Archie’s Law in Microsystems

Since 1942 Archie??s law is used every day to estimate, from electrical measurements, the quantity of oil present in oil fields. In this article, we perform the first experimental analysis of electric conductivity in well controlled models of porous media. We used microfluidic networks (called micromodels in the oil industry jargon), incorporating thousands of pores, with controlled wettability. Different electrode and pore geometries are considered. In all cases the evolution of the conductivity with the conductive fluid fraction (??saturation??) clearly reveals the presence of percolation thresholds, signaling that as the fraction of the conductive fluid decreases below some critical value, there are no more pathways involving only channels entirely filled with the conductive fluid that connect the electrodes. This behavior is observed in all cases, for all the network/electrode geometries and wetting properties we investigated, and is consequently likely to reflect a genuine behavior for microfluidic ??2D?? networks. The existing models??based on percolation theory or on mean field approach??reproduce correctly the structure of this behavior, but generally at a semi-quantitative level. The most successful case is obtained with the effective medium theory (EMT) model, with drainage and perpendicular electrodes. This outcome suggests that, despite the complexity of these systems, very simple models can describe correctly the physics of the system. Nonetheless, more precise modeling requires case-by-case studies. Our results are consistent with the current body of knowledge accumulated for decades on three-dimensional samples. The key point is that in 3D systems, owing to topological reasons, the threshold is extremely low in terms of water saturations. Archie??s law completely neglects the threshold effect. Nonetheless the percolation threshold should not be overlooked, and modeling should take this aspect systematically into account, as it is already done by several investigators.     

Hydraulic fracture in poro-hydro-elastic media

The prediction of fluid-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing, the so-called “fracking”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developing cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media that provides a rigorous approach to a diffusive crack modeling based on the introduction of a regularized crack surface. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. It shows that the quasi-static problem of elastically deforming, fluid-saturated porous media at fracture is related to a minimization principle for the evolution problem. The existence of this minimization principle for the coupled problem is advantageous with regard to a new unconstrained stable finite element design, while previous space discretizations of the saddle point principles are constrained by the LBB condition. This proposed formulation includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length.     

Phase-field modeling of hydraulic fracture

In this work a theoretical framework implementing the phase-field approach to fracture is used to couple the physics of flow through porous media and cracks with the mechanics of fracture. The main modeling challenge addressed in this work, which is a challenge for all diffuse crack representations, is on how to allow for the flow of fluid and the action of fluid pressure on the aggregate within the diffuse damage zone of the cracks. The theory is constructed by presenting the general physical balance laws and conducting a consistent thermodynamic analysis to constrain the constitutive relationships. Constitutive equations that reproduce the desired responses at the various limits of the phase-field parameter are proposed in order to capture Darcy-type flow in the intact porous medium and Stokes-type flow within open cracks. A finite element formulation for the solution of the governing model equations is presented and discussed. Finally, the theoretical and numerical model is shown to compare favorably to several important analytical solutions. More complex and interesting calculations are also presented to illustrate some of the advantageous features of the approach.     

A generalized mass transfer law unifying various particle transport mechanisms in dilute dispersions

A generalized mass transfer law for dilute dispersion of particles (or droplets) of any sizes suspended in a fluid has been described, which can be applied to turbulent or laminar flow. The generalized law reduces to the Fick’s law of diffusion in the limit of very small particles. Thus the study shows how the well-known and much-used Fick’s law of diffusion fits into the broader context of particle transport. The general expression for particle flux comprises a diffusive flux due to Brownian motion and turbulent fluctuation, a diffusive flux due to temperature gradient (thermophoresis plus stressphoresis) and a convective flux that arises primarily due to the interaction of particle inertia and the inhomogeneity of the fluid turbulence field (turbophoresis). Shear-induced lift force, electrical force, gravity, etc. also contribute to the convective flux. The present study includes the effects of surface roughness, and the calculations show that the presence of small surface roughness even in the hydraulically smooth regime significantly enhances deposition especially of small particles. Thermophoresis can have equally strong effects, even with a modest temperature difference between the wall and the bulk fluid. For particles of the intermediate size range, turbophoresis, thermophoresis and roughness are all important contributors to the overall deposition rate. The paper includes a parametric study of the effects of electrostatic forces due to mirror charging. The present work provides a unified framework to determine the combined effect of various particle transport mechanisms on mass transfer rate and the inclusion of other mechanisms not considered in this paper is possible.     

Mesh regularization for an ALE code based on the limitation of the Lagrangian mesh velocity

The Lagrangian approach is usually used for the simulation of flow with strong shock waves. Moreover, this approach is particularly well suited to treatment of material interfaces in the case of multimaterial flows.Unfortunately, this formulation leads to very large deformations in the mesh. The arbitrary Lagrangian‐Eulerian method overcomes this drawback by using a mesh regularization that is based on an analysis of cell geometry. The regularization step may be considered as a method used to correct the nonconvex and potentially tangled cells that constitute the mesh. In this paper, we present a new approach to mesh regularization. Instead of using a purely geometric criterion, we propose that the mesh evolution is computed on the basis of the flow vorticity. This approach is called the large Eddy limitation method, and it is aimed here to be used in finite volume direct arbitrary Lagrangian‐Eulerian methods. The large Eddy limitation method is general, which means that it is not restricted to applications in the finite volume framework dedicated to fluid flow simulation; for instance, it could also be naturally applied to the finite element framework.     

Wave propagation in a fractional viscoelastic Andrade medium: Diffusive approximation and numerical modeling

This study focuses on the numerical modeling of wave propagation in fractionally-dissipative media. These viscoelastic models are such that the attenuation is frequency-dependent and follows a power law with non-integer exponent within certain frequency regimes. As a prototypical example, the Andrade model is chosen for its simplicity and its satisfactory fits of experimental flow laws in rocks and metals. The corresponding constitutive equation features a fractional derivative in time, a non-local-in-time term that can be expressed as a convolution product whose direct implementation bears substantial memory cost. To circumvent this limitation, a diffusive representation approach is deployed, replacing the convolution product by an integral of a function satisfying a local time-domain ordinary differential equation. An associated quadrature formula yields a local-in-time system of partial differential equations, which is then proven to be well-posed. The properties of the resulting model are also compared to those of the Andrade model. The quadrature scheme associated with the diffusive approximation, and constructed either from a classical polynomial approach or from a constrained optimization method, is investigated. Finally, the benefits of using the latter approach are highlighted as it allows to minimize the discrepancy with the original model. Wave propagation simulations in homogeneous domains are performed within a split formulation framework that yields an optimal stability condition and which features a joint fourth-order time-marching scheme coupled with an exact integration step. A set of numerical experiments is presented to assess the overall approach. Therefore, in this study, the diffusive approximation is demonstrated to provide an efficient framework for the theoretical and numerical investigations of the wave propagation problem associated with the fractional viscoelastic medium considered.     

Advanced near-wall modeling for engine heat transfer

Recent developments in the engine heat transfer modeling tend to improve existing wall heat transfer models (temperature wall functions) which mostly rely on the standard or low-Re variants of k-ε turbulence model. Presently applied mesh resolutions already allow for first near-wall computational cells reaching the buffer or locally even viscous/conductive sub-layer, thus increasing the importance of more sophisticated modeling approach. As temperature gradient-induced density and fluid property variations become significant, wall heat transfer is strongly influenced by property variations (viscous/conductive sub-layer) and predictive capability of the turbulence model (buffer region), standard wall laws being inadequate anymore, even for attached boundary layers. The present approach relies on the k-ζ-f turbulence model and formulates a compressible wall function of Han and Reitz in the framework of hybrid wall treatment. The model is validated against spark ignition (SI) engine heat transfer measurements. Predicted wall heat flux evolutions on the cylinder head exhibit very good agreement with the experimental data, being superior to similar numerical predictions available in the published literature.     

Evaluation of the Darcy's law performance for two-fluid flow hydrodynamics in a particle debris bed using a lattice-Boltzmann model

In the present paper, multiphase flow dynamics in a porous medium are analyzed by employing the lattice-Boltzmann modeling approach. A two-dimensional formulation of a lattice-Boltzmann model, using a D2Q9 scheme, is used. Results of the FlowLab code simulation for single phase flow in porous media and for two-phase flow in a channel are compared with analytical solutions. Excellent agreement is obtained. Additionally, fluid-fluid interaction and fluid-solid interaction (wettability) are modeled and examined. Calculations are performed to simulate two-fluid dynamics in porous media, in a wide range of physical parameters of porous media and flowing fluids. It is shown that the model is capable of determining the minimum body force needed for the nonwetting fluid to percolate through the porous medium. Dependence of the force on the pore size, and geometry, as well as on the saturation of the nonwetting fluid is predicted by the model. In these simulations, the results obtained for the relative permeability coefficients indicate the validity of the reciprocity for the two coupling terms in the modified Darcy's law equations. Implication of the simulation results on two-fluid flow hydrodynamics in a decay-heated particle debris bed is discussed. Received on 1 December 1999     

Pressure field around a well in a stratified inhomogeneous bed

The steady axisymmetric flow of an incompressible fluid into a vertical well hydrodynamically perfectly drilled into a stratified inhomogeneous half-space consisting of three layers with different permeabilities is considered. The boundaries of the layers are assumed to be horizontal planes and the roof of the upper layer is assumed to be impermeable. The flow obeys a linear Darcy’s law. The pressure distribution on the well is assumed to be given, which is the main obstacle to finding an exact solution of the problem. Beginning with the classical studies of Muskat and Charnyi [1, 2], approximate solutions of such problems have been constructed as a superposition of flows generated by point sources with given intensities, distributed along the well axis in accordance with a fairly simple law. In the present study, this approach is used to obtain an integral equation for the source density distribution, which is then solved numerically. Comparison with the known exact solution for a thin elongated ellipsoid (“needle”) shows that this approach makes it possible to ensure an accuracy which at any rate is sufficient for applications.     

A mixed damage model for unsaturated porous media

The aim of this study is to present a framework for the modeling of damage in continuous unsaturated porous geomaterials. The damage variable is a second-order tensor. The model is formulated in net stress and suction independent state variables. Correspondingly, the strain tensor is split into two independent thermodynamic strain components. The proposed framework mixes micro-mechanical and phenomenological approaches. On the one hand, the effective stress concept of Continuum Damage Mechanics is used in order to compute the damaged rigidities. On the other hand, the concept of equivalent mechanical state is introduced in order to get a simple phenomenological formulation of the behavior laws. Cracking effects are also taken into account in the fluid transfer laws. To cite this article: C. Arson, B. Gatmiri, C. R. Mecanique 337 (2009).     

Hybrid continuum–coarse-grained modeling of erythrocytes

The red blood cell (RBC) membrane is a composite structure, consisting of a phospholipid bilayer and an underlying membrane-associated cytoskeleton. Both continuum and particle-based coarse-grained RBC models make use of a set of vertices connected by edges to represent the RBC membrane, which can be seen as a triangular surface mesh for the former and a spring network for the latter. Here, we present a modeling approach combining an existing continuum vesicle model with a coarse-grained model for the cytoskeleton. Compared to other two-component approaches, our method relies on only one mesh, representing the cytoskeleton, whose velocity in the tangential direction of the membrane may be different from that of the lipid bilayer. The finitely extensible nonlinear elastic (FENE) spring force law in combination with a repulsive force defined as a power function (POW), called FENE–POW, is used to describe the elastic properties of the RBC membrane. The mechanical interaction between the lipid bilayer and the cytoskeleton is explicitly computed and incorporated into the vesicle model. Our model includes the fundamental mechanical properties of the RBC membrane, namely fluidity and bending rigidity of the lipid bilayer, and shear elasticity of the cytoskeleton while maintaining surface-area and volume conservation constraint. We present three simulation examples to demonstrate the effectiveness of this hybrid continuum–coarse-grained model for the study of RBCs in fluid flows.     

An effective approach for modeling fluid flow in heterogeneous media using numerical manifold method

A major challenge of modeling fluid flow in heterogeneous media is to model the material interfaces, which may be arbitrarily oriented or intersected with Dirichlet, Neumann, or other boundaries, making it difficult to mesh and accurately satisfy the boundary constraints. In order to solve these problems, we derived a new continuous approach in the numerical manifold method (NMM). NMM is an ideal method to handle boundaries, considering its flexibility and efficiency with fixed mathematical mesh and its integration precision. With the two‐cover‐meshing system, we construct physical covers containing gradient jump terms defined as extended degrees of freedom to realize the refraction law across material interfaces. In the global equilibrium equations, the jump terms are naturally considered with the energy‐work seepage model. In this approach, high accuracy is expected from the newly constructed jump function together with simplex integration. Moreover, high mesh efficiency is realized by fixed triangular mathematical mesh with algorithms fully considering interfaces intersecting with Dirichlet, Neumann, or other boundaries and simplex integration on elements in arbitrary shapes. The new approach was coded into our NMM fluid flow model. We calculated examples involving fluid flow through a domain including (1) a single interface, (2) an idealized fault represented by multiple material interfaces, (3) intersected interfaces, and (4) an octagonal inclusion. We compared the simulated results to analytical solutions or results with denser mesh to test precision and efficiency and thereby proved that the new approach is accurate, efficient, and flexible, especially when considering intense geometric change or intersections. Copyright © 2015 John Wiley & Sons, Ltd.     

Derivation of the Forchheimer Law via Homogenization

In this paper we derive the Forchheimer law via the theory of homogenization. In particular, we study the nonlinear correction to Darcy's law due to inertial effects on the flow of a Newtonian fluid in rigid porous media. A general formula for this correction term is derived directly from the Navier–Stokes equation via homogenization. Unlike other studies based on the same approach that concluded for the nonlinear correction to be cubic in velocity for isotropic media, the present work shows that the nonlinear correction is quadratic. An example is constructed to illustrate our theory. In this example, the analytic solution to the Navier–Stokes equation is obtained and is utilized to show the validity of the quadratic correction. Both incompressible and compressible fluids are considered.     

Reactive Constrained Mixtures for Modeling the Solid Matrix of Biological Tissues

This article illustrates our approach for modeling the solid matrix of biological tissues using reactive constrained mixtures. Several examples are presented to highlight the potential benefits of this approach, showing that seemingly disparate fields of mechanics and chemical kinetics are actually closely interrelated and may be elegantly expressed in a unified framework. Thus, constrained mixture models recover classical theories for fibrous materials with bundles oriented in different directions or having different reference configurations, that produce characteristic fiber recruitment patterns under loading. Reactions that exchange mass among various constituents of a mixture may be used to describe tissue growth and remodeling, which may also alter the material’s anisotropy. Similarly, reactions that describe the breaking and reforming of bonds may be used to model free energy dissipation in a viscoelastic material. Therefore, this framework is particularly well suited for modeling biological tissues.     

Modeling the acidizing of a carbonate formation

Two approaches to the determination of the optimum conditions of the well bottom zone treatment in a carbonate formation are considered and compared. The first, bench approach is based on the investigation of the process of wormhole formation in rock cores, while the second approach is the mathematical modeling of the process within the framework of mechanics of multi-component seepage flows using averaged values of the reservoir porosity and permeability. A mathematical model of the process, that takes account for fluid flow and the chemical reaction between the acid and the rock matrix proceeding in the kinetic regime, is presented. Both approaches indicate the existence of an optimum rate of acid solution injection into the formation at a certain given value of the dimensionless Damköller number.