Rate Equations for Participate Processes in Porous Matrix
Ohen and Civan (1993) classified the indigenous particles that are exposed to solution in the pore space in two groups:
lump of total expansive (swelling, that is, total authigenic clay that is smectitic) and lump of total nonexpansive (nonswelling) particles, because of the difference of their rates of mobilization and sweepage from the pore surface. They considered that the particles in the flowing suspension are made of a combination of the indigenous particles of porous media entrained by the flowing suspension and the external particles introduced to the porous media via the injection of external fluids.
They considered that the particles of the flowing suspension can be redeposited and reentrained during their migration through porous media and the rates of mobilization of the redeposited particles should obey a different order of magnitude than the ndigenous particles of the porous media. Further, they assumed that the deposition of the suspended particles over the indigenous particles of the porous media blocks the indigenous particles and limits their contact and interaction with the flowing suspension in the pore space.
They considered that the swelling clays of the porous media can absorb water and swell to reduce the porosity until they are mobilized by the flowing suspension. The rate at which the various paniculate processes occur in porous media can be expressed by empirical equations. These equations can also be considered as the particulate material balance equations for the porous matrix. Here they are written as volume balance of particles.
The rate of surface deposition is proportional to the particle mass flux, uop, where Gp is the particle volume concentration in the flowing suspension, and the pore surface available for deposition that relates to (|>2/3 (Lichtner, 1992; Civan, 1995, 1996); kd is a deposition rate constant; a is a stationary deposition factor expressing deposition at stationary conditions; and zd is the volume fraction of the particles in the bulk media retained at the pore surface. Thus, the surface deposition rate equation is given by:
Pore Filling After Pore Throat Plugging
As stated by Chang and Civan (1991, 1992, 1997) and Ochi and Vernoux (1998), pore throats act like gates connecting the pores and create a "gate or valve effect," indicated by a severe reduction of permeability as they are plugged by particles and shut off. Let 8r represent the volume fraction of the particles in the bulk media captured and retained behind the pore throats. The pore filling following the pore throat plugging leads to an internal cake formation at a rate proportional to the particle flux, uap, and the pore volume, (|), available.
and k, = 0 otherwise (8-16) tcr represents the critical time when the pore throats are first jammed by particles. This time is similar to the screen-factor. Himes et al. (1991) define the screen-factor as:
A screen-factor value is the time for a given volume of a solution to pass through a network of five lOO-U.S.-mesh screens stacked together and normalized to the time taken for the carrier fluid alone (usually water). A higher screen-factor value means less mobility and poor injectivity. A value of one indicates equal mobility to the carrier fluid.
and Pcr is the critical value below which pore throat blocking can occur. One of the factors affecting the particle migration through a pore throat is the particle size relative to the pore throat size. The hydraulic tube diameter is given by the Carman-Kozeny equation:
The pore throat diameter can be estimated as a fraction, /, of the hydraulic tube diameter according to (Ohen and Civan, 1990, 1993):
Then, the ration of the particle to pore throat diameters can be approximated by:
King and Adegbesan (1997) state that the ratio of the median particle diameter to pore throat diameter is given by (Dullien, 1979):
A comparison of Eqs. 8-20 and 21 implies that, even if / = 1.0, Eq. 8-21 is applicable for tight porous media with a porosity of the order of 4 = 0.04. The value of the parameter Fs or its reciprocal (3 indicates that the flow of a particulate suspension into porous media may lead to one of the following phenomena (King and Adegbesan, 1997):
a. P<3, external filtercake formation
b. 3<(3<7, internal filtercake formation
c. p>7, negligible filtercake involvement
Pautz et al. (1989) point out that these rules-of-thumb have been derived based on experimental observations. The values 3 and 7 denote the critical values or Pcr. Note these values are very close to the values of 2 and 6 Gruesbeck and Collins (1982) for bridging of particles in perforations. Civan (1990, 1996) determined (3cr empirically by correlating between two dimensionless numbers. In the pore throat plugging process, the mean pore throat diameter, Dt, mean particle diameter, Dp, particle mass concentration, cp, viscosity of suspension, |o,, and the interstitial velocity of suspension, V = M/<|), are the important quantities. Therefore, a dimensional analysis among these variables leads to two dimensionless groups (Civan, 1996). The first is an aspect ratio representing the critical pore throat to particle diameter ratio necessary for plugging given by:
The relationship between (3cr and Rep can be developed using experimental data. Inferred by the Gruesbeck and Collins (1982) data for perforation plugging, and by Rushton (1985) and Civan (1990, 1996), such a relationship is expected to obey the following types of expressions:
where A, B, and C are some empirical parameters.
The above formulation is a simplistic approach. In reality, the pore and pore throat sizes are distribution functions, which vary by damage or stimulation. This can be considered by the methods developed by Ohen and Civan (1993) and Chang and Civan (1997).
Dislodgment and Redeposition of Particles at Pore Throats
Gruesbeck and Collins (1982) observed that the effluent particle concentration tended to fluctuate during constant flow rate experiments. Such phenomena did not occur during constant pressure difference experiments, which are more representative of the producing well conditions. They explain this behavior by consecutive dislodgment and formation of plugs at the pore throats. They postulate that, in heterogeneous systems, when a suspension of particles of various sizes flow through a porous media made of a wide range of grain sizes, narrow pathways are likely to be plugged first, diverting the flow to wider pathways, which transfer
the particles to the effluent more effectively. However, as the flow paths are plugged, the pressure difference across the porous media may exceed the critical stress necessary to break some of the plugs. Therefore, these plugs break and release particles into the flowing media increasing its particle concentration. Subsequently, the deposition process progresses to form new plugs during which the flowing media particle concentration decreases. Gruesbeck and Collins (1982) also observed a similar phenomena in systems of homogeneous grain sizes subjected to a constant rate injection of a suspension of particles. Millan-Arcia and Civan (1992) have reported frequent fluctuations in the effluent fluid concentrations and pH during injection of brine into sandstone.
Colloidal Release and Mobilization
Colloidal mobilization is a result of the physico-chemical reactions that involve electro-kinetic forces, zeta potential, and ionic strength (Wojtanowicz et al., 1987). Let ep denote the volume fraction of porous media occupied by the particles available for mobilization over the pore surface. The rate of colloidal expulsion or mobilization of particles at the pore surface is proportional to the excess critical salt concentration (ccr - c),
and the amount of the unblocked particles at the pore surface available for mobilization, £r\e.
a is the volumetric expansion coefficient for swelling clays as ot = 0 for nonswelling particles; ccr is the critical salt concentration; and r\e is the fraction of the unblocked particles approximated by (Civan et al., 1989; Ohen and Civan, 1993; Civan, 1996):
A, is an empirical constant; 2-> p represents the total volume of various p types of particles retained within the pore space; kr is a particle release rate constant given by (Khilar and Fogler, 1987, 1983; Kia et al., 1987):
Hydraulic Erosion and Mobilization
proportional to the excess pore wall shear stress, (Tw-Tcr), and the amount of the unblocked particles available for mobilization at the pore surface (Gruesbeck and Collins, 1982; Khilar and Fogler, 1987; Cernansky and Siroky, 1985; Civan, 1992, 1996).
Particle Transfer Across Fluid-Fluid Interfaces
The driving force for particle transfer between two fluid phases is the wettability of the fluid phases relative to the wettability of the particles. Particles prefer to be in the phase that wets them (Muecke, 1979)Civan, 1994). But, mixed-wet particles tend to remain on the interface where they are most stable (Ivanov et al., 1986). In the region involving the interface between wetting and nonwetting phases, it can be postulated that particles A in a weaker wettability phase-1 first move to the interface and then migrate from the interface to a stronger wettability phase-2 according to the following consecutive processes (Civan, 1996):
Nonwetting phase - 1 —» Interface —> Wetting phase - 2 (8-42) Therefore, the following power-law rate expressions can be proposed:
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