scholarly journals Multiscale Model Reduction of the Unsaturated Flow Problem in Heterogeneous Porous Media with Rough Surface Topography

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 904
Author(s):  
Denis Spiridonov ◽  
Maria Vasilyeva ◽  
Eric T. Chung ◽  
Yalchin Efendiev ◽  
Raghavendra Jana
Keyword(s):  
Porous Media ◽  
Boundary Conditions ◽  
Surface Topography ◽  
Rough Surface ◽  
Heterogeneous Porous Media ◽  
Coarse Grid ◽  
Basis Functions ◽  
Richards Equation ◽  
Small Scale ◽  
Fine Grid

In this paper, we consider unsaturated filtration in heterogeneous porous media with rough surface topography. The surface topography plays an important role in determining the flow process and includes multiscale features. The mathematical model is based on the Richards’ equation with three different types of boundary conditions on the surface: Dirichlet, Neumann, and Robin boundary conditions. For coarse-grid discretization, the Generalized Multiscale Finite Element Method (GMsFEM) is used. Multiscale basis functions that incorporate small scale heterogeneities into the basis functions are constructed. To treat rough boundaries, we construct additional basis functions to take into account the influence of boundary conditions on rough surfaces. We present numerical results for two-dimensional and three-dimensional model problems. To verify the obtained results, we calculate relative errors between the multiscale and reference (fine-grid) solutions for different numbers of multiscale basis functions. We obtain a good agreement between fine-grid and coarse-grid solutions.

scholarly journals Upscaled Lattice Boltzmann Method for Simulations of Flows in Heterogeneous Porous Media

Geofluids ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jun Li ◽  
Donald Brown
Keyword(s):  
Porous Media ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Effective Permeability ◽  
Heterogeneous Porous Media ◽  
Coarse Grid ◽  
Force Model ◽  
Fine Grid ◽  
Reduced Order ◽  
Boltzmann Method

An upscaled Lattice Boltzmann Method (LBM) for flow simulations in heterogeneous porous media at the Darcy scale is proposed in this paper. In the Darcy-scale simulations, the Shan-Chen force model is used to simplify the algorithm. The proposed upscaled LBM uses coarser grids to represent the average effects of the fine-grid simulations. In the upscaled LBM, each coarse grid represents a subdomain of the fine-grid discretization and the effective permeability with the reduced-order models is proposed as we coarsen the grid. The effective permeability is computed using solutions of local problems (e.g., by performing local LBM simulations on the fine grids using the original permeability distribution) and used on the coarse grids in the upscaled simulations. The upscaled LBM that can reduce the computational cost of existing LBM and transfer the information between different scales is implemented. The results of coarse-grid, reduced-order, simulations agree very well with averaged results obtained using a fine grid.

scholarly journals Contribution to Numerical Modeling of Water Flow in Variably Saturated, Heterogeneous Porous Media

2015 ◽  
Vol 28 (3) ◽  
pp. 179-197 ◽  
Author(s):  
Adrien Wanko ◽  
Gabriela Tapia ◽  
Robert Mosé
Keyword(s):  
Porous Media ◽  
Boundary Conditions ◽  
Water Flow ◽  
Numerical Test ◽  
Model Performance ◽  
Pressure Head ◽  
Soil Heterogeneity ◽  
Heterogeneous Porous Media ◽  
Richards Equation ◽  
Initial And Boundary Conditions

Abstract Solutions to the Richards equation for water flow in variably saturated porous media are the focus of this paper. Working with field conditions, the extreme variability and complexity of soil, initial and boundary conditions can make the flow problem difficult to solve. This paper proposes to improve the computational efficiency of the mixed hybrid finite element (MHFE) method coupled with the variables transformation. The transform variables were introduced in order to simulate problems with convergence difficulty attributed to the presence of sharp wetting fronts. Furthermore, for better convergence behaviour, a technique that switches between the mixed-form and the pressure-head based form of the Richard’s equation was applied. Special attention was given to the top boundary conditions dealing with ponding or evaporation problems. In order to avoid non-physical oscillation problems, a mass condensation scheme was implemented in the model. Performance indicators in time and error of different options of the numerical model are defined, analyzed and classified. Thus, for each test case, a suitable numerical method that identifies which form of the Richards equation is best suited, the relevance of the switching technique as well as the utility of the transformation of the primary variable is possible. The results for the 1D Numerical test cases that have been performed matched those from the literature results. For evaporation and infiltration problem’s, the number of iterations needed to get the solution decrease when using the method of transformed pressure. Finally, knowing the soil heterogeneity, initial and boundary conditions, an agglomerative hierarchical clustering allows to analyze the need or not to transform variables and to use other options.

scholarly journals MEASUREMENT OF PREFERENTIAL FLOW DURING INFILTRATION AND EVAPORATION IN POROUS MEDIA

2017 ◽  
Vol 43 (4) ◽  
pp. 1831
Author(s):  
A. Papafotiou ◽  
C. Schütz ◽  
P. Lehmann ◽  
P. Vontobel ◽  
D. Or ◽  
...  
Keyword(s):  
Porous Media ◽  
Water Distribution ◽  
Preferential Flow ◽  
Unsaturated Flow ◽  
Coupling Effect ◽  
Heterogeneous Systems ◽  
Heterogeneous Porous Media ◽  
Richards Equation ◽  
Hydraulic Coupling ◽  
Experimental Findings

Infiltration and evaporation are governing processes for water exchange between soil and atmosphere. In addition to atmospheric supply or demand, infiltration and evaporation rates are controlled by the material properties of the subsurface and the interplay between capillary, viscous and gravitational forces. This is commonly modeled with semi-empirical approaches using continuum models, such as the Richards equation for unsaturated flow. However, preferential flow phenomena often occur, limiting or even entirely suspending the applicability of continuum-based models. During infiltration, unstable fingers may form in homogeneous or heterogeneous porous media. On the other hand, the evaporation process may be driven by the hydraulic coupling of materials with different hydraulic functions found in heterogeneous systems. To analyze such preferential flow processes, water distribution was monitored in infiltration and evaporation lab experiments using neutron transmission techniques. Measurements were performed in 2D and 3D, using homogeneous and heterogeneous setups. The experimental findings demonstrate the fingering effect in infiltration and how it is influenced by the presence of fine inclusions in coarse background material. During evaporation processes, the hydraulic coupling effect is found to control the evaporation rate, limiting the modeling of water balances between soil and surface based on surface information alone.

scholarly journals A Computational Workflow for Flow and Transport in Fractured Porous Media Based on a Hierarchical Nonlinear Discrete Fracture Modeling Approach

Energies ◽  
2020 ◽  
Vol 13 (24) ◽  
pp. 6667
Author(s):  
Wenjuan Zhang ◽  
Waleed Diab ◽  
Hadi Hajibeygi ◽  
Mohammed Al Kobaisi
Keyword(s):  
Boundary Condition ◽  
Porous Media ◽  
Boundary Conditions ◽  
Large Scale ◽  
Fractured Porous Media ◽  
Small Scale ◽  
Flow And Transport ◽  
Fracture Length ◽  
Discrete Fracture ◽  
Computational Workflow

Modeling flow and transport in fractured porous media has been a topic of intensive research for a number of energy- and environment-related industries. The presence of multiscale fractures makes it an extremely challenging task to resolve accurately and efficiently the flow dynamics at both the local and global scales. To tackle this challenge, we developed a computational workflow that adopts a two-level hierarchical strategy based on fracture length partitioning. This was achieved by specifying a partition length to split the discrete fracture network (DFN) into small-scale fractures and large-scale fractures. Flow-based numerical upscaling was then employed to homogenize the small-scale fractures and the porous matrix into an equivalent/effective single medium, whereas the large-scale fractures were modeled explicitly. As the effective medium properties can be fully tensorial, the developed hierarchical framework constructed the discrete systems for the explicit fracture–matrix sub-domains using the nonlinear two-point flux approximation (NTPFA) scheme. This led to a significant reduction of grid orientation effects, thus developing a robust, applicable, and field-relevant framework. To assess the efficacy of the proposed hierarchical workflow, several numerical simulations were carried out to systematically analyze the effects of the homogenized explicit cutoff length scale, as well as the fracture length and orientation distributions. The effect of different boundary conditions, namely, the constant pressure drop boundary condition and the linear pressure boundary condition, for the numerical upscaling on the accuracy of the workflow was investigated. The results show that when the partition length is much larger than the characteristic length of the grid block, and when the DFN has a predominant orientation that is often the case in practical simulations, the workflow employing linear pressure boundary conditions for numerical upscaling give closer results to the full-model reference solutions. Our findings shed new light on the development of meaningful computational frameworks for highly fractured, heterogeneous geological media where fractures are present at multiple scales.

scholarly journals Complexity Reduction of Multiphase Flows in Heterogeneous Porous Media

SPE Journal ◽  
2016 ◽  
Vol 21 (01) ◽  
pp. 144-151 ◽  
Author(s):  
Mehdi Ghommem ◽  
Eduardo Gildin ◽  
Mohammadreza Ghasemi
Keyword(s):  
Porous Media ◽  
Model Reduction ◽  
Flow Behavior ◽  
Computational Cost ◽  
Heterogeneous Porous Media ◽  
Production Optimization ◽  
Dynamic Mode ◽  
Two Phase ◽  
Fine Grid ◽  
Mode Decomposition

Summary In this paper, we apply mode decomposition and interpolatory projection methods to speed up simulations of two-phase flows in heterogeneous porous media. We propose intrusive and nonintrusive model-reduction approaches that enable a significant reduction in the size of the subsurface flow problem while capturing the behavior of the fully resolved solutions. In one approach, we use the dynamic mode decomposition. This approach does not require any modification of the reservoir simulation code but rather post-processes a set of global snapshots to identify the dynamically relevant structures associated with the flow behavior. In the second approach, we project the governing equations of the velocity and the pressure fields on the subspace spanned by their proper-orthogonal-decomposition modes. Furthermore, we use the discrete empirical interpolation method to approximate the mobility-related term in the global-system assembly and then reduce the online computational cost and make it independent of the fine grid. To show the effectiveness and usefulness of the aforementioned approaches, we consider the SPE-10 benchmark permeability field, and present a numerical example in two-phase flow. One can efficiently use the proposed model-reduction methods in the context of uncertainty quantification and production optimization.

Multiscale modeling of acoustic wave propagation in 2D media

Geophysics ◽  
2014 ◽  
Vol 79 (2) ◽  
pp. T61-T75 ◽  
Author(s):  
Richard L. Gibson ◽  
Kai Gao ◽  
Eric Chung ◽  
Yalchin Efendiev
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Heterogeneous Media ◽  
Coarse Grid ◽  
Basis Functions ◽  
Seismic Survey ◽  
Fine Scale ◽  
Acoustic Wave Propagation ◽  
Fine Grid

Conventional finite-difference methods produce accurate solutions to the acoustic and elastic wave equation for many applications, but they face significant challenges when material properties vary significantly over distances less than the grid size. This challenge is likely to occur in reservoir characterization studies, because important reservoir heterogeneity can be present on scales of several meters to ten meters. Here, we describe a new multiscale finite-element method for simulating acoustic wave propagation in heterogeneous media that addresses this problem by coupling fine- and coarse-scale grids. The wave equation is solved on a coarse grid, but it uses basis functions that are generated from the fine grid and allow the representation of the fine-scale variation of the wavefield on the coarser grid. Time stepping also takes place on the coarse grid, providing further speed gains. Another important property of the method is that the basis functions are only computed once, and time savings are even greater when simulations are repeated for many source locations. We first present validation results for simple test models to demonstrate and quantify potential sources of error. These tests show that the fine-scale solution can be accurately approximated when the coarse grid applies a discretization up to four times larger than the original fine model. We then apply the multiscale algorithm to simulate a complete 2D seismic survey for a model with strong, fine-scale scatterers and apply standard migration algorithms to the resulting synthetic seismograms. The results again show small errors. Comparisons to a model that is upscaled by averaging densities on the fine grid show that the multiscale results are more accurate.

scholarly journals Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media

Fluids ◽  
2021 ◽  
Vol 6 (8) ◽  
pp. 298
Author(s):  
Aleksei Tyrylgin ◽  
Maria Vasilyeva ◽  
Dmitry Ammosov ◽  
Eric T. Chung ◽  
Yalchin Efendiev
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heterogeneous Media ◽  
Coupled System ◽  
Coarse Grid ◽  
Basis Functions ◽  
Fine Grid ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Element Method

In this paper, we consider the poroelasticity problem in fractured and heterogeneous media. The mathematical model contains a coupled system of equations for fluid pressures and displacements in heterogeneous media. Due to scale disparity, many approaches have been developed for solving detailed fine-grid problems on a coarse grid. However, some approaches can lack good accuracy on a coarse grid and some corrections for coarse-grid solutions are needed. In this paper, we present a coarse-grid approximation based on the generalized multiscale finite element method (GMsFEM). We present the construction of the offline and online multiscale basis functions. The offline multiscale basis functions are precomputed for the given heterogeneity and fracture network geometry, where for the construction, we solve a local spectral problem and use the dominant eigenvectors (appropriately defined) to construct multiscale basis functions. To construct the online basis functions, we use current information about the local residual and solve coupled poroelasticity problems in local domains. The online basis functions are used to enrich the offline multiscale space and rapidly reduce the error using residual information. Only with appropriate offline coarse-grid spaces can one guarantee a fast convergence of online methods. We present numerical results for poroelasticity problems in fractured and heterogeneous media. We investigate the influence of the number of offline and online basis functions on the relative errors between the multiscale solution and the reference (fine-scale) solution.

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