Random Porous Media Research Articles

Modeling Immiscible Two‐Phase Flow in Rough Fractures From Capillary to Viscous Fingering

AbstractWe develop an efficient computational model for simulating fluid invasion patterns emerging in variable aperture fractures. This two‐dimensional model takes into account the effect of capillary force on the fluid‐fluid interfaces and viscous pressure drop in both fluid phases. The pressure distribution is solved at each time step based on mass balance and local cubic law, considering an imposed pressure jump condition at the fluid‐fluid interface. This pressure jump corresponds to the Laplace pressure which includes both terms related to the out‐of‐plane (aperture‐spanning) curvature and to the in‐plane curvature. Simulating a configuration that emulates viscous fingering in two‐dimensional random porous media confirms that the model accounts properly for the role of viscous forces. Furthermore, direct comparison with previously obtained experimental results shows that the model reproduces the observed drainage patterns in a rough fracture reasonably well. The evolutions of tip location, the inlet pressures, and the invading phase fractal dimensions are analyzed to characterize the transition from capillary fingering to viscous fingering regimes. A radial injection scenario of immiscible invasion is also studied with varying modified capillary number and viscosity ratio, showing displacement patterns ranging from capillary fingering to viscous fingering to stable displacement. Such simulations using two contact angles show that the invading phase becomes more compact when the wetting condition changes from strong to weak drainage, as already observed in 2‐D porous media. The model can be used to bridge the gap in spatial scales of two‐phase flow between pore‐scale modeling approaches and the continuum Darcy‐scale models.

Non-dimensional lattice Boltzmann simulations on pore scale double diffusive natural convection in an enclosure filled with random porous media

Double diffusive natural convection in an enclosure filled with random porous media has been simulated by a Non-dimensional Lattice Boltzmann method (NDLBM). The random structures with different porosities are generated by a Controllable Structure Generation Scheme (CSGS). The NDLBM and the CSGS with mesoscopic scale governing parameters enable the direct simulations of transient velocity, temperature and concentration fields in pore scale for random media of a wide range of porosities. Validations of the parallel codes have been done by comparing to natural convection in a pure fluid filled enclosure obtained by various methods. Distributions of pore scale velocities, temperatures, and concentrations are quantitatively shown. Effects of the governing parameters on the macroscopic Nusselt, Sherwood and Biot numbers are presented. The changes of physical properties including solid fluid interface effects are also discussed.

Relative permeability of three immiscible fluids in random porous media determined by the lattice Boltzmann method

In this article, three immiscible fluids flowing in 3D porous media were simulated by the pseudopotential-based multiple relaxation time lattice Boltzmann model for multicomponent/multiphase flow, and the effects of capillary number (Ca), wettability and viscosity ratio (M) on the relative permeability were analysed. The simulation results show that the relative permeability of the three fluids increases with increasing Ca, whereas wettability influences the effect of Ca on the wetting phase (WP) relative permeability. The enhancement of porous wettability in the three-phase system leads to a decrease in the WP relative permeability and an increase in the relative permeability of the other two fluids. When M > 1, the nonwetting phase (NWP) has a lubricating effect and its relative permeability increases with increasing M in wet porous media, whereas the other two fluids are insensitive to M variations compared with the NWP.

On Local Fourier Analysis of Multigrid Methods for PDEs with Jumping and Random Coefficients

In this paper, we propose a novel nonstandard local Fourier analysis (LFA) variant for accurately predicting the multigrid convergence of problems with random and jumping coefficients. This LFA method is based on a specific basis of the Fourier space rather than the commonly used Fourier modes. To show the utility of this analysis, we consider, as an example, a simple cell-centered multigrid method for solving a steady-state single phase flow problem in a random porous medium. We successfully demonstrate the predictive capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis could be used to estimate a priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method.

Linear Stability Analysis of Transient Electrodeposition in Charged Porous Media: Suppression of Dendritic Growth by Surface Conduction

We study the linear stability of transient electrodeposition in a charged random porous medium, whose pore surface charges can be of any sign, flanked by a pair of planar metal electrodes. Discretization of the linear stability problem results in a generalized eigenvalue problem for the dispersion relation that is solved numerically, which agrees well with the analytical approximation obtained from a boundary layer analysis valid at high wavenumbers. Under galvanostatic conditions in which an overlimiting current is applied, in the classical case of zero surface charges, the electric field at the cathode diverges at Sand’s time due to electrolyte depletion. The same phenomenon happens for positive charges but earlier than Sand’s time. However, negative charges allow the system to sustain an overlimiting current via surface conduction past Sand’s time, keeping the electric field bounded. Therefore, at Sand’s time, negative charges greatly reduce surface instabilities and suppress dendritic growth, while zero and positive charges magnify them. We compare theoretical predictions for overall surface stabilization with published experimental data for copper electrodeposition in cellulose nitrate membranes and demonstrate good agreement between theory and experiment. We also apply the stability analysis to how crystal grain size varies with duty cycle during pulse electroplating.

A data‐driven model order reduction approach for Stokes flow through random porous media

AbstractDirect numerical simulation of Stokes flow through a random, impermeable rigid body matrix by finite elements requires meshes fine enough to resolve the pore‐size scale and is thus a computationally expensive task. It is well known that on a larger scale, Stokes flow can accurately be approximated by Darcy's law [1] with an effective diffusivity field depending on viscosity and the matrix phase topology. We propose a fully probabilistic Darcy type reduced‐order model which, based on only a few tens of full‐order model runs, is capable to learn a fine‐scale topology to effective diffusion map which is maximally predictive for fine‐scale response. After training, the model provides sharp predictive distributions based on repeated evaluation of the much cheaper Darcy equation only.

Intrinsic and apparent gas permeability of heterogeneous and anisotropic ultra-tight porous media

Accurate prediction of unconventional gas production requires deep understanding of the permeability of complex rock samples. Several predictive expressions of permeability, which include either simplifications of the porous media structure or the flow mechanisms, have been proposed recently. The main objective of this research is to quantify the impact of solid matrix complexity on both intrinsic and apparent permeability. To this end, numerous two-dimensional random porous media structures are constructed using the quartet structure generation set algorithm. Parametric and statistical analysis reveals the importance of the specific surface area of pores, tortuosity, heterogeneity and degree of anisotropy. Special focus is given to the directional dependency of the permeability on isotropic and anisotropic geometries, considering the great impact of anisotropy on the laboratory evaluation of permeability data and the anisotropic nature of shale rocks. Simulation results, for the same value of porosity, clearly indicate the drastic improvement of permeability due to the reduction of specific surface area of pores and their height to width ratio. This suggests that rock matrix complexity has significant impact on permeability and should not be neglected while forming permeability formulations for porous media. Finally, the results of the apparent permeability, obtained by solving the gas kinetic equation, are taken into consideration to demonstrate the enhancement ratio, slip factor and their correlation with the aforementioned parameters. Semi-analytical expressions for intrinsic and apparent permeability, considering continuum and slip flow respectively, are derived. The proposed formulations, suitable for both isotropic and anisotropic structures, have the advantage of not entailing any numerical or experimental data as input.

Mobilization of Crude Oil in Porous Media With Oil-Soluble Surfactant Delivered by Hydrosoluble Micelles

In this work, an oil-soluble surfactant was studied to enhance crude oil mobilization in a cryolite-packed miniature bed. The cryolite packed bed provided a transparent, random porous medium for observation at the microscopic level. In the first part of the paper, oil-soluble surfactants, Span 80 and Eni-surfactant (ES), were dissolved directly into the crude oil. The porous medium was imbued with the crude oil (containing the surfactants), and de-ionized water was the flooding phase; in this experiment, the system containing ES had the best performance. Subsequently, sodium dodecyl sulfate (SDS), a hydrosoluble surfactant, was used to solubilize the ES, with the SDS acting as a carrier for the ES to the contaminated porous media. Finally, the SDS/ES micellar solutions were used in oil-removal tests on the packed bed. Grayscale image analysis was used to quantify the oil recovery effectiveness for the flooding experiments by measuring the white pixel percentage in the packed bed images. The SDS/ES flooding mixture had a better performance than the SDS alone.

Uncertainty Quantification for Flow and Transport in Highly Heterogeneous Porous Media Based on Simultaneous Stochastic Model Dimensionality Reduction

Groundwater flow models are usually subject to uncertainty as a consequence of the random representation of the conductivity field. In this paper, we use a Gaussian process model based on the simultaneous dimension reduction in the conductivity input and flow field output spaces in order quantify the uncertainty in a model describing the flow of an incompressible liquid in a random heterogeneous porous medium. We show how to significantly reduce the dimensionality of the high-dimensional input and output spaces while retaining the qualitative features of the original model, and secondly how to build a surrogate model for solving the reduced-order stochastic model. A Monte Carlo uncertainty analysis on the full-order model is used for validation of the surrogate model.

Old and New Approaches Predicting the Diffusion in Porous Media

An accurate quantitative description of the diffusion coefficient in a porous medium is essential for predictive transport modeling. Well-established relations, such as proposed by Buckingham, Penman, and Millington–Quirk, relate the scalar diffusion coefficient to the porous medium’s porosity. To capture the porous medium’s structure in more detail, further models include fitting parameters, geometric, or shape factors. Some models additionally account for the tortuosity, e.g., via Archie’s law. A validation of such models has been carried out mainly via experiments relating the proposed description to a specific class of porous media (by means of parameter fitting). Contrary to these approaches, upscaling methods directly enable calculating the full, potentially anisotropic, effective diffusion tensor without any fitting parameters. As input only the geometric information in terms of a representative elementary volume is needed. To compute the diffusion–porosity relations, supplementary cell problems must be solved numerically and their (flux) solutions must be integrated. We apply this approach to provide easy-to-use quantitative diffusion–porosity relations that are based on representative single grain, platy, blocky, prismatic soil structures, porous networks, or random porous media. As a discretization method, we use the discontinuous Galerkin method on structured grids. To make the relations explicit, interpolation of the obtained data is used. We furthermore compare the obtained diffusion–porosity relations with the well-established relations mentioned above and also with the well-known Voigt–Reiss or Hashin–Shtrikman bounds. We discuss the ranges of validity and further provide the explicit relations between the diffusion and surface area and comment on the role of a tortuosity–porosity relation.

Stagnation Points as Loci of Solute Concentration Extrema at the Evaporative Surface of a Random Porous Medium

Evaporation of a saline solution from a porous medium often leads to the precipitation of salt at the surface of the porous medium. It is commonly observed that the crystallized salt does not form everywhere at the porous medium surface but at some specific locations. This is interpreted at the signature of spatial variations in the salt concentration at the surface of the porous medium prior to the onset of crystallization. We explore numerically the link between the ion concentration spatial variations at the surface and porous medium heterogeneities considering strongly anisotropic short-range correlated permeability Gaussian fields corresponding to a vertical layering perpendicular to the top evaporative surface for the case of the evaporation–wicking situation. It is shown that the ion concentration extrema at the surfaces correspond to stagnation points with minima corresponding to divergent stagnation points and maxima to convergent stagnation points. Counter-intuitively, the ion concentration maxima are shown to correspond to permeability minima. However, the ion concentration absolute maximum does not necessarily always correspond to the permeability absolute minimum. More generally, the study emphasizes the key role played by the impact of heterogeneities on the velocity field induced in the medium by the evaporation process. It is also shown that the number of ion mass fraction maxima at the porous medium surface is generally much lower than the naive prediction based on the number of correlation lengths spanning the medium.

Phase Equilibria of Polydisperse Square-Well Chain Fluid Confined in Random Porous Media: TPT of Wertheim and Scaled Particle Theory.

Extension of Wertheim's thermodynamic perturbation theory and its combination with scaled particle theory is proposed and applied to study the liquid-gas phase behavior of polydisperse hard-sphere square-well chain fluid confined in the random porous media. Thermodynamic properties of the reference system, represented by the hard-sphere square-well fluid in the matrix, are calculated using corresponding extension of the second-order Barker-Henderson perturbation theory. We study effects of polydispersity and confinement on the phase behavior of the system. While polydispersity causes increase of the region of phase coexistence due to the critical temperature increase, confinement decreases the values of both critical temperature and critical density making the region of phase coexistence smaller. This effect is enhanced with the increase of the size ratio of the fluid and matrix particles. The increase of the average chain length at fixed values of polydispersity and matrix density shifts the critical point to a higher temperature and a slightly lower density.

Evaluation of the path integral for flow through random porous media.

We present a path integral formulation of Darcy's equation in one dimension with random permeability described by a correlated multivariate lognormal distribution. This path integral is evaluated with the Markov chain Monte Carlo method to obtain pressure distributions, which are shown to agree with the solutions of the corresponding stochastic differential equation for Dirichlet and Neumann boundary conditions. The extension of our approach to flow through random media in two and three dimensions is discussed.

Experimental analysis of the flow near the boundary of random porous media

The aim of this work is to experimentally examine flow over and near random porous media. Different porous materials were chosen to achieve porosity ranging from 0.95 to 0.99. In this study, we report the detailed velocity measurements of the flow over and near random porous material inside a rectangular duct using a planar particle image velocimetry (PIV) technique. By controlling the flow rate, two different Reynolds numbers were achieved. We determined the slip velocity at the interface between the porous media and free flow. Values of the slip velocity normalized either by the maximum flow velocity or by the shear rate at the interface and the screening distance K1/2 were found to depend on porosity. It was also shown that the depth of penetration inside the porous material was larger than the screening length using Brinkman’s prediction. Moreover, we examined a model for the laminar coupled flow over and inside porous media and analyzed the permeability of a random porous medium. This study provided detailed analysis of flow over and at the interface of various specific random porous media using the PIV technique. This analysis has the potential to serve as a first step toward using random porous media as a new passive technique to control the flow over smooth surfaces.

Three-Dimensional Lattice Boltzmann Simulations of Single-Phase Permeability in Random Fractal Porous Media with Rough Pore–Solid Interface

Single-phase permeability k has intensively been investigated over the past several decades by means of experiments, theories and simulations. Although the effect of surface roughness on fluid flow and permeability in single pores and fractures as well as networks of fractures was studied previously, its influence on permeability in a random mass fractal porous medium constructed of pores of different sizes remained as an open question. In this study, we, therefore, address the effect of pore–solid interface roughness on single-phase flow in random fractal porous media. For this purpose, we apply a mass fractal model to construct porous media with a priori known mass fractal dimensions $$2.579 \le D_{\mathrm{m}} \le 2.893$$ characterizing both solid matrix and pore space. The pore–solid interface of the media is accordingly roughened using the Weierstrass–Mandelbrot approach and two parameters, i.e., surface fractal dimension $$D_{\mathrm{s}}$$ and root-mean-square (rms) roughness height. A single-relaxation-time lattice Boltzmann method is applied to simulate single-phase permeability in the corresponding porous media. Results indicate that permeability decreases sharply with increasing $$D_{\mathrm{s}}$$ from 1 to 1.1 regardless of $$D_{\mathrm{m}}$$ value, while k may slightly increase or decrease, depending on $$D_{\mathrm{m}}$$ , as $$D_{\mathrm{s}}$$ increases from 1.1 to 1.6.

Hindrance Factor Expression for Diffusion in Random Mesoporous Adsorbents Obtained from Pore-Scale Simulations in Physical Reconstructions

Hindered diffusion of solutes is the rate-limiting step in many processes for which random porous media play a central role as providers of adsorbing or reactive interfaces. The key to an optimized layout of these processes is the knowledge of the overall diffusive hindrance factor H(λ) = Deff,H(λ)/Dm, which quantifies the degree to which diffusion through a material (represented by the effective diffusion coefficient Deff,H) is hindered compared with diffusion in the bulk liquid (represented by Dm) in dependence of λ, the ratio of solute size to mean pore size. To arrive at an adequate hindrance factor expression for random mesoporous silica, we use electron tomography to physically reconstruct the mesopore space of three macro-mesoporous silica monoliths. The samples share the same general mesopore shape and topology at varied mean feature size, as established by morphological analysis, and serve as realistic models in pore-scale simulations of hindered diffusion. From a large set of Deff,H(λ) values fo...

Lattice Boltzmann simulation of asymptotic longitudinal mass dispersion in reconstructed random porous media

To research macroscopic mass transport characteristics of porous media, a lattice Boltzmann method (LBM) approach was utilized to calculate asymptotic longitudinal mass dispersion. In this study, a D2Q9 model with multi‐relaxation‐time (MRT) collision operator, which is appropriate for incompressible flow with a high Péclet number without refining the lattice, was chosen. With respect to the microstructure of porous media, random placement (RP) method was applied to obtain randomly positioned particles. Based on the exhausted numerical results presented in the study, a new correlation of longitudinal mass dispersion was established. By comparing with available experimental data in the literature, reasonable agreements are observed in a wide porosity range from 0.3 to 0.7, indicating the validity of the proposed correlation. © 2018 American Institute of Chemical Engineers AIChE J, 64: 2770–2780, 2018

Upper Bounds on the Permeability of Random Porous Media

Upper bounds on the permeability of random porous media are presented, which improve significantly on existing bounds. The derived bounds rely on a variational formulation of the upscaling problem from a viscous flow at the pore scale, described by Stokes equation, to a Darcy formulation at the macroscopic scale. A systematic strategy to derive upper bounds based on trial force fields is proposed. Earlier results based on uniform void or interface force fields are presented within this unified framework, together with a new proposal of surface force field and a combination of them. The obtained bounds feature detailed statistical information on the pore morphology, including two- and three-point correlation functions of the pore phase, the solid–fluid interface and its local orientation. The required spatial correlation functions are explicitly derived for the Boolean model of spheres, in which the solid phase is modelled as the union of penetrable spheres. Existing and new bounds are evaluated for this model and compared to full-field simulations on representative volume elements. For the first time, bounds allow to retrieve the correct order of magnitude of permeability for a wide range of porosity and even improve on some estimates. However, none of the bounds reproduces the non-analytic behaviour of the permeability–porosity curve at low solid concentration.

Hydraulic conductivities identification via Ensemble Kalman Filtering with transformed data considering the risk of systematic bias

In subsurface hydrology, Ensemble Kalman Filtering (EnKF) has been coupled with groundwater flow and transport models to solve the inverse problem. Several extensions of the EnKF have been proposed to improve its performance when dealing with non-multi-Gaussian random field models of the hydraulic conductivity. One such variant is the EnKF with transformed data (tEnKF), which uses Gaussian anamorphosis within a conditioning step. Although this transformation has been used in the past to identify hydraulic conductivities, previous studies have ignored the risk of introducing a systematic bias in the spatiotemporal evolution of the hydraulic head field during the forecast steps that the update steps may not correct over time. This paper proposes that in order to evaluate the performance of tEnKFs, applications in synthetically generated random porous media should take into account this risk by incorporating prior knowledge with a multi-Gaussian conductivity correlation structure, and by adopting a reference field with asymmetric correlation structure. As an example of this application, hydraulic conductivities using the tEnKF were identified by solving a one-dimensional, single phase flow problem in a continuous random porous medium. Common concepts in Geostatistics are used to explain the hypothesis underlying both EnKF and tEnKF and to establish a clear link between the tEnKF and the stochastic simulation of conditional random fields.

Viscoelastic flow simulations in random porous media

We investigate creeping flow of a viscoelastic fluid through a three dimensional random porous medium using computational fluid dynamics. The simulations are performed using a finite volume methodology with a staggered grid. The no slip boundary condition on the fluid-solid interface is implemented using a second order finite volume immersed boundary (FVM-IBM) methodology [1]. The viscoelastic fluid is modeled using a FENE-P type model. The simulations reveal a transition from a laminar regime to a nonstationary regime with increasing viscoelasticity. We find an increased flow resistance with increase in Deborah number even though shear rheology is shear thinning nature of the fluid. By choosing a length scale based on the permeability of the porous media, a Deborah number can be defined, such that a universal curve for the flow transition is obtained. A study of the flow topology shows how in such disordered porous media shear, extensional and rotational contributions to the flow evolve with increased viscoelasticity. We correlate the flow topology with the dissipation function distribution across the porous domain, and find that most of the mechanical energy is dissipated in shear dominated regimes instead, even at high viscoelasticity.