Multiresolution wavelet coarsening and analysis of transport in heterogeneous media

Ebrahimi, Fatemeh; Sahimi, Muhammad

doi:10.1016/s0378-4371(02)01199-8

Cited by 31 publications

(18 citation statements)

References 15 publications

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“…We limit comparisons to other approaches developed for binary media that use information on the modal conductances, proportions and geometry of the phases to calculate effective conductivity. Other techniques that require full knowledge of a fine scale field in order to complete the upscaling such as renormalization [37], anisotropic effective medium approximation [38] and wavelet coarsening [39] are not considered here.…”

Section: An Analytical Subgrid Model For Porous Binary Random Mediamentioning

confidence: 99%

Bayesian data assimilation for stochastic multiscale models of transport in porous media.

Marzouk

Waanders

Parno

et al. 2011

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We investigate Bayesian techniques that can be used to reconstruct field variables from partial observations. In particular, we target fields that exhibit spatial structures with a large spectrum of lengthscales. Contemporary methods typically describe the field on a grid and estimate structures which can be resolved by it. In contrast, we address the reconstruction of grid-resolved structures as well as estimation of statistical summaries of subgrid structures, which are smaller than the grid resolution. We perform this in two different ways (a) via a physical (phenomenological), parameterized subgrid model that summarizes the impact of the unresolved scales at the coarse level and (b) via multiscale finite elements, where specially designed prolongation and restriction operators establish the interscale link between the same problem defined on a coarse and fine mesh. The estimation problem is posed as a Bayesian inverse problem. Dimensionality reduction is performed by projecting the field to be inferred on a suitable orthogonal basis set, viz. the Karhunen-Loève expansion of a multiGaussian. We first demonstrate our techniques on the reconstruction of a binary medium consisting of a matrix with embedded inclusions, which are too small to be grid-resolved. The reconstruction is performed using an adaptive Markov chain Monte Carlo method. We find that the posterior distributions of the inferred parameters are approximately Gaussian. We exploit this finding to reconstruct a permeability field with long, but narrow embedded fractures (which are too fine to be grid-resolved) using scalable ensemble Kalman filters; this also allows us to address larger grids. Ensemble Kalman filtering is then used to estimate the values of hydraulic conductivity and specific yield in a model of the High Plains Aquifer in Kansas. Strong conditioning of the spatial structure of the parameters and the non-linear aspects of the water table aquifer create difficulty for the ensemble Kalman filter. We conclude with a demonstration of the use of multiscale stochastic finite elements to reconstruct permeability fields. This method, though computationally intensive, is general and can be used for multiscale inference in cases where a subgrid model cannot be constructed.

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Section: An Analytical Subgrid Model For Porous Binary Random Mediamentioning

confidence: 99%

Bayesian data assimilation for stochastic multiscale models of transport in porous media.

Marzouk

Waanders

Parno

et al. 2011

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“…The wavelet upscaling of the single-phase permeabilities was developed in our previous papers [26,60,82] which, for completeness, is briefly described here. We consider 2D models of oil reservoirs (or other types of porous media); its extension to 3D which poses no conceptual difficulty over what we describe here.…”

Section: Wavelet Scale Up Of the Single-phase Permeabilitiesmentioning

confidence: 99%

“…To make the identification, two thresholds, ε s and ε d , are introduced [26,60] and are set as a fraction of their corresponding largest values throughout the grid. The scale coefficient of each block is then examined.…”

Section: Wavelet Scale Up Of the Single-phase Permeabilitiesmentioning

confidence: 99%

“…In addition to analyzing the seismic data which were one of the prime motivations for developing the WT, the applications also include numerical simulation of flow and transport equations [62], upscaling of a conductivity or permeability field [26,60,82], pressure transient analysis [5,25,28,47,87], upscaling of miscible displacements in heterogeneous porous media [27], reservoir characterization [11,55,59,63,66], identification of the fracture density along wells in oil and gas reservoirs [81], as well as problems involving turbulent transport and chemical reactions in the atmosphere [38]; for a recent review, see [78,80].…”

Section: Wavelet Scale Up Of the Single-phase Permeabilitiesmentioning

confidence: 99%

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Upscaling of the permeability by multiscale wavelet transformations and simulation of multiphase flows in heterogeneous porous media

Rasaei

Sahimi

2008

Comput Geosci

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We describe a new approach for simulation of multiphase flows through heterogeneous porous media, such as oil reservoirs. The method, which is based on the wavelet transformation of the spatial distribution of the single-phase permeabilities, incorporates in the upscaled computational grid all the relevant data on the permeability, porosity, and other important properties of a porous medium at all the length scales. The upscaling method generates a nonuniform computational grid which preserves the resolved structure of the geological model in the near-well zones as well as in the high-permeability sectors and upscales the rest of the geological model. As such, the method is a multiscale one that preserves all the important information across all the relevant length scales. Using a robust front-detection method which eliminates the numerical dispersion by a high-order total variation diminishing method (suitable for the type of nonuniform upscaled grid that we generate), we obtain highly accurate results with a greatly reduced computational cost. The speedup in the computations is up to over three orders of magnitude, depending on the degree of heterogeneity of the model. To demonstrate the accuracy and efficiency of our methods, five distinct models (including one with fractures) of heterogeneous porous media are considered, and two-phase flows in the models are studied, with and without the capillary pressure.

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“…In some sense, the WT method, which has been developed by our group over the past few years (Mehrabi and Sahimi 1997;Ebrahimi and Sahimi 2002Sahimi 2008, 2009a,b), is also based on the minimization of the local variations in the permeabilities. The WTs represent relatively recent developments in mathematics (Daubechies 1988(Daubechies , 1992Press et al 1992), which have found a broad and diverse set of applications, including data processing, reservoir engineering and simulations, and upscaling (Sahimi 2003).…”

Section: Introductionmentioning

confidence: 99%

Upscaled Unstructured Computational Grids for Efficient Simulation of Flow in Fractured Porous Media

et al. 2009

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Discrete fracture modeling (DFM) is currently the most promising approach for modeling of naturally fractured reservoirs and simulation of multiphase fluid flow therein. In contrast with the classical double-porosity/double permeability models, in the DFM approach all the interactions and fluid flow in and between the fractures and within the matrix are modeled in a unified manner, using the same computational grid. There is no need for computing the shape factors, which are crucial to the accuracy of the double-porosity models. We have exploited this concept in order to develop a new method for the generation of unstructured computational grids. In the new approach the geological model (GM) of the reservoir is first generated, using square or cubic grid blocks. The GM is then upscaled using a method based on the multiresolution wavelet transformations that we recently developed. The upscaled grid contains a distribution of the square or cubic blocks of various sizes. A map of the blocks' centers is then used with an optimized Delauney triangulation method and the advancingfront technique, in order to generate the final unstructured triangulated grid suitable for use in any general reservoir simulator with any number of fluid phases. The new method also includes an algorithm for generating fractures that, contrary to the previous methods, does not require modifying their paths due to the complexities that may arise in spatial distribution of the grid blocks. It also includes an effective partitioning of the simulation domain that results in large savings in the computation times. The speed-up in the computations with the new upscaled unstructured grid is about three orders of magnitude over that for the initial GM. Simulation of waterflooding indicates that the agreement between the results obtained with the GM and the upscaled unstructured grid is excellent. The method is equally 123 196 M. Sahimi et al. applicable to the simulations of multiphase flow in unfractured, but highly heterogeneous, reservoirs.

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Multiresolution wavelet coarsening and analysis of transport in heterogeneous media

Cited by 31 publications

References 15 publications

Bayesian data assimilation for stochastic multiscale models of transport in porous media.

Bayesian data assimilation for stochastic multiscale models of transport in porous media.

Upscaling of the permeability by multiscale wavelet transformations and simulation of multiphase flows in heterogeneous porous media

Upscaled Unstructured Computational Grids for Efficient Simulation of Flow in Fractured Porous Media

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