Behnam Jafarpour

Transform-domain sparsity regularization for inverse problems in geosciences

Wehavedevelopedanewregularizationapproachforestimatingunknownspatialfields,suchasfaciesdistributionsorporosity maps. The proposed approach is especially efficient for fields that have a sparse representation when transformed into a complementary function space e.g., a Fourier space. Sparse transform representations provide an accurate characterization of the originalfieldwitharelativelysmallnumberoftransformedvariables.We use a discrete cosine transformDCTto obtain sparse representations of fields with distinct geologic features, such as channels or geologic formations in vertical cross section. Lowfrequency DCT basis elements provide an effectively reduced subspace in which the sparse solution is searched. The low-dimensional subspace is not fixed, but rather adapts to the data. The DCT coefficients are estimated from spatial observations with a variant of compressed sensing. The estimation procedure minimizes an l2-norm measurement misfit term while maintainingDCTcoefficientsparsitywithanl1-normregularizationterm. When measurements are noise-dominated, the performance of this procedure might be improved by implementing it in two steps — one that identifies the sparse subset of important transform coefficients and one that adjusts the coefficients to give a best fit to measurements. We have proved the effectiveness of this approach for facies reconstruction from both scatteredpointmeasurementsandarealobservations,forcrosswelltraveltime tomography, and for porosity estimation in a typical multiunit oilfield.Where we have tested our sparsity regularization approach, it has performed better than traditional alternatives.

A simultaneous perturbation stochastic approximation algorithm for coupled well placement and control optimization under geologic uncertainty

Development of subsurface energy and environmental resources can be improved by tuning important decision variables such as well locations and operating rates to optimize a desired performance metric. Optimal well locations in a discretized reservoir model are typically identified by solving an integer programming problem while identification of optimal well settings (controls) is formulated as a continuous optimization problem. In general, however, the decision variables in field development optimization can include many design parameters such as the number, type, location, short-term and long-term operational settings (controls), and drilling schedule of the wells. In addition to the large number of decision variables, field optimization problems are further complicated by the existing technical and physical constraints as well as the uncertainty in describing heterogeneous properties of geologic formations. In this paper, we consider simultaneous optimization of well locations and dynamic rate allocations under geologic uncertainty using a variant of the simultaneous perturbation and stochastic approximation (SPSA). In addition, by taking advantage of the robustness of SPSA against errors in calculating the cost function, we develop an efficient field development optimization under geologic uncertainty, where an ensemble of models are used to describe important flow and transport reservoir properties (e.g., permeability and porosity). We use several numerical experiments, including a channel layer of the SPE10 model and the three-dimensional PUNQ-S3 reservoir, to illustrate the performance improvement that can be achieved by solving a combined well placement and control optimization using the SPSA algorithm under known and uncertain reservoir model assumptions.

Wavelet Reconstruction of Geologic Facies From Nonlinear Dynamic Flow Measurements

The discrete wavelet transform (DWT) that is widely used in compressing natural images is considered for an effective representation of the geological facies in subsurface flow and transport inverse modeling problems. The inference of the heterogeneous hydraulic rock properties from the scattered dynamic measurements of the flow rates and pressures is a frequently encountered ill-posed inverse problem in subsurface characterization. To better pose this inverse problem, the original grid-based description of the spatial facies maps is replaced with a small number of DWT coefficients that are estimated from indirect nonlinear dynamic measurements. The compressed description of the facies in the wavelet domain after removing the unresolvable high-frequency components leads to an inverse problem with fewer parameters to resolve and improved geologic facies continuity. The main difficulty in the application of the DWT to inverse problems is the lack of sufficient data to resolve higher frequency detail coefficients. Prior information and sensitivity of the flow response to variation in the DWT coefficients are used to infer the location and value of the significant DWT coefficients. The results suggest that the large-scale geologic facies description that control the global flow pattern can be successfully inferred from the dynamic measurements in a reduced wavelet domain. While the flow data may contain information about significant DWT coefficients, a limited observability in ill-posed inverse problems may not allow the identification of these coefficients and the corresponding local spatial features. Therefore, an effective exploitation of the space-frequency localization advantage of the wavelets over the Fourier bases may not be available in solving ill-posed inverse problems.

A variable-control well placement optimization for improved reservoir development

Determination of well locations and their operational settings (controls) such as injection/production rates in heterogeneous subsurface reservoirs poses a challenging optimization problem that has a significant impact on the recovery performance and economic value of subsurface energy resources. The well placement optimization is often formulated as an integer-programming problem that is typically carried out assuming known well control settings. Similarly, identification of the optimal well settings is usually formulated and solved as a control problem in which the well locations are fixed. Solving each of the two problems individually without accounting for the coupling between them leads to suboptimal solutions. Here, we propose to solve the coupled well placement and control optimization problems for improved production performance. We present an alternating iterative solution of the decoupled well placement and control subproblems where each subproblem (e.g., well locations) is resolved after updating the decision variables of the other subproblem (e.g., solving for the control settings) from previous step. This approach allows for application of well-established methods in the literature to solve each subproblem individually. We show that significant improvements can be achieved when the well placement problem is solved by allowing for variable and optimized well controls. We introduce a well-distance constraint into the well placement objective function to avoid solutions containing well clusters in a small region. In addition, we present an efficient gradient-based method for solving the well control optimization problem. We illustrate the effectiveness of the proposed algorithms using several numerical experiments, including the three-dimensional PUNQ reservoir and the top layer of the SPE10 benchmark model.

Effective solution of nonlinear subsurface flow inverse problems in sparse bases

Identification of spatially variable hydraulic rock properties such as permeability and porosity is necessary for accurate prediction of fluid flow displacement in subsurface environments. The estimation of these properties from dynamic flow data usually involves solving a highly underdetermined nonlinear inverse problem in which a limited set of measurements is combined with prior knowledge to estimate the unknown parameters. The overwhelming number of unknowns, relative to available data, leads to many parameter combinations that explain the data equally well, but fail to predict the flow behavior in the reservoir. To improve solution non-uniqueness and numerical stability, additional information is typically incorporated into the solution procedure. One way to regularize underdetermined inverse problems is to demand certain structural properties from the solution that are usually derived from the physics of the problem. In this paper, we exploit the compact representation of spatially correlated geologic formations in sparse bases to facilitate the reconstruction of rock hydraulic property distributions from flow measurements that are nonlinearly related to unknown parameters. By formulating the solution in a compressive basis such as the wavelet or Fourier, we show that minimizing a data misfit cost function augmented with an additive or multiplicative regularization term that promotes sparse solutions, the reconstruction results can be improved. Convergence to a relevant sparse solution is adaptively carried out through an iteratively reweighting algorithm in the transform domain. We evaluate the performance of our inversion algorithm using a set of two-phase waterflooding experiments in an oil reservoir where nonlinear dynamic flow data are integrated to infer the spatial distribution of rock permeability.

Efficient Permeability Parameterization With the Discrete Cosine Transform

The inverse estimation of permeability fields (history matching) is commonly performed by replacing the original set of unknown spatially discretized permeabilities with a smaller (lower dimensionality) group of unknowns that captures the most important features of the field. This makes the inverse problem better posed by reducing redundancy. The Karhunen-Loeve Transform (KLT) is a classical option for deriving low dimensional parameterizations for history matching applications. The KLT can provide an accurate characterization of complex permeability fields but it can be computationally demanding. In many respects this approach provides a benchmark that can be used to evaluate the performance of more computationally efficient alternatives. The KLT requires knowledge of the permeability covariance function and can give poor results when this matrix does not adequately describe the actual permeability field. By contrast, the Discrete Cosine Transform (DCT) provides a robust parameterization alternative that does not require specification of covariances or other statistics. It is computationally efficient and in many cases is almost as accurate as the KLT. The DCT is able to accommodate prior information, if desired. Here we describe the DCT approach and compare its performance to the KLT for a set of geologically relevant examples. Introduction Reservoir characterization is generally based on localized borehole and outcrop observations that are interpolated to give regional descriptions of uncertain geological properties such as permeability. The interpolation process introduces uncertainty in the permeability field that translates directly into uncertainty about reservoir behavior. Incorporation of dynamic measurements during the production phase, i.e. history matching, provides a way to reduce permeability uncertainty. History matching identifies the permeability values that provide the best match, in terms of a specified performance measure, to observations of dynamic production variables such as bottom-hole pressure and fluid rates. This process can increase the accuracy and usefulness of model predictions if the estimated permeabilities provide a reasonable description of the true field. It is generally accepted that history matching methods work best when they incorporate geologically realistic facies information. Realistic facies representations should account for depositional continuity and connectivity since these properties have a significant effect on fluid flow within the reservoir [1]. When the permeability field is characterized by finely discretized block values the history matching problem can be ill-posed and result in non-unique solutions [2,3]. Ill-posed problems can produce reservoir models that honor observed measurements but provide incorrect predictions. Moreover, if estimated block permeabilities are not constrained to preserve facies connectivity, they may yield geologically inconsistent and unrealistic permeability fields. In order to deal with illposedness and to respect geological facies it is desirable to adopt a parametric description of permeability that is lowdimensional while also able to preserve important geological features and their connectivity. Several parameterization approaches with varying complexity have been proposed and implemented for reservoir history matching problems. A simple zonation approach is used by [4] in which an aggregate of block properties are assembled and assigned a single value. Adaptive versions of this approach have been adopted to perform the history matching in multiple steps with increasing resolution [5,6]. Other multi-resolution techniques have also been proposed for parameterization and history matching at different scales [7,8]. A particularly powerful parametrization approach suitable for history matching is the Karhunen-Loeve Transform (KLT), named after Karhunen [9] and Loeve [10]. This approach represents the permeability in any given block with a linear expansion (or transform) composed of the weighted eigenvectors of a specified block permeability covariance matrix. This matrix can, in turn, be derived from a specified continuous permeability covariance function. In practice, the covariances used to derive the KLT basis functions are often derived from permeability measurements. When this is done the KLT is data-dependent (i.e. its characterization of permeability depends on correlation properties of a particular SPE 106453 Efficient Permeability Parameterization with the Discrete Cosine Transform B. Jafarpour, SPE, D. B. McLaughlin, Massachusetts Institute of Technology

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.

A sparse Bayesian framework for conditioning uncertain geologic models to nonlinear flow measurements

We present a Bayesian framework for reconstructing hydraulic properties of rock formations from nonlinear dynamic flow data by imposing sparsity on the distribution of the parameters in a sparse transform basis through Laplace prior distribution. Sparse representation of the subsurface flow properties in a compression transform basis (where a compact representation is often possible) lends itself to a natural regularization approach, i.e. sparsity regularization, which has recently been exploited in solving ill-posed subsurface flow inverse problems. The Bayesian estimation approach presented here allows for a probabilistic treatment of the sparse reconstruction problem and has its roots in machine learning and the recently introduced relevance vector machine algorithm for linear inverse problems. We formulate the Bayesian sparse reconstruction algorithm and apply it to nonlinear subsurface inverse problems where solution sparsity in a discrete cosine transform is assumed. The probabilistic description of solution sparsity, as opposed to deterministic regularization, allows for quantification of the estimation uncertainty and avoids the need for specifying a regularization parameter. Several numerical experiments from multiphase subsurface flow application are presented to illustrate the performance of the proposed method and compare it with the regular Bayesian estimation approach that does not impose solution sparsity. While the examples are derived from subsurface flow modeling, the proposed framework can be applied to nonlinear inverse problems in other imaging applications including geophysical and medical imaging and electromagnetic inverse problem.

Inference of permeability heterogeneity from joint inversion of transient flow and temperature data

Characterization of the rock permeability distribution in compartmentalized deep aquifers, enhanced geothermal systems, and hydrocarbon reservoirs is important for predicting the flow and transport behavior in these formations. Reliable prediction of the fluid flow and transport processes can, in turn, lead to effective development of the subsurface energy and environmental resources. In deep formations where thermal gradients are significant, the transient temperature data can provide valuable information about the permeability distribution with depth and about the vertical fluid displacement. This paper examines the importance of temperature data in resolving the distribution of permeability with depth by jointly, and individually, integrating the transient temperature and flow data. We demonstrate that when estimating permeability distributions in deep geothermal reservoirs, incorporating temperature data can increase the resolution of the permeability distribution profile with depth. To illustrate the importance of temperature measurements, we adopt a coupled transient heat and fluid flow as a forward model to predict the heat and fluid transport in a geothermal reservoir and develop an adjoint model for efficient computation of the gradient information for model calibration. We perform a series of numerical experiments for integration of flow and pressure data alone, temperature data alone, and flow and pressure jointly with temperature data. In each case, we apply the maximum A-posteriori (MAP) method and the randomized maximum likelihood (RML) method for inversion and uncertainty quantification. Analysis of the sensitivity of temperature and production data to heterogeneous permeability distributions reveals that the temperature of fluid, even when measured at the surface, is sensitive to the permeability distribution in the vertical extent of the reservoir. Hence, temperature measurements can be augmented with flow-related data to enhance the resolution of the estimated permeability field with depth.

Prior model identification during subsurface flow data integration with adaptive sparse representation techniques

Construction of predictive reservoir models invariably involves interpretation and interpolation between limited available data and adoption of imperfect modeling assumptions that introduce significant subjectivity and uncertainty into the modeling process. In particular, uncertainty in the geologic continuity model can significantly degrade the quality of fluid displacement patterns and predictive modeling outcomes. Here, we address a standing challenge in flow model calibration under uncertainty in geologic continuity by developing an adaptive sparse representation formulation for prior model identification (PMI) during model calibration. We develop a flow-data-driven sparsity-promoting inversion to discriminate against distinct prior geologic continuity models (e.g., variograms). Realizations of reservoir properties from each geologic continuity model are used to generate sparse geologic dictionaries that compactly represent models from each respective prior. For inversion initially the same number of elements from each prior dictionary is used to construct a diverse geologic dictionary that reflects a wide range of variability and uncertainty in the prior continuity. The inversion is formulated as a sparse reconstruction problem that inverts the flow data to identify and linearly combine the relevant elements from the large and diverse set of geologic dictionary elements to reconstruct the solution. We develop an adaptive sparse reconstruction algorithm in which, at every iteration, the contribution of each dictionary to the solution is monitored to replace irrelevant (insignificant) elements with more geologically relevant (significant) elements to improve the solution quality. Several numerical examples are used to illustrate the effectiveness of the proposed approach for identification of geologic continuity in practical model calibration problems where the uncertainty in the prior geologic continuity model can lead to biased inversion results and prediction.