Modelling and Applications of Transport Phenomena in Porous Media

Transport phenomenain porous media are encounteredin various disciplines, e. g. , civil engineering, chemical engineering, reservoir engineering, agricul­ tural engineering and soil science. In these disciplines, problems are en­ countered in which various extensive quantities, e. g. , mass and heat, are transported through a porous material domain. Often, the void space of the porous material contains two or three fluid phases, and the various ex­ tensive quantities are transported simultaneously through the multiphase system. In all these disciplines, decisions related to a system's development and its operation have to be made. To do so a tool is needed that will pro­ vide a forecast of the system's response to the implementation of proposed decisions. This response is expressed in the form of spatial and temporal distributions of the state variables that describe the system's behavior. Ex­ amples of such state variables are pressure, stress, strain, density, velocity, solute concentration, temperature, etc. , for each phase in the system, The tool that enables the required predictions is the model. A model may be defined as a simplified version of the real porous medium system and the transport phenomena that occur in it. Because the model is a sim­ plified version of the real system, no unique model exists for a given porous medium system. Different sets of simplifying assumptions, each suitable for a particular task, will result in different models.

1 EIGHT LECTURES ON MATHEMATICAL MODELLING OF TRANSPORT IN POROUS MEDIA.- 1.1 Lecture One: Introduction.- 1.1.1 Porous medium.- 1.1.2 Modelling process.- 1.1.3 Selecting the size of an REV.- 1.2 Lecture Two: Microscopic Balance Equations.- 1.2.1 Velocity and flux.- 1.2.2 The general balance equation.- 1.2.3 Particular balance equations.- 1.2.4 Averaging rules.- 1.3 Lecture Three: Macroscopic Balance Equations.- 1.3.1 General balance equation.- 1.3.2 Particular cases.- 1.3.3 Stress in porous media.- 1.4 Lecture Four: Advective Flux.- 1.4.1 Advective flux of a single fluid that occupies the entire void space.- 1.4.2 Particular cases.- 1.4.3 Multiphase flow.- 1.5 Lecture Five: Complete Transport Model.- 1.5.1 Boundary conditions.- 1.5.2 Content of a complete model.- 1.6 Lecture Six: Modelling Mass Transport of a Single Fluid Phase Under Isothermal Conditions.- 1.6.1 Basic mass balance equations.- 1.6.2 Stationary nondeformable solid skeleton.- 1.6.3 Deformable porous medium.- 1.6.4 Boundary conditions.- 1.6.5 Complete mathematical model.- 1.7 Lecture Seven: Diffusive Flux.- 1.7.1 Diffusive mass flux.- 1.7.2 Diffusive heat flux.- 1.8 Lecture Eight: Modelling Contaminant Transport.- 1.8.1 The Phenomenon of dispersion.- 1.8.2 Fluxes.- 1.8.3 Sources and Sinks.- 1.8.4 Mass balance equation for a single component.- 1.8.5 Balance equations with immobile liquid.- 1.8.6 Balance equations for radionuclide decay chain.- 1.8.7 Two multicomponent phases.- 1.8.8 Boundary Conditions.- 1.8.9 Complete Mathematical Model.- References.- List of Main Symbols.- 2 MULTIPHASE FLOW IN POROUS MEDIA Th. DRACOS Swiss Federal Inst. of Technology (E.T.H.) Zurich, Switzerland.- 2.1 Capillary Pressure.- 2.1.1 Interfacial tension, contact angle and wettability.- 2.1.2 Interfacial curvature and capillary pressure.- 2.1.3 Equilibrium between a liquid and its vapor.- 2.1.4 Microscopic domain.- 2.1.5 Macroscopic space.- 2.1.6 Phase distribution in the pore space.- 2.2 Flow Equations for Immiscible Fluids.- 2.3 Mass Balance Equations.- 2.4 Simultaneous Flow of Two Fluids having a Small Density Difference.- 2.5 Measurement of the relations pc?i(S?i), and kr,?i(S?i).- 2.6 Mathematical descripton of the relations between pc,wSwand k,r,w.- 2.7 Complete Statement of Multiphase Flow Problems.- 2.8 Solute transport in multiphase flow through porous media.- References.- List of Main Symbols.- 3 PHASE CHANGE PHENOMENA AT LIQUID SATURATED SELF HEATED PARTICULATE BEDS J-M. BUCHLIN and A. STUBOS von Karman Institute for Fluid Dynamics Rhode Saint Genèse B-1640, Belgium.- 3.1 Introduction.- 3.2 Preboiling Phenomenology.- 3.3 Boiling regime and dryout heat flux.- 3.4 Constitutive Relationships-Bed Disturbances.- 3.4.1 Introduction.- 3.4.2 Bed permeability.- 3.4.3 Relative permeabilities and passabilities.- 3.4.4 Capillary pressure.- 3.4.5 Bed structural changes.- 3.5 Conclusions.- A. Zero-Dimensional Model.- B. Fractional downward heat flux by conduction.- C. Sub cooled zone thickness at the top of the bed.- References.- List of Main Symbols.- 4 HEAT TRANSFER IN SELF-HEATED PARTICLE BEDS SUBMERGED IN LIQUID COOLANT KENT MEHR and JORGEN WÜRTZ Commission of the European Communities Joint Research Centre, Ispra, Italy.- 4.1 The PAHR Scenario.- 4.2 Specific PAHR Phenomena.- 4.2.1 Bed characteristics.- 4.2.2 Heat conduction.- 4.2.3 Boiling debris beds.- 4.2.4 Dryout.- 4.2.5 Downward boiling.- 4.2.6 Unsteady state.- 4.2.7 Channeling.- 4.3 PAHR-2D.- 4.3.1 Basic equations.- 4.3.2 Spatial discretisation.- 4.3.3 Boundary conditions.- 4.3.4 Time integration.- 4.3.5 Solution procedure.- 4.3.6 D10 Post test calculation.- 4.3.7 Steep power ramp.- 4.4 In-pile experiments.- 4.4.1 Boiling.- 4.4.2 Dryout.- 4.4.3 Bed disturbance.- References.- List of Main Symbols.- 5 PHYSICAL MECHANISMS DURING THE DRYING OF A POROUS MEDIUM CH. MOYNE, CH. BASILICO, J. CH. BATSALE and A._DEGIOVANNI. Laboratoire d’Energétique et de Mécanique Théorique et Appliquée U.A. C.N.R.S. 875, Ecoles des Mines, Nancy, France.- 5.1 General Aspects of the Drying Process.- 5.1.1 The three drying periods.- 5.1.2 The characteristic drying curve concept.- 5.1.3 Conclusion.- 5.2 A General Model for Simultaneous Heat and Mass Transfer in a Porous Medium.- 5.2.1 The fundamental hypotheses of the model.- 5.2.2 Phenomenological laws.- 5.2.3 Conservation laws.- 5.2.4 System to be solved.- 5.2.5 Numerical solution.- 5.3 Application to Drying.- 5.3.1 High temperature convective drying.- 5.3.2 Low temperature convective drying.- 5.3.3 The diffusion model.- 5.3.4 Receding drying front.- 5.4 Conclusions.- References.- List of Main Symbols.- 6 STOCHASTIC DESCRIPTION OF POROUS MEDIA G. DE MARSILY Ecole des Mines de Paris, l’Université Pierre et Marie Curie Paris, France.- 6.1 Definition of Properties of Porous Media: The Example of Porosity.- 6.2 Stochastic Approach to Permeability and Spatial Variability.- 6.3 Stochastic Partial Differential Equations.- 6.3.1 Properties of stochastic partial differential equations.- 6.3.2 Spectral methods.- 6.3.3 The method of perturbations.- 6.3.4 Simulation method (Monte-Carlo).- 6.4 Example of stochastic solution to the transport equation.- 6.5 The problem of estimation of a RF by kriging.- 6.6 The intrinsic hypothesis: definition of the variogram.- 6.6.1 The intrinsic hypothesis.- 6.6.2 Determination of the variogram.- 6.6.3 Behaviour of the variogram for large h.- 6.6.4 Behaviour close to the origin.- 6.6.5 Anisotropy in the variogram.- 6.7 Conclusions.- References.- List of Main Symbols.