1988Oliver, D. S., Cunha, L. B., & Reynolds, A. C. (1997). Markov chain Monte Carlo methods for conditioning a permeability field to pressure data. Mathematical Geology, 29(1), 61-91. doi:10.1007/bf02769620RamaRao, B. S., LaVenue, A. M., De Marsily, G., & Marietta, M. G. (1995). 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Simultaneous identification of a contaminant source and hydraulic conductivity via the restart normal-score ensemble Kalman filter

[EN] Detecting where and when a contaminant entered an aquifer from observations downgradient of the source is a difficult task; this identification becomes more challenging when the uncertainty about the spatial distribution of hydraulic conductivity is accounted for. In this paper, we have implemented an application of the restart normal-score ensemble Kalman filter (NS-EnKF) for the simultaneous identification of a contaminant source and the spatially variable hydraulic conductivity in an aquifer. The method is capable of providing estimates of the spatial location, initial release time, the duration of the release and the mass load of a point-contamination event, plus the spatial distribution of hydraulic conductivity together with an assessment of the estimation uncertainty of all the parameters. The method has been applied in synthetic aquifers exhibiting both Gaussian and non-Gaussian patterns. The identification is made possible by assimilating in time both piezometric head and concentration observations from an array of observation wells. The method is demonstrated in three different synthetic scenarios that combine hydraulic conductivities with unimodal and bimodal histograms, and releases in high and low conductivity zones. The results prove that the specific implementation of the EnKF is capable of recovering the source parameters with some uncertainty and of recovering the main patterns of heterogeneity of the hydraulic conductivity fields by assimilating a sufficient number of state variable observations. The proposed approach is an important step towards contaminant source identification in real aquifers, which may have logconductivity spatial distributions with either Gaussian or non-Gaussian features, yet, it is still far from practical applications since the transport parameters, the external sinks and sources and the initial and boundary conditions are assumed known.Financial support to carry out this work was received from the Spanish Ministry of Economy and Competitiveness through project CGL2014-59841-P. The authors acknowledge the Associate Editor, and the anonymous reviewers for their thoughtful and constructive comments.Xu, T.; Gómez-Hernández, JJ. (2018). Simultaneous identification of a contaminant source and hydraulic conductivity via the restart normal-score ensemble Kalman filter. Advances in Water Resources. 112:106-123. https://doi.org/10.1016/j.advwatres.2017.12.011S10612311

A Pilot Point Guided Pattern Matching Approach to Integrate Dynamic Data into Geological Modeling

Methods based on multiple-point statistics (MPS) have been routinely used to characterize complex geological formations in the last decade. These methods use the available static data (for example, measured conductivities) for conditioning. Integrating dynamic data (for example, measured transient piezometric head data) into the same framework is challenging because of the complex non-linear relationship between the dynamic response and geology. The Ensemble PATtern (EnPAT) search method was recently developed as a promising technique to handle this problem. In this approach, a pattern is postulated to be composed of both parameter and state variables, and then, parameter values are sequentially (point-wise) simulated by directly sampling the matched pattern from an ensemble of training images of both geologic parameters and state variables. As a consequence, the updated ensemble of realizations of the geological parameters preserve curvilinear structures (i.e., non-multiGaussanity) as well as the complex relationship between static and dynamic data. Moreover, the uncertainty of flow and transport predictions can be assessed using the updated ensemble of geological models. In this work, we further modify the EnPAT method by introducing the pilot-point concept into the algorithm. More specifically, the parameter values at a set of randomly selected pilot point locations are simulated by the pattern searching procedure, and then a faster MPS method is used to complete the simulation by conditioning to the previously simulated pilot point values. This pilot point guided MPS implementation results in lower computational cost and more accurate inference of the parameter field. In addition, in some situations where there is sparsity of measured geologic static data, the EnPAT algorithm is extended to work only with the dynamic data. We employed a synthetic example to demonstrate the effectiveness of pilot points in the implementation of EnPAT, and also the capability of dynamic data to identify complex geologic structures when measured conductivity data are not available.The first three authors gratefully acknowledge the financial support by DOE through project DE-FE0004962. The fourth author acknowledges the financial support by the Spanish Ministry of Science and Innovation through project CGL2011-23295. The authors also wish to thank Wolfgang Nowak as well as two anonymous reviewers for their comments, which helped improving the final version of the manuscript.Li, L.; Srinivasan, S.; Zhou, H.; Gómez-Hernández, JJ. (2013). A Pilot Point Guided Pattern Matching Approach to Integrate Dynamic Data into Geological Modeling. Advances in Water Resources. 62(Part A):125-138. https://doi.org/10.1016/j.advwatres.2013.10.008S12513862Part

A pattern-search-based inverse method

Uncertainty 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.The authors gratefully acknowledge the financial support from the Ministry of Science and Innovation, project CGL2011-23295. The first author also acknowledges the scholarship provided by the China Scholarship Council (CSC [2007] 3020). The authors would like to thank Gregoire Mariethoz (University of New South Wales) and Philippe Renard (University of Neuchatel) for their enthusiastic help in answering questions about the direct sampling algorithm. Gregoire Mariethoz and two anonymous reviewers are also thanked for their comments during the reviewing process, which helped improving the final paper.Zhou ., H.; Gómez-Hernández, JJ.; Li ., L. (2012). A pattern-search-based inverse method. 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