Flow In Heterogeneous Porous Media Research Articles

Modeling gas flow in low-permeability formations: An efficient combination of mixed finite elements and high order time integration schemes

Numerical simulation of gas flow in low permeability formations is a challenging task due to the high nonlinearity induced by (i) the compressibility of the gas, (ii) the Klinkenberg slippage effect and (iii) the Langmuir adsorption of the gas on the pore surface. Because of these nonlinearities, modeling gas flow in low permeability formations requires a great deal of computational effort. In this work, we develop an efficient numerical model using advanced spatial and temporal discretization methods for a simultaneous solution of the coupled equations of gas flow, cubic Peng-Robinson equation of state, slippage effect and Langmuir adsorption. The spatial discretization is performed with the lumped hybrid formulation of the mixed finite element method which is well adapted for fluid flow in heterogeneous porous media. The time integration is performed with high-order methods via the method of lines (MOL) which allows large time steps and efficient solution of the nonlinear system of equations. Numerical experiments, performed for gas extraction in a heterogeneous domain, point out the high efficiency of the new model since it can be until 10 times more efficient than the classical first-order time discretization method. Results of global sensitivity analysis show that the intrinsic gas-phase permeability and the Klinkenberg factor are the most influential parameters controlling the pumping rate in the case of gas extraction in a homogeneous domain.

A lattice Boltzmann approach to mathematical modeling of myocardial perfusion.

A mathematical model of myocardial perfusion based on the lattice Boltzmann method (LBM) is proposed and its applicability is investigated in both healthy and diseased cases. The myocardium is conceptualized as a porous material in which the transport and mass transfer of a contrast agent in blood flow is studied. The results of myocardial perfusion obtained using LBM in 1D and 2D are confronted with previously reported results in the literature and the results obtained using the mixed-hybrid finite element method. Since LBM is not suitable for simulating flow in heterogeneous porous media, a simplified and computationally efficient 1D-analog approach to 2D diseased case is proposed and its applicability discussed.

Ensemble variational Bayesian approximation for the inversion and uncertainty quantification of Darcy flows in heterogeneous porous media with random parameters

The inversion and uncertainty quantification of physical parameters are important for many science and engineering problems. For nonlinear parameter inversion problems of flows in hydrocarbon reservoirs, estimating the true posterior probability distribution function for the random parameter is difficult when the dimension of the parameter is relatively high. Inversion methods such as gradient-based or heuristic optimization methods searching for the maximum a posterior are generally point-estimate, while uncertainty quantification using MCMC is computationally expensive. In the current study, a new ensemble variational Bayesian approximation method is developed to estimate the posterior probability density using an ensemble of realizations regarded as a trial sample distribution. The objective is to optimize the trial sample distribution such that the evidence lower bound is maximized indicating that the distance between the trial and true posterior distribution is minimized. A subspace is built using principle component analysis based on finite samples to obtain a diagonal positive-definite covariance matrix, and ensemble-based data assimilation is used to optimize the trial distribution efficiently. Heterogeneous 2D and 3D test cases of transient single- and two-phase Darcy flows are presented for validation.

Probabilistic assessment of scalar transport under hydrodynamically unstable flows in heterogeneous porous media

Quantitative predictions of scalar transport in natural porous media is a nontrivial task given the presence of multi-scale spatial heterogeneity in the permeability field. Due to data scarcity, the structural map of the permeability field is subject to uncertainty and therefore, model predictions are uncertain. For such reasons, probabilistic models of flow and transport in natural porous media are required in risk assessment and to provide reliable decision making under uncertainty. Further complexities arise when the viscosity of the injected solute differs from that of the ambient fluid. Under the presence of viscosity contrast, hydrodynamic instabilities give rise to viscous fingering, which induces additional disorder in both velocity and solute concentration fields. This work examines the combined role of viscous fingering and permeability heterogeneity in the probabilistic description of transport predictions. In particular, we focus on metrics that are important for risk analysis, such as the solute plume’s early arrival times and the maximum concentration observed at a given location. We propose to use the Projection Pursuit Adaptation (PPA) method in the Polynomial Chaos Expansion (PCE) framework to quantify uncertainty in transport model predictions. The PPA method is a data-driven approach that optimally represents a given quantity of interest in a low-dimensional manifold. Unlike other dimension reduction techniques in uncertainty quantification, the PPA method utilizes non-linear information of the quantity of interest to identify the low-dimensional manifold, thereby increasing the likelihood of finding a more accurate lower-dimensional space. Moreover, the PPA model converges to the physical solution in a mean squared sense with respect to the polynomial order, enabling the construction of a converged model even with limited available data. Then, the PPA results are compared to Monte Carlo simulations using the same amount of data. This comparison illustrates that while Monte Carlo simulations are able to capture low-order statistics, they struggle to represent more detailed probability density functions. Our results show how the combined effect of permeability heterogeneity and viscosity contrast can enhance the mobility of the solute plume.

The Transition from Darcy to Nonlinear Flow in Heterogeneous Porous Media: I—Single-Phase Flow

Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient ∇P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\ abla }}P$$\\end{document} and the fluid velocity v. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude ωz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _z$$\\end{document} of the vorticity is nearly zero. As Re increases, however, so also does ωz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _z$$\\end{document}, and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocity v, given by, -∇P=(μ/Ke)v+βnρ|v|2v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-{\\varvec{\ abla }}P=(\\mu /K_e)\ extbf{v}+\\beta _n\\rho |\ extbf{v}|^2\ extbf{v}$$\\end{document}, provides accurate representation of the numerical data, where μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu$$\\end{document} and ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho$$\\end{document} are the fluid’s viscosity and density, Ke\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_e$$\\end{document} is the effective Darcy permeability in the linear regime, and βn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _n$$\\end{document} is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation.

Semi-analytical Model to Predict Dynamic Capillary Pressure–Saturation Relationship for Flows in Heterogeneous Porous Media

The capillary pressure defines pressure difference between non-wetting and wetting fluids. The capillary pressure is part of the flow governing equations, and its definition can have a profound impact on the nature of fluids displacement in a multiphase flow environment. Conventionally, capillary pressure–saturation relationships are determined under equilibrium conditions which signify that all the fluid–fluid interfaces that exist at the pore scale maintain a static configuration at a certain instant in time. However, there exist experimental and numerical evidences that state that the dynamic nature of fluid flows indeed plays a prominent role in defining the trends of the capillary pressure–saturation relationships. In this paper, we develop a first of a kind semi-analytical model to predict the capillary pressure–saturation curves during drainage displacement by integrating the dynamics of fluid flow based on fundamental laws of fluid mechanics. The proposed semi-analytical model can potentially be incorporated into existing multiphase flow simulators to rapidly compute the capillary pressure at various saturations of the flow medium under dynamic flow conditions. The presented semi-analytical model has been validated against experimental and numerical data sets available in the literature at various flow conditions and considering different sets of fluid properties. We noticed a satisfactory match of the results predicted by the proposed semi-analytical model against the literature data. After performing a holistic sensitivity analysis, we notice that the properties of the porous medium, and the fluid–solid interactions play a significant role in defining the trends of the capillary pressure–saturation curves.

Finite analytical method on corner‐point grid for fluid flow in heterogeneous porous media

AbstractThe three‐dimensional (3D) corner‐point grid system, as an industrial standard, has been widely used in oil reservoir geological modeling and numerical simulation due to its versatility compared to traditional orthogonal grids. The finite analytical method (FAM) is proposed for corner‐point grid systems in this article, which can greatly improve the simulation accuracy, especially for strongly heterogeneous porous media. Based on the approximate analytical solution around the edge of the corner‐point cells, the internodal transmissibility of two adjacent cells is recalculated to replace the traditional one. The traditional internodal transmissibility is calculated based on the harmonic mean of the permeabilities of the two adjacent cells, which will greatly underestimate the transmissibility and lead to an uncontrollable error as the strength of heterogeneity increases. The proposed FAM can calculate the internodal transmissibility rather accurately, and its accuracy is irrelevant to the strength of the heterogeneity, which is the main advantage of FAM. Numerical tests show that the simulation results of the proposed FAM converge much faster than the traditional method as the cells are refined. The proposed FAM can be easily extended to the simulation of multiphase flow and be easily embedded in commercial reservoir numerical simulation software.

Modeling density-driven flow and solute transport in heterogeneous reservoirs with micro/macro fractures

In this paper, a numerical model is presented to simulate the density-driven flow in heterogeneous porous media with micro- and macro-fractures. The micro-fractures (like fissures) with relatively small fracture lengths compared to the reservoir are modeled using an equivalent continuum model. The macro-fractures (like faults) with large fracture lengths are modeled using the extended finite element method (XFEM). Governing equations are based on the mass conservation equation for both the matrix and fracture domains together with the proper constitutive laws for the fluid velocity, dispersive velocity of solute, and equation of state for density of fluid phase. Several parameter studies are performed to investigate the effects of fracture characteristics on the solute spread in the medium. It is demonstrated that in the absence of macro-fractures, the influence of micro-fractures’ aperture on the solute spread is fully substantial and predictable. However, in the presence of macro-fractures, there is a region where the macro-fracture performs like a vacuum diminishing the influence of nearby micro-fractures. Finally, a real hydrological problem of the Gödöllő Hills reservoir located in Hungary with five different layers and eight faults, which is full of saline water is numerically modeled to investigate the effect of pressure gradient, anisotropy, and density difference on the solute transport. It is illustrated that in the top layers of the Gödöllő Hills reservoir, the gravitational force and vertical permeability are mostly accountable for the solute flow across the faults, whereas in the bottom layers, the horizontal pressure gradient is primarily responsible for the solute transport.

A mixed pressure-velocity formulation to model flow in heterogeneous porous media with physics-informed neural networks

Current implementations of Physics Informed Neural Networks (PINNs) can experience convergence problems in simulating fluid flow in porous media with highly heterogeneous domains. The objective of this work is to develop an accurate implementation of PINNs to model fluid flow in heterogeneous porous media that avoids alternative approaches in the literature that tend to impose unphysical continuity of the hydraulic conductivity field. The proposed implementation is inspired by the mixed formulation of the governing equations, which separates the mass continuity equation and Darcy's law. This methodology has been used with the mixed finite element method and is known to provide accurate pressure and velocity fields in highly heterogeneous domains. In this work, a similar methodology is applied to PINNs, where the training loss function is based on the decoupled continuity equation and Darcy's law. The separation of the continuity equation and Darcy's law allows for calculating the residual term in the learning loss function without evaluating the spatial derivatives of the discontinuous hydraulic conductivity and associated non-differentiable functions. This approach provides more accurate automatic differentiation of the neural networks for pressure and velocity fields. The new implementation of PINNs can be applied to simulate flow in porous media, regardless of the type and level of heterogeneity with a discontinuous hydraulic conductivity distribution. A variety of structures of the new implementation are investigated. Numerical experiments show that the structure based on different neural networks for the pressure and each component of the velocity field provides the highest accuracy with equivalent training time as other structures. The proposed methodology presents a novel approach to broaden the scope of PINNs in modeling fluid flow in porous media, while preserving the precise representation of the domain's discontinuous features.

A novel deep learning-based automatic search workflow for CO2 sequestration surrogate flow models

Numerical simulation can significantly enhance subsurface resource utilisation's efficiency and economic benefits by multiphase flow in heterogeneous porous media. However, numerical simulation brings enormous computational demands and time consumption due to high-dimensional nonlinearity, heterogeneity, and coupling of multiple physical processes. Surrogate models can accelerate the establishment of complex models without sacrificing accuracy. However, creating well-performing surrogate models requires extensive human intervention and trial-and-error processes, even for cross-domain experts. This study proposes an automated surrogate flow model workflow based on deep learning called Surrogate Flow Model Search (SFMS). By incorporating neural architectures and loss functions into joint hyperparameter optimisation, SFMS automates many complex and time-consuming tasks involved in model development, such as designing neural architectures and loss functions. The automated surrogate model construction workflow enables researchers to develop high-quality surrogate models without extensive deep-learning expertise. We demonstrate the effectiveness of SFMS using saline aquifer CO2 injection as an example. The results show that SFMS can automatically generate a highly accurate surrogate flow model in a short time (<1300 s), capable of accurately predicting 120-time steps under different well controls and placements with a low average relative error (<0.4%). Therefore, SFMS can significantly reduce the time and effort required to develop accurate and reliable surrogate models, providing a new approach to building surrogate models.

Investigation of pore geometry influence on fluid flow in heterogeneous porous media: A pore-scale study

Brazilian pre-salt reservoirs are renowned for their intricate pore networks and vuggy nature, posing significant challenges in modeling and simulating fluid flow within these carbonate reservoirs. Despite possessing excellent petrophysical properties, such as high porosity and permeability, these reservoirs typically exhibit a notably low recovery factor, sometimes falling below 10%. Previous research has indicated that various enhanced oil recovery (EOR) methods, such as water alternating gas (WAG), can substantially augment the recovery factor in pre-salt reservoirs, resulting in improvements of up to 20%. Nevertheless, the fluid flow mechanism within Brazilian carbonate reservoirs, characterized by complex pore geometry, remains unclear. Our study examines the behavior of fluid flow in a similar heterogeneous porous material, utilizing a plug sample obtained from a vugular segment of a Brazilian stromatolite outcrop, known to share analogies with certain pre-salt reservoirs. We conducted single-phase and multi-phase core flooding experiments, complemented by medical-CT scanning, to generate flow streamlines and evaluate the efficiency of water flooding. Subsequently, micro-CT scanning of the core sample was performed, and two cross-sections from horizontal and vertical plates were constructed. These cross-sections were then employed as geometries in a numerical simulator, enabling us to investigate the impact of pore geometry on fluid flow. Analysis of the pore-scale modeling and experimental data unveiled that the presence of dead-end pores and vugs results in a significant portion of the fluid remaining stagnant within these regions. Consequently, the injected fluid exhibits channeling-like behavior, leading to rapid breakthrough and low areal swept efficiency. Additionally, the numerical simulation results demonstrated that, irrespective of the size of the dead-end regions, the pressure variation within the dead-end vugs and pores is negligible. Despite the stromatolite's favorable petrophysical properties, including relatively high porosity and permeability, as well as the presence of interconnected large vugs, the recovery factor during water flooding remained low due to early breakthrough. These findings align with field data obtained from pre-salt reservoirs, providing an explanation for the observed low recovery factor during water flooding in such reservoirs.

Prediction of pore-scale flow in heterogeneous porous media from periodic structures using deep learning

Data-driven deep learning models are emerging as a promising method for characterizing pore-scale flow through complex porous media while requiring minimal computational power. However, previous models often require extensive computation to simulate flow through synthetic porous media for use as training data. We propose a convolutional neural network trained solely on periodic unit cells to predict pore-scale velocity fields of complex heterogeneous porous media from binary images without the need for further image processing. Our model is trained using a range of simple and complex unit cells that can be obtained analytically or numerically at a low computational cost. Our results show that the model accurately predicts the permeability and pore-scale flow characteristics of synthetic porous media and real reticulated foams. We significantly improve the convergence of numerical simulations by using the predictions from our model as initial guesses. Our approach addresses the limitations of previous models and improves computational efficiency, enabling the rigorous characterization of large batches of complex heterogeneous porous media for a variety of engineering applications.

Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media

This work focuses on the development of a two-step field-split nonlinear preconditioner to accelerate the convergence of two-phase flow and transport in heterogeneous porous media. We propose a field-split algorithm named Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps: first, we apply a preconditioning step to update pressure and saturations nonlinearly by solving approximately two subproblems in a sequential fashion; then, we apply a global step relying on a Newton update obtained by linearizing the system at the preconditioned state. Using challenging test cases, FSMSN is compared to an existing field-split preconditioner, Multiplicative Schwarz Preconditioned for Inexact Newton (MSPIN), and to standard solution strategies such as the Sequential Fully Implicit (SFI) method or the Fully Implicit Method (FIM). The comparison highlights the impact of the upwinding scheme in the algorithmic performance of the preconditioners and the importance of the dynamic adaptation of the subproblem tolerance in the preconditioning step. Our results demonstrate that the two-step nonlinear preconditioning approach—and in particular, FSMSN—results in a faster outer-loop convergence than with the SFI and FIM methods. The impact of the preconditioners on computational performance–i.e., measured by wall-clock time–will be studied in a subsequent publication.

A physics-informed convolutional neural network for the simulation and prediction of two-phase Darcy flows in heterogeneous porous media

The physics-informed neural network (PINN) is a general deep learning framework for simulating flows with limited or no labeled data. In the current study, we develop a physics-informed convolutional neural network (PICNN) for simulating transient two-phase Darcy flows in heterogeneous reservoir models with source/sink terms in the absence of labeled data, where the finite volume method (FVM) is adopted to approximate the PDE residual in the loss function such that flux continuity between neighboring cells of different properties is defined rigorously, and a well model is adopted to approximate the high pressure gradient near sources or sinks. The implicit-pressure explicit-saturation (IMPES) scheme is employed such that only a single CNN needs to be trained per time step. Dirichlet boundary conditions are not a mandatory requirement for PICNN-based implicit solver but act as labeled data that can help enhance accuracy. The proposed approach is validated in homogeneous and heterogeneous reservoirs and aspects including efficiency and accuracy are discussed. In addition, we demonstrate that the CNN structure can be trained as a data-driven surrogate for two-phase Darcy flows given sufficient labeled samples.

The MLS-based numerical manifold method for Darcy flow in heterogeneous porous media

The moving least squares (MLS) interpolation gets rid of the shackles of meshes, while the numerical manifold method (NMM) can solve both continuity and discontinuity problems in a unified way. The MLS-based NMM (abbreviated as MLS-NMM) inherits the individual merits of MLS and NMM. In the present work, the MLS-NMM is extended to solve groundwater flow problems in heterogeneous porous media. To accurately simulate the flow velocity field at and in the vicinity of the material interfaces, the refraction law is introduced into the proposed model as a posteriori condition. Several examples are employed to verify the correctness and accuracy of the proposed model. The results indicate that the MLS-NMM can model the Darcy flow in heterogeneous porous media with high accuracy, as well as satisfy the refraction law rigorously. In addition, the smooth flow velocity field can be obtained directly without the need for extra post-processing.

A Single Mesh Approximation for Modeling Multiphase Flow in Heterogeneous Porous Media

A Single Mesh Approximation for Modeling Multiphase Flow in Heterogeneous Porous Media

USING PHYSICS-INFORMED NEURAL NETWORKS TO SOLVE FOR PERMEABILITY FIELD UNDER TWO-PHASE FLOW IN HETEROGENEOUS POROUS MEDIA

Physics-informed neural networks (PINNs) have recently been applied to a wide range of computational physical problems. In this paper, we use PINNs to solve an inverse two-phase flow problem in heterogeneous porous media where only sparse direct and indirect measurements are available. The forward two-phase flow problem is governed by a coupled system of partial differential equations (PDEs) with initial and boundary conditions. As for inverse problems, the solutions are assumed to be known at scattered locations but some coefficients or variable functions in the PDEs are missing or incomplete. The idea is to train multiple neural networks representing the solutions and the unknown variable function at the same time such that both the underlying physical laws and the measurements can be honored. The numerical results show that our proposed method is able to recover the incomplete permeability field in different scenarios. Moreover, we show that the method can be used to forecast the future dynamics with the same format of loss function formulation. In addition, we employ a neural network structure inspired by the deep operator networks (DeepONets) to represent the solutions which can potentially shorten the time of the training process.

A hybrid pressure approximation in the control volume finite element method for multiphase flow and transport in heterogeneous porous media

We present a new hybrid pressure formulation applied to the control volume finite element (CVFE) method to model multiphase flow and transport in highly heterogeneous porous media. The formulation effectively captures sharp saturation changes in the presence of discontinuous material properties by employing a discontinuous pressure approximation at material interfaces. The heterogeneous porous medium is divided into sub-domains within which material properties are uniform or smoothly varying. By construction, the resultant control volume dual mesh is restricted within a sub-domain. The artificial mass leakage across material property boundaries observed in classical CVFE methods is therefore circumvented. The approach applies the robust continuous pressure approximation in the rest of the computational domain; the discontinuous approximation is applied only at the sub-domain boundaries. The discontinuous parameters necessary to achieve mass conservative solutions, locally and globally, are described. We demonstrate the accuracy and efficiency of the new approach by comparison with the classical continuous CVFE method on various examples of heterogeneous domains as well as establishing the convergence of the numerical method. The proposed hybrid formulation significantly outperforms the accuracy and efficiency of classical CVFE methods that use the same order of approximation for modeling multiphase flow in heterogeneous porous media.

A numerical investigation into the influence of the surfactant injection technique on the foam flow in heterogeneous porous media

We propose a sequential algorithm to investigate the influence of surfactant injection on foam-induced mobility reduction in two-phase liquid–gas flows in highly heterogeneous porous media. A foam model in local equilibrium is adopted in this context, assuming non-Newtonian foam behavior. A fractional flow formulation based on global pressure that includes the surfactant in the liquid phase is adopted, resulting in a system that couples elliptic and hyperbolic equations. These equations are split into two sub-systems that group equations of the same kind and are solved employing a sequential numerical algorithm that combines a locally conservative hybrid finite element method to approximate the hydrodynamics of the system with a central-upwind finite volume method for the transport equations. The resulting spatial discretized system is approximated in time by applying a multi-step, implicit finite difference method. The hydrodynamics and hyperbolic problems are solved using a staggered algorithm in different time scales. Through comparison with experimental data and classical numerical methods, we validated the proposed methodology and demonstrated its ability to produce stable approximations with reduced numerical dissipation and computational cost. Then, we apply the sequential algorithm to compare the co-injection of the water–gas–surfactant with the Surfactant Alternating Gas (SAG) technique in two-dimensional flows in highly heterogeneous media. The numerical results suggest a better sweep efficiency using the SAG injection technique when compared with the co-injection approach for the equivalent amount of injected surfactant.

Method of Distributions for Two‐Phase Flow in Heterogeneous Porous Media

AbstractMultiscale heterogeneity and insufficient characterization data for a specific subsurface formation of interest render predictions of multi‐phase fluid flow in geologic formations highly uncertain. Quantification of the uncertainty propagation from the geomodel to the fluid‐flow response is typically done within a probabilistic framework. This task is computationally demanding due to, for example, the slow convergence of Monte Carlo simulations (MCS), especially when computing the tails of a distribution that are necessary for risk assessment and decision‐making under uncertainty. The frozen streamlines method (FROST) accelerates probabilistic predictions of immiscible two‐phase fluid flow problems; however, FROST relies on MCS to compute the travel‐time distribution, which is then used to perform the transport (phase saturation) computations. To alleviate this computational bottleneck, we replace MCS with a deterministic equation for the cumulative distribution function (CDF) of travel time. The resulting CDF‐FROST approach yields the CDF of the saturation field without resorting to sampling‐based strategies. Our numerical experiments demonstrate the high accuracy of CDF‐FROST in computing the CDFs of both saturation and travel time. For the same accuracy, it is about 5 and 10 times faster than FROST and MCS, respectively.