Darcy Flow Problem Research Articles

An augmented Lagrangian-based preconditioning technique for a class of block three-by-three linear systems

We propose an augmented Lagrangian-based preconditioner to accelerate the convergence of Krylov subspace methods applied to linear systems of equations with a block three-by-three structure such as those arising from mixed finite element discretizations of the coupled Stokes–Darcy flow problem. We analyze the spectrum of the preconditioned matrix and we show how the new preconditioner can be efficiently applied. Numerical experiments are reported to illustrate the effectiveness of the preconditioner in conjunction with flexible GMRES for solving linear systems of equations arising from a 3D test problem.

Global sensitivity analysis using multi-resolution polynomial chaos expansion for coupled Stokes–Darcy flow problems

Determination of relevant model parameters is crucial for accurate mathematical modelling and efficient numerical simulation of a wide spectrum of applications in geosciences. The conventional method of choice is the global sensitivity analysis (GSA). Unfortunately, at least the classical Monte-Carlo based GSA requires a high number of model runs. Response surfaces based techniques, e.g. arbitrary Polynomial Chaos (aPC) expansion, can reduce computational effort, however, they suffer from the Gibbs phenomena and high hardware requirements for higher accuracy. We introduce GSA for arbitrary Multi-Resolution Polynomial Chaos (aMR-PC) which is a localized aPC based data-driven polynomial discretization. The aMR-PC allows to reduce the Gibbs phenomena by construction and to achieve higher accuracy by means of localization also for lower polynomial degrees. We apply these techniques to perform the sensitivity analysis for the Stokes–Darcy problem which describes fluid flow in coupled free-flow and porous-medium systems. We consider the Stokes equations in the free-flow region, Darcy’s law in the porous-medium domain and the classical interface conditions across the fluid–porous interface including the conservation of mass, the balance of normal forces and the Beavers–Joseph condition for the tangential velocity. This coupled problem formulation contains four uncertain parameters: the exact location of the interface, the permeability, the Beavers–Joseph slip coefficient and the uncertainty in the boundary conditions. We carry out the sensitivity analysis of the coupled model with respect to these parameters using the Sobol indices on the aMR-PC expansion and conduct the corresponding numerical simulations.

On the geometry transferability of the hybrid iterative numerical solver for differential equations

The discovery of fast numerical solvers prompted a clear and rapid shift towards iterative techniques in many applications, especially in computational mechanics, due to the increased necessity for solving very large linear systems. Most numerical solvers are highly dependent on the problem geometry and discretization, facing issues when any of these properties change. The newly developed Hybrid Iterative Numerical Transferable Solver (HINTS) combines a standard solver with a neural operator to achieve better performance, focusing on a single geometry at a time. In this work, we explore the “T” in HINTS, i.e., the geometry transferability properties of HINTS. We first propose to directly employ HINTS built for a specific geometry to a different but related geometry without any adjustments. In addition, we propose the integration of an operator level transfer learning with HINTS to even further improve the convergence of HINTS on new geometries and discretizations. We conduct numerical experiments for a Darcy flow problem and a plane-strain elasticity problem. The results show that both the direct application of HINTS and the transfer learning enhanced HINTS are able to accurately solve these problems on different geometries. In addition, using transfer learning, HINTS is able to converge to machine zero even faster than the direct application of HINTS.

Gated Linear Model induced U-net for surrogate modeling and uncertainty quantification

We propose a novel deep learning based surrogate model for solving high-dimensional uncertainty quantification and uncertainty propagation problems. The proposed deep learning architecture is developed by integrating the well-known U-net architecture with the Gaussian Gated Linear Network (GGLN) and referred to as the Gated Linear Network induced U-net or GLU-net. The proposed GLU-net treats the uncertainty propagation problem as an image to image regression and hence, is extremely data efficient. Additionally, it also provides estimates of the predictive uncertainty. The network architecture of GLU-net is less complex with 44% fewer parameters than the contemporary works. We illustrate the performance of the proposed GLU-net in solving the Darcy flow problem under uncertainty under the sparse data scenario. We consider the stochastic input dimensionality to be up to 4225. Benchmark results are generated using the vanilla Monte Carlo simulation. We observe the proposed GLU-net to be accurate and extremely efficient even when no information about the structure of the inputs is provided to the network. Case studies are performed by varying the training sample size and stochastic input dimensionality to illustrate the robustness of the proposed approach.

Finite volume element methods for a multi-dimensional fracture model

This study concerns the Darcy flow problem in a two-dimensional fractured porous domain, in which the fracture is regarded as a one-dimensional interface, interacting with the surrounding media. In this paper, a finite volume element method (FVEM) is first proposed for the multi-dimensional fracture model, and error estimates for the pressure with optimal convergence are discussed. On this basis, a two-grid FVEM is developed for decoupling the multi-domain fracture model by a coarse grid approximation to the interface coupling conditions, and theoretical analysis demonstrates that approximation accuracy does not deteriorate under the two-grid decoupling technique. Finally, numerical experiments for FVEM and two-grid FVEM are presented to confirm the accuracy of theoretical analysis.

Randomized maximum likelihood based posterior sampling

Minimization of a stochastic cost function is commonly used for approximate sampling in high-dimensional Bayesian inverse problems with Gaussian prior distributions and multimodal posterior distributions. The density of the samples generated by minimization is not the desired target density, unless the observation operator is linear, but the distribution of samples is useful as a proposal density for importance sampling or for Markov chain Monte Carlo methods. In this paper, we focus on applications to sampling from multimodal posterior distributions in high dimensions. We first show that sampling from multimodal distributions is improved by computing all critical points instead of only minimizers of the objective function. For applications to high-dimensional geoscience inverse problems, we demonstrate an efficient approximate weighting that uses a low-rank Gauss-Newton approximation of the determinant of the Jacobian. The method is applied to two toy problems with known posterior distributions and a Darcy flow problem with multiple modes in the posterior.

An overlapping local projection stabilization for Galerkin approximations of Stokes and Darcy flow problems

An a priori analysis for a generalized local projection stabilized finite element approximation of the Stokes, and the Darcy flow equations are presented in this paper. A first-order conforming P1c finite element space is used to approximate both the velocity and pressure. It is shown that the stabilized discrete bilinear form satisfies the inf-sup condition in the generalized local projection norm. Moreover, a priori error estimates are established in a mesh-dependent norm as well as in the L2-norm for the velocity and pressure. The optimal and quasi-optimal convergence properties are derived for the Stokes and the Darcy flow problems. Finally, the derived estimates are numerically validated with appropriate examples.

Coupling Arbogast–Correa and Bernardi–Raugel elements to resolve coupled Stokes–Darcy flow problems

This paper presents a finite element method for solving coupled Stokes–Darcy flow problems by combining the classical Bernardi–Raugel finite elements and the recently developed Arbogast–Correa (AC) spaces on quadrilateral meshes. The novel weak Galerkin methodology is employed for discretization of the Darcy equation. Specifically, piecewise constant approximants separately defined in element interiors and on edges are utilized to approximate the Darcy pressure. The discrete weak gradients of these shape functions and the numerical Darcy velocity are established in the lowest order AC space. The Bernardi–Raugel elements (BR1,Q0) are used to discretize the Stokes equations. These two types of discretizations are combined at an interface, where kinematic, normal stress, and the Beavers–Joseph–Saffman (BJS) conditions are applied. Rigorous error analysis along with numerical experiments demonstrate that the method is stable and has optimal-order accuracy.

Constraint energy minimizing generalized multiscale discontinuous Galerkin method

Numerical simulation of flow problems and wave propagation in heterogeneous media has important applications in many engineering areas. However, numerical solutions on the fine grid are often prohibitively expensive, and multiscale model reduction techniques are introduced to efficiently solve for an accurate approximation on the coarse grid. In this paper, we propose an energy minimization based multiscale model reduction approach in the discontinuous Galerkin discretization setting. The main idea of the method is to extract the non-decaying component in the high conductivity regions by identifying dominant modes with small eigenvalues of local spectral problems, and define multiscale basis functions in coarse oversampled regions by constraint energy minimization problems. The multiscale basis functions are in general discontinuous on the coarse grid and coupled by interior penalty discontinuous Galerkin formulation. The minimal degree of freedom in representing high-contrast features is achieved through the design of local spectral problems, which provides the most compressed local multiscale space. We analyze the method for solving Darcy flow problem and show that the convergence is linear in coarse mesh size and independent of the contrast, provided that the oversampling size is appropriately chosen. Numerical results are presented to show the performance of the method for simulation on flow problem and wave propagation in high-contrast heterogeneous media.

English

We examine different approaches to an efficient solution of the stochastic Galerkin (SG) matrix equations coming from the Darcy flow problem with different, uncertain coefficients in apriori known subdomains. The solution of the SG system of equations is usually a very challenging task. A relatively new approach to the solution of the SG matrix equations is the reduced basis (RB) solver, which looks for a low-rank representation of the solution. The construction of the RB is usually done iteratively and consists of multiple solutions of systems of equations. We examine multiple approaches and their modifications to the construction of the RB, namely the reduced rational Krylov subspace method and Monte Carlo sampling approach. We also aim at speeding up the process using the deflated conjugate gradients (DCG). We test and compare these methods on a set of problems with a varying random behavior of the material on subdomains as well as different geometries of subdomains.

Combined face based and nodal based discretizations on hybrid meshes for non-isothermal two-phase Darcy flow problems

In the last 20 years many discretization schemes have been developed to approximate the Darcy fluxes on polyhedral cells in heterogeneous anisotropic porous media. Among them, we can distinguished cell based approaches like the Two Point Flux Approximation (TPFA) or the Multi Point Flux Approximation (MPFA) schemes, face based approaches like the Hybrid Finite Volume (HFV) scheme belonging to the family of Hybrid Mimetic Mixed methods and nodal based discretizations like the Vertex Approximate Gradient (VAG) scheme. They all have their own drawbacks and advantages which typically depend on the type of cells and on the anisotropy of the medium. In this work, we propose a new methodology to combine the VAG and HFV discretizations on arbitrary subsets of cells or faces in order to choose the best suited scheme in different parts of the mesh. In our approach the TPFA discretization is considered as an HFV discretization for which the face unknowns can be eliminated. The coupling strategy is based on a node to face interpolation operator at the interfaces which must be chosen to ensure the consistency, the coercivity and the limit conformity properties of the combined discretization. The convergence analysis is performed in the gradient discretization framework and convergence is proved for arbitrary cell or face partitions of the mesh. For face partitions, an additional stabilisation local to the cell is required to ensure the coercivity while for cell partitions no additional stabilisation is needed. The framework preserves at the interface the discrete conservation properties of the VAG and HFV schemes with fluxes based on local to each cell transmissibility matrices. This discrete conservative form allows to naturally extend the VAG and HFV discretizations of two-phase Darcy flow models to the combined VAG–HFV schemes. The efficiency of our approach is tested for single phase and immiscible two-phase Darcy flows on 3D meshes using a combination of the HFV and VAG discretizations as well as for non-isothermal compositional liquid gas Darcy flows on a vertical 2D cross-section of the Bouillante geothermal reservoir (Guadeloupe) using a combination of the TPFA and VAG discretizations.

Numerical Homogenization of Elliptic PDEs with Similar Coefficients

We consider a sequence of elliptic partial differential equations (PDEs) with different but similar rapidly varying coefficients. Such sequences appear, for example, in splitting schemes for time-dependent problems (with one coefficient per time step) and in sample based stochastic integration of outputs from an elliptic PDE (with one coefficient per sample member). We propose a parallelizable algorithm based on Petrov--Galerkin localized orthogonal decomposition that adaptively (using computable and theoretically derived error indicators) recomputes the local corrector problems only where it improves accuracy. The method is illustrated in detail by an example of a time-dependent two-pase Darcy flow problem in three dimensions.

Permeability from 3D Porous Media Images: a Fast Two-Step Approach

An efficient methodology to calculate absolute permeability of porous media using a two-step algorithm is developed. In the first step, the creeping flow equations over the pore space are translated into a Darcy flow problem with the pore space being represented by appropriately chosen local flow conductivities. In the next step, a combined renormalization group and multi-level iterative Laplace solver approach is used to upscale the local conductivities to obtain the effective permeability for the full domain. The accuracy and computational efficiency of the proposed two-step local conductivity–Laplace scheme (LC-LAP) are tested against a FFT (fast Fourier transform) accelerated solver which uses a semi-implicit method for the pressure-linked equation (SIMPLE-FFT) and against a solver that features the GPGPU implementation of the multiple-relaxation-time lattice Boltzmann method (MRT-LBM). A detailed comparison is made by computing permeabilities from all three methods over model geometries and digitized images obtained from micron-scale-resolution computerized tomography (micro-CT) of sandstone rocks of varying porosities and heterogeneity levels. We observe an agreement between our method and either benchmark methods (SIMPLE-FFT and MRT-LBM) that is similar to the agreement between both benchmarks. On the samples tested, the computational performance advantage of the LC-LAP approach ranges from 10- to 40-fold compered to SIMPLE-FFT and 8- to 25-fold compared to MRT-LBM. The proposed method is suitable for fast computations and for computations over very large volumes (due to much lower memory and compute resource requirements) for determining single-phase permeabilities of medium- to high-permeability rocks.

A deposition model coupling Stokes’ and Darcy’s equations with nonlinear deposition

In this work we investigate a filtration process whereby particulate is deposited in the flow domain, causing the porosity of the region to decrease. The fluid flow is modeled as a coupled Stokes–Darcy flow problem and the deposition (in the Darcy domain) is modeled using a nonlinear equation for the porosity. Existence and uniqueness of a solution to the governing equations is established. Additionally, the nonnegativity and boundedness of the porosity is shown. A finite element approximation scheme that preserves the nonnegativity and boundedness of the porosity is investigated. Accompanying numerical experiments support the analytical findings.

The Stochastic Galerkin Method for Darcy Flow Problem with Log-Normal Random Field Coefficients

This article presents a study of the Stochastic Galerkin Method (SGM) applied to the Darcy flow problem with a log-normally distributed random material field given by a mean value and an autocovariance function. We divide the solution of the problem into two parts. The first one is the decomposition of a random field into a sum of products of a random vector and a function of spatial coordinates; this can be achieved using the Karhunen-Loeve expansion. The second part is the solution of the problem using SGM. SGM is a simple extension of the Galerkin method in which the random variables represent additional problem dimensions. For the discretization of the problem, we use a finite element basis for spatial variables and a polynomial chaos discretization for random variables. The results of SGM can be utilised for the analysis of the problem, such as the examination of the average flow, or as a tool for the Bayesian approach to inverse problems.

Inverse Problem Solution using Bayesian Approach with Application to Darcy Flow Material Parameters Estimation

Standard numerical methods for solving inverse problems in partial differential equations do not reflect a possible inaccuracy in observed data. However, in real engineering applications we cannot avoid uncertainties caused by measurement errors. In the Bayesian approach every unknown or inaccurate value is treated as a random variable. This paper presents an application of the Bayesian inverse approach to the reconstruction of a porosity field as a parameter of the Darcy flow problem. However, this framework can be applied to a wide range of problems that involve some amount of uncertainty. Here the material field is modeled as a Gaussian random field, which is expressed as a function of several random variables. The information about these random variables is given by the resulting posterior distribution, which is then studied using the Cross-Entropy method and samples are generated using the Metropolis-Hastings algorithm.

Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media

Fully implicit time-space discretizations applied to the two-phase Darcy flow problem lead to the systems of nonlinear equations, which are traditionally solved by some variant of Newton's method. The efficiency of the resulting algorithms heavily depends on the choice of the primary unknowns since Newton's method is not invariant with respect to a nonlinear change of variable. In this regard the role of capillary pressure/saturation relation is paramount because the choice of primary unknowns is restricted by its shape. We propose an elegant mathematical framework for two-phase flow in heterogeneous porous media resulting in a family of formulations, which apply to general monotone capillary pressure/saturation relations and handle the saturation jumps at rocktype interfaces. The presented approach is applied to the hybrid dimensional model of two phase water-gas Darcy flow in fractured porous media for which the fractures are modeled as interfaces of co-dimension one. The problem is discretized using an extension of Vertex Approximate Gradient scheme. As for the phase pressure formulation, the discrete model requires only two unknowns by degree of freedom.

Efficient Reduced Basis Methods for Saddle Point Problems with Applications in Groundwater Flow

Reduced basis methods (RBMs) are recommended to reduce the computational cost of solving parameter-dependent PDEs in scenarios where many choices of parameters need to be considered, for example, in uncertainty quantification (UQ). A reduced basis is constructed during a computationally demanding offline (or set-up) stage that allows the user to obtain cheap approximations for parameter choices of interest, online. In this paper we consider RBMs for parameter-dependent saddle point problems, in particular the one that arises in the mixed formulation of the Darcy flow problem in groundwater flow modeling. We apply a discrete empirical interpolation method (DEIM) to approximate the inverse of the diffusion coefficient, which depends nonaffinely on the system parameters. We develop an efficient RBM that exploits the DEIM approximation and combine it with a sparse grid stochastic collocation mixed finite element method (SCMFEM) to construct a surrogate solution, which then allows for efficient forward UQ. Through numerical experiments we demonstrate that significant computational savings can be made when we use the RB-DEIM-SCMFEM scheme over standard high fidelity methods. For groundwater flow problems, we provide a thorough cost assessment of the new method and show how the size of the reduced basis, and hence, the extent of the savings, depends on the statistical properties of the input parameters.

A multiscale flux basis for mortar mixed discretizations of Stokes–Darcy flows

A multiscale flux basis algorithm is developed for the Stokes–Darcy flow problem. The method is based on a non-overlapping domain decomposition algorithm, where the global problem is reduced to a coarse scale mortar interface problem that is solved by an iterative solver. Subdomain solves are required at each interface iteration, so the cost for the method without a multiscale basis can be high when the number of subdomains or the condition number of the interface problem is large. The proposed algorithm involves precomputing a multiscale flux basis, which consists of the flux (or velocity trace) response from each mortar degree of freedom. It is computed by each subdomain independently before the interface iteration begins. The subdomain solves required at each iteration are substituted by a linear combination of the multiscale basis. This may lead to a significant reduction in computational cost since the number of subdomain solves is fixed, depending only on the number of mortar degrees of freedom associated with a subdomain. Several numerical examples are carried out to demonstrate the efficiency of the multiscale flux basis implementation for large-scale Stokes–Darcy problems.

Similarity Solutions for Laminar Boundary Layer Flow of Darcian Fluid over Horizontal Plate with a Convective Boundary Condition

This paper considers the problem of hydrodynamics and thermal boundary layers of Darcy flow over horizontal surface embedded in a porous medium. The solutions of such problems for the cases of uniform surface temperature and variable surface temperature have been studied and analysed in many papers. This paper, however, attempts to find similarity solutions for the Darcy flow problem with a convective boundary condition at the plate surface. It is found that the solution is possible when the heat transfer coefficient is proportional to x–2/3. The numerical solutions thus obtained are analyzed for a range of values of the parameter characterizing the hot fluid convection process. Analytical expressions are provided for local surface heat flux and total surface heat transfer rate while the flow variables are discussed graphically.