Hierarchical Porous Media Research Articles

High-order models for convection–diffusion-reaction transport in multiscale porous media

Developing highly accurate models for predicting the convection–diffusion-reaction (CDR) transport in hierarchical porous media with strong heterogeneities on multiple scales is crucial but not yet available. In this work, an innovative high-order multiscale computational framework is developed to capture the local and global variation characteristics of flow fields and reactant concentration at multiple scales. The homogenized solutions and macro-meso high-order solutions are established by the formal two-scale asymptotic analysis. By directly expanding the mesoscopic cell functions to the microscopic levels, the three-scale high-order models are built by assembling the meso-micro high-order expansions of mesoscopic cell functions and macro-meso low/high-order models. The present approaches follow the reverse thought process of the reiteration homogenization method, and provide a very innovative way to develop highly accurate and efficient solutions for the CDR coupling problems in multiscale porous media. The effectiveness and accuracy of the proposed multiscale models are validated by several representative cases.

Moment analysis for predicting effective transport properties in hierarchical retentive porous media

We report on a new homogenization approach to solve, with drastically improved speed and accuracy, the general advection-diffusion equation in hierarchical porous media with localized diffusion and adsorption/desorption processes, thus opening the way to a much deeper understanding of the band broadening process in chromatographic systems. The proposed robust and efficient moment-based approach allows us to compute the exact local and integral concentration moments and hence provides exact solutions for the effective velocity and dispersion coefficients of migrating solute particles. Innovative to the proposed method is also that it not only produces the exact effective transport parameters of the long-time asymptotic solution, but also their entire transient. The analysis of the transient behaviour can be used, for example, to properly identify the time and length scales needed to achieve the macro-transport conditions. If the hierarchical porous media can be represented as the periodic repetition of a unit lattice cell, the method only requires the solution of the time-dependent advection-diffusion equations for the zeroth order and first-order exact local moments, exclusively on the unit cell. This implies an enormous reduction of the computational efforts and a significant improvement of the accuracy of the results when compared to the direct numerical simulation (DNS) approaches which require flow domains that are long enough to achieve steady-state conditions, and hence often cover tens to hundreds of unit cells. The reliability of the proposed method is verified by comparing its predictions with DNS results, in one, two and three dimensions, in both transient and asymptotic conditions. The influence of top and bottom no-slip walls on the separation performance of chromatographic columns with micromachined porous and nonporous pillars is discussed in detail.

Tuning capillary flow in porous media with hierarchical structures

Immiscible fluid–fluid displacement in porous media is of great importance in many engineering applications, such as enhanced oil recovery, agricultural irrigation, and geologic CO2 storage. Fingering phenomena, induced by the interface instability, are commonly encountered during displacement processes and somehow detrimental since such hydrodynamic instabilities can significantly reduce displacement efficiency. In this study, we report a possible adjustment in pore geometry, which aims to suppress the capillary fingering in porous media with hierarchical structures. Through pore-scale simulations and theoretical analysis, we demonstrate and quantify the combined effects of wettability and hierarchical geometry on displacement patterns, showing a transition from fingering to compact mode. Our results suggest that with a higher porosity of the second-order porous structure, the displacement can stay compact across a wider range of wettability conditions. Combined with our previous work on viscous fingering in such media, we can provide a complete insight into the fluid-fluid displacement control in hierarchical porous media, across a wide range of flow conditions from capillary- to viscous-dominated modes. The conclusions of this work can benefit the design of microfluidic devices and tailoring porous media for better fluid displacement efficiency at the field scale.

Hyperpolarised xenon MRI and time-resolved X-ray computed tomography studies of structure-transport relationships in hierarchical porous media

Catalysed diesel particulate filter (DPF) monoliths are hierarchical porous solids, as demonstrated by mercury porosimetry. Establishing structure-transport relationships, including assessing the general accessibility of the catalyst, is challenging, and, thus, a comprehensive approach is necessary. Contributions, from each porosity level, to transport have been established using hyperpolarised (hp) xenon-129 magnetic resonance imaging (MRI) of gas dispersion within DPF monoliths at variable water saturation, since X-ray Computerised-Tomography, and 1H and 2H NMR methods, have shown that porosity levels dry out progressively. At high saturation, hp 129Xe MRI showed gas transport between the channels of the monolith is predominantly taking place at channel wall intersections with high macroporosity. The walls themselves make a relatively small contribution to through transport due to the distribution of the micro-/meso-porous washcoat layer away from intersections. Only at low saturation, when the smallest pores are opened, do hp 129Xe MR images became strongly affected by relaxation. This observation indicates accessibility of paramagnetic (catalytic) centres for gases arises only once the smallest pores are open.

Efficient Prediction of Multidomain Flow and Transport in Hierarchically Structured Porous Media

AbstractStructural hierarchy is a fundamental characteristic of natural porous media. Yet it provokes one of the grand challenges for the modeling of fluid flow and transport since pore‐scale structures and continuum‐scale domains often coincide independent of the observation scale. Common approaches to represent structural hierarchy build, for example, on a multidomain continuum for transport or on the coupling of the Stokes equations with Darcy's law for fluid flow. These approaches, however, are computationally expensive or introduce empirical parameters that are difficult to derive with independent observations. We present an efficient model for fluid flow based on Darcy's law and the law of Hagen‐Poiseuille that is parameterized based on the explicit pore space morphology obtained, for example, by X‐ray μ‐CT and inherently permits the coupling of pore‐scale and continuum‐scale domain. We used the resulting flow field to predict the transport of solutes via particle tracking across the different domains. Compared to experimental breakthrough data from laboratory‐scale columns with hierarchically structured porosity built from solid glass beads and microporous glass pellets, an excellent agreement was achieved without any calibration. Furthermore, we present different test scenarios to compare the flow fields resulting from the Stokes‐Brinkman equations and our approach to comprehensively illustrate its advantages and limitations. In this way, we could show a striking efficiency and accuracy of our approach that qualifies as general alternative for the modeling of fluid flow and transport in hierarchical porous media, for example, fractured rock or karstic aquifers.

Pore-scale study of effects of macroscopic pores and their distributions on reactive transport in hierarchical porous media

For applications of reactive transport in porous media, optimal porous structures should possess both high surface area for reactive sites loading and low mass transport resistance. Hierarchical porous media with a combination of pores at different scales are designed for this purpose. Using the lattice Boltzmann method, pore-scale numerical studies are conducted to investigate diffusion-reaction processes in 2D hierarchical porous media generated by self-developed reconstruction scheme. Complex interactions between porous structures and reactive transport are revealed under different conditions. Simulation results show that adding macropores can greatly enhance the mass transport, but at the same time reduce the reactive surface, leading to complex change trend of the total reaction rate. Effects of gradient distribution of macropores within the porous medium are also investigated. It is found that a front-loose, back-tight (FLBT) hierarchical structure is desirable for enhancing mass transport, increasing total reaction rate, and improving catalyst utilization. On the whole, from the viewpoint of reducing cost and improving material performance, hierarchical porous structures, especially gradient structures with the size of macropores gradually decreasing along the transport direction, are desirable for catalyst application.

Large-scale model of flow in heterogeneous and hierarchical porous media

Heterogeneous porous structures are very often encountered in natural environments, bioremediation processes among many others. Reliable models for momentum transport are crucial whenever mass transport or convective heat occurs in these systems. In this work, we derive a large-scale average model for incompressible single-phase flow in heterogeneous and hierarchical soil porous media composed of two distinct porous regions embedding a solid impermeable structure. The model, based on the local mechanical equilibrium assumption between the porous regions, results in a unique momentum transport equation where the global effective permeability naturally depends on the permeabilities at the intermediate mesoscopic scales and therefore includes the complex hierarchical structure of the soil. The associated closure problem is numerically solved for various configurations and properties of the heterogeneous medium. The results clearly show that the effective permeability increases with the volume fraction of the most permeable porous region. It is also shown that the effective permeability is sensitive to the dimensionality spatial arrangement of the porous regions and in particular depends on the contact between the impermeable solid and the two porous regions.

Multicomponent competitive monovalent cation exchange in hierarchical porous media with multimodal reactive mineral facies

This paper presents a stochastic model for multicomponent competitive monovalent cation exchange in hierarchical porous media. Reactive transport in porous media is highly sensitive to heterogeneities in physical and chemical properties, such as hydraulic conductivity (K), and cation exchange capacity (CEC). We use a conceptual model for multimodal reactive mineral facies and develop a Eulerian-based stochastic theory to analyze the transport of multiple cations in heterogeneous media with a hierarchical organization of reactive minerals. Numerical examples investigate the retardation factors and dispersivities in a chemical system made of three monovalent cations (Na+, K+, and Cs+). The results demonstrate how heterogeneity influences the transport of competitive monovalent cations, and highlight the importance of correlations between K and CEC. Further sensitivity analyses are presented investigating how the dispersion and retardation of each cation are affected by the means, variances, and integral scales of K and CEC. The volume fraction of organic matter is shown to be another important parameter. The Eulerian stochastic framework presented in this work clarifies the importance of each system parameters on the migration of cation plumes in formations with hierarchical organization of facies types. Our stochastic approach could be used as an alternative to numerical simulations for 3D reactive transport in hierarchical porous media, which become prohibitively expensive for the multicomponent applications considered in this work.

ChemInform Abstract: Multiscale Adsorption and Transport in Hierarchical Porous Materials

AbstractReview: 172 refs.

Averaged model for momentum and dispersion in hierarchical porous media.

Hierarchical porous media are multiscale systems, where different characteristic pore sizes and structures are encountered at each scale. Focusing the analysis to three pore scales, an upscaling procedure based on the volume-averaging method is applied twice, in order to obtain a macroscopic model for momentum and diffusion-dispersion. The effective transport properties at the macroscopic scale (permeability and dispersion tensors) are found to be explicitly dependent on the mesoscopic ones. Closure problems associated to these averaged properties are numerically solved at the different scales for two types of bidisperse porous media. Results show a strong influence of the lower-scale porous structures and flow intensity on the macroscopic effective transport properties.

A comprehensive multiscale moisture transport analysis: From porous reference silicates to cement-based materials

Natural and manufactured disordered systems are ubiquitous and often involve hierarchical structures. This structural organization optimizes defined physical properties at several scales from molecular to representative volumes where the usual homogenization approach becomes efficient. For studying a particular physical property on these systems it is thus required to use a general method of analysis based on the joint application of complementary techniques covering the whole set of time-and length-scales. Here we review a comprehensive multiscale method presented for analyzing the three-dimensional moisture transport in hierarchical porous media such as synthesized reference silicates and cement-based materials. Several techniques (NMR spectroscopy, relaxometry, diffusometry, X-ray micro-tomography, conductivity…) have been used to evidence the interplay between the different scales involved in this transport process. This method allows answering the general opened questions concerning the scale dependence of such a moisture transport in cement-based materials. We outline the main results of the multiscale techniques applied on reference porous silicates allowing separating the impact of geometry, hydric state and wettability on the moisture transport. Based on this approach, we prove that this transport at micro- and meso-scale is determinant to modify the moisture at macro-scale during setting or for hardened cement-based materials.

A Note on Upscaling Retardation Factor in Hierarchical Porous Media with Multimodal Reactive Mineral Facies

We present a model for upscaling the time-dependent effective retardation factor, \({\widetilde{R}}(t)\), in hierarchical porous media with multimodal reactive mineral facies. The model extends the approach by Deng et al. (Chemosphere 91(3):248–257, 2013) in which they expanded a Lagrangian-based stochastic theory presented by Rajaram (Adv Water Resour 20(4):217–230, 1997) in order to describe the scaling effect of \(\widetilde{R}(t)\). They used a first-order linear approximation in deriving their model to make the derivation tractable. Importantly, the linear approximation is known to be valid only to variances of 0.2. In this note, we show that the model can be derived with a higher-order approximation, which allows for representing variances from 0.2 to 1.0. We present the derivation and use the resulting model to recalculate \(\widetilde{R}(t)\) for the scenario examined by Deng et al. (2013).

Upscaling retardation factor in hierarchical porous media with multimodal reactive mineral facies

Aquifer heterogeneity controls spatial and temporal variability of reactive transport parameters and has significant impacts on subsurface modeling of flow, transport, and remediation. Upscaling (or homogenization) is a process to replace a heterogeneous domain with a homogeneous one such that both reproduce the same response. To make reliable and accurate predictions of reactive transport for contaminant in chemically and physically heterogeneous porous media, subsurface reactive transport modeling needs upscaled parameters such as effective retardation factor to perform field-scale simulations. This paper develops a conceptual model of multimodal reactive mineral facies for upscaling reactive transport parameters of hierarchical heterogeneous porous media. Based on the conceptual model, covariance of hydraulic conductivity, sorption coefficient, flow velocity, retardation factor, and cross-covariance between flow velocity and retardation factor are derived from geostatistical characterizations of a three-dimensional unbounded aquifer system. Subsequently, using a Lagrangian approach the scale-dependent analytical expressions are derived to describe the scaling effect of effective retardation factors in temporal and spatial domains. When time and space scales become sufficiently large, the effective retardation factors approximate their composite arithmetic mean. Correlation between the hydraulic conductivity and the sorption coefficient can significantly affect the values of the effective retardation factor in temporal and spatial domains. When the temporal and spatial scales are relatively small, scaling effect of the effective retardation factors is relatively large. This study provides a practical methodology to develop effective transport parameters for field-scale modeling at which remediation and risk assessment is actually conducted. It does not only bridge the gap between bench-scale measurements to field-scale modeling, but also provide new insights into the influence of hierarchical mineral distribution on effective retardation factor.

Commentary on Russo [1991], Serrano [1990, 1998], and other applications of the water‐content‐based form of Richards' Equation to heterogeneous soils

Commentary on <i>Russo</i> [1991], <i>Serrano</i> [1990, 1998], and other applications of the water‐content‐based form of Richards' Equation to heterogeneous soils

Multiple logarithmic oscillations for statistical fractals on ultrametric spaces with application to recycle flows in hierarchical porous media

The interference of logarithmic oscillations characteristic to statistical fractal and ultrametric structures is analyzed. We introduce an ultrametric model consisting in a hierarchy of branches for which the probability distribution ξ̃(n) of the total number, n, of branches has a long tail of the inverse power law type ξ̃(n) ∼ A(ln n)n -(1 + H) as n → ∞ where H is a fractal exponent characteristic for the ultrametric structure and A(ln n) is a periodic function of ln n. A comparison is made with a statistical fractal derived by means of the Shlesinger-Hughes stochastic renormalization approach. We consider a positive random variable, X, selected from a narrow unimodal probability density with finite moments. A process of stochastic amplification of the random variable is introduced which leads to a probability density of the amplified variable with a long tail: P̃(X) dX ∼ dXX-(1 + ℋ)B[ln (X/Xm)] as X → ∞ where ℋ is another fractal exponent, Xm is a cutoff value of the random variable X and B[ln (X/Xm)] is a periodic function of ln (X/Xm). Finally, the interaction between these two types of logarithmic oscillations is analyzed by assuming that the stochastic amplification of the random variable X takes place on an ultrametric structure. The final probability density of X, P*(X) dX, displays a phenomenon of amplitude modulation P*(X) dX ∼ d[ln (X/Xm)] [ln (X/Xm)](-1 + H)Ξ as X → ∞ where Ξ = Ξ(1) + Ξ(2) is made up of two additive contributions: a periodic function Ξ(1)[ln (X/Xm)] of ln (X/Xm) and a superposition Ξ(2) {ln (X/Xm), ln [ln (X/Xm)]} of periodic functions of ln (X/Xm) modulated by much slower periodic functions in ln [ln (X/Xm)]. The model is applied for the analysis of recycle flows in porous media. We assume that a hierarchical structure of pores exists which corresponds to an ultrametric space and that the recycle flow leads to a stochastic amplification of the residence time of fluid elements in the system. We show that the probability density of the residence time has an asymptotic behavior similar to that of P*(X) dX and investigate the possibilities of measurement of multiple logarithmic oscillations by means of tracer experiments. A physical interpretation of multiple logarithmic oscillations is given in the case of flow systems: they are generated by two delayed feedback processes occurring in two different logarithmic time scales, ln t and ln ln t. The fast delayed feedback process in ln t is given by the recycle flows corresponding to a given level of the porous structure whereas the slow delayed feedback process in ln ln t is due to the exchange of fluid among the different levels of the hierarchical structure of pores.

Dynamics of Fluids in Hierarchical Porous Media.

Dynamics of Fluids in Hierarchical Porous Media.