Adjoint Multiscale Mixed Finite Elements

Krogstad, Stein; Hauge, Vera Louise; Gulbransen, Astrid Fossum

doi:10.2118/119112-pa

Cited by 26 publications

(20 citation statements)

References 18 publications

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“…For both the Cartesian and the flowbased coarse grids, the coarse-scale saturation error is the largest initially and decays towards water breakthrough; the qualitative behaviour of the fine-scale error is almost identical, and the corresponding curves are therefore not reported. It may come as a surprise how well both the static Cartesian and the flow-based grid capture the water-cut curve for layer 1, given the large initial error, but this result is in correspondence with previous observations [1,15] both for Cartesian and corner-point models. For layer 37, we also observe how using a static flow-based grid gives a significant Fig.…”

Section: Dynamically Adaptive Gridsupporting

confidence: 86%

“…Another example of flow-based gridding on cornerpoint grids was presented by Krogstad et al [15], who used such grids to accelerate forward simulations in a production optimization workflow. In the next section, we will give a more quantitative study of the gridding methods introduced in the previous section when applied together with the multiscale transport solver (Eq.…”

Section: Example 3 (Facies Model) We Consider a Rectangular Domain Wimentioning

confidence: 99%

“…The coarse-scale solver is effectively an upwind solver and hence has less coarse-scale numerical diffusion. In a multiscale setting, the results above are interesting for the following reason: Krogstad et al [15] have previously demonstrated that the combination of a multiscale flow solver and a flow-adapted grid can be very efficient if one can avoid communication through the underlying fine grid and only store fine-scale fluxes at the coarse interfaces and use precomputed mappings to move saturations from the transport grid to the multiscale flow solver. Replacing fine-scale fluxes on each coarse-grid interface by net fluxes will decrease the communication need even further and hence increase the efficiency of the overall solver.…”

Section: Multiscale Versus Coarse-scale Solvermentioning

confidence: 99%

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Flow-based coarsening for multiscale simulation of transport in porous media

2011

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Geological models are becoming increasingly large and detailed to account for heterogeneous structures on different spatial scales. To obtain simulation models that are computationally tractable, it is common to remove spatial detail from the geological description by upscaling. Pressure and transport equations are different in nature and generally require different strategies for optimal upgridding. To optimize the accuracy of a transport calculation, the coarsened grid should generally be constructed based on a posteriori error estimates and adapt to the flow patterns predicted by the pressure equation. However, sharp and rigorous estimates are generally hard to obtain, and herein we therefore consider various ad hoc methods for generating flow-adapted grids. Common for all is that they start by solving a single-phase flow problem once and then continue to form a coarsened grid by amalgamating cells from an underlying fine-scale grid. We present several variations of the original method. First, we discuss how to include a priori information in the coarsening process, e.g. to adapt to special geological features or to obtain less irregular grids in regions where flowadaption is not crucial. Second, we discuss the use of bi-directional versus net fluxes over the coarse blocks and show how the latter gives systems that better represent the causality in the flow equations, which can be exploited to develop very efficient nonlinear solvers. Finally, we demonstrate how to improve simulation accuracy by dynamically adding local resolution near strong saturation fronts.

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Section: Dynamically Adaptive Gridsupporting

confidence: 86%

Section: Example 3 (Facies Model) We Consider a Rectangular Domain Wimentioning

confidence: 99%

Section: Multiscale Versus Coarse-scale Solvermentioning

confidence: 99%

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Flow-based coarsening for multiscale simulation of transport in porous media

2011

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“…7) for saturation. In addition, an adjoint code for the multiscale method combined with the flow-based coarsening approach is included, see [22].…”

Section: Example 7 (Primary Production From a Gas Reservoir)mentioning

confidence: 99%

Open-source MATLAB implementation of consistent discretisations on complex grids

et al. 2011

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Accurate geological modelling of features such as faults, fractures or erosion requires grids that are flexible with respect to geometry. Such grids generally contain polyhedral cells and complex grid-cell connectivities. The grid representation for polyhedral grids in turn affects the efficient implementation of numerical methods for subsurface flow simulations. It is well known that conventional two-point flux-approximation methods are only consistent for K-orthogonal grids and will, therefore, not converge in the general case. In recent years, there has been significant research into consistent and convergent methods, including mixed, multipoint and mimetic discretisation methods. Likewise, the so-called multiscale methods based upon hierarchically coarsened grids have received a lot of attention. The paper does not propose novel mathematical methods but instead presents an open-source Matlab® toolkit that can be used as an efficient test platform for (new) discretisation and solution methods in reservoir simulation. The aim of the toolkit is to support reproducible research and simplify the development, verification and validation and testing and comparison of new discretisation and solution methods on general unstructured grids, including in particular corner point and 2.5D PEBI grids. The toolkit consists of a set of data structures and routines for creating, manipulating and visualising petrophysical data, fluid models and (unstructured) grids, including support for industry standard input formats, as well as routines for computing single and multiphase (incompressible) flow. We review key features of the toolkit and discuss a generic mimetic formulation that includes many known discretisation methods, including both the standard two-point method as well as consistent and convergent multipoint and mimetic methods. Apart from the core routines and data structures, the toolkit contains addon modules that implement more advanced solvers and functionality. Herein, we show examples of multiscale methods and adjoint methods for use in optimisation of rates and placement of wells.

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“…Also, even though the mathematical framework presented by [30] and [19] does not limit the derivation of the adjoint equations to any particular solution strategy, no explicit discussion on how it can be applied to sequentially coupled system of equations was presented. A multiscale adjoint method applied to life-cycle optimization is presented by [20], in which a sequential solution of flow and transport is employed, such that, consequentially, the adjoint model also follows a sequential solution strategy. However, in that work, the discussion is focused on the promising computational savings provided by multiscale simulation and not so much detail is given as to what extent the gradient computation itself can impose challenges.…”

Section: Introductionmentioning

confidence: 99%

Computing derivative information of sequentially coupled subsurface models

2018

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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 gradientbased 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.

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Adjoint Multiscale Mixed Finite Elements

Cited by 26 publications

References 18 publications

Flow-based coarsening for multiscale simulation of transport in porous media

Flow-based coarsening for multiscale simulation of transport in porous media

Open-source MATLAB implementation of consistent discretisations on complex grids

Computing derivative information of sequentially coupled subsurface models

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