Multi-Phase and Species Systems in Porous Media
The reservoir formation is considered in three parts:
1 the stationary or deforming solid phase containing
a. porous matrix made from detrital grains, minerals and clays, and
b. the immobile materials attached to the pore surface including authigenic or diagenetic minerals and clays; various deposits; scale forming recipitates such as wax, asphaltene, sulfur, and gels; trapped gas, connate water and residual oil;
2 the flowing or mobile fluid phases including
a. gas,
b. oil,
c. brine, and
d. chemicals used for improved recovery;
3 various types of species that the solid and fluid phases may contain.
Typical species are:
1 ions including the anions such as C1-, HCOɝ-, COɝ-², SО4-² and the cations such as K+, NA+, Ca+², Ba+², Mg+²;
2 molecules such as СH4, CO2, H2S, N2, molasses, polymers, surfactants, paraffin, asphaltene, and resins;
3 pseudocomponents such as gas, oil, and brine with prescribed compositions;
4 particulates such as minerals, clays, sand, gels, paraffin, asphaltene, sulfur, precipitates, crystalline matter, mud fines, debris, and bacteria; and
5 associates such as the pairs of ions and molecules, coagulates of various particulates, micelles, and microemulsions. The characteristics of the particulates play an important role in formation damage. Based on their characteristics, particles can be classified as:
1 indigenous, in-situ generated, or externally introduced;
2 dissolved or nondissolved;
3 water-wet, mixed-wet, or oil-wet;
4 deformable (soft) or nondeformable (hard);
5 sticky or nonsticky;
6 swelling or nonswelling;
7 organic or inorganic;
8 reactive or inert;
9 biological or nonbiological;
10 growing or nongrowing; and
11 associating or nonassociating.
The species content of a system can be expressed in a variety of alternative ways:
The relationship between volume fraction and volume ratio is given by:
In the following formulations, the various phases (solid and fluid) in porous media are denoted by j, s denotes the solid phase, n is the total number of phases, &• is the volume fraction of the j'h phase in porous media, fy is the porosity of porous media, and Sj is the saturation or volume fraction of jth phase in the pore space. The following equations can be written:
The density and velocity of a mixture is given by the volume fraction weighted averages, respectively, as:
Therefore, the density and velocity of a mixture are variable when the composition varies even if the constituents are incompressible. For incompressible systems, it is more convenient to use volumetric balance equations.
Several other relationships, which may be convenient to use in the formulation of the transport phenomena in porous media. The volume flux , My, and the velocity , Vy, of a phase j are related.
Multi-Species and Multi-Phase Macroscopic Transport Equations
The macroscopic description of transport in porous media is obtained by elemental volume averaging (Slattery, 1972). The formulations of the macroscopic equations of conservations in porous media have been carried out by many researchers. A detailed review of these efforts is presented by Whitaker (1999). The mass balances of various phases are given by (Civan, 1996, 1998):
where urj is the fluid flux relative to the solid phase, t is the time and V is the divergence operator. Pj is the phase density, ṁj is the net mass rate of the phase j added per unit volume of phase j. Dj is the hydraulic dispersion coefficient which has been omitted in the petroleum engineering literature. The species i mass balance equations for the water, oil, gas and solid phases are given by:
in whch wtj is the mass fraction of species į in the jth phase, jy denotes the spontaneous or dispersive mass flux of species i in the jth phase given by modifying the equation by Olson and Litton (1992):
where Dij is the coefficient of dispersion of species i in the jth phase, k is the Boltzmann constant, and T is temperature. The first term represents the ordinary dispersive transport by concentration gradient. For particulate species of relatively large sizes the first term may be neglected. The second term represents the dispersion induced by the gradient of the potential interaction energy, Φij. When the particles are subjected to uniform interaction potential field then the second term drops out. The third term represents the induced dispersion of bacterial species by substrate or nutrient, 5, concentration gradient due to the chemotaxis phenomena (Chang et al., 1992). Dsj is the substrate dispersion coefficient. Incorporating Eq. 7-33 into Eq. 7-34 leads to the following alternative form:
Considering the possibility of the inertial flow effects due to the narrowing of pores by formation damage, the Forchheimer (1901) equation is used for the momentum balance. Although more elaborate forms of the macroscopic equation of motion are available, Blick and Civan (1988) have shown that Forchheimer's equation is satisfactory for all practical purposes. The Forchheimer equation for multi-dimensional and multiphase fluids flow can be written for the jth phase as (Civan, 1994; Tutu et al., 1983; Schulenberg and Miiller, 1987):
where the first term is the fluid-content-dependent potential or simply the negative of the "effective stress" due to the interactions of the fluid with the pore surface, g is the gravitational acceleration, g(z-z0) is the potential of fluid due to gravity, z is the positive upward distance measured from a reference at z0 , and Q is the overburden potential, which is the work of a vertical displacement due to the addition of fluid into porous media (Smiles and Kirby, 1993). K and |3 denote the Darcy or laminar permeability and the non-Darcy or inertial flow coefficient tensors, respectively. Krj and βrj; are the relative permeability and relative inertial flow coefficient, respectively. Eq. 7-38 can be written as, for convenience.
The permeability and inertial flow coefficient for porous materials are determined by means of laboratory core flow data and thus correlated empirically (Civan and Evans, 1998). Liu et al. (1995) give:
qj and qja denote the external and interface heat transfer to the phase j per unit volume of phase y; &y is the thermal conductivity of phase j, Note that the enthalpy Hj and internal energy Uj per unit mass of phase j are related according to:
When the system is at thermal equilibrium (i.e. Tw = T0 = Tg = Ts = T} then Eq. 7-48 can be written for each phase and then added to obtain the total energy balance equation as:
Invoking Eq. 7-33, Eq. 7-46 can be written in an alternative form as:
The equation of motion given by Chase and Willis (1992), for deforming porous matrix can be written as following:
The superscript a denotes a quantity associated with the dividing surface, which is moving at a macroscopic velocity of uα, and na is the unit vector normal to the dividing surface. rαs, rαs and rαij are the rates of addition of mass of the porous matrix, the th phase, and the species i in the jth phase, respectively. [I ... I] denotes a jump in a quantity across a dividing surface defined by:
where the signs + and - indicate the post and fore sides, respectively, of the dividing surface.
References
Blick, E. F., & Civan, F., "Porous Media Momentum Equation for Highly Accelerated Flow," SPE Reservoir Engineering, Vol. 3, No. 3, 1988, pp. 1048-1052.
Chang, M.-M., Bryant, R. S., Stepp, A. K., & Bertus, K. M. "Modeling and Laboratory Investigations of Microbial Oil Recovery Mechanisms in Porous Media," Topical Report No. NIPER-629, FC22-83FE60149, U.S. Department of Energy, Bartlesville, Oklahoma, 1992, p. 27.
Chase, G. G., & Willis, M. S., "Compressive Cake Filtration," Chem. Eng. Sci., Vol. 47, 1992, pp. 1373-1381.
Civan, F., "Waterflooding of Naturally Fractured Reservoirs—An Efficient Simulation Approach," SPE Production Operations Sympsoium, March 21-23, 1993, Oklahoma City, Oklahoma, pp. 395-407.
Civan, F. Predictability of Formation Damage: An Assessment Study and Generalized Models, Final Report, U.S. DOE Contract No. DE-AC22- 90-BC14658, April 1994.
Civan, F. "A Multi-Phase Mud Filtrate Invasion and Well Bore Filter Cake Formation Model," SPE Paper No. 28709, Proceedings of the SPE International Petroleum Conference & Exhibition of Mexico, October 10-13, 1994, Veracruz, Mexico, pp. 399-412.
Civan, F. "A Multi-Purpose Formation Damage Model," SPE 31101 paper, Proceedings of the SPE Formation Damage Symposium, Lafayette, LA, February 14-15, 1996, pp. 311-326.
Civan, F., "Quadrature Solution for Waterflooding of Naturally Fractured Reservoirs," SPE Reservoir Evaluation and Engineering, April 1998, pp. 141-147.
Civan, F, & Evans, R. D., "Determining the Parameters of the Forchheimer Equation from Pressure-Squared vs. Pseudopressure Formulations," SPE Reservoir Evaluation and Engineering, February 1998, pp. 43-46.
Forchheimer, P., "Wasserbewegung durch Boden," Zeitz. ver. Deutsch Ing. Vol. 45, 1901, pp. 1782-1788.
Liu, X., Civan, F., & Evans, R. D. "Correlation of the Non-Darcy Flow Coefficient, J. of Canadian Petroleum Technology, Vol. 34, No. 10, 1995, pp. 50-54.
Olson, T. M., & Litton, G. M. "Colloid Deposition in Porous Media and an Evaluation of Bed-Media Cleaning Techniques," Chapter 2, pp. 14-25, in Transport and Remediation of Subsurface Contaminants, Colloidal, Interfacial, and Surfactant Phenomena, Sabatini, D. A. and R. C. Knox (Eds.), ACS Symposium Series 491, American Chemical Society, Washington, DC (1992).
Schulenberg, T., & Miiller, U., "An Improved Model for Two-Phase Flow Through Beds of Coarse Particles," Int. J. Multiphase Flow, Vol. 13, No. 1, 1987, pp. 87-97.
Slattery, J. C. Momentum, Energy and Mass Transfer in Continua, McGraw- Hill Book Co., New York, 1972, pp. 191-197.
Smiles, D. E., & Kirby, J. M., "Compressive Cake Filtration—A Comment," Chem. Eng. ScL, Vol. 48, No. 19, 1993, pp. 3431-3434.
Tutu, N. K., Ginsberg, T., & Chen, J. C., "Interfacial Drag for Two-Phase Flow Through High Permeability Porous Beds," Interfacial Transport Phenomena, Chen, J. C. & Bankoff, S. G., (eds.), ASME, New York, pp. 37-44.
Whitaker, S., The Method of Volume Averaging, Kluwer Academic Publishers, Boston, 1999, 219 p.
