Transport Problems In Porous Media Research Articles

Learning generic solutions for multiphase transport in porous media via the flux functions operator

Traditional numerical schemes for simulating multiphase flow and transport in porous media can be computationally expensive, and continues to be an active area of research. Advances in machine learning for scientific computing have the potential to help speed up the simulation time in many scientific and engineering fields. DeepONet has recently emerged as a powerful tool for accelerating the solution of partial differential equations (PDEs) by learning operators (mapping between function spaces) of PDEs. In this work, we learn the mapping between the space of flux functions of the Buckley-Leverett PDE and the space of solutions (saturations). We use Physics-Informed DeepONets (PI-DeepONets) to achieve this mapping without any paired input-output observations, except for a set of given initial or boundary conditions; ergo, eliminating the expensive data generation process. By leveraging the underlying physical laws via soft penalty constraints during model training, in a manner similar to Physics-Informed Neural Networks (PINNs), and a unique deep neural network architecture, the proposed PI-DeepONet model can predict the solution accurately given any type of flux function (concave, convex, or non-convex) while achieving up to four orders of magnitude improvements in speed over traditional numerical solvers. Moreover, the trained PI-DeepONet model demonstrates excellent generalization qualities, rendering it a promising tool for accelerating the solution of transport problems in porous media.

A level-set immersed boundary method for reactive transport in complex topologies with moving interfaces

A simulation framework based on the level-set and the immersed boundary methods (LS-IBM) has been developed for reactive transport problems in porous media involving a moving solid-fluid interface. The interface movement due to surface reactions is tracked by the level-set method, while the immersed boundary method captures the momentum and mass transport at the interface. The proposed method is capable of accurately modeling transport near evolving boundaries in Cartesian grids. The framework formulation guarantees second order accuracy in space. Since the interface velocity is only defined at the moving boundary, an interface velocity propagation method is also proposed. The method can be applied to other moving interface problems of the “Stefan” type. Here, we validate the proposed LS-IBM both for flow and transport close to an immersed object with reactive boundaries as well as for crystal growth. The proposed method provides a powerful tool to model more realistic problems involving moving reactive interfaces in complex domains.

Features of transport in non-Gaussian random porous systems

The goal of this work is to employ a semi-analytical framework to investigate key features associated with the transport behavior of an inert solute in non-Gaussian random fields. We focus our analysis on the transport dynamics of a solute plume through a porous medium characterized by spatially heterogeneous non-Gaussian log-conductivity fields, Y. We rest on a stochastic Lagrangian framework to provide semi-analytical formulations to evaluate the statistical moments and cumulative distribution function (CDF) of solute concentration. The heterogeneous structure of the log-conductivity field is modeled as a Generalized Sub-Gaussian process. This model has been shown to capture non-Gaussian and scale-dependent features displayed by several variables, including key parameters of porous media. Our results suggest that the effects of non-Gaussianity in Y on solute concentration statistics are more pronounced at locations near the solute source zone and at early times. The impact of the analyzed non-Gaussian nature of the field of Y is also significant at the lower tails of the distribution. We also explore conditions under which when the concentration CDF in Generalized Sub-Gaussian Y fields can be approximated by the widely used beta distribution. Furthermore, the methodology used in this work is an alternative to the commonly used numerical Monte Carlo method and can be employed as a benchmark tool in computational stochastic mass transport problems in porous media.

New Finite Volume–Multiscale Finite-Element Model for Solving Solute Transport Problems in Porous Media

AbstractThis paper proposes a finite volume–Yeh’s finite-element–multiscale finite-element model (FVYMSFEM) for solute transport simulations, which is extended from its elliptic groundwater-flow eq...

DuMux 3 – an open-source simulator for solving flow and transport problems in porous media with a focus on model coupling

We present version 3 of the open-source simulator for flow and transport processes in porous media DuMux. DuMux is based on the modular C++ framework Dune (Distributed and Unified Numerics Environment) and is developed as a research code with a focus on modularity and reusability. We describe recent efforts in improving the transparency and efficiency of the development process and community-building, as well as efforts towards quality assurance and reproducible research. In addition to a major redesign of many simulation components in order to facilitate setting up complex simulations in DuMux, version 3 introduces a more consistent abstraction of finite volume schemes. Finally, the new framework for multi-domain simulations is described, and three numerical examples demonstrate its flexibility.

Simulation of reactive transport in porous media using radial point collocation method

In this paper, a meshfree Radial Point Collocation method (RPCM) for modelling of reactive transport involving first-order decay and adsorption is presented. Unlike conventional grid-based methods such as Finite Element (FEM) and Finite Difference Methods (FDM), it does not require nodal connectivity information for domain representation. The proposed method does not necessitate operator splitting for accommodating reactions as it is done in FEM and FDM. The model domain is divided into intersecting circular shaped local support domains. This technique solves the ill-conditioning problem associated with globally supported system and allows use of more number of nodes which increases the accuracy of models. The multi-quadrics radial basis function (MQ-RBF) is used for the approximation/interpolation of solute concentration values in the local support domain. 1D and 2D models are developed and verified against benchmark analytical solutions. Sensitivity analyses are performed for shape parameters of the MQ-RBF, support domain size and nodal density. Two case studies are presented with regular and irregular shaped domains and results are compared with FEM. The results demonstrate advantages of the proposed model over grid-based methods while handling adsorption and first-order decay reactions. This study shows that the proposed model is effective for simulating reactive transport problems in porous media.

An approximate Jacobian nonlinear solver for multiphase flow and transport

We present an approximate Jacobian approach for solving nonlinear, multiphase flow and transport problems in porous media. A backward Euler time discretization scheme is used prior to spatial discretization with a lowest order mixed finite element method (MFEM). This results in a fully implicit nonlinear algebraic system of equations. Conventionally, an exact Jacobian construction is employed during the Newton linearization to obtain a linear system of equations after spatial and temporal discretization. This fully coupled, monolithic linear system, usually in pressure and saturation (or concentration) unknowns, requires specialized preconditioners such as constrained pressure residual (CPR) or two stage preconditioner. These preconditioners operate on the linear system to decouple pressure and saturation (or concentration) degrees of freedom (DOF) in order to use existing linear solvers for positive definite (PD) matrices such as GMRES or AMG, to name a few. In this work, we present an alternative to two-stage preconditioning (or CPR) for solving the aforementioned monolithic system after Newton linearization. This approach relies upon a decoupling approximation for the pressure-saturation (or concentration) block sub-matrices, during Newton linearization, to obtain block diagonal sub-matrices. The resulting linear system is easily reduced, trivially eliminating these diagonal sub-matrices, to obtain a system in pressure DOF circumventing the need for specialized preconditioners. Further, the linear system has lesser DOF owing to the elimination of saturation (or concentration) unknowns. This nonlinear solver is demonstrated to be as accurate as the exact Jacobian approach, measured in terms of convergence of nonlinear residual to a desired tolerance for both methods. Our numerical results indicate a consistent computational speedup by a factor of approximately 1.32 to 4.0 for the two-phase flow model formulation under consideration. This is related to the DOF of the linear systems for the approximate and exact Jacobian approaches. For multicomponent flow and transport, this speedup is expected to be directly proportional to the number of concentration degrees of freedom. A number of field scale numerical simulations are also presented to demonstrate the efficacy of this approach for realistic problems.

A space–time domain decomposition approach using enhanced velocity mixed finite element method

A space–time domain decomposition approach is presented as a natural extension of the enhanced velocity mixed finite element (EVMFE), introduced by Wheeler et al. in (2002) [26], for spatial domain decomposition. The proposed approach allows for different space–time discretizations on non-overlapping, subdomains by enforcing a mass continuity at non-matching interfaces to preserve local mass conservation inherent to the mixed finite element methods. To this effect, we consider three different model formulations: (1) a linear single phase flow problem, (2) a non-linear slightly compressible flow and tracer transport, and (3) a non-linear slightly compressible, multiphase flow and transport. We also present a numerical solution algorithm for the proposed domain decomposition approach where a monolithic (fully coupled in space and time) system is constructed that does not require subdomain iterations. This space–time EVMFE method accurately resolves advection–diffusion transport features, in a heterogeneous medium, while circumventing non-linear solver convergence issues associated with large time-step sizes for non-linear problems. Numerical results are presented for the aforementioned, three, model formulations to demonstrate the applicability of this approach to a general class of flow and transport problems in porous media.

Time-fractional particle deposition in porous media

In the percolation process where fluids carry small solid particles, particle deposition causes a real-time permeability change of the medium as the swarm of particles propagates along the medium. Then the permeability change influences percolation and deposition behaviors as a feedback. This fact triggers memory effect in the deposition dynamics, which means the particulate transport and deposition behaviors become history-dependent. In this paper, we conduct the time-fractional generalization of the classical phenomenological model of particle deposition in porous media to incorporate the memory effect. We tested and compared the effects of employing different types of fractional operators, i.e. the Riemann–Liouville type, the Hadamard type and the Prabhakar type. Numerical simulation results show that the system behaviors vary according to the change of distinct memory kernels in an expected way. We then discuss the physical meaning of the time-fractional generalization. It is shown that different types of fractional operators unanimously ground themselves on the local-Newtonian time transformation in a complex system, which is equivalent to a class of history integrals. By the introduction of various memory kernels, it enables the model to more powerfully fit and approximate observed data. Further, the fundamental meaning of this work is not to show which fractional operator is ‘better’, but to argue collectively the legitimacy and practicality of a non-Markovian particle deposition dynamics in porous media, and in fact it is admissible to a bunch of memory kernels which differ greatly from each other in functional forms. Hopefully the presented generalized mass conservation formalism offers a broader framework to investigate transport problems in porous media.

A unified a posteriori error estimate of local discontinuous Galerkin approximations for reactive transport problems

To solve reactive transport problems in porous media, local discontinuous Galerkin (LDG) approximations are investigated. Based on the duality technique and the residual error notations, a unified a posteriori error estimate in L2(L2) norm is obtained, which is usually used for guiding anisotropic and dynamic mesh adaptivity.

Solution of double nonlinear problems in porous media by a combined finite volume–finite element algorithm

The combined finite volume–finite element scheme for a double nonlinear parabolic convection-dominated diffusion equation which models the variably saturated flow and contaminant transport problems in porous media is extended. Whereas the convection is approximated by a finite volume method (Multi-Point Flux Approximation), the diffusion is approximated by a finite element method. The scheme is fully implicit and involves a relaxation-regularized algorithm. Due to monotonicity and conservation properties of the approximated scheme and in view of the compactness theorem we show the convergence of the numerical scheme to the weak solution. Our scheme is applied for computing two dimensional examples with different degrees of complexity. The numerical results demonstrate that the proposed scheme gives good performance in convergence and accuracy.

Simulation of flow and solute coupled 2‐D problems with velocity‐dependent dispersion coefficient based on the network method

AbstractThe governing equations of coupled density‐driven flow and solute transport problems in porous media, with velocity‐dependent dispersion coefficient, are strongly nonlinear and must be solved numerically. This contribution presents a network model, based on the network simulation method, capable of simulating the transient solution to this kind of problem efficiently and with a relatively low computational time. The mathematical model is formulated using the stream function and concentration variables. Simulation of the network model is carried out in the standard electric circuit simulation code, Pspice. The present model is first applied to simulate the original benchmark Henry problem, and the solution is compared with those obtained by other authors. A study of the grid size is also carried out. In addition, the modified version of Simpson and Clement of the Henry problem, as well as the groundwater flow in the closed desert basin of Pilot Valley, is studied using the proposed model. Copyright © 2011 John Wiley & Sons, Ltd.

Uncertainty quantification in modeling flow and transport in porous media

This special issue of Stochastic Environmental Research and Risk Assessment in honor of Shlomo P. Neuman presents recent developments in uncertainty quantification in predictions of flow and transport in heterogeneous porous media. It covers subjects as diverse as stochastic hydrogeology, aquifer characterization, effective (average) descriptions of subsurface processes, inverse modeling, and scaling theory, all of which benefited from the insightful contributions of Shlomo P. Neuman. We hope that this volume will become a reference point for students learning these subject, scientists pursuing innovative research, and professionals and water resources planners looking to quantify predicting uncertainty arising from the intrinsic complexity of the subsurface environment. S. P. Neuman is one of the pioneers in the use of stochastic modeling as a means of dealing with uncertainty in hydraulic and transport parameters of heterogeneous subsurface environments. This special issue contains a number of contributions in this field. Riva et al. introduce a stochastic approach to interpret field-scale tracer tests conducted in heterogeneous aquifers. By using Monte Carlo simulations, the authors demonstrate that directly linking porosity and permeability distribution to the spatial variability of soil particle sizes renders the best prediction of the observed heavy tailing of measured breakthrough curves. Zhang et al. compare the performance of Monte Carlo and stochastic collocation methods to evaluate the probability of the concentration of a conservative solute attaining a given threshold. Their analysis shows that Latin Hypercube sampling and Quasi Monte Carlo outperform standard Monte Carlo simulations, and that a sparse-grid collocation and a probabilistic collocation method provide an accurate and efficient alternative to Monte Carlo simulations. Broyda et al. develop partial differential equations satisfied by the probability density function (PDF) of the concentration of solutes undergoing heterogeneous reactions in porous media with uncertain chemical properties. The authors obtain a semi-analytical solution for the concentration PDF and demonstrate that its shape and evolution are significantly affected by both spatial variability of advective velocity and the Damkohler number. Tartakovsky presents a novel Lagrangian particle model based on smoothed particle hydrodynamics (SPH). His analysis reveals the advantages of Lagrangian methods for the stochastic analysis of miscible density-driven fluid flows in the presence of RayleighTaylor instability. When used in the context of Monte Carlo simulations, the method can provide reliable estimated of the probability of occurrence of rare events. Two papers are concerned with the estimation of hydrogeological parameters governing groundwater flow. Liu et al. present an approach to efficient and accurate parameter estimation for nonlinear flow and transport problems in porous media. They discuss the relative merits and weaknesses of different schemes, including Maximum a posteriori (MAP) and Monte Carlo based methods, used to parameterize a DNAPL dissolution/transport model. Methods to diagnose Markov Chain Monte Carlo (MCMC) samples are introduced, compared and discussed in the context of computationally demanding parameter estimation A. Guadagnini (&) Dipartimento di Ingeneria Idraulca, Ambientale, Infrastrutture Viarie e Rilevamento, Politecnico di Milano, Piazza L. Da Vinci, 32, 20133 Milan, Italy e-mail: alberto.guadagnini@polimi.it

A parallel global-implicit 2-D solver for reactive transport problems in porous media based on a reduction scheme and its application to the MoMaS benchmark problem

In this article, an approach for the efficient numerical solution of multi-species reactive transport problems in porous media is described. The objective of this approach is to reformulate the given system of partial and ordinary differential equations (PDEs, ODEs) and algebraic equations (AEs), describing local equilibrium, in such a way that the couplings and nonlinearities are concentrated in a rather small number of equations, leading to the decoupling of some linear partial differential equations from the nonlinear system. Thus, the system is handled in the spirit of a global implicit approach (one step method) avoiding operator splitting techniques, solved by Newton’s method as the basic algorithmic ingredient. The reduction of the problem size helps to limit the large computational costs of numerical simulations of such problems. If the model contains equilibrium precipitation-dissolution reactions of minerals, then these are considered as complementarity conditions and rewritten as semismooth equations, and the whole nonlinear system is solved by the semismooth Newton method.

On the use of large time steps with ELLAM for transport with kinetic reactions over heterogeneous domains

AbstractAn Eulerian Lagrangian localized adjoint method (ELLAM) is considered for the resolution of advection‐dominated transport problems in porous media. Contrary to standard Eulerian methods, ELLAM can use large time steps because the advection term is approximated accurately without any CFL restriction. However, it is shown in this article that special care must be taken for the approximation of the dispersive and reactive terms when large time steps are used over heterogeneous domains. An alternative procedure is proposed. It is based on an equivalent dispersion coefficient or an equivalent reaction rate when different zones are encountered during the tracking. Numerical experiments are performed with variable dispersion or variable reaction rates over space (including nonlinearity). When classical ELLAM require numerous time steps to handle heterogeneity, the alternative procedure is shown to perform with the same accuracy in a single time step. © 2009 American Institute of Chemical Engineers AIChE J, 2009

Application of a mixing-ratios based formulation to model mixing-driven dissolution experiments

We address the question of how one can combine theoretical and numerical modeling approaches with limited measurements from laboratory flow cell experiments to realistically quantify salient features of complex mixing-driven multicomponent reactive transport problems in porous media. Flow cells are commonly used to examine processes affecting reactive transport through porous media, under controlled conditions. An advantage of flow cells is their suitability for relatively fast and reliable experiments, although measuring spatial distributions of a state variable within the cell is often difficult. In general, fluid is sampled only at the flow cell outlet, and concentration measurements are usually interpreted in terms of integrated reaction rates. In reactive transport problems, however, the spatial distribution of the reaction rates within the cell might be more important than the bulk integrated value. Recent advances in theoretical and numerical modeling of complex reactive transport problems [De Simoni M, Carrera J, Sanchez-Vila X, Guadagnini A. A procedure for the solution of multicomponent reactive transport problems. Water Resour Res 2005;41:W11410. doi: 10.1029/2005WR004056, De Simoni M, Sanchez-Vila X, Carrera J, Saaltink MW. A mixing ratios-based formulation for multicomponent reactive transport. Water Resour Res 2007;43:W07419. doi: 10.1029/2006WR005256] result in a methodology conducive to a simple exact expression for the space–time distribution of reaction rates in the presence of homogeneous or heterogeneous reactions in chemical equilibrium. The key points of the methodology are that a general reactive transport problem, involving a relatively high number of chemical species, can be formulated in terms of a set of decoupled partial differential equations, and the amount of reactants evolving into products depends on the rate at which solutions mix. The main objective of the current study is to show how this methodology can be used in conjunction with laboratory experiments to properly describe the key processes that occur in a complex, geochemically-active system under chemical equilibrium conditions. We model three CaCO 3 dissolution experiments reported in Singurindy et al. [Singurindy O, Berkowitz B, Lowell RP. Carbonate dissolution and precipitation in coastal environments: Laboratory analysis and theoretical consideration. Water Resour Res 2004;40:W04401. doi: 10.1029/2003WR002651, Singurindy O, Berkowitz B, Lowell RP. Correction to Carbonate dissolution and precipitation in coastal environments: laboratory analysis and theoretical consideration. Water Resour Res 2005;41:W11701. doi: 10.1029/2005WR004433], in which saltwater and freshwater were mixed in different proportions. The integrated reaction rate within the cell estimated from the experiments are modeled independently by means of (a) a state-of-the-art reactive transport code, and (b) the uncoupled methodology of [12, 13], both of which use dispersivity as a single, adjustable parameter. The good agreement between the results from both methodologies demonstrates the feasibility of using simple solutions to design and analyze laboratory experiments involving complex geochemical problems.

On the Efficiency of the Direct Substitution Approach for Reactive Transport Problems in Porous Media

Nonlinear reactive transport problems can be solved using the Operator Splitting (OS) approach, where transport and reaction processes are separated or the Direct Substitution Approach (DSA) where chemical and transport equations are solved simultaneously. The OS techniques can be very attractive, but are known to introduce splitting errors with SNIA (Non Iterative OS) and have low convergence rate with SIA (Iterative OS). These problems are avoided with DSA which is more robust than OS schemes. On the other hand, DSA is more complicated and very demanding in terms of computing time and memory requirements. This can make DSA less efficient than OS schemes especially for fine discretizations and chemically simple problems. In this work, DSA, SIA and SNIA are combined with a new sparse direct (unifrontal/multifrontal) solver. The efficiency of this solver is not dependent on the matrix conditioning. The performance of the three approaches is studied for two transport problems with simple and difficult chemical reactions and for different number of unknowns. Results show that when combined with an efficient sparse direct solver, DSA is more efficient than SIA and SNIA even for chemically simple problems and large number of unknowns.

Efficient, process‐preserving modified Newton's method for solving reactive multicomponent transport problems in porous media

AbstractIn the context of hydrogeochemical multicomponent transport problems coupled via reaction rates the fully implicit solution of the (discretized) problem can be computationally expensive, but it avoids the consistency errors inherent in the popular operator splitting approaches which separate transport and reaction operators.The efficiency of the globally implicit approach is enhanced however by using Newton's method, analysing the reaction network, and modifying the Jacobian to decouple species equations without deteriorating the performance of the Newton iteration. This is possible in the case of slow kinetics and/or small concentrations. A flexible componentwise sparse matrix linear solver can adapt and change the coupling strategy potentially in every iteration.Numerical results of academical and realistic case studies are presented that demonstrate the potential of the approach. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multicomponent reaction modelling using a stochastic algorithm

The numerical simulation of multicomponent reactive transport has become a dynamic field of subsurface hydrology. Efficient simulation models are necessary which are capable of representing a multitude of parameters influencing geochemical reactions. The mathematical structure of reactive transport problems in porous media is determined by the decision in the chemical model part whether reactions are kinetically controlled or whether a local equilibrium assumption is made between various species involved in the geochemical reactions. The assumptions made in deterministic models are not hold true for reactions involving chemical species of low concentration. Here the emergent phenomenon evolving out of the random fluctuations of the species is difficult to be captured. The stochastic behaviour of the physicochemical processes can be modelled at mesoscopic level by application of stochastic algorithms. The proposed model was verified by comparing the results of Gillespie’s algorithms with a deterministic solution for PCE and BTEX degradation reactions.

Assessment of quantitative imaging of contaminant distributions in porous media

In this article an experimental setup designed to assist in the characterization of complex solute transport problems in porous media is described. Glass beads representing the medium are confined in a 2-D transparent Perspex box and a water flow transports a fluorescent dye. Under suitable illumination, the dye emits visible light which is collected by a CCD camera. The image acquired by this non-invasive optical technique is processed to estimate the 2-dimensional distribution of tracer concentrations by using an appropriate calibration curve that links fluorescent intensity and solute concentration. Details about the dye choice and discussion about photobleaching are reported. An analysis of the experimental error on the concentration profile is also presented. A few recent results of a study on contaminant plume within a homogenous porous matrix constituted by glass beads having mean diameter of 1 mm or 2 mm shows the performance of constructed model.