Fractional flow equation

The water cut at the watered-out was 0.98. The oil viscosity was 9 mPa s in the experiments by Wang et al. The water viscosity of 1 mPa s is assumed. According to the fractional flow equation that follows. 

The trapping number defined by Eq. 7.103 for an arbitrary dipping angle is consistent with the conventional Buckley-Leverett fractional flow theory. In the Buckley-Leverett fractional flow equation, the gravity term is multiplied by sina (Leverett, 1941). However, Figure 7.34 shows that the trapped residual saturation predicted by Eq. 7.103 is lower than the experimental data at the same trapping number. This figure compares the relationship between the... 

In several cases, a steady rate of filtration in never achieved. In such cases it is possible to describe the time dependence of filtration by introducing an efficiency factor fi representing the fraction of filtered particles remaining in the filter cake rather than being swept along by the bulk flow. Equation 16.7.4 then becomes... 

Zhou et al. [55], The most effective method to assess the capacity is the flow simulation which includes volumetric formulas and more reservoir parameters rather than other methods [56], Mass balance and constitutive relations are accounted in mathematical models to capacity assessment and dimensional analysis consists of fractional flow formulation with dimensionless assessment and analytical approaches [33], From the formulations demonstrated by Okwen and Stewart for analytical investigation, it can be deduced that the C02 buoyancy and injection rate have affected the storage capacity [57], Zheng et al. have indicated the equations employed in Japanese and Chinese methodology and have noted that some parameters in Japanese relation can be compared to the CSLF and DOE techniques [58]. 

In order to determine the required reactor volume one must relate the temperature (and thus k) and the local pressure P to the fraction conversion using an energy balance and conventional fluid flow equations. 

In the model described in this work every effort has been made to ensure that it embodies physically meaningful parameters. It is inevitcible, however, that some simplistic idealizations of the physical processes involved in GPC must be made in order to arrive at a system of equations which lends itself to mathematical solution. The parameters considered are, the axial dispersion, interstitial volume fraction, flow rate, gel particle size, column length, intra-particle diffusivity, accessible pore volume fraction and mass transfer between the bulk solution eind the gel particles. 

Deriving the compressible, transient form of the stagnation-flow equations follows a procudeure that is largely analogous to the steady-state or the constant-pressure situation. Beginning with the full axisymmetric conservation equations, it is conjectured that the solutions are functions of time t and the axial coordinate z in the following form axial velocity u = u(t, z), scaled radial velocity V(t, z) = v/r, temperature T = T(t, z), and mass fractions y = Yk(t,z). Boundary condition, which are applied at extremeties of the z domain, are radially independent. After some manipulation of the momentum equations, it can be shown that... 

A closed form similarity solution for the nonlinear time-dependent slow-flow equations has been used as the basis for a simple, time-dependent, analytic model of localized ignition which requires minimal chemical and physical input (8). As a fundamental part of the model, there are two constants which must calibrated the radii, or fraction of the time-dependent simi-... 

Volume fraction Flow resistance parameter Obstruction factor (rate equation)... 

For each compartment in the configuration, apply the mass-balance law to obtain the differential equation expressing the variation of amount per unit of time. In these expressions, constant or variable fractional flow rates k can be used. 

The deterministic model with random fractional flow rates may be conceived on the basis of a deterministic transfer mechanism. In this formulation, a given replicate of the experiment is based on a particular realization of the random fractional flow rates and/or initial amounts 0. Once the realization is determined, the behavior of the system is deterministic. In principle, to obtain from the assumed distribution of 0 the distribution of common approach is to use the classical procedures for transformation of variables. When the model is expressed by a system of differential equations, the solution can be obtained through the theory of random differential equations [312-314]. 

Besides the deterministic context, the predicted amount of material is subjected now to a variability expressed by the second equation. This expresses the random character of the fractional flow rate, and it is known as process uncertainty. Extensive discussion of these aspects will be given in Chapter 9. 

InPFcombustionmodelinginorderto accurately tackle the physical problem, the Euler approach has been employed to numerically investigate the flow characteristics, heat transfer, and species transport by solving the time-averaged global mass, momentum, energy, and species mass fractions conservahon equations. 

It is assumed that the foam remains at the hmiting capillary pressure independently of pressure gradient and gas and liquid flow rates. This implies that the water saturation remains at over a range of water fractional flows (approximately 0 < Iw < 0.2). The equations of multiphase flow can be manipulated to yield expressions for the pressure gradient and gas mobility when the water saturation eqnals S. ... 

Because the density variation in the flow equations is more important than in the chemical equations, p can be considered constant in the chemical equations. For constant D, and using the equation of mixture fraction (5.69), one sees the following results ... 

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