Effective Solution through the Streamline Technique and HT-Splitting for the 3D Dynamic Analysis of the Compositional Flows in Oil Reservoirs

In this article we suggest a new phenomenological mathematical model for the groundwater transport of colloid particles through porous media which is able to describe some significant effects experimentally observed but not captured within the framework of the classic approach. Our basic idea is to consider both the pure water and the colloid suspension as two thermodynamic phases. Using the network models of porous media, we simulated numerically the transport process at the pore-scale. By averaging the result derived, we have obtained the relative permeabilities for both phases, the percolation threshold for suspension flow, and the effective suspension viscosity. Due to specific laws of colloid particles repartition between various classes of pores, the relative permeability of suspension happens to be a highly nonlinear function of saturation, very far from the diagonal straight line. This determines a difference between the macroscale phase velocities. The suspension velocity is shown to be higher than that of water in major cases, only if the colloid particles are not too large. The suggested model predicts and describes in a closed form the effect of colloid transport facilitation observed experimentally.     

Three-Dimensional Analytical Models for Virus Transport in Saturated Porous Media

Analytical models for virus transport in saturated, homogeneous porous media are developed. The models account for three-dimensional dispersion in a uniform flow field, and first-order inactivation of suspended and deposited viruses with different inactivation rate coefficients. Virus deposition onto solid particles is described by two different processes: nonequilibrium adsorption which is applicable to viruses behaving as solutes; and colloid filtration which is applicable to viruses behaving as colloids. The governing virus transport equations are solved analytically by employing Laplace/Fourier transform techniques. Instantaneous and continuous/periodic virus loadings from a point source are examined.     

Mechanisms of Particle Transport Acceleration in Porous Media

Experimental data show that the groundwater transport of radionuclides in porous media is frequently facilitated when accompanied with colloid particles. This is usually explained by the size exclusion mechanism which implies that the particles move through the largest pores where the flow velocity is higher. We call attention to three other mechanisms which influence the colloid particle motion, while determining both the probable transport facilitation and retardation. First of all, it is shown that the transport facilitation may be significantly reduced and even transformed into a retardation due to the growth of the effective suspension viscosity (a friction-limited facilitation). Secondly, we will show that the transport of particles through the largest pores can be retarded due to a reduced connectivity of the large-pore cluster (a percolation-breakup retardation). Thirdly, we highlight the Fermi mechanism of acceleration known in statistical physics which is based on the elastic collisions between particles. All three effects are analyzed in terms of the velocity enhancement factor, by using statistical models of porous media in the form of a capillary bundle and a 3D capillary network. Optimal and critical regimes of velocity enhancement are quantified. Estimations show that for realistic parameters, the maximal facilitation of colloid transport is close to the experimentally observed data.     

Cotransport of Suspended Colloids and Nanoparticles in Porous Media

The objective of this study is to develop a model for cotransport of colloids and nanoparticles (NPs) in porous media under two particle capture mechanisms. The particle capture rate is proportional to the capture probability, which is a function of retained concentration, called the filtration function. Laboratory bench-scale experiments of individual transport of NPs and colloidal-size kaolinite clay particles through packed columns produced breakthrough curves (BTCs) that monotonically increased with time and stabilised at some value lower than the injected concentration. We discuss the filtration function that corresponds to BTCs stabilising at the concentration lower than the injected value. This so-called binary filtration function incorporates two particle capture mechanisms. The analytical transport model with a binary filtration function was capable to fit successfully BTCs obtained from individual transport experiments using kaolinite and NPs conducted by Chrysikopoulos et al. (Transp Porous Med 119(1):181–204, 2017). Assuming that the electrostatic particle–solid matrix interaction and the fraction of the solid matrix surface area occupied by a single attached particle (kaolinite or NP) are the same for individual transport of either kaolinite particles or NPs and for simultaneous cotransport of kaolinite particles and NPs, the proposed binary filtration function was extended for the cotransport case. Although the breakthrough data from cotransport experiments with kaolinite particles and NPs have six degrees of freedom, the developed cotransport model successfully matches the BTCs by tuning two constants only. This validates the developed model for cotransport of two colloidal populations with different attachments and straining rates.

     

Effects of Flow Velocity and Nonionic Surfactant on Colloid Straining in Saturated Porous Media Under Unfavorable Conditions

Knowledge of colloid straining mechanism in porous media is of importance for protecting groundwater from being contaminated by biocolloids (e.g., bacteria and protozoa) and by contaminants whose transport can be facilitated by mobile particles. This study examined effects of flow velocity on colloid straining in porous media under unfavorable chemical conditions. Saturated column experiments were conducted using glass beads as collector and a $3\,\mu \text{ m}$ carboxylate-modified polystyrene latex microsphere as model colloid. To unambiguously examine colloid straining mechanisms, attachment was minimized by extensively cleaning the collectors and adopting deionized water as solution. Results show that increasing flow velocity decreases colloid straining under unfavorable chemical conditions, in agreement with to theoretical finding in literature. This study additionally examined effects of nonionic surfactant (Triton X-100) on colloid straining in porous media under unfavorable chemical conditions. Results show that the addition of Triton X-100 decreases colloid straining and the decrease is enhanced by increasing the concentration of Triton X-100.     

Transport of Neutrally Buoyant and Dense Variably Sized Colloids in a Two-Dimensional Fracture with Anisotropic Aperture

The transport of monodisperse as well as polydisperse colloid suspensions in a two-dimensional, water saturated fracture with spatially variable and anisotropic aperture is investigated with a particle tracking model. Both neutrally buoyant and dense colloid suspensions are considered. Although flow and transport in fractured subsurface formations have been studied extensively by numerous investigators, the transport of dense, polydisperse colloid suspensions in a fracture with spatially variable and anisotropic aperture has not been previously explored. Simulated snapshots and breakthrough curves of ensemble averages of several realizations of a log-normally distributed aperture field show that polydisperse colloids exhibit greater spreading than monodisperse colloids, and dense colloids show greater retardation than neutrally buoyant colloids. Moreover, it is demonstrated that aperture anisotropy oriented along the flow direction substantially increases colloid spreading; whereas, aperture anisotropy oriented transverse to the flow direction retards colloid movement.     

Modeling the Chlorides Transport in Cementitious Materials By Periodic Homogenization

In this work, we develop a macroscopic model for diffusion–migration of ionic species in saturated porous media, based on periodic homogenization. The prior application is chloride transport in cementitious materials. The dimensional analysis of Nernst–Planck equation lets appear dimensionless numbers characterizing the ionic transfer in porous media. Using experimental data, these dimensionless numbers are linked to the perturbation parameter ${\varepsilon}$ . For a weak-imposed electrical field, or in natural diffusion, the asymptotic expansion of Nernst–Planck equation leads to a macroscopic model coupling diffusion and migration at the same order. The expression of the homogenized diffusion coefficient only involves the geometrical properties of the material microstructure. Then, parametric simulations are performed to compute the chloride diffusion coefficient through different complexity of the elementary cell to go on as close as possible to experimental diffusion coefficient of the two cement pastes tested.     

Linking Morphology of Porous Media to Their Macroscopic Permeability by Deep Learning

Flow, transport, mechanical, and fracture properties of porous media depend on their morphology and are usually estimated by experimental and/or computational methods. The precision of the computational approaches depends on the accuracy of the model that represents the morphology. If high accuracy is required, the computations and even experiments can be quite time-consuming. At the same time, linking the morphology directly to the permeability, as well as other important flow and transport properties, has been a long-standing problem. In this paper, we develop a new network that utilizes a deep learning (DL) algorithm to link the morphology of porous media to their permeability. The network is neither a purely traditional artificial neural network (ANN), nor is it a purely DL algorithm, but, rather, it is a hybrid of both. The input data include three-dimensional images of sandstones, hundreds of their stochastic realizations generated by a reconstruction method, and synthetic unconsolidated porous media produced by a Boolean method. To develop the network, we first extract important features of the images using a DL algorithm and then feed them to an ANN to estimate the permeabilities. We demonstrate that the network is successfully trained, such that it can develop accurate correlations between the morphology of porous media and their effective permeability. The high accuracy of the network is demonstrated by its predictions for the permeability of a variety of porous media.

     

The Physical Origin of Interfacial Coupling in Two-Phase Flow through Porous Media

Recently developed transport equations for two-phase flow through porous media usually have a second term that has been included to account properly for interfacial coupling between the two flowing phases. The source and magnitude of such coupling is not well understood. In this study, a partition concept has been introduced into Kalaydjian's transport equations to construct modified transport equations that enable a better understanding of the role of interfacial coupling in two-phase flow through natural porous media. Using these equations, it is demonstrated that, in natural porous media, the physical origin of interfacial coupling is the capillarity of the porous medium, and not interfacial momentum transfer, as is usually assumed. The new equations are also used to show that, under conditions of steady-state flow, the magnitude of mobilities measured in a countercurrent flow experiment is the same as that measured in a cocurrent flow experiment, contrary to what has been reported previously. Moreover, the new equations are used to explicate the mechanism by which a saturation front steepens in an unstabilized displacement, and to show that the rate at which a wetting fluid is imbibed into a porous medium is controlled by the capillary coupling parameter, . Finally, it is argued that the capillary coupling parameter, , is dependent, at least in part, on porosity. Because a clear understanding of the role played by interfacial coupling is important to an improved understanding of two-phase flow through porous media, the new transport equations should prove to be effective tools for the study of such flow.     

Numerical Calculation of Effective Diffusion in Unsaturated Porous Media by the TRT Lattice Boltzmann Method

Numerical models that solve transport of pollutants at the macroscopic scale in unsaturated porous media need the effective diffusion dependence on saturation as an input. We conducted numerical computations at the pore scale in order to obtain the effective diffusion curve as a function of saturation for an academic sphere packing porous medium and for a real porous medium where pore structure knowledge was obtained through X-ray tomography. The computations were performed using a combination of lattice Boltzmann models based on two relaxation time (TRT) scheme. The first stage of the calculations consisted in recovering the water spatial distribution into the pore structure for several fixed saturations using a phase separation TRT lattice Boltzmann model. Then, we performed diffusion computation of a non-reactive solute in the connected water structure using a diffusion TRT lattice Boltzmann model. Finally, the effective diffusion for each selected saturation value was estimated through inversion of a macroscopic classical analytical solution.     

Reliability of Algorithms Interpreting Topological and Geometric Properties of Porous Media for Pore Network Modelling

Pore network models (PNMs) offer a computationally efficient way to analyse transport in porous media. Their effectiveness depends on how well they represent the topology and geometry of real pore systems, for example as imaged by X-ray CT. The performance of two popular algorithms, maximum ball and watershed, is evaluated for three porous systems: an idealised medium with known pore throat properties and two rocks with different morphogenesis—carbonate and sandstone. It is demonstrated that while the extracted PNM simulates simple flow (permeability) with acceptable accuracy, their topological and geometric properties are significantly different. This suggests that such PNM may not serve more complex studies, such as reactive/convective transport of contaminants or bacteria, and further research is necessary to improve the interpretation of real pore spaces with networks. Linear topology–geometry relations are derived and presented to stimulate development of more realistic PNM.

     

Comparison of Kinetic and Hybrid-Equilibrium Models Simulating Colloid-Facilitated Contaminant Transport in Porous Media

The presence of colloidal particles in groundwater can enhance contaminant transport by reducing retardation effects and carrying them to distances further than predicted by a conventional advective/dispersive equation with normal retardation values. When colloids exist in porous media and affect contaminant migration, the system can best be simulated as a three-phase medium. Mechanisms of mass transfer from one phase to another by colloids and contaminants can be kinetic or equilibrium-based, depending on the sorption–desorption reaction rate between the aqueous and solid phases. When the rate of sorption between the water phase and the solid phase(s) is not much greater than the rate of change in contaminant concentration in the water phase, kinetic sorption models may better describe the phenomenon. In some cases of modeling one or more mass transfer processes, a useful simplification may be to introduce the local equilibrium assumption. In this study, the local equilibrium assumption for sorption processes on colloidal surfaces (hybrid equilibrium model) was compared with kinetic-based models. Sensitivity analyses were conducted to deduce the effect of major parameters on contaminant transport. The results obtained from the hybrid equilibrium model in predicting the transport of colloid-facilitated groundwater contaminant are very similar to those of the kinetic model, when the point of interest is not at contaminant and colloid source vicinities and the time of interest is sufficiently long for imposed sources.     

Experimental Investigation and Modelling of Colloidal Release in Porous Media

Gas production from underground storage reservoirs is sometimes associated with solid particles eroded from the rock matrix. This phenomenon often called sand production can cause damage to the storage equipments, leading the operator to choke the wells and prevent them from producing at full capacity. Colloid release is often associated as a precursor of larger solid production. Indeed, in sandstone storage sites, clay release induced by the presence of condensed water associated with the gas production in the near-wellbore region can be a forecast of intergranular cement erosion. The objective of this work is twofold: firstly to experimentally investigate colloidal particle detachment through ionic strength reduction (absence of salinity of the condensed water) in porous media and secondly to determine its evolution with time and to model it. Laboratory experiments with model systems are developed to reproduce the particle generation and their transport in porous media. The model porous medium is a packed column of two powders: silicon carbide particles of 50 μm and silica particles of 0.5 μm (3% by weight) initially mixed together. Brine flows at different concentrations are imposed through the porous sample and, at very low salt concentration, colloid silica particles are massively released from the medium. Experimental evolutions of the particle concentration with time are compared to solutions of the advection–dispersion equation including first-order source terms for colloid release. The dispersion coefficients of the porous medium have been determined with tracer tests. The experimental results exhibit a different behaviour at short- and long-time intervals and a model has been built to predict the colloid production evolution with the introduction of two different time scales for the eroded rate. The model can be used in a core test to evaluate the amount of detachable fines and the rate of erosion.     

A computational framework for fluid–porous structure interaction with large structural deformation

We study the effect of poroelasticity on fluid–structure interaction. More precisely, we analyze the role of fluid flow through a deformable porous matrix in the energy dissipation behavior of a poroelastic structure. For this purpose, we develop and use a nonlinear poroelastic computational model and apply it to the fluid–structure interaction simulations. We discretize the problem by means of the finite element method for the spatial approximation and using finite differences in time. The numerical discretization leads to a system of non-linear equations that are solved by Newton’s method. We adopt a moving mesh algorithm, based on the Arbitrary Lagrangian–Eulerian method to handle large deformations of the structure. To reduce the computational cost, the coupled problem of free fluid, porous media flow and solid mechanics is split among its components and solved using a partitioned approach. Numerical results show that the flow through the porous matrix is responsible for generating a hysteresis loop in the stress versus displacement diagrams of the poroelastic structure. The sensitivity of this effect with respect to the parameters of the problem is also analyzed.

     

Mobile and immobile fluid transport: Coupling framework

We introduce a solver method for mobile and immobile transport regions. The motivation is driven by transport processes in porous media (e.g. waste disposal, chemical deposition processes). We analyze the coupled transport‐reaction equation with mobile and immobile areas. We apply analytical methods, such as Laplace‐transformation, and for the numerical methods we apply Godunov's scheme, see (Mat. Sb. 1959; 47 :271–306; Finite Volume Methods for Hyperbolic Problems. Cambridge University Press: Cambridge, 2002). The method is based numerically on flux‐based characteristic methods and is an attractive alternative to the classical higher‐order TVD methods, see (J. Comput. Phys. 1993; 49 :357–393). In this paper, we will focus on the derivation of analytical solutions for general and special solutions of the characteristic methods that are embedded in a finite‐volume method. At the end of the paper, we illustrate the higher‐order method for different benchmark problems. Copyright © 2010 John Wiley & Sons, Ltd.     

On the Domain of Validity of Brinkman’s Equation

An increasing number of articles are adopting Brinkman’s equation in place of Darcy’s law for describing flow in porous media. That poses the question of the respective domains of validity of both laws, as well as the question of the value of the effective viscosity μ ^e which is present in Brinkman’s equation. These two topics are addressed in this article, mainly by a priori estimates and by recalling existing analyses. Three main classes of porous media can be distinguished: “classical” porous media with a connected solid structure where the pore surface S _p is a function of the characteristic pore size l _p (such as for cylindrical pores), swarms of low concentration fixed particles where the pore surface is a function of the characteristic particle size l _s , and fiber-made porous media at low solid concentration where the pore surface is a function of the fiber diameter. If Brinkman’s 3D flow equation is valid to describe the flow of a Newtonian fluid through a swarm of fixed particles or fibrous media at low concentration under very precise conditions (Lévy 1983), then we show that it cannot apply to the flow of such a fluid through classical porous media.     

Foreword of the Founding Editor of the Journal Transport in Porous Media,Jacob Bear

The intrinsic permeability is a crucial parameter to characterise and quantify fluid flow through porous media. However, this parameter is typically uncertain, even if the geometry of the pore structure is available. In this paper, we perform a comparative study of experimental, semi-analytical and numerical methods to calculate the permeability of a regular porous structure. In particular, we use the Kozeny–Carman relation, different homogenisation approaches (3D, 2D, very thin porous media and pseudo 2D/3D), pore-scale simulations (lattice Boltzmann method, Smoothed Particle Hydrodynamics and finite-element method) and pore-scale experiments (microfluidics). A conceptual design of a periodic porous structure with regularly positioned solid cylinders is set up as a benchmark problem and treated with all considered methods. The results are discussed with regard to the individual strengths and limitations of the used methods. The applicable homogenisation approaches as well as all considered pore-scale models prove their ability to predict the permeability of the benchmark problem. The underestimation obtained by the microfluidic experiments is analysed in detail using the lattice Boltzmann method, which makes it possible to quantify the influence of experimental setup restrictions.

     

Solute diffusion in fractured porous media with memory effects due to adsorption

Modelling of solute transport in fractured porous media is a subject of intensive research in many engineering disciplines, such as petroleum engineering, water resources management, civil engineering. Recent field and laboratory experiments show that, in presence of strong adsorption, the behaviour of solute penetrating into the fractured porous medium diverges from classical hypotheses, rendering impossible the adjustment of classical transport models. The aim of this paper is to develop a mathematical continuous model of solute transport, when strong adsorption of solute occurs on the grains of the porous matrix. The macroscopic model is obtained by upscaling the pore and the fracture behaviours, by using the multiple scale expansion method. We obtain a non-standard diffusion behaviour of solute which shows local non-equilibrium between transport in the fractures and in the porous matrix, as well as memory effects. To cite this article: J. Lewandowska et al., C. R. Mecanique 330 (2002) 879–884.     

Surface Permeability of Particulate Porous Media

The dispersion process in particulate porous media at low saturation levels takes place over the surface elements of constituent particles and, as we have found previously by comparison with experiments, can be accurately described by superfast nonlinear diffusion partial differential equations. To enhance the predictive power of the mathematical model in practical applications, one requires the knowledge of the effective surface permeability of the particle-in-contact ensemble, which can be directly related with the macroscopic permeability of the particulate media. We have shown previously that permeability of a single particulate element can be accurately determined through the solution of the Laplace–Beltrami Dirichlet boundary value problem. Here, we demonstrate how that methodology can be applied to study permeability of a randomly packed ensemble of interconnected particles. Using surface finite element techniques, we examine numerical solutions to the Laplace–Beltrami problem set in the multiply-connected domains of interconnected particles. We are able to directly estimate tortuosity effects of the surface flows in the particle ensemble setting.