Mohammadali Tarrahi

Integration of microseismic monitoring data into coupled flow and geomechanical models with ensemble Kalman filter

Hydraulic stimulation of low-permeability rocks in enhanced geothermal systems, shale resources, and CO2 storage aquifers can trigger microseismic events, also known as microearthquakes (MEQs). The distribution of microseismic source locations in the reservoir may reveal important information about the distribution of hydraulic and geomechanical rock properties. In this paper, we present a framework for conditioning heterogeneous rock permeability and geomechanical property distributions on microseismic data. To simulate the multiphysics processes in these systems, we combine a fully coupled flow and geomechanical model with the Mohr-Coulomb type rock failure criterion. The resulting multiphysics simulation constitutes the forecast model that relates microseismic source locations to reservoir rock properties. We adopt this forward model in an ensemble Kalman filter (EnKF) data assimilation framework to jointly estimate reservoir permeability and geomechanical property distributions from injection-induced microseismic response measurements. We show that integration of a large number of spatially correlated microseismic data with practical ensemble sizes can lead to severe underestimation of ensemble spread, and eventually ensemble collapse. To mitigate the variance underestimation issue, two low-rank data representation schemes are presented and discussed. In the first approach, microseismic data are projected onto a low-dimensional subspace defined by the left singular vectors of the perturbed observation matrix. The second method uses a coarser grid for representing the microseismic data. A series of numerical experiments is presented to evaluate the performance of the proposed methods and to illustrate their applicability for assimilating microseismic data into coupled flow and geomechanical forward models to estimate multiphysics rock properties.

Heterogeneous reservoir characterization using efficient parameterization through higher order SVD (HOSVD)

Parameter estimation through reduced-order modeling play a pivotal role in designing real-time optimization schemes for the Oil and Gas upstream sector through the closed-loop reservoir management framework. Reservoir models are in general complex, nonlinear, and large-scale, i.e., large number of states and unknown parameters. Consequently, model reduction techniques are of great interest in reducing the computational burden in reservoir modeling and simulation. Furthermore, de-correlating system parameters in all history matching and reservoir characterization problems is an important task due to its effects on reducing ill-posedness of the system. In this paper, we utilize the higher order singular value decomposition (HOSVD) to reparameterize reservoir characteristics, e.g. permeability, and perform several forward reservoir simulations by the resulted reduced order map as an input. To acquire statistical consistency we repeat all experiments for a set of 1000 samples using both HOSVD and Proper orthogonal decomposition (POD). In addition, we provide RMSE analysis for a better understanding in process of comparing HOSVD and POD. Results show that HOSVD provide a better performance in a RMSE point of view.

Fast linearized forecasts for subsurface flow data assimilation with ensemble Kalman filter

Ensemble methods present a practical framework for parameter estimation, performance prediction, and uncertainty quantification in subsurface flow and transport modeling. In particular, the ensemble Kalman filter (EnKF) has received significant attention for its promising performance in calibrating heterogeneous subsurface flow models. Since an ensemble of model realizations is used to compute the statistical moments needed to perform the EnKF updates, large ensemble sizes are needed to provide accurate updates and uncertainty assessment. However, for realistic problems that involve large-scale models with computationally demanding flow simulation runs, the EnKF implementation is limited to small-sized ensembles. As a result, spurious numerical correlations can develop and lead to inaccurate EnKF updates, which tend to underestimate or even eliminate the ensemble spread. Ad hoc practical remedies, such as localization, local analysis, and covariance inflation schemes, have been developed and applied to reduce the effect of sampling errors due to small ensemble sizes. In this paper, a fast linear approximate forecast method is proposed as an alternative approach to enable the use of large ensemble sizes in operational settings to obtain more improved sample statistics and EnKF updates. The proposed method first clusters a large number of initial geologic model realizations into a small number of groups. A representative member from each group is used to run a full forward flow simulation. The flow predictions for the remaining realizations in each group are approximated by a linearization around the full simulation results of the representative model (centroid) of the respective cluster. The linearization can be performed using either adjoint-based or ensemble-based gradients. Results from several numerical experiments with two-phase and three-phase flow systems in this paper suggest that the proposed method can be applied to improve the EnKF performance in large-scale problems where the number of full simulation is constrained.

Distributed Gauss-Newton optimization method for history matching problems with multiple best matches

Minimizing a sum of squared data mismatches is a key ingredient in many assisted history matching (AHM) workflows. A novel approach is developed to efficiently find multiple local minima of a data mismatch objective function, by performing Gauss-Newton (GN) minimizations concurrently while sharing information between dispersed regions in the reduced parameter space dynamically. To start, a large number of different initial parameter values (i.e., model realizations) are randomly generated and are used as initial search points and base-cases for each subsequent optimization. Predicted data for all realizations are obtained by simulating these search points concurrently, and relevant simulation results for all successful simulation jobs are recorded in a training data set. A local quadratic model around each base-case is constructed using the GN formulation, where the required sensitivity matrix is approximated by linear regression of nondegenerated points, collected in the training data set, that are closest to the given base-case. A new search point for each base-case is generated by minimizing the local quadratic approximate model within a trust region, and the training data set is updated accordingly once the simulation job corresponding to each search point is successfully completed. The base-cases are updated iteratively if their corresponding search points improve the data mismatch. Finally, each base-case will converge to a local minimum in the region of attraction of the initial base-case. The proposed approach is applied to different test problems with uncertain parameters being limited to hundreds or fewer. Most local minima of these test problems are found with both satisfactory accuracy and efficiency.

Distributed Gauss-Newton Method for History Matching Problems with Multiple Best Matches

A novel assisted-history-matching (AHM) approach is developed to efficiently find multiple local minima of the objective function, by performing Gauss-Newton (GN) minimizations concurrently and and sharing information from dispersed regions in parameter space dynamically. To start, a large number of different initial parameter values (i.e., model realizations) are randomly generated and are used as base-cases for each realization. Production data for all realizations are obtained by simulating these base-cases concurrently. A local quadratic model around each base-case is constructed using the GN formulation, where the required sensitivity-matrix is approximated by linear interpolation of non-degenerated points that are closest to the given base-case. New search points are generated by minimizing the local quadratic approximate models. The base-cases are updated iteratively if their corresponding search points improve the data mismatch. Finally, each base case will converge to a local minimum in the vicinity of the initial base-case. The proposed approach is applied to different test problems. Most local minima of these test problems are found with satisfactory accuracy. Compared to traditional AHM approaches using derivative-free optimization algorithms using multiple initial start values, the propose approach may achieve a reduction factor in computer resource usage that is proportional to the number of parameters.