Darcy averaging

Coarse (1.5 Darcy) Average (0.35 Darcy) Fine (0.06 Darcy)... 

For a single fluid flowing through a section of reservoir rock, Darcy showed that the superficial velocity of the fluid (u) is proportional to the pressure drop applied (the hydrodynamic pressure gradient), and inversely proportional to the viscosity of the fluid. The constant of proportionality is called the absolute permeability which is a rock property, and is dependent upon the pore size distribution. The superficial velocity is the average flowrate... 

Heterogeneity, nonuniformity and anisotropy are terms which are defined in the volume-average sense. They may be defined at the level of Darcy s law in terms of permeability. Permeability, however, is more sensitive to conductance, mixing and capillary pressure than to porosity. 

Heterogeneity, nonuniformity and anisotropy are defined as follows. On a macroscopic basis, they imply averaging over elemental volumes of radius e about a point in the media, where e is sufficiently large that Darcy s law can be applied for appropriate Reynolds numbers. In other words, volumes are large relative to that of a single pore. Further, e is the minimum radius that satisfies such a condition. If e is too large, certain nonidealities may be obscured by burying their effects far within the elemental volume. 

In the fluid flow model, simulation is based on Darcy s law for the steady flow of Newtonian fluids through porous media. This law states that the average... 

If calculated pressure drop from inlet to outlet of line system (not including control or hand valves) is greater than approximately 10%, but less than about 40% of the inlet pre.s.sure Pi (pounds per. square inch gauge), the Darcy equation will yield reasonable accuracy when using a specific volume based on the average of upstream (inlet) and do vnstream (outlet)... 

The flow velocity, pressure and dynamic viscosity are denoted u, p and fj and the symbol (...) represents an average over the fluid phase. Kim et al. used an extended Darcy equation to model the flow distribution in a micro channel cooling device [118]. In general, the permeability K has to be regarded as a tensor quantity accounting for the anisotropy of the medium. Furthermore, the description can be generalized to include heat transfer effects in porous media. More details on transport processes in porous media will be presented in Section 2.9. 

A fine aqueous suspension containing 1 lbm of solids per cubic foot of suspension is to be filtered in a constant pressure filter. It is desired to filter at an average rate of 100 gpm, and the filter cake must be removed when it gets 2 in. thick. What filter area is required Data —AP = 10 psi, p(wet cake) = 85 lbm/ft3, K (permeability) = 0.118 Darcy, a = 2 x 109ft 1. 

Darcy s law asserts that the average flow velocity, v, of a fluid through a packed bed is proportional to the pressure gradient, AP, across the bed and is inversely proportional to the length, /, of the bed. Thus,... 

In this equation ut should be interpreted as the volumetric flux density (directional flow rate per unit total area). The indexes range from 1 to 3, and repetition of an index indicates summation over that index according to the conventional summation convention for Cartesian tensors. The term superficial velocity is often used, but it is in our opinion that it is misleading because n, is neither equal to the average velocity of the flow front nor to the local velocity in the pores. The permeability Kg is a positive definite tensor quantity and it can be determined both from unidirectional and radial flow experiments [20], Darcy s law has to be supplemented by a continuity equation to form a complete set of equations. In terms of the flux density this becomes ... 

Advection is the transport of dissolved contaminant mass due to the bulk flow of groundwater, and is by far the most dominant mass transport process [2]. Thus, if one understands the groundwater flow system, one can predict how advection will transport dissolved contaminant mass. The speed and direction of groundwater flow may be characterized by the average linear velocity vector (v). The average linear velocity of a fluid flowing in a porous medium is determined using Darcy s Law [2] ... 

The hierarchy of equations thereby obtained can be closed by truncating the system at some arbitrary level of approximation. The results eventually obtained by various authors depend on the implicit or explicit hypotheses made in effecting this closure—a clearly unsatisfactory state of affairs. Most contributions in this context aim at calculating the permeability (or, equivalently, the drag) of a porous medium composed of a random array of spheres. The earliest contribution here is due to Brinkman (1947), who empirically added a Darcy term to the Stokes equation in an attempt to represent the hydrodynamic effects of the porous medium. The so-called Brinkman equation thereby obtained was used to calculate the drag exerted on one sphere of the array, as if it were embedded in the porous medium continuum. Tam (1969) considered the same problem, treating the particles as point forces he further assumed, in essence, that the RHS of Eq. (5.2a) was proportional to the average velocity and hence was of the explicit form... 

Fig. 10.17. Average capillary pressure as a function of gas fractional flow Us = 0.02 cm min 1 porous medium of 0.05 mm sand tube length L = 60 cm permeability 72 pm2 (darcy) porosity - 0.31 surfactant - 1% SiponateDS 10.

The population balance simulator has been developed for three-dimensional porous media. It is based on the integrated experimental and theoretical studies of the Shell group (38,39,41,74,75). As described above, experiments have shown that dispersion mobility is dominated by droplet size and that droplet sizes in turn are sensitive to flow through porous media. Hence, the Shell model seeks to incorporate all mechanisms of formation, division, destruction, and transport of lamellae to obtain the steady-state distribution of droplet sizes for the dispersed phase when the various "forward and backward mechanisms become balanced. For incorporation in a reservoir simulator, the resulting equations are coupled to the flow equations found in a conventional simulator by means of the mobility in Darcy s Law. A simplified one-dimensional transient solution to the bubble population balance equations for capillary snap-off was presented and experimentally verified earlier. Patzek s chapter (Chapter 16) generalizes and extends this method to obtain the population balance averaged over the volume of mobile and stationary dispersions. The resulting equations are reduced by a series expansion to a simplified form for direct incorporation into reservoir simulators. 

To simplify further the description of foam flow, bubble velocity is removed from the set of internal variables. It is assumed that a fraction, Xr, of all bubbles is carried with velocity, v(x,t TT,o) as flowing foam (f). This assumption is justified because flowing bubbles remain in contact most of the time, and their average velocity, , can be related to Darcy s velocity of the gas phase, = u /([Pg.328]

The average bubble velocity is related to Darcy s velocity of the gas phase through relation (A-11)... 

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)... 

Here, the superficial velocity, v, represents a fluid state, and the density, p, a fluid property which, for a compressible fluid, can be related to the pressure through an equation of state. The porosity, (p, which is defined as the void fraction within the media, is a macroscopic property of the porous material. Sources and/or sinks located within the physical system are represented using y/. Volume averaging the differential momentum balance for the same physical situation yields Darcy s law ... 

As discussed in Section 1, the absolute permeability is a constitutive macroscopic property which arises in the local volume-averaged momentum balance, which is Darcy s law (Eq. (2)). It is interesting that, while the porosity can be defined independently of the equation of continuity, the permeability is... 

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