Fine-scale Solution Research Articles

A 3-D extension of the Multiscale Control Volume method for the simulation of the single-phase flow in anisotropic and heterogeneous porous media

The level of detail on modern geological models requires higher resolution grids that may render the simulation of multiphase flow in porous media intractable. Moreover, these models may comprise highly heterogeneous media with phenomena taking place in different scales. Scale transferring techniques allow for the solution of such problems in a lower resolution scale at reduced computational cost. Among these techniques, the Multiscale Finite Volume (MsFV) method constructs a set of numerical operators in order to map quantities from the fine-scale mesh to a coarser one and vice-versa while maintaining flux conservation on both scales. However, the MsFV formulation, as originally stated, is only consistent on k-orthogonal grids since it uses a linear Two-point Flux Approximation (TPFA) method and may struggle to generate consistent primal-dual coarse grids pairs on unstructured grids. The Multiscale Restriction Smoothed-Basis method (MsRSB) improves on the MsFV by introducing a new iterative procedure to find the multiscale operators and the concept of support regions which reduces the method's complexity when applied to unstructured fine and coarse grids. The original version was only consistent on k-orthogonal fine grids due to the TPFA discretization, but filtering methods have been developed to also enable consistent multipoint schemes on the fine scale. Meanwhile, the Multiscale Control Volume method (MsCV) replaces the TPFA by the Multipoint Flux Approximation with a Diamond stencil (MPFA-D) scheme on the fine-scale while further enhancing the generation of the geometric entities to allow unstructured grids on the fine and coarse scales for two-dimensional simulations. In this work we propose an extension to three-dimensional geometries of both the MsCV and the algorithm to obtain the multiscale geometric entities based on the concept of a background grid, a coarser grid used as a proxy for the primal coarse grid. We modify the original MPFA-D method to use the very robust Global Least Squares (GLS) interpolation technique to obtain the required auxiliary nodal unknowns. We also introduce an enhanced version of the 3-D MsCV with the incorporation of the enhanced MsRSB (E-MsRSB) to enforce M-matrix properties and improve convergence. Finally, we employ the MsCV operators in a two-stage smoother to show how it can be used as a good iterative procedure to recover the fine-scale solution. We show that the 3-D MsCV method produces good results employing unstructured grids on both scales to handle the simulation of the single-phase flow in anisotropic, and heterogeneous porous media and that the smoothing procedure is able to accurately retrieve the fine-scale solution within a few iterations.

On the upscaling of the diffusion equation in a heterogeneous medium by the two different methods

Upscaling the flow equations appears in many studies related to diffusion, heat conductivity, and flows in porous media. Variable coefficients of the original fine-scale process description are substituted by averaged constant values. The different upscaling procedures have been suggested in the literature. The question arises, whether they result in the same or different upscaled models; moreover, whether the solutions of the coarse-scale equations provide a reasonably accurate description of the fine scale. In this work, we consider three sample 1D diffusion problems with periodic coefficients: diffusion with or without the external source, as well as diffusion with advection. These problems are upscaled by the two methods. The method of direct upscaling selects the averaged coefficients to provide the minimum difference between the solutions of the coarse-scale and fine-scale models. The method of continuous upscaling, developed previously, consists of the continuous averaging transformation between the distant scales. New expressions for the upscaled diffusion coefficients were derived for this case. It turns out that the direct upscaling results in multiple optimum parameters of the upscaled model. Meanwhile, continuous upscaling points at one of them. The coarse-scale approximation of a fine-scale solution may be unsatisfactory, even for the best choice of the upscaled coefficient. A numerical study demonstrates a nontrivial behavior of the diffusion coefficient under continuous upscaling, while it converges to an asymptotic value.

Efficient inverse-designed structural infill for complex engineering structures

Inverse design of high-resolution and fine-detailed 3D lightweight mechanical structures is notoriously expensive due to the need for vast computational resources and the use of very fine-scaled complex meshes. Furthermore, in designing for additive manufacturing, infill is often neglected as a component of the optimized structure. In this paper, both concerns are addressed using a so-called de-homogenization topology optimization procedure on complex engineering structures discretized by 3D unstructured hexahedrals.Using a rectangular-hole microstructure (reminiscent to the stiffness optimal orthogonal rank-3 multi-scale) as a base material for the multi-scale optimization, a coarse-scale optimized geometry can be obtained using homogenization-based topology optimization. Due to the microstructure periodicity, this coarse-scale geometry can be up-sampled to a fine single-scale physical geometry with optimized infill, with only a minor loss in structural performance and at a fraction of the cost of a fine-scale solution. The upsampling on 3D unstructured grids is achieved through stream surface tracing, aligning with the optimized local orientations. The periodicity of the physical geometry can be tuned, such that the material serves as a structural component and also as an efficient infill for additive manufacturing designs.The method is demonstrated through three examples of varying geometrical complexity. It achieves comparable structural performance to “brute force” state-of-the-art methods but stands out for its significant computational time reduction. By allowing multiple active layers, the mapped solution becomes more mechanically stable, leading to an increased critical buckling load factor without additional computational expense. The control of active layers also provides direct control over the internal structure, i.e., infill, ensuring that the infill is incorporated as a structural component and enhancing the manufacturability of the de-homogenization procedure. Furthermore, the proposed approach exhibits promising results, achieving volume fractions and weighted compliance values within 5% of the large-scale SIMP model, while demonstrating a computational efficiency improvement ranging from 10 times to over 250 times.

A variational multiscale method derived from an adaptive stabilized conforming finite element method via residual minimization on dual norms

This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a triangulation of the domain; we denote these spaces as coarse and enriched spaces. Building on the adaptive stabilized finite element method via residual minimization, we find a coarse-scale approximation in a continuous space by minimizing the residual on a dual discontinuous Galerkin norm; this process allows us to compute a robust error estimate to construct an on-the-fly adaptive method. We reinterpret the residual projection using the variational multiscale framework to derive a fine-scale approximation. As a result, on each mesh of the adaptive process, we obtain stable coarse- and fine-scale solutions derived from a symmetric saddle-point formulation and an a-posteriori error indicator to guide automatic adaptivity. We test our framework in several challenging scenarios for linear and nonlinear convection-dominated diffusion problems to demonstrate the framework’s performance in providing stability in the solution with optimal convergence rates in the asymptotic regime and robust performance in the pre-asymptotic regime. Lastly, we introduce a heuristic dual-term contribution in the variational form to improve the full-scale approximation for symmetric formulations (e.g., diffusion problem).

A transient global-local generalized FEM for parabolic and hyperbolic PDEs with multi-space/time scales

This paper presents a novel Generalized Finite Element Method with global-local enrichment (GFEMgl) to solve time-dependent parabolic and hyperbolic problems with subscale features in both space and time. Compared with currently available GFEMgl, the proposed method requires a solution of local problems to solve transient PDEs, possibly with a different time integrator from the global problem or other local problems. The fine-scale solution can be easily retrieved. The proposed method is tested for the solution of the heat, advection, and advection-diffusion equations whose accuracy, stability, scalability, and convergence are thoroughly analyzed. Compared to direct analysis, numerical results show that the proposed method has the following properties 1) the accuracy closely matches direct analysis result with a fine mesh; 2) critical time step size is loosened; 3) optimal convergence rate is achieved. Moreover, the proposed method allows local problems to be turned off if their solutions are not helpful for the current global time step, which can further improve its efficiency.

Adaptive multiscale extended finite element method (MS-XFEM) for the simulation of multiple fractures propagation in geological formations

In fractured geological formations, as a result of the in-situ stress changes, fractures can propagate or slide. This phenomenon can be efficiently modeled by the extended finite element method (XFEM) when there are only a few fractures present. However, geological reservoirs contain many fractures which can also cross and are densely populated. Therefore, the classical XFEM is too expensive to be applied for the simulation of propagating fractures in geological formations. To reduce the costs, typically, homogenization or upscaling is used. However, they result in inaccurate solutions, since no separation of scales exists in this process. To resolve this challenge, in this work, a multiscale XFEM (MS-XFEM) for propagating fractures is developed and presented. In each time step, given the current geometries of the fractures, local XFEM-based basis functions are constructed or adaptively updated. The adaptive update takes place in certain regions where fracture geometries are changed due to propagation. Using these basis functions, a very efficient FEM-based coarse-scale system is developed since it has no extra degrees of freedom (DOFs). Once the coarse-scale system is solved, its solution is prolonged to the fine-scale original resolution using the basis functions. This approximate fine-scale solution is then used to estimate the group of growing fractures tips and their growing angles. This allows for exploiting the locality of the propagation process fully while solving a global system. To control the error, an iterative procedure is also developed. Proof-of-concept test cases are presented to study the developed MS-XFEM algorithm. It is shown that MS-XFEM results are capable of predicting the propagating paths for complicated fracture patterns. As such, MS-XFEM casts a promising method for field-scale applications.

Online Adaptive Algorithm for Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method

In this research, we propose an online basis enrichment strategy within the framework of a recently developed constraint energy minimizing generalized multiscale discontinuous Galerkin method. Combining the technique of oversampling, one makes use of the information of the current residuals to adaptively construct basis functions in the online stage to reduce the error of multiscale approximation. A complete analysis of the method is presented, which shows the proposed online enrichment leads to a fast convergence from multiscale approximation to the fine-scale solution. The error reduction can be made sufficiently large by suitably selecting oversampling regions and the number of oversampling layers. Further, the convergence rate of the enrichment algorithm depends on a factor of exponential decay regarding the number of oversampling layers and a user-defined parameter. Numerical results are provided to demonstrate the effectiveness and efficiency of the proposed online adaptive algorithm.

Multiscale solver for multi-component reaction–diffusion systems in heterogeneous media

Coupled nonlinear system of reaction–diffusion equations describing multi-component (species) interactions with heterogeneous coefficients is considered. Finite volume method based approximation for the space is used to construct semi-discrete form for the computation of numerical solutions. Two techniques for time approximations, namely, a fully implicit (FI) and a semi-implicit (SI) schemes are examined. The fully implicit scheme is constructed using Newton’s method and leads to the coupled system of equations on each nonlinear and time iterations which is computationally rather expensive. In order to minimize the latter hurdle, an efficient and fast multiscale solver is proposed for reaction–diffusion systems in heterogeneous media. To construct fast solver, we apply a semi-implicit scheme that leads to an uncoupled system for each individual component. Problems in heterogeneous domains require a very fine grid for accurate solutions of large systems of equations at each time step iteration. Here we present a multiscale model reduction technique to reduce the size of the discrete system. Multiscale solver is based on the uncoupled operator of the problem and constructed by the use of Generalized Multiscale Finite Element Method (GMsFEM). In GMsFEM we use a diffusion part of the operator and construct multiscale basis functions by solving spectral problems in each local domain associated with the coarse grid nodes. We collect multiscale basis functions to construct a projection/prolongation matrix and generate reduced order model on the coarse grid for fast solution. Moreover, the prolongation operator is used to reconstruct a fine-scale solution and accurate approximation of the reaction part of the problem which then leads to a very accurate and computationally effective multiscale solver. We provide numerical results for two species competition test problems in two-dimensional domain with heterogeneous inclusions. Our computed solutions predict strong competition between the two species and demonstrate that diffusion can dictate the dominance of one species over the other in heterogeneous environments. We investigate the influence of number of the multiscale basis functions to the method accuracy and ability to work with different values of the diffusion coefficients.

Simplified Subgrid multiscale stabilized finite element method in the transient framework for Stokes equations

This article presents a novel stabilized finite element analysis for the transient Stokes model. The algebraic subgrid variational multiscale finite element scheme with dynamic subscales approach has been employed to arrive at the stabilized formulation. Both the coarse and the fine scale solutions are of time dependent nature and the unknown fine scale solution is completely eliminated in terms of the coarse scale solution during the derivation. This elimination results into the emergence of a new subgrid multiscale stabilized formulation in the transient framework. This formulation facilitates the theoretical derivations of the robustness properties of the scheme. The fully implicit backward Euler scheme has been applied for the time discretization. Here we have analysed the stability property of the approximate solution. As well as a detailed derivation of the a posteriori error estimate has been presented. The scheme is numerically validated for a benchmark problem and appropriate numerical experiments have been carried out to verify the theoretically established order of convergence results.

Multiband homogenization of metamaterials in real-space: Higher-order nonlocal models and scattering at external surfaces

This work develops a dynamic homogenization approach for metamaterials. It finds an approximate macroscopic homogenized equation with constant coefficients posed in space and time; however, the resulting homogenized equation is higher order in space and time. The homogenized equation can be used to solve initial-boundary-value problems posed on arbitrary non-periodic macroscale geometries with macroscopic heterogeneity, such as bodies composed of several different metamaterials or with external boundaries. First, considering a single band, the higher-order space derivatives lead to additional continuity conditions at the boundary between a homogeneous material and a metamaterial. These provide predictions of wave scattering in 1-d and 2-d that match well with the exact fine-scale solution; compared to alternative approaches, they provide a single equation that is valid over a broad range of frequencies, are easy to apply, and are much faster to compute. Next, the setting of two bands with a bandgap is considered. The homogenized equation has also higher-order time derivatives. Notably, the homogenized model provides a single equation that is valid over both bands and the bandgap. The continuity conditions are applied to wave scattering at a boundary, and show good agreement with the exact fine-scale solution. Using that the order of the highest time derivative is proportional to the number of bands considered, a nonlocal-in-time structure is conjectured for the homogenized equation in the limit of infinite bands. This suggests that homogenizing over finer length and time scales -- with the temporal homogenization being carried out through the consideration of higher bands in the dispersion relation -- is a mechanism for the emergence of macroscopic spatial and temporal nonlocality, with the extent of temporal nonlocality being related to the number of bands considered.

Triple mesh methods and their application to two-phase flow in porous media

The computational cost of simulation on fine-scale reservoir models can be prohibitively expensive. While upscaling typically helps in reducing cost, it also results in a reduction in accuracy. Two-scale approaches such as dual mesh methods (DMM) have been developed over time in an attempt to reduce simulation cost while maintaining near fine-scale levels of accuracy and resolution. While these two-scale methods can be very effective in improving accuracy, they can be quite expensive as well. This paper presents two multimesh methods, namely, the Triple Mesh Method (TMM) and the Extended Triple Mesh Method (ETMM). Both methods involve introducing a third grid at an intermediate scale between the coarse and fine scales. These involve two levels of upscaling of the grid properties and two levels of downscaling the solutions to the flow equation. TMM involves two successive local downscaling steps of the flow solution from the coarsest to the finest mesh. ETMM on the other hand involves two successive extended local downscaling steps of the flow solution from the coarsest to the finest mesh. Each downscaling step in ETMM involves the application of ‘Directional Oversampling’ first introduced in the extended dual mesh method EDMM. These two methods along with DMM and EDMM methods were tested on different waterflooding problems and results compared with the coarse-scale and fine-scale solutions. The results show both ETMM and TMM to be effective in error reduction and also more cost effective than their respective dual mesh alternatives.

Adaptive formulation for two-phase reactive transport in heterogeneous porous media

An adaptive scheme for the sequential formulation of reactive transport involving two fluid phases is presented. Instead of solving the transport problem on the global fine-scale model, we decompose the domain into coarse gridblocks that are aggregates of fine gridblocks, and we define and solve transport problem on each of the coarse blocks. The details of the prolongation operator that allows for accurate reconstruction of the fine-scale solution within a coarse gridblock are described. The prolongation operator is independent of the details of the chemical reactions involving the particular component being transported and does not interfere with the chemical module, providing flexibility of implementation and applicability to a wide range of conditions. Test cases include applications of the adaptive strategy for chains of homogeneous decay reactions, as well heterogeneous reactions that involve changes in porosity and the reactive surface area with variable time step size and changes in operational conditions.

Numerical Study of Soil-Thawing Effect of Composite Piles Using GMsFEM

During construction works, it is advisable to prevent strong thawing and an increase in the moisture content of the foundations of engineering structures in the summer. Since the density of water and ice differ, due to the difference bulging of the foundation sections can occur when it freezes back in winter. In this work, the effect of fiber-reinforced piles on the thermal field of the surrounding soil is investigated numerically; that is, the study of the influence of aggregates with high and low thermal-physical properties on the temperature of frozen soils is conducted. Basalt and steel fiber reinforcement are compared. The difficulty of this work is that the inclusions inside piles are too small compared to the pile itself. Therefore, to solve the Stefan problem, a generalized multiscale finite element method (GMsFEM) was used. In the GMsFEM, the usual conforming partition of the domain into a coarse grid was used. It allowed reducing problem size and, consequently, accelerating the calculations. Results of the multiscale solution were compared with fine-scale solution, the accuracy of GMsFEM was investigated, and the optimal solution parameters were defined. Therefore, GMsFEM was shown to be well suited for the designated task. Collation of basalt and steel fiber reinforcement showed a beneficial effect of high thermal conductive material inclusion on freezing of piles in winter.

Multiscale finite-difference method for frequency-domain acoustic wave modeling

Conventional finite-difference frequency-domain (FDFD) methods can describe wave attenuation and velocity dispersion more easily than time-domain methods. However, there are significant challenges associated with the computational costs for solving the linear system when frequency-domain methods are applied in models with large dimensions or fine-scale property variations. Direct-iterative solvers and parallel strategies attempt a trade-off between memory and time costs. We have followed the general framework of a heterogeneous multiscale method and developed a multiscale FDFD approach to solve the Helmholtz equation with lower memory and time costs. To achieve this, the discrete linear system approximating the Helmholtz equation is constructed on a coarse mesh, making its dimension much smaller than that of conventional methods. The coefficient matrix in the linear system of dimension-reduction captures fine-scale heterogeneity in the media by coupling fine- and coarse-scale meshes. Several test models are used to verify the accuracy of our multiscale method and investigate potential sources of error. Numerical results demonstrate that our method accurately approximates the wavefields of fine-scale solutions at low frequencies of the source and could produce solutions with small errors by reducing the size of the coarse-mesh cells at high frequencies as well. Comparisons of computational costs with conventional FDFD methods show that our multiscale method significantly reduces computation time and memory consumption.

Extended Dual Mesh Method with applications in two-phase flow in heterogeneous porous media

Modern geological models typically contain too many cells that make running reservoir simulations at times impractical or prohibitively costly. While upscaling helps with this problem, upscaled models typically suffer loss in accuracy. Different multiscale and dual mesh methods have been developed in an attempt to achieve fine-scale accuracy at reduced computational cost. This paper presents a dual-grid simulation method called Extended Dual Mesh Method (EDMM). EDMM is aimed at reducing homogenization and numerical dispersion errors inherent in upscaled models. In EDMM, velocities are first solved on the coarse scale. Fine-scale velocity fields are then computed by solving extended (oversampled) local flow problems with Neumann boundary conditions similar to Dual Mesh Method (DMM). These extended local problems are constructed such that local and global conservation is guaranteed. To achieve this, we propose a new concept of called Directional Oversampling (DO). DO ensures flux continuity between oversampled partitions therefore guaranteeing that the obtained global fine-scale velocity field is conservative. The resulting fine flux field is then used to compute fine-scale saturation. Also proposed is a new approach for calculating the coarse block interface transmissibilities using fine-scale-block mobilities thereby improving the coupling between the coarse-scale and fine-scale models. EDMM combines the simplicity of the DMM with the accuracy improvement of oversampling and is compatible with any upscaling technique. Examples were used to test the accuracy and robustness of EDMM and results were compared to the fine-scale, DMM and coarse-scale solutions. Results show that the method is a significant improvement on the DMM, much better than coarse grid results and comparable in quality to fine-scale solutions whilst maintaining speed comparable to DMM. Two error indicators, the water cut error and the breakthrough error, were employed in quantify accuracy of the different solution methods relative to the fine scale solution. The indicators show EDMM to be multiple times more accurate than DMM and in some examples orders of magnitudes more accurate. This work demonstrates EDMM to be accurate not only in predicting the water breakthrough at producer wells but also at predicting water-cut after breakthrough. • Extended Dual Mesh Method EDMM is a dual grid method with an extended local downscaling step. • Results from 3 examples show EDMM to be an improvement on upscaling as well as Dual Mesh Method in terms of accuracy. • EDMM is parallelizable and is compatible with any upscaling technique.

A variational multiscale immersed meshfree method for heterogeneous materials

We introduce an immersed meshfree formulation for modeling heterogeneous materials with flexible non-body-fitted discretizations, approximations, and quadrature rules. The interfacial compatibility condition is imposed by a volumetric constraint, which avoids a tedious contour integral for complex material geometry. The proposed immersed approach is formulated under a variational multiscale based formulation, termed the variational multiscale immersed method (VMIM). Under this framework, the solution approximation on either the foreground or the background can be decoupled into coarse-scale and fine-scale in the variational equations, where the fine-scale approximation represents a correction to the residual of the coarse-scale equations. The resulting fine-scale solution leads to a residual-based stabilization in the VMIM discrete equations. The employment of reproducing kernel (RK) approximation for the coarse- and fine-scale variables allows arbitrary order of continuity in the approximation, which is particularly advantageous for modeling heterogeneous materials. The effectiveness of VMIM is demonstrated with several numerical examples, showing accuracy, stability, and discretization efficiency of the proposed method.

Improving coarse-scale simulation models with a dual-porosity dual-permeability upscaling technique and a near-well approach

In the management of an oil field, the creation of a highly detailed fine-scale model to properly represent the key reservoir heterogeneities is important. This model better represents the main reservoir features but remains a significant challenge for dynamic reservoir simulations with impractically high computational costs, even for single numerical simulation runs. Therefore, the scaling up of a highly detailed model is a crucial step in reservoir engineering studies to enable field-scale simulation runs within a time-frame compatible with the needs of the industry. Upscaling must be carefully performed because it can introduce significant errors and bias to the numerical models due to truncation errors and the smoothing of sub-grid heterogeneities.This work focuses on improving the representation of small-scale heterogeneities in coarse models. It expands the dual-porosity and dual-permeability upscaling technique proposed (Rios et al., 2020) to be applied to 3D highly heterogeneous systems. Specific near-well treatment is considered to improve the overall flow distribution in the wells; with this procedure, the coarse model better represents the production and injection profiles throughout the wells. Therefore, the final coarse-scale model can represent the main reservoir heterogeneities and possible preferential pathways, while improving the definition of the wells’ productivity and injectivity indices.The proposed upscaling technique is applied to different reservoir simulation models with immiscible water flooding in 3D evaluations. Our approach was validated by two case studies: SPE-10 and a field-case heterogeneous reservoir with karst structures. The focus is to demonstrate that the division of the porous media into primary and secondary systems improves capillary and gravity representation of coarse models in 3D applications, while the near-well upscaling makes the flow distribution in the wells in agreement with the fine-scale solution. The resulting coarse-scale dual-permeability models are much faster than the reference fine models (between 0.025% and 1.5% of the fine-scale simulation time), more accurate than the traditional flow-based upscaling results, and can better reproduce the fine-scale responses in different upscale ratios (from 25 up to 1 700).

Shale gas permeability upscaling from the pore-scale

The effective permeability of large shale samples provides useful insights into shale gas production. However, its determination can only be achieved through upscaling, since the direct pore-scale simulation of gas flows in large rock samples is not feasible due to the high computational cost and absence of pore connectivity in sample images. Although the Brinkman formulation is widely used in the permeability upscaling of conventional rocks, how to choose the effective viscosity in this coarse-scale model is not clear. Moreover, its application in shale rocks, where the rarefaction effects are important so that the conventional Navier–Stokes equations are inadequate, is rare, and its accuracy has not been assessed. This study aims to address the above two problems, by comparing the Brinkman solutions of several two-dimensional and three-dimensional random porous media containing fractures with the fine-scale solutions of the Stokes and Boltzmann equations (for continuum and rarefied gas flows, respectively). It is found that the use of the fluid viscosity in the Brinkman model, instead of the controversial effective viscosity, leads to accurate results for the cases considered. Additionally, the macroscopic quantity in rarefied gas flows in shale rocks is well predicted by the Brinkman model for a wide range of gas rarefaction, since the error is found to be less than 7%, while neglecting the rarefaction effects leads to significant underestimation of effective permeability (up to 90% in the cases studied). Although heterogeneity and anisotropy of the porous medium increase the error of the effective permeability derived from the Brinkman model, generally speaking, the effective permeability extracted from this coarse-scale model compares favorably to its fine-scale counterpart.

Upscaling capillary pressure curves for numerical modeling of gravity-capillary driven flow

This work investigates upscaling of capillary pressure curves for modeling gravity segregation under the influence of capillary heterogeneity. We consider two flowing phases driven by gravity and capillary forces and seek the saturation spatial and temporal variation until equilibrium is reached. Existing upscaling methods, found in the literature, are applied to a number of cases. Resulting saturation solutions are compared to fine-scale simulations and the different methods are evaluated, showing in general that the popular capillary limit method produces the best results. However, a large number of cases are found to have significant errors. This leads to the conclusion that all existing methods are often inadequate and developing new methods should be considered. We therefore propose a new optimization-based upscaling method. Using this approach it is shown that capillary pressure can be upscaled to Brooks-Corey type functions and produce accurate upscaled simulations matching the fine-scale solutions. Optimization upscaling is computationally demanding and requires the fine-scale simulations for minimizing an objective function. However, it is shown that the upscaled curves can be applied to different permeability realizations to calculate ensemble average saturation solutions.

A Bayesian Numerical Homogenization Method for Elliptic Multiscale Inverse Problems

A new strategy based on numerical homogenization and Bayesian techniques for solving multiscale inverse problems is introduced. We consider a class of elliptic problems which vary at a microscopic scale, and we aim at recovering the highly oscillatory tensor from measurements of the fine scale solution at the boundary, using a coarse model based on numerical homogenization and model order reduction. Assuming a known micro structure, our aim is to recover a macroscopic scalar parameterization of the microscale tensor. We provide a rigorous Bayesian formulation of the problem, taking into account different possibilities for the choice of the prior measure. We prove well-posedness of the effective posterior measure and, by means of G-convergence, we establish a link between the effective posterior and the fine scale model. Several numerical experiments illustrate the efficiency of the proposed scheme and confirm the theoretical findings.