CNN-powered micro- to macro-scale flow modeling in deformable porous media
Institute of Mechanics and Computational Mechanics (IBNM), Leibniz University Hannover（莱布尼茨汉诺威大学力学与计算力学研究所（IBNM））；Institute of Applied Mechanics, University of Stuttgart（斯图加特大学应用力学研究所）；Institute of General Mechanics, RWTH Aachen University（亚琛工业大学普通力学研究所）
This work introduces a novel application for predicting the macroscopic intrinsic permeability tensor in deformable porous media, using a limited set of micro-CT images of real microgeometries. The primary goal is to develop an efficient, machine-learning (ML)-based method that overcomes the limitations of traditional permeability estimation techniques, which often rely on time-consuming experiments or computationally expensive fluid dynamics simulations. The novelty of this work lies in leveraging Convolutional Neural Networks (CNN) to predict pore-fluid flow behavior under deformation and anisotropic flow conditions. Particularly, the described approach employs binarized CT images of porous micro-structure as inputs to predict the symmetric second-order permeability tensor, a critical parameter in continuum porous media flow modeling. The methodology comprises four key steps: (1) constructing a dataset of CT images from Bentheim sandstone at different volumetric strain levels; (2) performing pore-scale simulations of single-phase flow using the lattice Boltzmann method (LBM) to generate permeability data; (3) training the CNN model with the processed CT images as inputs and permeability tensors as outputs; and (4) exploring techniques to improve model generalization, including data augmentation and alternative CNN architectures. Examples are provided to demonstrate the CNN's capability to accurately predict the permeability tensor, a crucial parameter in various disciplines such as geotechnical engineering, hydrology, and material science. An exemplary source code is made available for interested readers.
We formulate a numerical method for solving the two-phase flow poroelasticity equations. The scheme employs the interior penalty discontinuous Galerkin method and a sequential time-stepping method. The unknowns are the phase pressures and the displacement. Existence of the solution is proved. Three-dimensional numerical results show the accuracy and robustness of the proposed method.
Detailed understanding of the coupling between fluid flow and solid deformation in porous media is crucial for the development biomedical devices and novel energy technologies relating to a wide range of geological and biological processes. Well established models based on poroelasticity theory exist for describing coupled fluid-solid mechanics. However, these models are not adapted to describe systems with multiple fluid phases or 'hybrid-scale' systems containing both solid-free regions and porous matrices. To address this problem, we present a novel computational fluid dynamics approach bas…
Freezing of a deformable water-saturated porous medium. Part I: THM formulation, OpenGeoSys-6 implementation and benchmarking
BGE Technology GmbH（BGE技术有限公司）；UFZ Leipzig（莱比锡赫尔姆霍兹环境研究中心）；Technische Universität Bergakademie Freiberg（弗莱贝格矿业技术大学）；Technische Universität Dresden（德累斯顿工业大学）
We study reactive transport in a stressed porous media, where dissolution of the solid matrix causes two simultaneous, competing effects: pore enlargement (chemical deformation), and pore compaction due to mechanical weakening. A novel, mechanistic pore-scale model simulates flooding of a sample under fixed confining stress, showing that increasing stress inhibits permeability enhancement, increasing the injected volume required to reach a certain permeability, in agreement with recent experiments. We explain this behavior by stress concentration downstream, in the less dissolved (hence stiffe…
Many parts of biological organisms are comprised of deformable porous media. The biological media is both pliable enough to deform in response to an outside force and can deform by itself using the work of an embedded muscle. For example, the recent work (Ludeman et al., 2014) has demonstrated interesting 'sneezing' dynamics of a freshwater sponge, when the sponge contracts and expands to clear itself from surrounding polluted water. We derive the equations of motion for the dynamics of such an active porous media (i.e., a deformable porous media that is capable of applying a force to itself w…
In this letter, we show that pore-scale diffusiophoresis of colloidal particles along local salt gradients manifests in the macroscopic dispersion of particles in a porous medium. Despite is transient character, this microscopic phenomenon controls large-scale particle transport by altering their partitioning between transmitting and dead-end pores. It determines the distribution of residence and arrival times in the medium. Depending on the diffusiophoretic mobility, particles can be mobilized from or trapped in dead-end pores, which provides a means for the controlled manipulation of particl…
We develop constitutive equations for multi-component, multi-phase, macro-scale flow in a porous medium exposed to temperature-, composition-, and pressure -gradients. The porous medium is non-deformable. We define the pressure and the composition of the representative elementary volume (REV) in terms of the volume and surface averaged pressure and the saturation, and the respective driving forces from these variables. New contributions due to varying porosity or surface tension offer explanations for non-Darcy behavior. The interaction of a thermal and mechanical driving forces give thermal o…
Many applications of porous media research involves high pressures and, correspondingly, exchange of thermal energy between the fluid and the matrix. While the system is relatively well understood for the case of non-moving porous media, the situation when the elastic matrix can move and deform, is much more complex. In this paper we derive the equations of motion for the dynamics of a deformable porous media which includes the effects of friction forces, stresses, and heat exchanges between the media, by using the new methodology of variational approach to thermodynamics. This theory extends …
The flow of non-Newtonian fluids is ubiquitous in many applications in the geological and industrial context. We focus here on yield stress fluids (YSF), i.e. a material that requires minimal stress to flow. We study numerically the flow of yield stress fluids in 2D porous media on a macroscopic scale in the presence of local heterogeneities. As with the microscopic problem, heterogeneities are of crucial importance because some regions will flow more easily than others. As a result, the flow is characterized by preferential flow paths with fractal features. These fractal properties are charac…
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…
Using thermodynamic and variational principles we examine a basic phase field model for a mixture of two incompressible fluids in strongly perforated domains. With the help of the multiple scale method with drift and our recently introduced splitting strategy for Ginzburg-Landau/Cahn-Hilliard-type equations [Schmuck et al., Proc. R. Soc. A 468:3705-3724, 2012.], we rigorously derive an effective macroscopic phase field formulation under the assumption of periodic flow and a sufficiently large Péclet number. As for classical convection-diffusion problems, we obtain systematically diffusion-disp…
The electrolyte (comprising of solute ions and solvents) flow-through the porous media is frequently encountered in nature or in many engineering applications, such as the electrochemical systems, manufacturing of composites, geothermal engineering, soil pollution. In this study, we provide a new general theory for the electrochemical flows-through porous media. We use static method and set up two representative elementary volumes (REVs). One is the macroscopic REV of the mixture of the porous media and the electrolyte, while the other is the microscopic REV in the electrolyte fluid. The estab…
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…
Velocity distribution functions link the micro- and macro-level theories of fluid flow through porous media. Here we study them for the fluid absolute velocity and its longitudinal and lateral components relative to the macroscopic flow direction in a model of a random porous medium. We claim that all distributions follow the power exponential law controlled by an exponent $γ$ and a shift parameter $u_0$ and examine how these parameters depend on the porosity. We find that $γ$ has a universal value $1/2$ at the percolation threshold and grows with the porosity, but never exceeds 2.
In a recent paper by Lasseux, Valdés-Parada and Porter (J.~Fluid~Mech. \textbf{805} (2016) 118-146), it is found that the apparent gas permeability of the porous media is a nonlinear function of the Knudsen number. However, this result is highly questionable, because the adopted Navier-Stokes equations and the first-order velocity-slip boundary condition are first-order (in terms of the Knudsen number) approximations of the Boltzmann equation and the kinetic boundary condition for rarefied gas flows. Our numerical simulations based on the Bhatnagar-Gross-Krook kinetic equation and regularized …
We derive the entropy production for transport of multi-phase fluids in a non-deformable, porous medium exposed to differences in pressure, temperature, and chemical potentials. Thermodynamic extensive variables on the macro-scale are obtained by integrating over a representative elementary volume (REV). Using Euler homogeneity of the first order, we obtain the Gibbs equation for the REV. From this we define the intensive variables, the temperature, pressure and chemical potentials and, using the balance equations, derive the entropy production for the REV. The entropy production defines sets …
The macroscopic spreading and mixing of solute plumes in saturated porous media is ultimately controlled by processes operating at the pore scale. Whilst the conventional picture of pore-scale mechanical dispersion and molecular diffusion leading to persistent hydrodynamic dispersion is well accepted, this paradigm is inherently two-dimensional (2D) in nature and neglects important three-dimensional (3D) phenomena. We discuss how the kinematics of steady 3D flow at the porescale generate chaotic advection, involving exponential stretching and folding of fluid elements,the mechanisms by which i…
Single fluid porous medium systems are typically modeled at an averaged length scale termed the macroscale using Darcy's law. Standard approaches for modeling macroscale single fluid phase flow of non-Newtonian fluids extend Darcy's law, using an effective viscosity and assuming that the permeability is invariant. This approach results in a need to determine the effective viscosity for every fluid and flow rate. We use the thermodynamically constrained averaging theory (TCAT) to examine the formulation and closure of a macroscale model for non-Newtonian flow that is consistent with microscale …
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 contin…
In this paper, a pore-scale network modeling method, based on the flow continuity residual in conjunction with a Newton-Raphson non-linear iterative solving technique, is proposed and used to obtain the pressure and flow fields in a network of interconnected distensible ducts representing, for instance, blood vasculature or deformable porous media. A previously derived analytical expression correlating boundary pressures to volumetric flow rate in compliant tubes for a pressure-area constitutive elastic relation has been used to represent the underlying flow model. Comparison to a preceding eq…
Microscale turbulence in porous media is a new physical phenomenon that exhibits unique properties unlike those in classical turbulence flows. At low values of porosity, the surface forces on the solid obstacles compete with the inertial force of the fluid flow to result in the formation of flow instabilities. In this paper, we report the origin and mechanism of a symmetry-breaking phenomenon in periodic porous media that causes a deviation in the direction of the mean flow from that of the applied pressure gradient. Large Eddy Simulation (LES) is used to simulate turbulent flow in a homogeneo…
The flow of deformable particles, such as droplets, dragged by a fluid, through a network of narrow pores inside rocks or other porous media is key in a range of applications, from enhanced oil recovery and water filtration to lab on a chip sorting of cells. The collective dynamics and its impact on the flow are poorly understood. Here, using droplets as a prototype, we show that collective transport can occur for conditions under which a single particle would get trapped at a pore channel. When a series of droplets gets trapped, the fluids flow is affected significantly, leading to an increas…
Diverse processes rely on the viscous flow of polymer solutions through porous media. In many cases, the macroscopic flow resistance abruptly increases above a threshold flow rate in a porous medium---but not in bulk solution. The reason why has been a puzzle for over half a century. Here, by directly visualizing the flow in a transparent 3D porous medium, we demonstrate that this anomalous increase is due to the onset of an elastic instability. We establish that the energy dissipated by the unstable flow fluctuations, which vary across pores, generates the anomalous increase in flow resistanc…
Traditional models of electrokinetic transport in porous media are based on homogenized material properties, which neglect any macroscopic effects of microscopic fluctuations. This perspective is taken not only for convenience, but also motivated by the expectation of irrotational electro-osmotic flow, proportional to the electric field, for uniformly charged surfaces (or constant zeta potential) in the limit of thin double layers. Here, we show that the inherent heterogeneity of porous media generally leads to macroscopic vortex patterns, which have important implications for convective trans…
Porous asphalt (PA) is an open-graded porous material with a porosity of 20%, allowing fast drainage of rain and improving driving and acoustic conditions. However, the high porosity leads to significant contact with water resulting in a shorter life expectancy. To improve the durability and performance of PA, the distribution of water and its residence time have to be understood which entails capturing diverse multiphase phenomena. For these reasons, a numerical study is performed to analyze in detail the fluid transport mechanisms at play in PA, towards estimating the liquid distribution ins…
New insight into the contribution of the microscale vortex evolution towards convection heat transfer in porous media is presented in this paper. The objective is to determine how the microscale vortices influence convection heat transfer in turbulent flow inside porous media. The microscale temperature distribution is analyzed using flow visualization in 2D using streamlines and in 3D using Q structures. The pertinent observations are supplemented with the comparison of surface skin friction and heat transfer using: (1) surface skin friction lines and (2) joint PDF of pressure and skin fricti…
Shear stress is an important physical factor that regulates proliferation, migration and morphogenesis. In particular, the homeostasis of blood vessels is dependent on shear stress. To mimic this process ex vivo, efforts have been made to seed scaffolds with vascular and other cell types in the presence of growth factors and under pulsatile flow conditions. However, the resulting bioreactors lack information on shear stress and flow distributions within the scaffold. Consequently, it is difficult to interpret the effects of shear stress on cell function. Such knowledge would enable researchers…
This document is a presentation of the equations of simultaneous water and oil flow in deformable porous medium and linear poromechanics as well as the staggered solution algorithm to solve the coupled system of equations.
Derivation of macroscopic models for advection-diffusion processes in the presence of dominant heterogeneous (e.g., surface) reactions using homogenisation theory or volume averaging is often deemed unfeasible due to the strong coupling between scales that characterise such systems. In this work, we show how the upscaling can be carried out by applying and extending the methods presented in literature. The approach relies on the decomposition of the microscale concentration into a reactive component, given by the eigenfunction of the advection-diffusion operator, the associated eigenvalue whic…
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…
We derive and study a new hyperbolic two-phase model of a porous deformable medium saturated by a viscous fluid. The governing equations of the model are derived in the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems and by generalizing the unified hyperbolic model of continuum fluid and solid mechanics. Similarly to the unified model, the presented model takes into account the viscosity of the saturating fluid through a hyperbolic reformulation. The model accounts for such dissipative mechanisms as interfacial friction and viscous dissipation of the saturated flu…
We perform direct numerical simulations of the flow through a model of a deformable porous medium. Our model is a two-dimensional hexagonal lattice, with defects, of soft elastic cylindrical pillars, with elastic shear modulus $G$, immersed in a liquid. We use a two-phase approach: the liquid phase is a viscous fluid and the solid phase is modeled as an incompressible viscoelastic material, whose complete nonlinear structural response is considered. We observe that the Darcy flux ($q$) is a nonlinear function -- steeper than linear -- of the pressure-difference ($ΔP$) across the medium. Furthe…
Upscaling the effect of heterogeneities in porous media is crucial for macroscopic flow predictions, with widespread applications in energy and environmental settings. In this study, we derive expressions for the upscaled flow properties of a porous medium with a vertical heterogeneity, using a combination of asymptotic analysis and numerical simulations. Then, we use these upscaled expressions to describe the dynamic flooding of an aquifer, where the classic Buckley-Leverett formulation is modified to account for heterogeneities. In particular, we show that heterogeneities can modify flooding…
Pore-scale observation of vortical flow structures in porous media is a significant challenge in many natural and industrial systems. Vortical structure dynamics is believed to be the driving mechanism in the transition regime in porous media based on the pore Reynolds number, $Re_p$. To examine this assertion, a refractive-index matched randomly packed porous medium is designed to measure the scale of vortical flow structures in transition from unsteady laminar to turbulent using two-dimensional time-resolved Particle Image Velocimetry (PIV). Planar PIV data for $Re_p$ from 100 to 948 is used…
We examine the effects of horizontally layered heterogeneities on the spreading of two-phase gravity currents in a porous medium, with application to numerous environmental flows, most notably geological carbon sequestration. Geological heterogeneities, which are omnipresent within all reservoirs, affect the large-scale propagation of such flows through the action of small-scale capillary forces, yet the relationship between these small and large scales is poorly understood. Here, we derive a simple upscaled model for a gravity current under an impermeable cap rock, which we use to investigate…
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…
We construct an ensemble distribution to describe steady immiscible two-phase flow of two incompressible fluids in a porous medium. The system is found to be ergodic. The distribution is used to compute macroscopic flow parameters. In particular, we find an expression for the overall mobility of the system from the ensemble distribution. The entropy production at the scale of the porous medium is shown to give the expected product of the average flow and its driving force, obtained from a black-box description. We test numerically some of the central theoretical results.
Models that describe two-fluid flow in porous media suffer from a widely-recognized problem that the constitutive relationships used to predict capillary pressure as a function of the fluid saturation are non-unique, thus requiring a hysteretic description. As an alternative to the traditional perspec- tive, we consider a geometrical description of the capillary pressure, which relates the average mean curvature, the fluid saturation, the interfacial area between fluids, and the Euler characteristic. The state equation is formulated using notions from algebraic topology and cast in terms of me…
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.
Simulation of fluid flow in porous media has many applications, from the micro-scale (cell membranes, filters, rocks) to macro-scale (groundwater, hydrocarbon reservoirs, and geothermal) and beyond. Direct simulation of flow in porous media requires significant computational resources to solve within reasonable timeframes. An integrated method combining predictions of fluid flow (fast, limited accuracy) with direct flow simulation (slow, high accuracy) is outlined. In the tortuous flow paths of porous media, Deep Learning techniques based on Convolutional Neural Networks (CNNs) are shown to gi…
Theoretical and experimental investigations of water vapor interaction with porous materials are carried out both at the macro level and at the micro level. At the macro level, the influence of the arrangement structure of individual pores on the processes of water vapor interaction with porous material as a continuous medium is studied. At the micro level, it is very interesting to investigate the dependence of the characteristics of the water vapor interaction with porous media on the geometry and dimensions of the individual pore. In this paper, a study was carried out by means of mathemati…
From the definition of a generalized conformable spatial derivative, an exponential conformable function with three parameters $(a,b,α)$ is proposed for a viscous and an inertial-viscous steady-state Navier-Stokes 1D models, obtaining analytical solutions for both generalized conformable models. The conformable models' parameters are optimized to compare the viscous model to a Darcian 1D flow and the inertial-viscous model to a non-Darcian 1D model for a specific range of Darcy numbers $(1\times10^{-2}<Da\leq 1)$. Velocity profiles for the porous medium and the conformable model are computed a…
The intermittent burst dynamics during the slow drainage of a porous medium is studied experimentally. We have shown that this system satisfies a set of conditions known to be true for critical systems, such as intermittent activity with bursts extending over several time and length scales, self-similar macroscopic fractal structure and $1/f^α$ power spectrum. Additionally, we have verified a theoretically predicted scaling for the burst size distribution, previously assessed via numerical simulations. The observation of $1/f^α$ power spectra is new for porous media flows and, for specific bou…
Models of porous media are often applied to relatively small systems, which leads not only to system-size-dependent results, but also to phenomena that would be absent in larger systems. Here we investigate one such finite-size effect: anisotropy of the permeability tensor. We show that a non-zero angle between the external body force and macroscopic flux vector exists in three-dimensional periodic models of sizes commonly used in computer simulations and propose a criterion, based on the system size to the grain size ratio, for this phenomenon to be relevant or negligible. The finite-size ani…
The dynamics of the wetting front are considered during the imbibition of a fluid into a porous substrate through a circular drawing area. A mathematical model of this process, assuming incompressible Darcy flow, is presented, before the full finite element scheme for solving this set of equations is given allowing the reader to reproduce all presented results. Asymptotic analysis is performed revealing contradictions between the assumptions of Darcy's equation and the solutions it produces, along with qualitative results for the behaviour of the wetting front and macroscopic contact angles. V…
A two-phase model and its application to wavefields numerical simulation are discussed in the context of modeling of compressible fluid flows in elastic porous media. The derivation of the model is based on a theory of thermodynamically compatible systems and on a model of nonlinear elastoplasticity combined with a two-phase compressible fluid flow model. The governing equations of the model include phase mass conservation laws, a total momentum conservation law, an equation for the relative velocities of the phases, an equation for mixture distortion, and a balance equation for porosity. They…
