Porous media coupling

Gens, A. Olivella, S. 2000. Non-isothermal multiphase flow in deformable porous media. Coupled formulation and application to nuclear waste disposal. In Developments in Theoretical Geomechanics. Smith Carter eds., Balkema, Rotterdam, p. 619-640. 

The simple pore structure shown in Figure 2.69 allows the use of some simplified models for mass transfer in the porous medium coupled with chemical reaction kinetics. An overview of corresponding modeling approaches is given in [194]. The reaction-diffusion dynamics inside a pore can be approximated by a one-dimensional equation... 

Lichtner, P. C., 1988, The quasi-stationary state approximation to coupled mass transport and fluid-rock interaction in a porous medium. Geochimica et Cosmochimica Acta 52, 143-165. 

Abstract A general theoretical and finite element model (FEM) for soft tissue structures is described including arbitrary constitutive laws based upon a continuum view of the material as a mixture or porous medium saturated by an incompressible fluid and containing charged mobile species. Example problems demonstrate coupled electro-mechano-chemical transport and deformations in FEMs of layered materials subjected to mechanical, electrical and chemical loading while undergoing small or large strains. 

The coupling effects of various poromechanical processes on the response of a porous medium have been successfully addressed by Biot s theory of poroelasticity and its extensions [3,4,5,8,2], The chemical effects have also been addressed by considering interaction between the porous matrix and a pore fluid comprising of a solute and solvent [10, 7, 6], Comprehensive anisotropic poromechanics formulations and corresponding solutions for the inclined borehole problem have been presented [4—2], However, the coupled chemo-thermo-hydro-mechanical response of an anisotropic porous medium has not been addressed to date. 

The release of soluble species contained in a porous medium in contact with water is the result of complex and coupled phenomena. This includes water transfer in the porous medium up to saturation dissolution of the species in the pore water according to the local chemical context and transport of species in solution due to the effect of concentration gradients. 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

Core Floods. At present the strong coupling between droplet size and flow has major experimental consequences (1) flow experiments must be performed under steady-state conditions (since otherwise the results may be controlled by long-lived, uninterpretable transients) (2) in situ droplet sizes cannot be obtained from measurements on an injected or produced dispersion (because these can change at core faces and inside the core) and (3) care must be taken that pressure drops measured across porous media are not dominated by end effects. Likewise, since abrupt droplet size changes can occur inside a porous medium, if the flow appears to be independent of the injected droplet-size distribution, it is likely that a new distribution is quickly forming inside the medium (38). 

Forces applied to a water-saturated porous medium will cause stresses which result in strain (deformation). The stress, strain and groundwater pressure in a water-saturated porous medium are coupled, as first recognized by Biot (1941). Under the assumed stress conditions, the vertical normal component of total stress (o ) that acts downwards on a horizontal plane at any depth is caused by the weight of the overlying water-saturated rock. This stress is born by the solid matrix of the porous medium (o ) and by the pressure of the groundwater in the pores (p ) (e.g. Hubbert andRubey, 1959)... 

In order to include the coupling between the rugged laminar flow in a porous medium and the molecular diffusion, Horvath and Lin [50] used a model in which each particle is supposed to be surrounded by a stagnant film of thickness 5. Axial dispersion occurs only in the fluid outside this stagnant film, whose thickness decreases with increasing velocity. In order to obtain an expression for S, they used the Pfeffer and Happel "free-surface" cell model [52] for the mass transfer in a bed of spherical particles. According to the Pfeffer equation, at high values of the reduced velocity the Sherwood number, and therefore the film mass transfer coefficient, is proportional to... 

In the DGM model as presented by Mason and Malinauskas [11a] all the different contributions to the transport are taken into accoimt. The wall of the porous medium is considered as a very heavy component and so contributes to the momentum transfer. The model is schematically represented in Fig. 9.12 for a binary mixture (in analogy with an electriccd network). As can be seen from this figure, the flux contributions by Knudsen diffusion /k, and of molecular (continuum) diffusion of the mixture /m,i23re in series and so are coupled. The total flux of component i (i = 1,2) due to these contributions is /j km- Note that /k = /m,i2- The contribution of the viscous flow and of the surface diffusion are parallel with / km J d so are considered independent of each other (no coupling terms, e.g. no transport interaction between gas phase and surface diffusion). 

A new model that explicitly accounts for multiple sources of nonequilibrium influencing solute transport in porous media was presented by Brusseau et al. (1989c, 1990b). The multiprocess nonequilibrium (MPNE) model was designed to simulate solute transport in porous media where both transport-related and sorption-related nonequilibrium processes contribute to the observed nonequilibrium. The sorption dynamics of such systems was represented by two serially arranged bicontinuums coupled in parallel. A schematic of the model conceptualization is shown in Fig. 11-6, taken from Brusseau et al. (1989c). This conceptualization results in discretization of the porous medium into four sorption domains, where instantaneous sorption occurs in the first domain and rate-limited sorption occurs in the other three. 

The reactive chemical transport simulator TIICC (12,13) is being used to study effects on mass transport of precipitation/dissolution reactions. This paper sets forth the mathematical and numerical bases for the coupling of these reactions to mass diffusion in a porous medium and presents results of calculations to demonstrate consequences of the coupling. 

Despite our use of a capillary model to characterize a porous medium, most porous beds employed for chromatographic purposes are random and generally the medium is isotropic. In such media, the effective solute dispersivity still arises from the nonuniform pore velocity coupled with molecular diffusion... 

Though such an analysis leads to a relationship formally identical to Eq. (177) it furnishes no insight into the algebraic signs of the coefficients, nor does it suggest that the origin of these additional terms is due in part to the action of local couples within the porous medium. In the presence of screw-like properties it seems natural to extend Eq. (181) by writing... 

K Coupling, rotation, and translation dyadics (38), (39) Dyadic, triadic and tetradic, resistance coefficients, respectively, for a porous medium (footnote 19)... 

In Biot formalism, there exist several assumptions that restrict its generality and make true liquid-solid coupling impossible. Biot assumed that for a REV in a multiphasic porous medium, a single energy functional could be stipulated to define the energy state. It has been shown that for N continuous contiguous phases, N functionals are needed to fully describe behavior. (For example, simultaneous countercurrent flow of two immiscible liquids is evidence that at least two separate energy functionals are needed.)... 

Abstract This contribution deals with the modeling of coupled thermal (T), hydraulic (H) and mechanical (M) processes in subsurface structures or barrier systems. We assume a system of three phases a deformable fractured porous medium fully or partially saturated with liquid and a gas which remains at atmospheric pressure. Consideration of the thermal flow problem leads to an extensively coupled problem consisting of an elliptic and parabolic-hyperbolic set of partial differential equations. The resulting initial boundary value problems are outlined. Their finite element representation and the required solving algorithms and control options for the coupled processes are implemented using object-oriented programming in the finite element code RockFlow/RockMech. 

A porous medium composed by solid grains, water and gas is considered. Thermal, hydraulic and mechanical aspects are taken into account, including coupling between them in all possible directions. The problem is formulated in a multiphase and multispecies approach. The three phases are solid phase (s), liquid phase (/, water + air dissolved) and gas phase (g, mixture of dry air and water vapour). The three species are solid (-), water (w, as liquid or evaporated in the gas phase) and air (a, dry air, as gas or dissolved in the liquid phase). 

The MOTIF code is a three-dimensional finite-element code capable of simulating steady state or transient coupled/uncoupled variable-density, variable- saturation fluid flow, heat transport, and conservative or nonspecies radionuclide) transport in deformable fractured/ porous media. In the code, the porous medium component is represented by hexahedral elements, triangular prism elements, tetrahedral elements, quadrilateral planar elements, and lineal elements. Discrete fractures are represented by biplanar quadrilateral elements (for the equilibrium equation), and monoplanar quadrilateral elements (for flow and transport equations). 

A recent example is coupled T-H-M modelling of the Tunnel Sealing Experiment (TSX) in URL (Guo et al. 2002). To provide data for preliminary validation a surface laboratory experiment known as the Thermal Evaluation of Material Test (TEMT) was conducted. A steel vessel, 1.47 m long with an interior diameter of 0.74 m, was filled with a medium-grained sand and heated by circulating hot water. An array of thermistors monitored the evolution of temperature. Comparison between the physical test results and MOTIF simulation results. Figure 6, shows that MOTIF can be used to simulate convection dominated heat transfer in a porous medium. 

The theoretical formulation employed in COMPASS is based on considering an unsaturated soil as a three-phase porous medium consisting of solid, liquid and gas. A set of coupled governing differential equations can be developed to describe heat and moisture flow within a deformable porous media. The primary variables of the model are pore water pressure, /, pore air pressure, u , temperature, T and displacement, u. 

To calculate the pore pressure response due to a volume source we use the Green s function based on the effective Biot theory. We write the coupled system of equations directly from the constitutive relations given by Biot (1962). These are the total stress of the isotropic porous medium, the stress in the porous fluid, the momentum balance equation for total stress, and the generalized Darcy s law. Following Parra (1991) and Boutin et al. (1987), the coupled system of differential equations in the... 

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