Diagnostic Equations for Typical Cases
Wojtanowicz et al. (1987, 1988) have analyzed and developed diagnostic equations for two special cases of practical importance:
1 . Deposition of the externally introduced particles during the injection of a suspension of particles
2. Mobilization and subsequent deposition of the indigeneous particles of porous medium during the injection of a particle free solution
Deposition of Externally Introduced Particles
Three distinct permeability damage mechanisms are analyzed for a given injection fluid rate and particle concentration. Particles are retained mainly in the thin core section near the inlet face. In this region the concentration of the flowing phase is assumed the same as the injected fluid (i.e., pp/ - ( P p f ) . ). Gradual pore reduction by surface deposition occurs when the particles of the injected suspension are smaller than the pore constrictions. Assume that the surface deposition is the dominant mechanism compared to the entrainment, that is, kr»ke.
Then, the solution of Eqs. 10-35 and 36 yields:
A substitution of Eq. 10-37 into Eq. 10-29 leads to the following expression for the area occupied by the surface deposits
Substitution of Eqs. 10-10 and 38 into Eq. 10-28 yields the following diagnostic equation:
in which the empirical constant is given by
Single pore blocking occurs when the size of the particles in the injected fluid are comparable or bigger than the size of the pore constrictions. A substitution of Eqs. 10-12, 13, and 14 into Eq. 10-28 yields the following diagnostic equation:
Cake formation near the inlet face of the porous media occurs when the particles in the injected solutions are large relative to the pore size and at high a concentration. Combining Eqs. 10-22 and 17 yields the following diagnostic equation:
Mobilization and Subsequent Deposition of Indigeneous Particles
This case deals with the injection of a clear (particle free) solution into a porous media. A core is visualized as having two sections designated as the inlet and outlet sides. The particles of the porous media entrained by the flowing phase in the inlet part are recaptured and deposited at the outlet side of the core. Near the inlet port, the mobilization and entrainment of particles by the flowing phase is assumed to be the dominant mechanism compared to the particles retention (i.e., ke » kr}. Thus, dropping the particle retention term, Eqs. 10-35 and 36 yield the following solution for the mass of particles remaining on the pore surface
Substituting Eq. 10-45 and (pp/). =0, Eq. 10-32 yields the following expression for the particle concentration of the flowing phase passing from the inlet to the outlet side of the core as
Depending on the particle concentration and size of the flowing phase entering the core, the outlet side diagnostic equations for three permeability damage mechanisms mentioned previously are derived next.
Gradual Pore Reduction by Surface Deposition and Sweeping
Assume that the mass of the indigeneous or previously deposited particles on the pore surface is m*. Then, the area occupied by these particles is given by Eq. 10-29 as
Afg denotes the open flow area when all the deposits are removed. If simultaneous, gradual pore surface deposition and sweeping ar occurring near the outlet region, then both the entrainment and retention terms are considered equally important. Thus, substituting Eq. 10-46, Eq. 10-35 yields the following ordinary differential equation:
The solution of Eq. 10-49, subject to the initial condition mp=m*p (previously deposited particles), is obtained by the integration factor method as:
Then, the area occupied by the remaining particles is given by Eq. 10-29 as:
Eliminating Afo between Eqs. 10-48 and 53, substituting Eqs. 10-47 and 52 for A* and Ap, and then applying Eq. 10-10 for Af and A*f yields the following diagnostic equation:
Normally, mpg =m*p. Wojtanowicz et al. (1987) simplified Eq. 10-54 by substituting C7 = 0 when the mass of the particles initially available on the pore surface is small compared to the mass of particles deposited later (i.e., mJsO). If only pore sweeping occurs, then kr « ke. Thus, substitute kr = 0 in Eq. 10-56 to obtain C8 =0 and Eq. 10-54 becomes:
If only gradual surface deposition is taking place in the outlet region, then kr » ke . Therefore, dropping the particle retention term and substituting Eq. 10-46, Eqs. 10-35 and 36 for mpg = 0 are solved to obtain the amount of particles retained as:
Single Pore Blocking
If the permeability damage is solely due to single pore blocking, then substituting Eqs. 10-46, 12, 13 and 14 into Eq. 10-28 yields the following diagnostic equation:
If the permeability damage is by cake formation, then substituting Eqs. 10-46 and 22 into Eq. 10-17 yields the following diagnostic equation
A list of the diagnostic equations derived in this section are summarized in this article for convenience.
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