In data assimilation problems, various types of data are naturally linked to different spatial resolutions (e.g., seismic and electromagnetic data), and these scales are usually not coincident to the subsurface simulation model scale. Alternatives like upscaling/downscaling of the data and/or the simulation model can be used, but with potential loss of important information. Such alternatives introduce additional uncertainties which are not in the nature of the problem description, but the result of the post processing of the data or the geo-model. To address this issue, a novel multiscale (MS) data assimilation method is introduced. The overall idea of the method is to keep uncertain parameters and observed data at their original representation scale, avoiding upscaling/downscaling of any quantity. The method relies on a recently developed mathematical framework to compute adjoint gradients via a MS strategy in an algebraic framework. The fine-scale uncertain parameters are directly updated and the MS grid is constructed in a resolution that meets the observed data resolution. This formulation therefore enables a consistent assimilation of data represented at a coarser scale than the simulation model. The misfit objective function is constructed to keep the MS nature of the problem. The regularization term is represented at the simulation model (fine) scale, whereas the data misfit term is represented at the observed data (coarse) scale. The computational aspects of the method are investigated in a simple synthetic model, including an elaborate uncertainty quantification step, and compared to upscaling/downscaling strategies. The experiment shows that the MS strategy provides several potential advantages compared to more traditional scale conciliation strategies: (1) expensive operations are only performed at the coarse scale; (2) the matched uncertain parameter distribution is closer to the “truth”; (3) faster convergence behavior occurs due to faster gradient computation; and (4) better uncertainty quantification results are obtained. The proof-of-concept example considered in this paper sheds new lights on how one can reduce uncertainty within fine-scale geo-model parameters with coarse-scale data, without the necessity of upscaling/downscaling the data nor the geo-model. The developments demonstrate how to consistently formulate such a gradient-based MS data assimilation strategy in an algebraic framework which allows for implementation in available computational platforms.

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We introduce a semi-analytical iterative multiscale derivative computation methodology that allows for error control and reduction to any desired accuracy, up to fine-scale precision. The model responses are computed by the multiscale forward simulation of flow in heterogeneous porous media. The derivative computation method is based on the augmentation of the model equation and state vectors with the smoothing stage defined by the iterative multiscale method. In the formulation, we avoid additional complexity involved in computing partial derivatives associated to the smoothing step. We account for it as an approximate derivative computation stage. The numerical experiments illustrate how the newly introduced derivative method computes misfit objective function gradients that converge to fine-scale one as the iterative multiscale residual converges. The robustness of the methodology is investigated for test cases with high contrast permeability fields. The iterative multiscale gradient method casts a promising approach, with minimal accuracy-efficiency tradeoff, for large-scale heterogeneous porous media optimization problems.

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We present an efficient workflow that combines multiscale (MS) forward simulation and stochastic gradient computation - MS-StoSAG - for the optimization of well controls applied to waterflooding under geological uncertainty. A two-stage iterative Multiscale Finite Volume (i-MSFV), a mass conservative reservoir simulation strategy, is employed as the forward simulation strategy. MS methods provide the ability to accurately capture fine scale heterogeneities, and thus the fine-scale physics of the problem, while solving for the primary variables in a more computationally efficient coarse-scale simulation grid. In the workflow, the construction of the basis fuctions is performed at an offline stage and they are not reconstructed/updated throughout the optimization process. Instead, inaccuracies due to outdated basis functions are addressed by the i-MSFV smoothing stage. The Stochastic Simplex Approximate Gradient (StoSAG) method, a stochastic gradient technique is employed to compute the gradient of the objective function using forward simulation responses. Our experiments illustrate that i-MSFV simulations provide accurate forward simulation responses for the gradient computation, with the advantage of speeding up the workflow due to faster simulations. Speed-ups up to a factor of five on the forward simulation, the most computationally expensive step of the optimization workflow, were achieved for the examples considered in the paper. Additionally, we investigate the impact of MS parameters such as coarsening ratio and heterogeneity contrast on the optimization process. The combination of speed and accuracy of MS forward simulation with the flexibility of the StoSAG technique allows for a flexible and efficient optimization workflow suitable for large-scale problems.@en

We propose a novel adaptive, adjoint-based, iterative multiscale finite volume (i-MSFV) method. The method aims to reduce the computational cost of the smoothing stage of the original i-MSFV method by selectively choosing fine-scale sub-domains (or sub-set of primary variables) to solve for. The selection of fine-scale primary variables is obtained from a goal-oriented adjoint model. An adjoint-based indicator is utilized as a criterion to select the primary variables having the largest errors. The Lagrange multipliers from the adjoint model can be interpreted as sensitivities of the objective function value with respect to deviations from the constraints. In case of adjoining the porous media flow equations with Lagrange multipliers, this implies that the multipliers are the sensitivities of the objective function with respect to the residuals of the flow equations, i.e., to the residual error that remains after approximately solving linear equations with the aid of an iterative solver. This allow us to recognize at which locations the solution contains more errors. More specifically, we propose a modification to the i-MSFV method to adaptively reduce the size of the fine-scale system that must be smoothed. The aim is to make the fine-scale smoothing stage less computationally demanding. To that end, we introduce a goal-oriented, adjoint-based fine-scale system reduction criterion. We demonstrate the performance of our method via single-phase, incompressible flow simulation models with challenging geological settings and using a history-matching like misfit objective function as the goal. The performance of the newly introduced method is compared to the original i-MSFV method. We investigate the adaptivity versus accuracy of the method and demonstrate how the solution accuracy varies by varying the number of unknowns selected to be smoothed. It is shown that the method can provide accurate solutions at reduced computational cost. The proof-of-concept applications indicate that the method deserves further investigations.@en

The exploitation of subsurface resources is, inevitably, surrounded by uncertainty. Limited knowledge on the economical, operational, and geological setting are just a few instances of sources of uncertainty. From the geological point of view, the currently available technology is not able to provide the description of the fluids and rock properties at the necessary level of detail required by the mathematical models utilized in the exploitation decision-making process. However, even if a full, accurate description of the subsurface was available, the outcome of such hypothetical mathematical model would likely be computationally too expensive to be evaluated considering the currently available computational power, hindering the decision making process. Under this reality, geoscientists are consistently making effort to improve the mathematical models, while being inherently constrained by uncertainty, and to find more efficient ways to computationally solve these models. Closed-loop Reservoir Management (CLRM) is a workflow that allows the continuous update of the subsurface models based on production data from different sources. It relies on computationally demanding optimization algorithms (for the assimilation of production data and control optimization) which require multiple simulations of the subsurface model. One important aspect for the successful application of the CLRM workflow is the definition of a model that can both be run multiple times in a reasonable timespan and still reasonably represent the underlying physics. Multiscale (MS) methods, a reservoir simulation technique that solves a coarser simulation model, thus increasing the computational speed up, while still utilizing the fine-scale representation of the reservoir, figures as an accurate and efficient simulation strategy. This thesis focuses on the development of efficient algorithms for subsurface models optimization by taking advantage of multiscale simulation strategies. It presents (1) multiscale analytical derivative computation strategies to efficiently and accurately address the optimization algorithms employed in the CLRM workflow and (2) novel strategies to handle the mathematical modeling of subsurface management studies from a multiscale perspective. On the latter, we specifically address a more fundamental multiscale aspect of data assimilation studies: the assimilation of observations from a distinct spatial representation compared to the simulation model scale. As a result, this thesis discusses in detail the development of mathematical models and algorithms for the derivative computation of subsurface model responses and their application into gradient-based optimization algorithms employed in the data assimilation and life-cycle optimization steps of CLRM. The advantages are improved computational efficiency with accuracy maintenance and the ability to address the subsurface management from a multiscale view point not only from the forward simulation perspective, but also from the inverse modeling side.@en

A generic framework for the computation of derivative information required for gradient-based optimization using sequentially coupled subsurface simulation models is presented. The proposed approach allows for the computation of any derivative information with no modification of the mathematical framework. It only requires the forward model Jacobians and the objective function to be appropriately defined. The flexibility of the framework is demonstrated by its application in different reservoir management studies. The performance of the gradient computation strategy is demonstrated in a synthetic water-flooding model, where the forward model is constructed based on a sequentially coupled flow-transport system. The methodology is illustrated for a synthetic model, with different types of applications of data assimilation and life-cycle optimization. Results are compared with the classical fully coupled (FIM) forward simulation. Based on the presented numerical examples, it is demonstrated how, without any modifications of the basic framework, the solution of gradient-based optimization models can be obtained for any given set of coupled equations. The sequential derivative computation methods deliver similar results compared to FIM methods, while being computationally more efficient.@en

In data assimilation problems, various types of data are naturally linked to different spatial resolutions (e.g. seismic and electromagnetic data), and these scales are usually not coincident to the subsurface simulation model scale. Alternatives like down/upscaling of the data and/or the simulation model can be used, but with potential loss of important information. To address this issue, a novel Multiscale (MS) data assimilation method is introduced. The overall idea of the method is to keep uncertain parameters and observed data at their original representation scale, avoiding down/upscaling of any quantity. The method relies on a recently developed mathematical framework to compute adjoint gradients via a MS strategy. The fine-scale uncertain parameters are directly updated and the MS grid is constructed in a resolution that meets the observed data resolution. The advantages of the technique are demonstrated in the assimilation of data represented at a coarser scale than the simulation model. The misfit objective function is constructed to keep the MS nature of the problem. The regularization term is represented at the simulation model (fine) scale, whereas the data misfit term is represented at the observed data (coarse) scale. The performance of the method is demonstrated in synthetic models and compared to down/upscaling strategies. The experiments show that the MS strategy provides advantages 1) on the computational side – expensive operations are only performed at the coarse scale; 2) with respect to accuracy – the matched uncertain parameter distribution is closer to the “truth”; and 3) in the optimization performance – faster convergence behaviour due to faster gradient computation. In conclusion, the newly developed method is capable of providing superior results when compared to strategies that rely on the up/downscaling of the response/observed data, addressing the scale dissimilarity via a robust, consistent MS strategy. @en

An efficient multiscale (MS) gradient computation method for subsurface flow management and optimization is introduced. The general, algebraic framework allows for the calculation of gradients using both the Direct and Adjoint derivative methods. The framework also allows for the utilization of any MS formulation that can be algebraically expressed in terms of a restriction and a prolongation operator. This is achieved via an implicit differentiation formulation. The approach favors algorithms for multiplying the sensitivity matrix and its transpose with arbitrary vectors. This provides a flexible way of computing gradients in a form suitable for any given gradient-based optimization algorithm. No assumption w.r.t. the nature of the problem or specific optimization parameters is made. Therefore, the framework can be applied to any gradient-based study. In the implementation, extra partial derivative information required by the gradient computation is computed via automatic differentiation. A detailed utilization of the framework using the MS Finite Volume (MSFV) simulation technique is presented. Numerical experiments are performed to demonstrate the accuracy of the method compared to a fine-scale simulator. In addition, an asymptotic analysis is presented to provide an estimate of its computational complexity. The investigations show that the presented method casts an accurate and efficient MS gradient computation strategy that can be successfully utilized in next-generation reservoir management studies.

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A multiscale gradient computation method for multiphase flow in heterogeneous porous media is developed. The method constructs multiscale primal and dual coarse grids, imposed on the given fine-scale computational grid. Local multiscale basis functions are computed on (dual-) coarse blocks, constructing an accurate map (prolongation operator) between coarse- and fine-scale systems. While the expensive operations involved in computing the gradients are performed at the coarse scale, sensitivities with respect to uncertain parameters (e.g., grid block permeabilities) are expressed in the fine scale via the partial derivatives of the prolongation operator. Hence, the method allows for updating of the geological model, rather than the dynamic model only, avoiding upscaling and the inevitable loss of information. The formulation and implementation are based on automatic differentiation (AD), allowing for convenient extensions to complex physics. An IMPES coupling strategy for flow and transport is followed, in the forward simulation. The flow equation is computed using a multiscale finite volume (MSFV) formulation and the transport equation is computed at the fine scale, after reconstruction of mass conservative velocity field. To assess the performance of the method, a synthetic multiphase flow test case is considered. The multiscale gradients are compared against those obtained from a fine-scale reference strategy. Apart from its computational efficiency, the benefits of the method include flexibility to accommodate variables expressed at different scales, specially in multiscale data assimilation and reservoir management studies.@en

In this work, the application of tensor methodologies for computer-assisted history matching of channelized reservoirs is explored. A tensor-based approach is used for the parameterization of petrophysical parameters to reduce the dimensionality of the parameter estimation problem. Building on the work of Afra and Gildin (2013); Afra et.al. (2014); Afra and Gildin (2016), permeability fields of multiple model realizations are collected in a tensor form which is subsequently decomposed to derive a low-dimensional representation of the dominant spatial structures in the models. This representation then is used to estimate an identifiable reduced set of parameters using an ensemble Kalman filter (EnKF) strategy. This approach is attractive for the parameter estimation of permeabilities because it increases the ability to represent channelized structures in the updates resulting in an improved predictive capacity of the history-matched models. In particular, channel continuity is better preserved than with a Principal Component Analysis (PCA) parameterization.@en

We present an efficient multiscale (MS) gradient computation that is suitable for reservoir management studies involving optimization techniques for, e.g., computer-assisted history matching or life-cycle production optimization. The general, algebraic framework allows for the calculation of gradients using both the Direct and Adjoint derivative methods. The framework also allows for the utilization of any MS formulation in the forward reservoir simulation that can be algebraically expressed in terms of a restriction and a prolongation operator. In the implementation, extra partial derivative information required by the gradient methods is computed via automatic differentiation. Numerical experiments demonstrate the accuracy of the method compared against those based on fine-scale simulation (industry standard). @en