We extend the two-scale expansion approach of periodic homogenization to include time scales and thus can tackle the full instationary Navier-Stokes-Cahn-Hilliard model at the pore scale as microscale. Time scale separation allows us to keep microscale dynamics, responsible e.g. for hysteresis, and arrive at a numerically tractable micro-macro model including coupled generalized Darcy's laws.
A stochastic approach to the filling dynamics of an open topology porous structure permeated with a perfectly wetting fluid is presented. From the discrete structure of the disordered voids network with only nearest neighbors links, we derive the "microscopic" (at the pores scale) dynamical equations governing the filling dynamics of the coupled pores and the fluid pressure dynamics. The model yields two fundamental consequences. The first consequence regards the emergence of Darcy's law and the dependence of the predicted permeability with the voids network topology. The second one is the pre…
Motivated by casting of fresh concrete in reinforced concrete structures, we introduce a numerical model of a steady-state non-Newtonian fluid flow through a porous domain. Our approach combines homogenization techniques to represent the reinforced domain by the Darcy law with an interface coupling of the Stokes and Darcy flows through the Beavers-Joseph-Saffman conditions. The ensuing two-scale problem is solved by the Finite Element Method with consistent linearization and the results obtained from the homogenization approach are verified against fully resolved direct numerical simulations.
We present gravimetrical and optical imaging experiments on the capillarity-driven imbibition of silicone oils in monolithic silica glasses traversed by 3D networks of pores (mesoporous Vycor glass with 6.5 nm or 10 nm pore diameters). As evidenced by a robust square-root-of-time Lucas-Washburn (L-W) filling kinetics, the capillary rise is governed by a balance of capillarity and viscous drag forces in the absence of inertia and gravitational effects over the entire experimental times studied, ranging from a few seconds up to 10 days. A video on the infiltration process corroborates a collecti…
In this paper we study a Darcy-scale mathematical model for biofilm formation in porous media. The pores in the core are divided into three phases: water, oil, and biofilm. The water and oil flow are modeled by an extended version of Darcy's law and the substrate is transported by diffusion and convection in the water phase. Initially there is biofilm on the pore walls. The biofilm consumes substrate for production of biomass and modifies the pore space which changes the rock permeability. The model includes detachment of biomass due to water flux and death of bacteria, and is implemented in M…
A homogenization approach is proposed for the treatment of porous wall boundary conditions in the computation of compressible viscous flows. Like any other homogenization approach, it eliminates the need for pore-resolved fluid meshes and therefore enables practical flow simulations in computational fluid domains with porous wall boundaries. Unlike alternative approaches however, it does not require prescribing a mass flow rate and does not introduce in the computational model a heuristic discharge coefficient. Instead, it models the inviscid flux through a porous wall surrounded by the flow a…
We study the dynamics of flow-networks in porous media using a pore-network model. First, we consider a class of erosion dynamics assuming a constitutive law depending on flow rate, local velocities, or shear stress at the walls. We show that depending on the erosion law, the flow may become uniform and homogenized or become unstable and develop channels. By defining an order parameter capturing these different behaviors we show that a phase transition occurs depending on the erosion dynamics. Using a simple model, we identify quantitative criteria to distinguish these regimes and correctly pr…
This work is concerned with the intricate interplay between node or pore pressures and connection or throat conductivities in flow or pore networks. A setting similar to pore networks is given by fracture networks. Recently, a non-local generalization of Darcy's law for flow and transport in porous media was presented in the context of unbounded or periodic pore networks. In this work, we first outline a robust method for the extraction of the hydraulic conductivity distribution, which is at the heart of the non-local Darcy formulation. Second, a theory for mean pressure and flow in bounded ne…
A new approach to the modelling of wetting fronts in porous media on the Darcy scale is developed, based on considering the types (modes) of motion the menisci go through on the pore scale. This approach is illustrated using a simple model case of imbibition of a viscous incompressible liquid into an isotropic porous matrix with two modes of motion for the menisci, the wetting mode and the threshold mode. The latter makes it necessary to introduce an essentially new technique of conjugate problems that allows one to link threshold phenomena on the pore scale with the motion on the Darcy scale.…
In this article we investigate the permeability of a porous medium as given in Darcy's law. The permeability is described by an effective hydraulic pore radius in the porous medium, the fluctuation in local hydraulic pore radii, the length of streamlines, and the fractional volume conducting flow. The effective hydraulic pore radius is related to a characteristic hydraulic length, the fluctuation in local hydraulic radii is related to a constriction factor, the length of streamlines is characterized by a tortuosity, and the fractional volume conducting flow from inlet to outlet is described by…
An open-sourced multiphase Darcy-Brinkman approach is proposed to simulate two-phase flow in hybrid systems containing both solid-free regions and porous matrices. This micro-continuum model is rooted in elementary physics and volume averaging principles, where a unique set of partial differential equations is used to represent flow in both regions and scales. The crux of the proposed model is that it tends asymptotically towards the Navier-Stokes volume-of-fluid approach in solid-free regions and towards the multiphase Darcy equations in porous regions. Unlike existing multiscale multiphase s…
The modeling of flow and heat transfer in porous media systems have always been a challenge and, the extended Darcy transport models for flow and equilibrium and non-equilibrium energy models for heat transfer are being used for macro-level analysis, however, the limitations of these models are subjected to porous geometry. The forced convective flow of an incompressible viscous fluid through a channel filled with four different types of porous geometries constructed using the Triply-Periodic-Minimal-Surface (or TPMS) model, are presented in this study. Four TPMS lattice shapes namely; Diamond…
A upscaled lattice Boltzmann method (LBM) for flow simulations in heterogeneous porous media, at both pore and Darcy scales, is proposed in this paper. In the micro-scale simulations, we model flows using LBM with the modified Guo et al. algorithm where we replace the force model with a simple Shan-Chen force model. The proposed upscaled LBM uses coarser grids to represent the effects of the fine-grid (pore-scale) simulations. For the upscaled LBM, effective properties and reduced-order models are proposed as we coarsen the grid. The effective properties are computed using solutions of local p…
Darcy's law and the Brinkman equation are two main models used for creeping fluid flows inside moving permeable particles. For these two models, the time derivative and the nonlinear convective terms of fluid velocity are neglected in the momentum equation. In this paper, a new momentum equation including these two terms are rigorously derived from the pore-scale microscopic equations by the volume-averaging method, which can reduces to Darcy's law and the Brinkman equation under creeping flow conditions. Using the lattice Boltzmann equation method, the macroscopic equations are solved for the…
Natural convection in porous media is a fundamental process for the long-term storage of CO2 in deep saline aquifers. Typically, details of mass transfer in porous media are inferred from the numerical solution of the volume-averaged Darcy-Oberbeck-Boussinesq (DOB) equations, even though these equations do not account for the microscopic properties of a porous medium. According to the DOB equations, natural convection in a porous medium is uniquely determined by the Rayleigh number. However, in contrast with experiments, DOB simulations yield a linear scaling of the Sherwood number with the Ra…
Pushing two fluids with different density one against the other causes the development of the Rayleigh-Taylor instability at their interface, which further evolves in a complex mixing layer. In porous media, this process is influenced by the viscous resistance experienced while flowing through the pores, which is described by the Darcy's law. Here, we investigate the mixing properties of the Darcy-Rayleigh-Taylor system in the limit of large Péclet number by means of direct numerical simulations in three and two dimensions. In the mixing zone, the balance between gravity and viscous forces res…
Electrolyte filling is a time-critical step during battery manufacturing that also affects the battery performance. The underlying physical phenomena during filling mainly occur on the pore scale and are hard to study experimentally. In this paper, a computational approach, i.e.\ the lattice Boltzmann method, is used to study the filling process and corresponding pore-scale phenomena in 3D lithium-ion battery cathodes. The electrolyte flow through the nanoporous binder is simulated using a homogenization approach. Besides the process time, the influence of structural and physico-chemical prope…
Wettability is a pore-scale property that impacts the relative movement and distribution of fluids in a porous medium. There are reservoir fluids that provoke the surface within pores to undergo a wettability change. This wettability change, in turn, alters the dynamics of relative permeabilities at the Darcy scale. Thus, modeling the impact of wettability change in relative permeabilities is essential to understand fluids interaction in porous media. In this study, we include time-dependent wettability change into the relative permeability--saturation relation by modifying the existing relati…
The lattice Boltzmann method can be used to simulate flow through porous media with full geometrical resolution. With such a direct numerical simulation, it becomes possible to study fundamental effects which are difficult to assess either by developing macroscopic mathematical models or experiments. We first evaluate the lattice Boltzmann method with various boundary handling of the solid-wall and various collision operators to assess their suitability for large scale direct numerical simulation of porous media flow. A periodic pressure drop boundary condition is used to mimic the pressure dr…
A two-scale model is presented to simulate the dynamic ion transport and adsorption processes in porous electrodes used for capacitive deionization (CDI). At the pore scale, the Stokes equation governing water flow in porous CDI electrodes is solved using the lattice Boltzmann method and Nernst-Planck equations describing ion transport is solved using the finite volume method. The ion adsorption process is considered at the surface of carbon electrodes. At the continuum scale, Darcy equation and advection-diffusion equation governing water flow and solute transport through CDI cells are solved…
Historically, the dominant conceptual paradigm of porous media flow, solute mixing and transport was based on steady two-dimensional flows in heterogeneous porous media. Although it is now well recognised that novel transport phenomena can arise in unsteady and/or three-dimensional flows at both the pore- or Darcy-scales, appropriate methods for analysis and understanding of these more complex flows have not been widely employed. In this primer we advocate for methods borrowed from dynamical systems (chaos) theory, which aim to uncover the \emph{Lagrangian kinematics} of these flows: namely ho…
Physically consistent coupling conditions at the fluid-porous interface with correctly determined effective parameters are necessary for accurate modeling and simulation of various applications. To describe single-fluid-phase flows in coupled free-flow and porous-medium systems, the Stokes/Darcy equations are typically used together with the conservation of mass across the interface, the balance of normal forces and the Beavers-Joseph condition on the tangential velocity. The latter condition is suitable for flows parallel to the interface but not applicable for arbitrary flow directions. More…
In this paper, we derive upscaled equations for modelling biofilm growth in porous media. The resulting macro-scale mathematical models consider permeable multi-species biofilm including water flow, transport, detachment and reactions. The biofilm is composed of extracellular polymeric substances (EPS), water, active bacteria and dead bacteria. The free flow is described by the Stokes and continuity equations and the water flux inside the biofilm by the Brinkman and continuity equations. The nutrients are transported in the water phase by convection and diffusion. This pore-scale model include…
We derive a homogenized macroscopic model for fluid flows over ordered homogeneous porous surfaces. The unconfined free-flow is described by the Navier-Stokes equation, and the Darcy equation governs the seepage flow within the porous domain. Boundary conditions that accurately capture mass and momentum transport at the contact surface between these two domains are derived using the multiscale homogenization technique. In addition to obtaining the generalized version of the widely used Beavers-Joseph slip condition for tangential velocities, the present work provides an accurate formulation fo…
Transport and particulate processes are ubiquitous in environmental, industrial and biological applications, often involving complex geometries and porous media. In this work we present a general population balance model for particle transport at the pore-scale, including aggregation, breakage and surface deposition. The various terms in the equations are analysed with a dimensional analysis, including a novel collision-induced breakage mechanism, and split into one- and two-particles processes. While the first are linear processes, they might both depend on local flow properties (e.g. shear).…
Hypothesis Control of capillary flow through porous media has broad practical implications. However, achieving accurate and reliable control of such processes by tuning the pore size or by modification of interface wettability remains challenging. Here we propose that the flow of liquid by capillary penetration can be accurately adjusted by tuning the geometry of porous media and develop numerical method to achieve this. Methodologies On the basis of Darcys law, a general framework is proposed to facilitate the control of capillary flow in porous systems by tailoring the geometric shape of por…
The typical temporal resolution used in modern simulations significantly exceeds characteristic time scales at which the system is driven. This is especially so when systems are simulated over time-scales that are much longer than the typical temporal scales of forcing factors. We investigate the impact of space-time upscaling on reactive transport in porous media driven by time-dependent boundary conditions whose characteristic time scale is much smaller than that at which transport is studied or observed at the macroscopic level. The focus is on transport of a reactive solute undergoing diff…
