Model Considering the Clayey Formation Swelling and Indigeneous and External Particles
Civan et al. (1989) and Ohen and Civan (1991, 1993) considered the formation damage by clayey formation swelling and migration of externally injected and indigeneous particles. They assumed constant physical properties of the particles and the carrier fluid in the suspension. They also considered the effect of fluid acceleration during the narrowing of the flow passages by formation damage. Ohen and Civan (1993) classified the indigeneous particles that are exposed to solution in the pore space in two groups: lump of total expansive (swelling, i.e. 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 indigeneous 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 articles 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 indigeneous particles of the porous media. Further, they assumed that the deposition of the suspended particles over the indigeneous particles of the porous media blocks the indigeneous 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. They assumed that permeability reduction is a result of the porosity reduction by net particle deposition and formation swelling and by formation plugging by size exclusion. The Ohen and Civan (1993) formulation is applicable for dilute and concentrated suspensions, whereas, Gruesbeck and Collins' (1982) model applies to dilute suspensions. The mass balance equations for the total water (flowing plus absorbed) in porous media and the total particles (suspended plus deposited) in porous media are given, respectively, by:
Thus, adding Eqs. 10-133 and 134 yields the total mass balance equation for the water and particles in porous media as:
In Eqs. 10-133- through 135, ɸ is the instantaneous porosity, pw and pp are the densities of water and particles, u is the volumetric flux of the flowing suspension of particles, ɛw, ɛp, and ɛ*p represent the volume fraction of porous media containing the absorbed water, particles deposited from the flowing suspension, and the indigeneous particles in the pore space, respectively, and σw and σp denote the volume fractions of the water and particles, respectively, in the flowing suspension. Thus,
a volumetric weighted sum of the densities of the water and particles by:
For simplification purposes, assume constant densities for the water and particles. However, note that the density of suspension is not a constant, because it is variable by the particle and water volume fractions based on Eq. 10-137. Therefore, Eqs. 10-134 and 135 can be expressed, respectively, as:
Considering the rapid flow of suspension as the flow passages narrow during the formation damage, the Forchheimer equation is used as the momentum balance equation:
in which θ is the inclination angle and Z0 is a reference level. Νnd is the non-Darcy number given by
The inertial flow coefficient, β can be estimated by the Liu et al. (1995) correlation:
Brinkman's application of Einstein's equation is used to estimate the viscosity of the suspension:
The particle volume fraction and the flow potential can be calculated by solving Eqs. 10-138 and 145 simultaneously, using an appropriate numerical method such as the finite difference method used by Ohen and Civan (1993), subject to the initial and boundary conditions given by:
The volumetric rate of water absorption is estimated by (Civan et al., 1989):
where t is the actual contact time of flowing water with the porous media and B is an absorption rate constant. The porosity change by clayey formation swelling is estimated by (Civan and Knapp, 1987; Ohen and Civan, 1990, 1993):
where A, is the swelling coefficient determined by an appropriate empirical correlation such as by those given by Seed et al. (1962) and Nayak and Christensen (1970). The volume balance of particles (indigeneous and/or external types) of the flowing suspension deposited in porous media is given as the difference of the deposition by the pore surface and pore throat deposition processes and the re-entrainment rates by the colloidal and hydrodynamic processes as (Civan, 1996, 1996):
Let single and double primes denote the nonswelling and swelling clays. The volume balances of the nonmobilized indigeneous nonswelling and swelling clays remaining in porous media is given in terms of the colloidal and hydrodynamic mobilization rates, respectively, by:
where a is the expansion coefficient of a unit clay volume, estimated by:
in which αs is the expansion coefficient at saturation. The initial conditions are given by:
The instantaneous permeability is estimated by means of the modified Kozeny-Carman equation
The linear flow model presented above can be converted to the radial flow model by the application of the transformation given by (Ohen and Civan, 1991):
Model Assisted Analysis of Experimental Data
Without the theoretical analysis and understanding, laboratory work can be premature, because the analyst may not exactly know what to look for and what to measure. The theoretical analysis of various processes involved in formation damage provide a scientific guidance in designing the experimental tests and helps in selecting a proper, meaningful set of variables that should be measured. Having studied the various issues involving formation damage by fines migration, we are prepared to conduct laboratory experiments in a manner to extract useful information. Here, the analysis of experimental data by means of the mathematical models developed in this chapter is illustrated by several examples taken from the literature.
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