Description of Fluid and Species Transport
The preceding treatment of the porous media impairment phenomena implies that the suspended particle and dissolved species concentrations may be different in the plugging and nonplugging pathways. Then, separate sets of balance equations are required for the plugging and nonplugging pathways. Consequently, the numerical solution would require highly intensive computational effort. However, this problem can be conveniently circumvented by assuming that there is hydraulic interaction between these pathways (i.e., they are not isolated from each other).
1. The mass balances are considered for the following four pseudocomponents
c. Suspended paraffins and asphaltenes
d. Dissolved paraffins and asphaltenes
2. Total thermal equilibrium energy balance is considered
3. Non-Newtonian fluid description using the Rabinowitsch-Mooney equation is resorted
4. The Forhheimer equation for the non-Darcy flow description is used
5. The average flow is defined as a volume fraction weighted linear sum of the flow through the plugging and nonplugging paths according to Gruesbeck and Collins (1982)
6. The average permeability is defined as a volume weighted linear sum of the permeabilities of the plugging and nonplugging paths according to Gruesbeck and Collins (1982)
a. In the plugging paths, a snowball deposition effect is represented by an exponential decay function
b. In the nonplugging paths, a gradual pore size reduction, represented by the power law function, is considered
7. The precipitation of the asphaltene and paraffin is predicted, applying Chung's (1992) thermodynamic model for non-ideal solutions to determine the cloud point and the quantity of the precipitates to be formed.
The total mass balance of the gas component is given by:
The first, second, and third terms respectively represent the accumulation, transport, and well production. Assuming that the oil component exists only in the liquid phase and does not vaporize into the gas phase, the oil component mass balance is given by:
for which Ring et al. (1994) assumed woL = 1.0. Considering that organic precipitates only exist in the liquid phase, because it is the wetting phase for these particles, the suspended paraffin and asphaltene particle mass balances are expressed by:
If the particle density, pp, is assumed constant, and the suspended particle content is expressed by the volume fraction of organic substance (paraffin or asphaltene), σpL, according to Eq. 14-95, then Eq. 14-94 can be simplified as:
Note that both Ring et al. (1994) and Civan (1996) neglected the term on the right, representing the dispersion of particles. The mass balances of the paraffin and asphaltene dissolved in oil is given by:
S is the saturation, p is the density, t is the time, x is distance, u is the volume flux, σp L is the volume fraction of the organic precipitates in the liquid phase, wp,L denotes the mass fraction, xiL is the mole fraction of organic dissolved in the oil, Mi is the molecular weight and DiL is the dispersion coefficient. 3ԑ,/3t represents the volume rate of retention of organic deposits in porous media determined according to Eqs. 14-68, 69, and 76. Assuming that the various phases are at thermal equilibrium at a temperature of Tv = TL = Ts = T , the total porous media energy balance is given by:
where U and H are the internal energy and enthalpy, respectively, q is the energy loss, p is pressure, k denotes the thermal conductivity, and T is temperature. Ring et al. (1994) simplified Eq. 14-98 as:
The first, second, and third terms represent the accumulation, convection, and conduction heat transfer. The last terms represent the heat carried away by production at wells, heat losses into formation surrounding the reservoir and the external heat losses. The deposition of organic precipitates in porous media reduces the flow passages causing the fluids to accelerate. Therefore, Darcy's law is modified as following, considering the inertial effects, according to the Forchheimer equation (Civan, 1996):
where K is the permeability, PJ is the fluid pressure, and the non-Darcy number is given by:
where β is the inertial flow coefficient, and py and |iiy denote the density and viscosity of a fluid phase J. Note that the formulations presented here are applicable for jnultidimensional cases encountered in the field if 3/3x is replaced by V • and a vector-tensor notation is applied.
The source terms appearing on the right of Eqs. 14-92 through 99 are considered a sum of the internal (rock-fluid and fluid-fluid interactions) and external (wells) sources. When the oil is supersaturated, the internal contribution to the source terms in Eq. 14-94 is determined as the excess quantity of organic content of oil above the organic solubility at saturation conditions determined by Chung's (1992) thermodynamic model:
Civan (1995) carried out case studies similar to Ring et al. (1994) using the Sutton and Roberts data (1974). Figure 14-51 shows a comparison of the predicted and measured permeability impairments by paraffin deposition for below and above bubble point pressure cases. Note that, above the bubble point pressure, only the liquid phase exists and there is more severe formation damage. Whereas, below the bubble point pressure, both the liquid and vapor phases exist and there is less severe formation damage.
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