Darcy velocity

The system states (dependent variables) are the pressure, p, and the superficial (Darcy) velocity, v. The density, p, and viscosity, p, are fluid properties, and g is the acceleration of gravity. The porosity, < )(z), and permeability, fc(z), represent the macroscopic properties of the media. Both are spatially dependent and are represented as continuous functions of position z, as explicitly noted. While the per-... 

Three 1-D column studies were conducted to investigate the effects of EtOH addition (0, 5, and 10% wt) on the micellar solubilization of residual PCE by 4% Tween 80 in a uniform porous medium. The applied Darcy velocities ranged from 3. 3 to 4.9 cm/hr. Periods of flow interruption (8 and 15 hours) were utilized to evaluate PCE mass transfer under no flow conditions. A summary of the experimental conditions and physical... 

Column Experiment Darcy Velocity If e.ss Flow Interrupt l e.ml... 

As the concentration of EtOH increased from 0 to 10%, the effective steady-state mass transfer coefficient declined from 0.17 1/hr to 0.11 1/hr, which was due in part to the change in Darcy velocity. Using correlations developed for Ke,ss as a function of Darcy velocity and alcohol concentration (Taylor, 1999), the effect of EtOH concentration can be evaluated at a single, representative Darcy velocity. For example, using a Darcy velocity of 4.0 cm/hr, the value of Ke,ss would be 0.14, 0.13, and 0.13 for 4% Tween 80, 4% Tween 80 + 5% EtOH and 4% Tween 80 + 10% EtOH, respectively. Thus, the addition of EtOH to 4% Tween 80 had no discemable influence on the effective steady-state mass transfer coefficient. It should be recognized, however, that although the mass transfer coefficient remained essentially unchanged, the steady-state concentration of PCE in the column effluent (C") and the cumulative PCE mass recovery increased substantially as a result of EtOH addition (Table 2). This behavior can be explained by the fact that the equilibrium solubility of PCE (C" sat) increased by more than 50%, from 26,900 mg/L to 42,300 mg/L, with the addition of 10% EtOH. 

Figure 5. Digital images of surfactant fronts for a) Box A flushed with 4% Tween at a Darcy velocity of 4.7 cm/hr (0.4 pore volumes) and b) Box B flushed with 4% Tween 80 + 5% EtOH at a Darcy velocity of 4.8 cm/hr (0.5 pore volumes).

To evaluate the effects of cosolvent on surfactant delivery and PCE recovery, Box B was flushed with 4% Tween 80 + 5% EtOH at a Darcy velocity of 4.8 cm/hr. The surfactant/cosolvent mixture, which had a density of 0.994 g/cm3, was also representative of a neutral buoyancy flood solution (Shook et al, 1998). It is important to recognize that "neutral buoyancy" refers to density of flushing fluid after solubilization of the DNAPL. Thus, the initial density of the surfactant formulation must be less than that of the resident aqueous phase. Figure 5b shows the location and shape of the 4% Tween 80 + 5% EtOH front after flushing Box B with 0.5 pore volumes of solution. The lower density of the 4% Tween 80 + 5% EtOH solution (0.994 g/cm3) relative to the density of resident pore water (0.998 g/cm3) caused the injected solution to flow preferentially along the top of Box B (Figure 5b). This effect can become severe at low flow rates (Taylor, 1999). The... 

The water permeability k, is obtained from pressure aiHfiDcin versus Darcy velocity Vj curves for different... 

Figure 8.6 Temispack reconstruction of the present-day distribution of overpressures induced by compaction disequilibrium for two assumed conditions of fault permeability a. faults are assumed to be permeable b. faults are assumed to be impermeable (arrows Darcy velocity) (from Burrus et al., 1991, Geological Society Special Publication no. 59, Fig. 7, p. 97. Reprinted by permission).

W = width of source area parallel to groundwater flow direction (cm) Ugw = groundwater Darcy velocity (cm/yr)... 

Here, the longitndinal dispersion coefficient Dl is nsed, and v is the interstitial velocity eqnal to the Darcy velocity, n, divided by the porosity, ( ). Similarly, the mass that leaves from the right side is... 

Note that Eq. 2.66 is derived from the interstitial velocities v and Vq. Obviously, Eq. 2.66 also holds for Darcy velocities. 

Here, the constants, K (mPa s ) and n (dimensionless), are the consistency index and the exponent, respectively Up is the Darcy velocity (m/s) of the polymer-containing water phase k is the average permeability in m is the water phase relative permeability S is water satnration (fraction) is porosity (fraction) is the viscosity at infinite shear rate and C is an empirical constant. Note that Eq. 5.23 is made more general by including the nonunit water saturation, Sw, and the water relative permeability, k,w, as was done previously by Hirasaki and Pope (1974). To consider the polymer permeability reduction fector Ffa explicitly (to be discussed later), we should divide the permeability k by Ekr, and n, is substitnted for Up. Then Eq. 5.23 becomes... 

Here, Cei is a constant. Different researchers used different values for the constant Ceh for example, Marshall and Mentzner (1964) considered Cei to be 1 or 0.5 Sadowski and Bird (1965) considered both Cei and equal to 1. Also in this formula, u is the Darcy velocity in m/s, is the porosity in fraction, and dp is the grain particle diameter. 

It is difficult to use a single equation to describe the viscosity in the entire shear rate range. In developing a comprehensive model for apparent viscosity, Papp, Delshad et al. (2008) assumed that its dependence on Darcy velocity (or equivalent shear rate) consists of two parts the shear-viscosity-dominant part, Psii, and elongational-viscosity-dominant part, Peii... 

Then the apparent viscosity expression that covers the entire range of Darcy velocity is... 

In this equation, ( ) is the porosity in fraction, and u is the Darcy velocity of the displacing fluid. The velocity used by Abrams (1975) is v/[( )(Soi - Sm)]. He also modified the capillary number by multiplying the viscosity ratio (liw/lio)°-" ... 

The difference between Eqs. 7.84 and 7.85 is that the Darcy velocity is used in Eq. 7.84, whereas the pressure gradient and permeability are used in Eq. 7.85. When core flood tests were run at a constant velocity (thus, a constant v a/a), it appeared that the oil recovery increased for those rocks with lower permeabilities (Taber et al., 1973). Clearly, the higher recovery was not a... 

We also assume that these boundaries are impermeable, so that the Darcy velocity satisfies the kinematic condition u7 n = 0. No other boundary conditions can be satisfied by the velocity, as explained in Subsection C. 1. Finally, the thermal energy equation is unchanged from its usual form,... 

A uniform and constant Darcy velocity of 15 myear-1 and effective porosity of 0.3 were used along the entire cross-section. The tailings fluid has relatively high chloride concentrations (0.016 mol L 1) compared with the background water has only 0.0007 molL-1. It is widely believed that Cl- acts as a conservative solute in most aquifer systems, and thus its distribution can be used to retrieve dispersivity for the aquifer. By trial and error, a longitudinal dispersivity between 10 to 15 m appears to fit the concentration differences best in monitor wells sampled in September, 1994. It... 

Coupled models are computationally expensive. In order to develop a manageable model, the expansion of chemical reactions often comes at the expense of physical hydrogeology. Commonly, a uniform and constant Darcy velocity was assumed (e.g., Zhu et al., 2001a). Often, the transport problem is reduced to one or two dimensions. 

Transient Displacement. Experimental displacement results for the simultaneous injection of aqueous surfactant solution and nitrogen into a core initially saturated with a surfactant solution are shown in Figures 12 and 13. Darcy velocities relative to the exit pressure of 4.8 MPa are 0.43 m/day (1.4 ft/day) for gas and 0.046 m/day (0.15 ft/day) for liquid yielding a gas fractional flow or foam quality of 90%. Figure 12 provides the transient liquid saturation profiles. Experimental data points are connected by dashed lines. Time is expressed nondimensionally in pore volumes, PV, which is the ratio of total volumetric flow rate (at exit pressure) multiplied by elapsed time and divided by the void volume of the core. 

In that same publication, Heller et al. (22) also designed an experiment to determine the steady-state mobility of a foam consisting of surfactant brine and either isooctane at ambient pressure or liquid C02 at about 1000 psia (6.9 MPa). In their reported experiments, the measured relative mobilities of both isooctane and dense C02 ranged from somewhat less than 0.1 to 0.7 cP-1, showing an increase in relative mobility with increasing flow rate for combined Darcy velocities from 15 up to 250 ft/day (4.6 to 76 m/day). 

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