Random Porous Media Research Articles

Generalized modeling of spontaneous imbibition based on Hagen-Poiseuille flow in tortuous capillaries with variably shaped apertures.

Spontaneous imbibition of wetting liquids in porous media is a ubiquitous natural phenomenon which has received much attention in a wide variety of fields over several decades. Many traditional and recently presented capillary-driven flow models are derived based on Hagen-Poiseuille (H-P) flow in cylindrical capillaries. However, some limitations of these models have motivated modifications by taking into account different geometrical factors. In this work, a more generalized spontaneous imbibition model is developed by considering the different sizes and shapes of pores, the tortuosity of imbibition streamlines in random porous media, and the initial wetting-phase saturation. The interrelationships of accumulated imbibition weight, imbibition rate and gas recovery and the properties of the porous media, wetting liquids, and their interactions are derived analytically. A theoretical analysis and comparison denote that the presented equations can generalize several traditional and newly developed models from the literature. The proposed model was evaluated using previously published data for spontaneous imbibition measured in various natural and engineered materials including different rock types, fibrous materials, and silica glass. The test results show that the generalized model can be used to characterize the spontaneous imbibition behavior of many different porous media and that pore shape cannot always be assumed to be cylindrical.

Sub-Doppler resonances in the backscattered light from random porous media infused with Rb vapor

We report on the observation of sub-Doppler resonances on the back-scattered light from a random porous glass medium with rubidium vapor filling its interstices. The sub-Doppler spectral lines are the consequence of saturated absorption where the incident laser beam saturates the atomic medium and the back-scattered light probes it. Some specificities of the observed spectra reflect the transient atomic evolution under confinement inside the pores. Simplicity, robustness and potential miniaturization are appealing features of this system as a spectroscopic reference.

Computational homogenization of plastic porous media with two populations of voids

The macroscopic yield response of random porous media containing two populations of spherical voids is investigated via large volume computations. The computed yield surface is compared to analytical criteria recently developed for the above-mentioned porous media. To overcome the observed discrepancies, the analytical models are modified by introducing additional parameters which are numerically derived. These parameters depend neither on the void volume fraction nor on the void size. Large volume computations are also used to build the yield surface for porous media containing a single population of randomly distributed spherical voids. The results show that, for an identical fraction of porosities, there is no significant difference between a double and a single population of voids. This result implies that the yield response of porous media containing two populations of voids may be also described by analytical criteria developed in the case of a single population.

Effective conductivity in random porous media with convex and non-convex porosity

A fundamental study on the effective conductivity in random porous media is presented in this paper. Porous media are distinguished based on the convexity of the insulating porous phase: microstructures with convex porosity are representative of foams and sponges, while non-convex porosity includes granular and particulate materials. This distinction allows us to identify two limit cases within which the effective conductivity of an isotropic porous medium lies. The effective conductivity is evaluated through a random walk method, applied on structures numerically reconstructed with packing algorithms which use spheres to represent convex elements. Simulation results are compared with theoretical relations and experimental data in a range of 0–40% in porosity, wherein the solid phase is entirely connected and not suspended in the porous phase.This study shows that, given the porosity, structures with convex porosity are more conducting than structures with non-convex pores, because the bottlenecks for conduction are narrower in the second case. These situations identify two limit cases for the effective conductivity in a random isotropic porous medium, as confirmed by experimental data of structures with partly convex and partly non-convex porosity. In addition, for convex porosity the effective conductivity is larger when pores are isolated, attaining the Hashin–Shtrikman upper bound, while when pores are overlapped the effective conductivity is well described by the Archie power law.

A Hybrid HDMR for Mixed Multiscale Finite Element Methods with Application to Flows in Random Porous Media

Stochastic modeling has become a popular approach to quantifying uncertainty in flows through heterogeneous porous media. In this approach the uncertainty in the heterogeneous structure of material properties is often parametrized by a high-dimensional random variable, leading to a family of deterministic models. The numerical treatment of this stochastic model becomes very challenging as the dimension of the parameter space increases. To efficiently tackle the high-dimensionality, we propose a hybrid high-dimensional model representation (HDMR) technique, through which the high-dimensional stochastic model is decomposed into a moderate-dimensional stochastic model, in the most active random subspace, and a few one-dimensional stochastic models. The derived low-dimensional stochastic models are solved by incorporating the sparse-grid stochastic collocation method with the proposed hybrid HDMR. In addition, the properties of porous media, such as permeability, often display heterogeneous structure across multiple spatial scales. To treat this heterogeneity we use a mixed multiscale finite element method (MMsFEM). To capture the nonlocal spatial features (i.e., channelized structures) of the porous media and the important effects of random variables, we can hierarchically incorporate the global information individually from each of the random parameters. This significantly enhances the accuracy of the multiscale simulation. Thus, the synergy of the hybrid HDMR and the MMsFEM reduces the dimension of the flow model in both the stochastic and physical spaces, and hence significantly decreases the computational complexity. We analyze the proposed hybrid HDMR technique and the derived stochastic MMsFEM. Numerical experiments are carried out for two-phase flows in random porous media to demonstrate the efficiency and accuracy of the proposed hybrid HDMR with MMsFEM.

New Scheme of Finite Difference Heterogeneous Multiscale Method to Solve Saturated Flow in Porous Media

A new finite difference scheme, the development of the finite difference heterogeneous multiscale method (FDHMM), is constructed for simulating saturated water flow in random porous media. In the discretization framework of FDHMM, we follow some ideas from the multiscale finite element method and construct basic microscopic elliptic models. Tests on a variety of numerical experiments show that, in the case that only about a half of the information of the whole microstructure is used, the constructed scheme gives better accuracy at a much lower computational time than FDHMM for the problem of aquifer response to sudden change in reservoir level and gives comparable accuracy at a much lower computational time than FDHMM for the weak drawdown problem.

Geometric Models for Isotropic Random Porous Media: A Review

Models for random porous media are considered. The models are isotropic both from the local and the macroscopic point of view; that is, the pores have spherical shape or their surface shows piecewise spherical curvature, and there is no macroscopic gradient of any geometrical feature. Both closed-pore and open-pore systems are discussed. The Poisson grain model, the model of hard spheres packing, and the penetrable sphere model are used; variable size distribution of the pores is included. A parameter is introduced which controls the degree of open-porosity. Besides systems built up by a single solid phase, models for porous media with the internal surface coated by a second phase are treated. Volume fraction, surface area, and correlation functions are given explicitly where applicable; otherwise numerical methods for determination are described. Effective medium theory is applied to calculate physical properties for the models such as isotropic elastic moduli, thermal and electrical conductivity, and static dielectric constant. The methods presented are exemplified by applications: small-angle scattering of systems showing fractal-like behavior in limited ranges of linear dimension, optimization of nanoporous insulating materials, and improvement of properties of open-pore systems by atomic layer deposition of a second phase on the internal surface.

Phase separation of binary nonadditive hard sphere fluid mixture confined in random porous media

I analyze the fluid-fluid phase separation of nonadditive hard sphere fluid mixture absorbed in random porous media. An equation of state is derived by using the perturbation theory to this complex system with quenched disorders. The results of this theory are in good agreement with those obtained from semi-grand canonical ensemble Monte Carlo simulations. The contact value of the fluid-fluid radial distribution functions of the reference which is the key point of the perturbation process is derived as well, the comparison against Monte Carlo simulations shows that it has an excellent accuracy.

Fluids in porous media: The case of neutral walls

The bulk phase behavior of a fluid is typically altered when the fluid is brought into confinement by the walls of a random porous medium. Inside the porous medium, phase-transition points are shifted, or may disappear altogether. A crucial determinant is how the walls interact with the fluid particles. In this work, we consider the situation whereby the walls are neutral with respect to the liquid and vapor phases. In order to realize the condition of strict neutrality, we use a symmetric binary mixture inside a porous medium that interacts identically with mixture species. Monte Carlo simulations are then used to obtain the phase behavior. Our main finding is that, in the presence of the porous medium, a liquid-vapor critical point still exists. At the critical point, the distribution of the order parameter remains scale invariant, but self-averaging is violated. These findings provide further evidence that random confinement by neutral walls induces critical behavior of the random Ising model (i.e., Ising models with dilution type disorder, where the disorder couples to the energy).

Multilevel Monte Carlo for two phase flow and Buckley-Leverett transport in random heterogeneous porous media

Monte Carlo (MC) is a well known method for quantifying uncertainty arising for example in subsurface flow problems. Although robust and easy to implement, MC suffers from slow convergence. Extendi...

Hard convex body fluids in random porous media: Scaled particle theory

The scaled particle theory (SPT) is applied to describe thermodynamic properties of hard convex body (HCB) fluid in random porous media. To this purpose we extended the SPT2 approach, which has been developed previously for a hard sphere fluid in random porous matrices. The analytical expressions for the chemical potential and the pressure of an HCB fluid in an HCB or an overlapping hard convex body (OHCB) matrix are obtained and analyzed. Within the SPT2 approach a series of new approximations are proposed. The grand canonical Monte Carlo (GCMC) simulations are performed to verify an accuracy of the SPT2 and the developed approaches. The obtained analytical expressions include three types of porosity: the geometrical porosity ϕ0, the probe particle porosity ϕ and the porosity ϕ⁎ defined by the maximum packing fraction of a fluid in a given matrix. The effect of nonsphericity parameters of matrix and fluid particles on the thermodynamic properties of a fluid is discussed.

Ferrofluid Permeation into Three-Dimensional Random Porous Media: A Numerical Study Using the Lattice Boltzmann Method

Colloidal suspensions containing magnetic nanoparticles placed in appropriate carrier liquids present strong magnetic dipoles. These suspensions, in general, exhibit normal liquid behaviour coupled with super paramagnetic properties. This leads to the possibility of remotely controlling the flow of such liquids with a moderate-strength external magnetic field. In this study, we numerically investigate the viability of controlling and steering a base-fluid with magnetic-sensitive nanoparticles into randomly structured fibrous porous media. Three dimensional flow simulations are performed using the lattice Boltzmann method. The simulation results for the flow front are presented, and the effect of the magnetic field strength on the rate of ferrofluid penetration is discussed. It is shown that the porosity of the porous medium and the size of the fibres have a strong effect on the ferrofluid penetration rate.

A nonparametric belief propagation method for uncertainty quantification with applications to flow in random porous media

A probabilistic graphical model approach to uncertainty quantification for flows in random porous media is introduced. Model reduction techniques are used locally in the graph to represent the random permeability. Then the conditional distribution of the multi-output responses on the low dimensional representation of the permeability field is factorized into a product of local potential functions. An expectation–maximization algorithm is used to learn the nonparametric representation of these potentials using the given input/output data. We develop a nonparametric belief propagation method for uncertainty quantification by employing the loopy belief propagation algorithm. The nonparametric nature of our model is able to capture non-Gaussian features of the response. The proposed framework can be used as a surrogate model to predict the responses for new input realizations as well as our confidence on these predictions. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed framework for solving uncertainty quantification problems in flows through porous media using stationary and non-stationary permeability fields.

Direct simulation of structured wall bounded packed beds using hybrid FE/FV methods

Direct numerical simulations are conducted for regular or structured porous media represented by packed beds of spheres enclosed in square cross-section channels for different channel-to-sphere diameter ratios (Ds/dp=1 to ∞) using hybrid finite element/volume (FE/FV) methods. The hybrid flow solver methodology is based on the pressure correction or projection method, which involves a fractional step approach to obtain an intermediate velocity field by solving the original momentum equations with the matrix-free, implicit, cell-centered finite volume method. The poisson equation resulting from the fractional step approach is then solved by node based Galerkin finite element method for an auxiliary variable which is closely related to pressure and is used to update the velocity field and pressure field.The hybrid flow solver has been extended for applications in structured packed beds for both infinite and also wall bounded porous media. The packed spheres representing the porous media are shrunk by 2% dp so that there is a finite, but very small spacing between spheres and also between sphere and wall to avoid highly skewed elements. Studies are conducted to test our novel hybrid FE/FV solver for porous media flows and also to analyze wall effects in detail for regular, structured porous media in Simple Cubic (SC) configuration. Three dimensional numerical simulations are conducted for different channel to sphere diameter ratios (Ds/dp) and also for infinite porous media over wide range of Reynolds numbers (1⩽ReE⩽5000) so that the dimensionless pressure drop or friction factor dependence on the Reynolds number and Ds/dp ratio is clearly understood. Results are validated with available empirical correlations and experimental data. Friction factor correlations for random beds are suitable for structured beds only at very low Reynolds numbers (ReE⩽20), but deviate significantly beyond that due to the channeling present in regular porous media which becomes increasingly significant at higher Reynolds numbers. For structured beds, wall influence was observed to be significant on the friction factor calculation in both laminar and turbulent regimes. Compared to infinite beds, friction factors increased as the channel-to-sphere diameter ratio (Ds/dp) decreased in the entire flow regime. This is unlike the behavior for random porous media where walls have contrasting effects at lower and higher Reynolds numbers when compared to the infinite random beds. Dimensionless pressure drop and friction factor results obtained from our simulations for a wide range of ReE and Ds/dp are curve-fitted to provide generalized correlations for predicting the friction factors in Simple Cubic (SC) structured packed beds.

Global transmission and reflection coefficients of the ultrasonic component in porous media: A new approach

The purpose of this paper is to propose a new approach to determining the global ultrasonic transmission and reflection coefficients in a random porous medium. Boundary conditions at the interface of the medium were used to determine local transmission and reflection coefficients. A study of the behavior of different waves inside the medium was carried out to derive a new global formulation that takes interior phenomena into consideration. All the results were obtained independently of the geometrical and physical characteristics of the medium so that the method can be applied to any two-phase porous medium. This study is based on normal incidence ultrasonic wave propagation.

Microfluidics of imbibition in random porous media

A free-energy lattice Boltzmann approach has been used to perform simulations of liquid penetration into random porous media. We focus our study on the effects of microstructures, particularly microtopography, on liquid penetration driven by capillary force and external pressure. For this purpose we set up a model structure that consists of a network of interconnected capillaries with varying pore geometries. The results showed that the discontinuities in the solid-surface curvature, as are present as corners on the capillary surfaces, have strong influences on liquid penetration through their pinning effects and interactions with local geometry.

A stochastic dimension reduction multiscale finite element method for groundwater flow problems in heterogeneous random porous media

Summary In this paper we present a stochastic dimension reduction multiscale finite element method for solving groundwater flow problems in heterogeneous random porous media. The stochastic conductivity is parameterized in a stochastic space using a truncated Karhunen–Loeve expansion with finite random variables. This often results in high stochastic dimensionality due to small correlation length and large variance of the inherent covariance function. In order to treat the high-dimensional stochastic problems efficiently, a truncated high-dimensional model representation (HDMR) technique is proposed that decomposes the high-dimensional stochastic problem into a moderate-dimensional stochastic problem and a few one-dimensional stochastic problems. The derived low-dimensional stochastic problems are solved by incorporating a sparse grid stochastic collocation method (SGSCM) with the proposed HDMR. This leads to a set of uncoupled deterministic problems at the sparse grid collocation points. To capture the small-scale heterogeneities in physical space, a multiscale finite element method (MsFEM) is utilized to solve these deterministic problems. Since the SGSCM and MsFEM are implemented in the different spaces, the integrated approach inherits the advantages from both of the SGSCM and the MsFEM, in which each low-dimensional stochastic problem is decoupled each other at small number of stochastic collocation points and is solved on a coarse spatial mesh. To demonstrate the efficiency and accuracy of the proposed stochastic MsFEM with the HDMR technique, a few numerical experiments are carried out for the transient water flow problems in heterogeneous random media with different correlation lengths and various degrees of spatial variabilities.

Fluids in random porous media: Scaled particle theory

The scaled particle theory (SPT) is applied to describe thermodynamic properties of a hard sphere (HS) fluid in random porous media. To this purpose, we extended the SPT2 approach, which has been developed previously. The analytical expressions for the chemical potential of an HS fluid in HS and overlapping hard sphere (OPH) matrices, sponge matrix, and hard convex body (HCB) matrix are obtained and analyzed. A series of new approximations for SPT2 are proposed. The grand canonical Monte Carlo (GGMC) simulations are performed to verify an accuracy of the SPT2 approach combined with the new approximations. A possibility of mapping between thermodynamic properties of an HS fluid in random porous media of different types is discussed. It is shown that thermodynamic properties of a fluid in the different matrices tend to be equal if the probe particle porosities and the specific surface pore areas of considered matrices are identical. The obtained results for an HS fluid in random porous media as reference systems are used to extend the van der Waals equation of state to the case of a simple fluid in random porous media. It is observed that a decrease of matrix porosity leads to lowering of the critical temperature and the critical density of a confined fluid, while an increase of a size of matrix particles causes an increase of these critical parameters.

An Algorithm for 3D Pore Space Reconstruction from a 2D Image Using Sequential Simulation and Gradual Deformation with the Probability Perturbation Sampler

The construction of a faithful 3D pore space model of a porous medium that could reproduce the macroscopic behavior of that medium is of great interest in various fields including medicine, material science, hydrology and petroleum engineering. A computationally efficient algorithm is developed that uses the probability perturbation method and sequential multiple-point statistics simulations to generate 3D stochastic and equiprobable representations of random porous media when only a 2D thin section image is available. By employing the probability perturbation method as a gradual deformation technique, the pore patterns of a single 2D image are deformed to generate a series of 2D stochastically simulated images. The 3D pore structure is then generated by simply stacking the 2D-simulated images. The quality of the 3D reconstruction is critically dependent on the rate of deformation and a simple general procedure for choosing this parameter is presented. Various criteria such as porosity, two-point auto-correlation function, multiple-point connectivity function, local percolation probability, absolute permeability obtained by lattice-Boltzmann method (LBM), formation factor and two-phase relative permeability calculations are used to validate the results. The method is tested on two random porous solids; Berea Sandstone and synthetic Silica, for which directly measured 3D micro-CT images are available. The stochastically reconstructed 3D pore space preserves the low- and high-order spatial statistics, the macroscopic flow properties and the microstructure of the 3D micro-CT images.

Analysis of stochastic mimetic finite difference methods and their applications in single-phase stochastic flows

Stochastic modeling has become a widely accepted approach to quantify uncertainty in applications where diffusion plays a central role. In many of these applications the geometry is complex, and important properties of the underlying deterministic continuum model need to be captured accurately. To address these problems we present a stochastic mimetic finite difference (MFD) method for diffusion equations with random input data. Specifically, we use the MFD methodology for the spatial approximation to ensure the necessary accuracy and robustness is achieved. To treat the high-dimensionality of the stochastic approximation efficiently, we use a stochastic collocation method. We consider the stochastic MFD approximation in hybrid form, and perform a rigorous analysis of its semi-discretization and full discretization for the pressure, flux and Lagrange multipliers. Convergence rates are developed for statistical moments of the quantities of interest. Numerical results are presented for single-phase flow in random porous media, and support the efficiency of the stochastic MFD.