A 3-dimensional well model in the flow transport through porous media

In petroleum extraction and exploitation, the well is usually treated as a point or line source, due to its radius is much smaller comparing with the scale of the whole reservoir. Especially, in 3-dimensional situation, the well is regarded as a line source. In this paper, we analyze the modeling error for this treatment for steady flows through porous media and present a new algorithm for line-style well to characterize the wellbore flow potential. We also provide a numerical example to demonstrate the effectiveness of the proposed method.     

A stochastic model for fronts in porous media

We analyze a stochastic model for the motion of fronts in two-phase fluids and derive upscaled equations for the capillary pressure. This extends results of [11], where the same law for the capillary pressure was derived under an assumption on typical explosion patterns. With the work at hand we remove that assumption and show that in the stochastic case the upscaled equations hold almost surely.     

An analytical model for carrier-facilitated solute transport in weakly heterogeneous porous media

Carrier-facilitated solute transport in heterogeneous aquifers is studied within a Lagrangian framework. Dissolved solutes and carriers are advected by steady random groundwater flow, which is modeled by Darcy's law with uncertain hydraulic conductivity that is treated as a stationary random space function. We derive general expressions for the spatial moments of the dissolved concentration and the concentration associated with the carrier phase. In order to reduce the computational effort, we use previously derived solutions for the flow field. This enables us to obtain closed-form solutions for the spatial moments of the two concentration fields. The mass and center of gravity of the two propagating plumes depend only on the mean velocity field and chemical/degradation processes. The higher (second and third) moments are affected by the coupling between reactions (sorption/desorption and degradation) among the three phases (i.e., dissolved, carrier and sorbed concentrations) and the aquifer’s heterogeneity. We investigate the potentially enhancing effect of carriers by comparing spatial moments of the two propagating plumes. The forward/backward mass transfer rates between the liquid and carrier phases, and the degradation coefficients are identified as critical parameters. The carrier's role is most prominent when detachment from carrier sites is slow, provided that degradation on the carriers is smaller than that in the liquid phase.     

Stochastic transport theory for porous media

The type of porous media considered in the present study consists of a randomly structured solid or -phase in which the existing voids permit the transport of fluid or-phase, when a material sample is subjected to an external load. By introducing a system functional characterizing the constraint flow of the-phase and a set of control parameters, the solution of the random flow process is shown to be a strong Markov or diffusion process in the velocity space or subspace of the more general probabilistic function space. The analysis is illustrated by experimental observations on a particular porous medium, i.e. Al-Glycol-Resin composite and the obtained results by the use of scanning-electron microscopy and quantitative stereology.     

Exact solutions for two-phase colloidal-suspension transport in porous media

Two-phase transport of colloids and suspensions occurs in numerous areas of chemical, environmental, geo-, and petroleum engineering. The main effects are particle capture by the rock and altering the flux by changing the suspended and retained concentrations. Multiple mechanisms of suspended particle capture are discussed. The mathematical model for m independent particle-capture mechanisms is considered, resulting in an (m + 2) × (m + 2) system of partial differential equations. Using the stream-function as an independent variable instead of time splits the system into an (m + 1) × (m + 1) auxiliary system, containing only concentrations and one lifting hydrodynamic equation for an unknown phase saturation. Introduction of the concentration potential linked with retention concentrations yields an exact solution of the auxiliary problem. The exact formulae allow for predicting the profiles and breakthrough histories for the suspended and retained concentrations, and phase saturations. The solution shows that for small retained concentrations, the suspended concentration is in a steady-state behind the concentration front, where all the retained concentrations are proportional to the mass of suspended particles that passed via a given reservoir cross-section. The maximum penetration depths for suspended and retained particles are the same and are equal to those for a single-phase flow.     

A multicomponent flow model in deformable porous media

We propose a model for multicomponent flow of immiscible fluids in a deformable porous medium accounting for capillary hysteresis. Oil, water, and air in the soil pores offer a typical example of a real situation occurring in practice. We state the problem within the formalism of continuum mechanics as a slow diffusion process in Lagrange coordinates. The balance laws for volumes, masses, and momentum lead to a degenerate parabolic PDE system. In the special case of a rigid solid matrix material and three fluid components, we prove under further technical assumptions that the system is mathematically well posed in a small neighborhood of an equilibrium.     

Discontinuous Galerkin relaxation algorithm for solute transport in porous media

We consider a degenerate parabolic convection dominated equation which models the transport of contaminant in porous media. The numerical scheme is fulfilled by combining the DG – Discontinuous Galerkin Method with and an efficient relaxation algorithm that recently developed. Numerical results show the efficiency of our scheme. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A model of multiphase flow and transport in porous media applied to gas migration in underground nuclear waste repository

We prove existence of solutions for a new model of two compressible and partially miscible phase flow in porous media, applied to gas migration in an underground nuclear waste repository. This model, modeling fully and partially water saturated situations, consist of a coupled system of quasilinear parabolic partial differential equations. We seek a new set of variables in order to obtain a system which belongs to the class of equations considered by Alt and Luckhaus such that it would be possible to use their existence theorem. A simulation of a numerical test case is performed in order to numerically demonstrate the ability of this model to take in account the appearance of one phase. To cite this article: F. Smaï, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Newton method for reactive solute transport with equilibrium sorption in porous media

We present a mass conservative numerical scheme for reactive solute transport in porous media. The transport is modeled by a convection-diffusion-reaction equation, including equilibrium sorption. The scheme is based on the mixed finite element method (MFEM), more precisely the lowest-order Raviart-Thomas elements and one-step Euler implicit. The underlying fluid flow is described by the Richards equation, a possibly degenerate parabolic equation, which is also discretized by MFEM. This work is a continuation of Radu et al. (2008) and Radu et al. (2009) [1] and [2] where the algorithmic aspects of the scheme and the analysis of the discretization method are presented, respectively. Here we consider the Newton method for solving the fully discrete nonlinear systems arising on each time step after discretization. The convergence of the scheme is analyzed. In the case when the solute undergoes equilibrium sorption (of Freundlich type), the problem becomes degenerate and a regularization step is necessary. We derive sufficient conditions for the quadratic convergence of the Newton scheme.     

Asymptotic transport models for heat and mass transfer in reactive porous media

We propose an approach for deriving in a rigorous but formal way a family of models of mass and heat transfer in reactive porous media. At a microscopic level we set a model coupling the Boltzmann equation in the gas phase, the heat equation on the solid phase and appropriate interface condititons. An asymptotic expansion leads to a system of coupled diffusion equations where the effective diffusion tensors are defined from the microscopic geometry of the material through auxiliary problems. The ellipticity of the diffusion operator is addressed. To cite this article: P. Charrier, B. Dubroca, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A multiphase single relaxation time lattice Boltzmann model for heterogeneous porous media

The lattice Boltzmann (LB) method has been shown to be a highly efficient numerical method for solving fluid flow in confined domains such as pipes, irregularly shaped channels or porous media. Traditionally the LB method has been applied to flow in void regions (pores) and no flow in solid regions. However, in a number of scenarios, this may not suffice. That is partial flow may occur in semi-porous regions. Recently gray-scale LB methods have been applied to model single phase flow in such semi-porous materials. Voxels are no longer completely void or completely solid but somewhere in between. We extend the single relaxation time LB method to model multiphase, immiscible flow (e.g., gas and liquid or water and oil) in a semi-porous medium. We compare the solution to test cases and find good agreement of the model as compared to analytical solutions. We then apply the model to real porous media and recover both capillary and viscous flow regimes. However, some deficiencies in the single relaxation time LB method applied to multiphase flow are uncovered and we describe methods to overcome these limitations.     

Two-dimensional modeling of microscale transport and biotransformation in porous media

We develop a model for simulating the growth of a biofilm in a tortuous tube. The solutions to the Navier-Stokes equations and the advection-diffusion equation are calculated numerically using finite differences. These solutions are then coupled with a biofilm growth model. © 1994 John Wiley & Sons, Inc.     

Numerical analysis of a method for high Peclet number transport in porous media

A variationally consistent eddy viscosity discretization is presented in [W.J. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput. 133 (2002) 147-157] for the stationary convection diffusion problem. This discretization is extended to the evolutionary problem in [N. Heitmann, Subgridscale stabilization of time-dependent convection dominated diffusive transport, J. Math. Anal. Appl. 331 (2007) 38-50] with a near optimal error bound. In the following, we couple this discretization with the porous media problem. We present a comprehensive analysis of stability and error for the velocity field derived from the porous media problem. Next, using a backward Euler approximation for the time derivative we follow the inherited error in velocity through the coupling with the convection diffusion problem. The method is shown to be stable and the error near optimal and independent of the diffusion coefficient, ?.     

Simulation of a compositional model for multiphase flow in porous media

In this article, we consider the simulation of a compositional model for three‐dimensional, three‐phase, multicomponent flow in a porous medium. This model consists of Darcy's law for volumetric flow velocities, mass conservation for hydrocarbon components, thermodynamic equilibrium for mass interchange between phases, and an equation of state for saturations. A discretization scheme based on the block‐centered finite difference method for pressures and compositions is developed. Numerical results are reported for the benchmark problem of the third comparative solution project (CSP) organized by the society of petroleum engineers (SPE). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Analysis of flow and thermal field in nanofluid using a single phase thermal dispersion model

Flow and thermal field in nanofluid is analyzed using single phase thermal dispersion model proposed by Xuan and Roetzel [Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707]. The non-dimensional form of the transport equations involving the thermal dispersion effect is solved numerically using semi-explicit finite volume solver in a collocated grid. Heat transfer augmentation for copper–water nanofluid is estimated in a thermally driven two-dimensional cavity. The thermo-physical properties of nanofluid are calculated involving contributions due to the base fluid and nanoparticles. The flow and heat transfer process in the cavity is analyzed using different thermo-physical models for the nanofluid available in literature. The influence of controlling parameters on convective recirculation and heat transfer augmentation induced in buoyancy driven cavity is estimated in detail. The controlling parameters considered for this study are Grashof number (10³ < Gr < 10⁵), solid volume fraction (0 < ? < 0.2) and empirical shape factor (0.5 < n < 6). Simulations carried out with various thermo-physical models of the nanofluid show significant influence on thermal boundary layer thickness when the model incorporates the contribution of nanoparticles in the density as well as viscosity of nanofluid. Simulations incorporating the thermal dispersion model show increment in local thermal conductivity at locations with maximum velocity. The suspended particles increase the surface area and the heat transfer capacity of the fluid. As solid volume fraction increases, the effect is more pronounced. The average Nusselt number from the hot wall increases with the solid volume fraction. The boundary surface of nanoparticles and their chaotic movement greatly enhances the fluid heat conduction contribution. Considerable improvement in thermal conductivity is observed as a result of increase in the shape factor.     

Conformal symmetric model of the porous media

We present a conformal theory for the random fractal fields. As an example, the density of the porous matter is considered. The equation that expresses density in terms of a nonfractal field is evaluated. Assuming the hypothesis of scale and conformal symmetry for the latter, we derive the correlation functions for density. The log-normal conformal model is studied.     

Homogenization of a three-phase flow model in porous media

We give homogenization results for an immiscible and incompressible three-phase flow model in a heterogeneous petroleum reservoir with periodic structure, including capillary effects. We consider a model which leads to a coupled system of partial differential equations which includes an elliptic equation and two nonlinear degenerate parabolic equations of convection–diffusion types. Using two-scale convergence, we get an homogenized model which governs the global behavior of the flow. The determination of effective properties require the numerical resolution of local problems in a standard cell.     

An equivalent pipe network model for free surface flow in porous media

Free surface flow analysis in porous media is challenging in many practical applications with strong non-linearity. An equivalent pipe network model is proposed for the simulation and evaluation of free surface flow in porous media. On the basis of representative elementary volume with homogeneous pore-scale patterns, the pore space of the homogeneous isotropic porous media is conceptualized as a collection of capillary tubes. According to Hagen-Poiseulle's law and flux equivalence principle, equivalent hydraulic parameters and unified governing formulations for the pipe network model are deduced. The two-dimensional free surface flow problem is reduced to a one-dimensional problem of pipe networks and a one-dimensional procedure based on the finite element method is then developed by introducing a continuous penalized Heaviside function. The proposed equivalent pipe network model is verified with results from numerical solutions and laboratory-measured data available in the literature, and good agreements are obtained. The proposed equivalent pipe network model is shown to be effective in analyzing the free surface flow in porous media. The numerical results also indicate that the proposed equivalent pipe network model has weak sensitivity of the mesh size and penalty parameters.     

A cellular-automata method for studying porous media properties

A cellular-automata (CA) approach for investigating properties of porous media with tortuous channels and different smoothness of pore walls is proposed. This approach is aimed at combining two different CA models: the first one is intended for constructing the morphology of a porous material; the second, for simulating a fluid flow through it. The porous media morphology is obtained as a result of evolution of a cellular automaton, forming a “steady pattern.” The result is then used for simulating a fluid flow through a porous medium by applying the Lattice Gas CA model. The method has been tested on a small fragment of a porous material and implemented for investigating a carbon electrode of a hydrogen fuel cell on a multiprocessor cluster.