A homogenised model for flow, transport and sorption in a heterogeneous porous medium

A major challenge in flow through porous media is to better understand the link between microstructure and macroscale flow and transport. For idealised microstructures, the mathematical framework of homogenisation theory can be used for this purpose. Here, we consider a two-dimensional microstructure comprising an array of obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium. We use homogenisation via the method of multiple scales to systematically upscale a novel problem involving cells of varying area to obtain effective continuum equations for macroscale flow and transport. The equations are characterised by the local porosity, a local anisotropic flow permeability, an effective local anisotropic solute diffusivity and an effective local adsorption rate. These macroscale properties depend non-trivially on the two degrees of microstructural geometric freedom in our problem: obstacle size and obstacle spacing. We exploit this dependence to construct and compare scenarios where the same porosity profile results from different combinations of obstacle size and spacing. We focus on a simple example geometry comprising circular obstacles on a rectangular lattice, for which we numerically determine the macroscale permeability and effective diffusivity. We investigate scenarios where the porosity is spatially uniform but the permeability and diffusivity are not. Our results may be useful in the design of filters or for studying the impact of deformation on transport in soft porous media.

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Reduced Model for Properties of Multiscale Porous Media with Changing Geometry

Computation ◽

10.3390/computation9030028 ◽

2021 ◽

Vol 9 (3) ◽

pp. 28 ◽

Cited By ~ 1

Author(s):

Malgorzata Peszynska ◽

Joseph Umhoefer ◽

Choah Shin

Keyword(s):

Porous Media ◽

Reactive Transport ◽

Multiple Scales ◽

Pore Space ◽

Type Model ◽

Crystal Formation ◽

Pore Filling ◽

Reduced Models ◽

Flow And Transport ◽

The Impact

In this paper, we consider an important problem for modeling complex coupled phenomena in porous media at multiple scales. In particular, we consider flow and transport in the void space between the pores when the pore space is altered by new solid obstructions formed by microbial growth or reactive transport, and we are mostly interested in pore-coating and pore-filling type obstructions, observed in applications to biofilm in porous media and hydrate crystal formation, respectively. We consider the impact of these obstructions on the macroscopic properties of the porous medium, such as porosity, permeability and tortuosity, for which we build an experimental probability distribution with reduced models, which involves three steps: (1) generation of independent realizations of obstructions, followed by, (2) flow and transport simulations at pore-scale, and (3) upscaling. For the first step, we consider three approaches: (1A) direct numerical simulations (DNS) of the PDE model of the actual physical process called BN which forms the obstructions, and two non-DNS methods, which we call (1B) CLPS and (1C) LP. LP is a lattice Ising-type model, and CLPS is a constrained version of an Allen–Cahn model for phase separation with a localization term. Both LP and CLPS are model approximations of BN, and they seek local minima of some nonconvex energy functional, which provide plausible realizations of the obstructed geometry and are tuned heuristically to deliver either pore-coating or pore-filling obstructions. Our methods work with rock-void geometries obtained by imaging, but bypass the need for imaging in real-time, are fairly inexpensive, and can be tailored to other applications. The reduced models LP and CLPS are less computationally expensive than DNS, and can be tuned to the desired fidelity of the probability distributions of upscaled quantities.

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Diffusion limited mixing in heterogeneous porous media

10.5194/egusphere-egu21-15721 ◽

2021 ◽

Author(s):

Mayumi Hamada ◽

Pietro de Anna

Keyword(s):

Porous Medium ◽

Porous Media ◽

Flow Field ◽

Fast Diffusion ◽

Pore Scale ◽

Mixing Processes ◽

Local Flow ◽

Gaussian Shape ◽

Diffusion Limited ◽

The Impact

A pore-scale description of the transport and mixing processes is particularly relevant when looking at biological and chemical reactions. For instance, a microbial population growth is controlled by local concentrations of nutrients and oxygen, and chemical reaction are driven by molecular-scale concentration gradients. The heterogeneous flow field typically found in porous media results from the contrast of velocities that deforms and elongates the mixing fronts between solutes that often evolves through a lamella-like topology. For continuous Darcy type flow field a novel framework that describes the statistical distribution of concentration being transported was recently developed (Le Borgne et al., JFM 2015). In this model, concentrations in each lamella are distributed as a Gaussian-like profile which experiences diffusion in the transverse direction while the lamella is elongated by advection along the local flow direction. The evolving concentration field is described as the superposition of each lamella. We hypothesize that this novel view, while perfectly predicting the distribution of concentration for Darcy scale mixing processes, will breakdown when the processes description is at the pore scale. Indeed the presence of solid and impermeable boundaries prevents lamella concentration to diffuse freely according to the a Gaussian shape, and therefore changes the mixing front profile, the lamella superposition and elongation rules. Previous work (Hamada et al, PRF, 2020) demonstrated that the presence of solid boundaries leads to an enhanced diffusion and thus fast homogenization of concentrations. In a purely diffusive process the local mixing time is reduced by a factor of ten with respect to the continuous case and concentration gradient are dissipated exponentially fast while a power law decrease is observed in continuous medium. To investigate the impact of these mechanisms on mixing we developed an experimental set-up to visualize and quantify the displacement of a conservative tracer in a synthetic porous medium. The designed apparatus allows to obtain high resolution concentration measurements at the pore scale. We show that the resulting mixing measures, computed in terms of concentration probability density function and dilution index values, diverge qualitatively and quantitatively from what happens in a continuous domain. These observations suggest that description of pore-scale diffusion-limited mixing requires model that takes into account the confined nature of porous medium, otherwise we will tend to overestimate concentration value and neglect the fast diffusion dynamic taking place at microscopic level.

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Modelling and simulation of waves in three-layer porous media

Nonlinear Processes in Geophysics ◽

10.5194/npg-20-1023-2013 ◽

2013 ◽

Vol 20 (6) ◽

pp. 1023-1030 ◽

Cited By ~ 3

Author(s):

S. R. Pudjaprasetya

Keyword(s):

Porous Medium ◽

Porous Media ◽

Gravity Waves ◽

Wave Reflection ◽

Single Layer ◽

Reflection And Transmission ◽

Weakly Nonlinear ◽

Model Equations ◽

The Impact ◽

Wave Transmission Coefficient

Abstract. The propagation of gravity waves in an emerged three-layer porous medium is considered in this paper. Based on the assumption that the flow can be described by Darcy's Law, an asymptotic theory is developed for small-amplitude long waves. This leads to a weakly nonlinear Boussinesq-type diffusion equation for the wave height, with coefficients dependent on the conductivities and depths of each layer. In the limit of equal conductivities of all layers, the equation reduces to the single-layer result recorded in the literature. The model equations are numerically integrated in the case of an incident monochromatic wave hitting the layers. The results exhibit dissipation and also a downstream net height rise at infinity. Wave transmission coefficient in three-layer porous media with conductivity of mangrove is discussed. Numerically, propagation of an initial solitary wave through a porous medium shows the emergence of wave reflection and transmission that both evolve as permanent waves. Additionally we examine the impact of a solitary gravity wave on a porous medium breakwater.

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Adhesion of Colloids and Bacteria to Porous Media: A Critical Review

Reviews of Adhesion and Adhesives ◽

10.7569/raa.2019.097314 ◽

2019 ◽

Vol 7 (4) ◽

pp. 417-460 ◽

Cited By ~ 1

Author(s):

Runwei Li ◽

Changfu Wei ◽

Hefa Cheng ◽

Gang Chen

Keyword(s):

Porous Medium ◽

Porous Media ◽

Bacterial Adhesion ◽

Fate And Transport ◽

Theoretical Models ◽

Water Interface ◽

Subsurface Soil ◽

Air Water Interface ◽

Surface Physicochemical Properties ◽

The Impact

Adhesion of colloids and bacteria to various surfaces is important for a variety of environmental phenomena including microbial biofouling and contamination prevention. Under saturated conditions, both colloids and bacteria have the opportunity to attach to porous medium surfaces. Under water unsaturated conditions or in the presence of the air-water interface, besides the porous medium surfaces, colloids and bacteria can also attach to the air-water interface, including the air-water-solid threephase interface. The magnitudes of adhesion of colloids and bacteria are correlated to the interactions of the colloids and bacteria with the surfaces, which are a function of their surface physicochemical properties. In this review, adhesion theories are revisited and adhesion of colloids and bacteria to porous media and the air-water interface is discussed. The interaction forces are quantified using various theoretical models including the DLVO models and used to interpret related adhesion. The impact of surfactants on colloid and bacterial adhesion is also discussed. The review also includes the implementation of the adhesion theory in interpreting colloid and bacterial fate and transport in the subsurface soil.

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Understanding hydraulic fracturing: a multi-scale problem

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2015.0426 ◽

2016 ◽

Vol 374 (2078) ◽

pp. 20150426 ◽

Cited By ~ 53

Author(s):

J. D. Hyman ◽

J. Jiménez-Martínez ◽

H. S. Viswanathan ◽

J. W. Carey ◽

M. L. Porter ◽

...

Keyword(s):

Hydraulic Fracturing ◽

Multiple Scales ◽

Element Model ◽

Fracture Network ◽

Working Fluid ◽

Pore Scale ◽

Network Modelling ◽

Flow And Transport ◽

Fractured Well ◽

The Impact

Despite the impact that hydraulic fracturing has had on the energy sector, the physical mechanisms that control its efficiency and environmental impacts remain poorly understood in part because the length scales involved range from nanometres to kilometres. We characterize flow and transport in shale formations across and between these scales using integrated computational, theoretical and experimental efforts/methods. At the field scale, we use discrete fracture network modelling to simulate production of a hydraulically fractured well from a fracture network that is based on the site characterization of a shale gas reservoir. At the core scale, we use triaxial fracture experiments and a finite-discrete element model to study dynamic fracture/crack propagation in low permeability shale. We use lattice Boltzmann pore-scale simulations and microfluidic experiments in both synthetic and shale rock micromodels to study pore-scale flow and transport phenomena, including multi-phase flow and fluids mixing. A mechanistic description and integration of these multiple scales is required for accurate predictions of production and the eventual optimization of hydrocarbon extraction from unconventional reservoirs. Finally, we discuss the potential of CO 2 as an alternative working fluid, both in fracturing and re-stimulating activities, beyond its environmental advantages. This article is part of the themed issue ‘Energy and the subsurface’.

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Influence of pore geometry on motility and trapping of metal reducing bacteria

10.5194/egusphere-egu2020-4226 ◽

2020 ◽

Author(s):

Lazaro J. Perez ◽

Nicole L. Sund ◽

Rishi Parashar ◽

Andrew E. Plymale ◽

Dehong Hu ◽

...

Keyword(s):

Porous Medium ◽

Porous Media ◽

Large Scale ◽

Ad Hoc ◽

Pore Space ◽

Practical Applications ◽

Bacterial Migration ◽

Reducing Bacteria ◽

The Impact ◽

Metal Reducing Bacteria

Diverse processes such as bioremediation, biofertilization, and microbial drug delivery rely on bacterial migration in porous media. However, how pore-scale confinement alters bacterial motility is unknown due to the inherent heterogeneity in porous media. As a result, models of migration are limited and often employ ad hoc assumptions. We aim to determine the impact of pore confinement in the spreading dynamics of two populations of motile metal reducing bacteria by directly visualizing individual Acidovorax and Pelosinus in an unconfined liquid medium and in a microfluidic chip containing regular placed pillars. We observe that the length of runs of the two species differs from the unconfined and confined medium. Results show that bacteria in the confined medium display a systematic shorter jumps due to grain obstacles when compared to the open porous medium. Close inspection of the trajectories reveals that cells are intermittently and transiently trapped, which produces superdiffusive motion at early and subdiffusion behavior at late times, as they navigate through the confined pore space. While in the open medium, we observe a linearly increasing variance with respect to time for Acidovorax, and for Pelosinus the variance increases at a much faster rate showing super diffusive behavior at early times. At late times, the rate of growth in spreading increases for Acidovorax while it reduces for Pelosinus. We finally discuss that the paradigm of run-and-tumble motility is dramatically altered in the confined porous medium and its practical applications of these effects on large-scale transport.

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Closed-Form Solution of Radial Transport of Tracers in Porous Media Influenced by Linear Drift

Energies ◽

10.3390/en12010029 ◽

2018 ◽

Vol 12 (1) ◽

pp. 29 ◽

Cited By ~ 2

Author(s):

Lateef Akanji ◽

Gabriel Falade

Keyword(s):

Porous Medium ◽

Porous Media ◽

Closed Form ◽

Oil Recovery ◽

Closed Form Solution ◽

Radial Transport ◽

Tracer Concentration ◽

Tracer Injection ◽

Linear Drift ◽

The Impact

A new closed-form analytical solution to the radial transport of tracers in porous media under the influence of linear drift is presented. Specifically, the transport of tracers under convection–diffusion-dominated flow is considered. First, the radial transport equation was cast in the form of the Whittaker equation by defining a set of transformation relations. Then, linear drift was incorporated by considering a coordinate-independent scalar velocity field within the porous medium. A special case of low-intensity tracer injection where molecular diffusion controls tracer propagation but convection with linear velocity drift plays a significant role was presented and solved in Laplace space. Furthermore, a weak-form numerical solution of the nonlinear problem was obtained and used to analyse tracer concentration behaviour in a porous medium, where drift effects predominate and influence the flow pattern. Application in enhanced oil recovery (EOR) processes where linear drift may interfere with the flow path was also evaluated within the solution to obtain concentration profiles for different injection models. The results of the analyses indicated that the effect of linear drift on the tracer concentration profile is dependent on system heterogeneity and progressively becomes more pronounced at later times. This new solution demonstrates the necessity to consider the impact of drift on the transport of tracers, as arrival times may be significantly influenced by drift intensity.

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Impact of Synthetic Porous Medium Geometric Properties on Solute Transport Using Direct 3D Pore-Scale Simulations

Geofluids ◽

10.1155/2019/6810467 ◽

2019 ◽

Vol 2019 ◽

pp. 1-13

Author(s):

Paolo Roberto Di Palma ◽

Nicolas Guyennon ◽

Andrea Parmigiani ◽

Christian Huber ◽

Falk Heβe ◽

...

Keyword(s):

Porous Medium ◽

Porous Media ◽

Effective Diffusion ◽

Volume Averaging ◽

Transport Processes ◽

Hydrodynamic Dispersion ◽

Pore Scale ◽

Geometric Properties ◽

Advection Diffusion ◽

The Impact

Transport processes in porous media have been traditionally studied through the parameterization of macroscale properties, by means of volume-averaging or upscaling methods over a representative elementary volume. The possibility of upscaling results from pore-scale simulations, to obtain volume-averaging properties useful for practical purpose, can enhance the understanding of transport effects that manifest at larger scales. Several studies have been carried out to investigate the impact of the geometric properties of porous media on transport processes for solute species. However, the range of pore-scale geometric properties, which can be investigated, is usually limited to the number of samples acquired from microcomputed tomography images of real porous media. The present study takes advantage of synthetic porous medium generation to propose a systematic analysis of the relationships between geometric features of the porous media and transport processes through direct simulations of fluid flow and advection-diffusion of a non-reactive solute. Numerical simulations are performed with the lattice Boltzmann method on synthetic media generated with a geostatistically based approach. Our findings suggest that the advective transport is primarily affected by the specific surface area and the mean curvature of the porous medium, while the effective diffusion coefficient scales as the inverse of the tortuosity squared. Finally, the possibility of estimating the hydrodynamic dispersion coefficient knowing only the geometric properties of porous media and the applied pressure gradient has been tested, within the range of tested porous media, against advection-diffusion simulations at low Reynolds (<10-1) and Peclet numbers ranging from 101 to 10-2.

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Conservative solute transport with periodic velocity and sinusoidal source concentration in semi-infinite porous media: An analytical solution

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v6i1.1818 ◽

2014 ◽

Vol 6 (1) ◽

pp. 1024-1031

Author(s):

R R Yadav ◽

Gulrana Gulrana ◽

Dilip Kumar Jaiswal

Keyword(s):

Porous Medium ◽

Porous Media ◽

Laplace Transformation ◽

Seepage Velocity ◽

Transformation Technique ◽

Numerical Examples ◽

One Dimensional ◽

Porous Domain ◽

Conservative Solute Transport ◽

Source Concentration

The present paper has been focused mainly towards understanding of the various parameters affecting the transport of conservative solutes in horizontally semi-infinite porous media. A model is presented for simulating one-dimensional transport of solute considering the porous medium to be homogeneous, isotropic and adsorbing nature under the influence of periodic seepage velocity. Initially the porous domain is not solute free. The solute is initially introduced from a sinusoidal point source. The transport equation is solved analytically by using Laplace Transformation Technique. Alternate as an illustration; solutions for the present problem are illustrated by numerical examples and graphs.

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