Porous media single-phase flow

Fio. 9. Streamlines showing steady-state velocity profile during single-phase flow in a reconstructed porous medium. 

The effects of the pressure drawdown on the oil-production rate are shown in Figure 2. Both the experimental data and theoretical calculations based on the Darcy s law are shown. Initially the oil-production rate increased linearly with the increasing pressure drawdown across the porous medium, in agreement with Darcy s law for single-phase flow through porous media. At low drawdown pressures, the dissolved gas remained largely in solution, and therefore, the oil was flowing as a... 

In the two-medium treatment of the single-phase flow and heat transfer through porous media, no local thermal equilibrium is assumed between the fluid and solid phases, but it is assumed that each phase is continuous and represented with an appropriate effective total thermal conductivity. Then the thermal coupling between the phases is approached either by the examination of the microstructure (for simple geometries) or by empiricism. When empiricism is applied, simple two-equation (or two-medium) models that contain a modeling parameter hsf (called the interfacial convective heat transfer coefficient) are used. As is shown in the following sections, only those empirical treatments that contain not only As/but also the appropriate effective thermal conductivity tensors (for both phases) and the dispersion tensor (in the fluid-phase equation) are expected to give reasonably accurate predictions. 

Single-phase fluid flow in porous media is a well-studied case in the literature. It is important not only for its application, but the characterization of the porous medium itself is also dependent on the study of a single-phase flow. The parameters normally needed are porosity, areal porosity, tortuosity, and permeability. For flow of a constant viscosity Newtonian fluid in a rigid isotropic porous medium, the volume averaged equations can be reduced to the following the continuity equation,... 

Shear Factor, F. The shear factor F is a generalized hydrodynamic resistivity of porous media. It appears in the momentum equation 63 and is needed to solve the problem of single-phase flow in porous media. The shear factor F can be related to the pressure drop of a unidirectional flow without bounding wall effects, that is, in a one-dimensional medium through equation 21. In this section, we give a detailed account for the derivation of the expressions for fv and F. 

It should be noted that most of water is stored between the yams rather than within them. In the other words, all the water can be accommodated by the pores within the yams, and it seems likely that the water is chiefly located there. It should be noted that pores of different sizes are distributed within a fabric (Fig. 1). By a porous medium we mean a material contained a solid matrix with an interconnected void. The interconnectedness of the pores allows the flow of fluid through the fabric. In the simple situation ( single phase flow ) the pores is saturated by a single fluid. In two-phase flow a liquid and a gas share the pore space. As it is shown clearly in Fig.l, in fabrics the distribution of pores with respect to shape and size is irregular. On the pore scale (the microscopic scale) the flow quantities (velocity, pressure, etc.) will clearly be irregular. 

Homogeneous Models. The basic assumption in these models is that the emulsion is a continuum, single-phase liquid that is, its microscopic features are unimportant in describing the physical properties or bulk flow characteristics. It ignores interactions between the droplets in the emulsions and the rock surface. The emulsion is considered to be a single-phase homogeneous fluid, and its flow in a porous medium is modeled by using well-documented concepts of Newtonian and non-Newtonian fluid flow in porous media (26, 38). 

Darcy s law (Darcy, 1856) is a phenomenological law that is valid for the viscous flow of a single-phase fluid (e.g. groundwater flow) through porous media in any direction. This basic law of fluid flow is a macroscopic law providing averaged descriptions of the actual microscopic flow behaviour of the fluids over some representative elementary volume of the porous medium. For isothermal and isochemical subsurface conditions, the law can be written as (Hubbert, 1953)... 

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.)... 

Analysis of time-dependent consolidation requires the solution of Biot s consolidation equations coupled to the equations describing flow. The transient hydro-mechanical coupling between pore pressure and volumetric strain for a linear elastic, mechanically isotropic porous medium and fully saturated with a single fluid phase (i.e. water), is given by the fluid continuity equation ... 

Foam with large and less stable bubbles is less likely to flow as a single fluid. Mast and Fried deduced that foam is propogated inside a porous medium by the breaking and reforming of foam bubbles. The gas flows as a discontinuous phase while toe liquid is transported as a free phase via toe film network. Nahid proposed that toe gas flow could be treated according to Darcy s law if a correction factor for the gas permeability is used. 

In the model developed by Iliuta et al. [120], a cocurrent two-phase trickle flow through a porous medium of uniform initial porosity and single-sized catalytic particles is considered (Figure 5.7a and b). Two-phase flow is assumed unidirectional, and both the flowing phases are assumed as viscous Newtonian. The gas/liquid + fine/porous medium + fine multiphase system is viewed as a system of three interpenetrated continua (i) a flowing gas, (ii) a dilute pseudohomogeneous suspension consisting of liquid and fine particles, and (iii) a stationary... 

The prepreg fibre bed is typically assumed to be an elastic porous medium with incompressible and inextensible fibres and fully saturated with the resin. The resin is assumed to flow in the pores between the fibres, and the fibre mass in the laminate remains constant during cure. The governing equations of the system must describe the behaviour of the composite constituents the fibre bed and the resin. Firstly, the equilibrium of forces on the representative element is considered. Secondly, the mass conservation for the representative element must be satisfied. For a porous medium saturated with a single phase fluid, the total stress tensor a,) is separated into two parts as (tensile stresses are considered positive) ... 

Flow through a porous medium is described by Darcy s law, which, in its basic form, is only valid for a single flnid flow and was extended to a second fluid by Wyckoff Botset (1936). This yields equation 10 for the flow velocity of the fluid and gas phase, respectively. 

The numerical simulation of single-phase fluid flow in fibrous porous medium is considered in this paper. The object of the study is the square domain which includes 4 cylinders (see Fig. 1). Planar flow that perpendicular to the axes of cylinders is considered in this paper. This model is based on the numerical solution of the Navier-Stokes equations for incompressible fluid flow ... 

The general development of capillary bundle/non-Newtonian flow models usually takes the following pattern the Darcy velocity, u, for single-phase steady-state flow of fluid through a porous medium of length L is given by rearranging Equation 6.1 as follows ... 

The rheological description of foam flow in porous media has been treated in different ways. one approach has been to use the single-phase fluid viscosities to calculate relative permeabilities to each fluid on the basis of experimental measurements of flow rates and pressure drop in foam flow through a porous medium. 

Another approach to describing foam rheology has been to calculate an apparent foam viscosity from flow-rate and pressure-drop measurements made during flow through a porous medium, Darcy s law is used with the rock permeability, or water relative permeability if an oil phase is present, to calculate apparent foam viscosity. Results show that the apparent viscosity calculated in this manner is a strong function of foam quality, decreasing approximately linearly as foam quality increases. Ifiis approach essentially treats the foam as a single phase. 

Physical systems are modelled by postulating a relationship between three objects - the properties, 95, the state and the auxiliary data, ipa- The auxiliary data and the properties together are referred to as input. The state is sometimes called the output. The properties characterise the unchanging aspects of the system the state characterises the aspects of the system that respond to different selections of the auxiliary data. The auxiliary data corresponds to those aspects of the system that are under human, or other, control. An example is that of single phase fluid flow in a porous medium the properties are the permeability, the state is the pressure and the flux. The auxiliary data are the boundary conditions imposed on the flow system. In a time dependent problem, the auxiliary data will also include the initial conditions. 

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