For convenience in the modeling, Pang and Sharma (1994) divide the entire filtration process into two phases:

1 the initial internal cake filtration, and

2 the later external cake filtration.

They separate these two filtration phases by a "transition time" after which the particle migration into porous formation becomes negligible and an external filter cake begins forming over the injection well formation face. Sharma and Pang (1997) assumed that transition from the internal to the external cake filtration occurs when the porosity, ɸ, of the formation face decreases to a minimum critical value, ɸ, by particle deposition, below which particle invasion into porous media is not possible. Their models apply for single phase water flow in the near wellbore region. Hence, the effect of the oil-water two-phase flow during the initial water injection period is neglected because this initial period is relatively short.

## Transition Time

Wennberg and Sharma (1997) estimate the transition time based on the expressions given by Iwasaki (1937) for particle deposition rate and the filtration coefficient, respectively, as:

where b is an empirical constant and X,0 is the filtration coefficient without particle deposition. Although more sophisticated expressions are available in the literature (See Chapter 8), they used Eq. 19-15 for simplicity. Thus, invoking Eq. 19-15 into Eq. 19-14 yields the following expression, similar to the rate equation for particle deposition in pluggable pathways given by Gruesbeck and Collins (1982):

They obtain the analytic solution of Eq. 19-16 for constant flow rate and suspension particle concentration for two cases as the following:

At the transition time, the porosity attains the minimum critical porosity, ɸ , and the maximum critical volume fraction of the deposited particles becomes a* = ɸ0 - ɸ* . Under these conditions, Eqs. 19-17 and 18 can be used to obtain the following expressions, respectively, for the transition time:

## Internal Filtration Models

Pang and Sharma (1994, 1995) and Sharma et al. (1997) have pursued their derivations in terms of the following variables:

Here, v represents the interstitial velocity of the fluid phase and n denotes the fraction of the pore space occupied by the particle deposits in porous media. X' is the product of the deposition rate constant, K, and the interstitial velocity of the flowing suspension. In the following, their formulations are presented in a manner consistent with the rest of the presentation of this chapter. The damage of a core plug by the injection of a dilute particulate suspension can be described by means of the volumetric balance equations of the suspended and deposited particles in porous media, given, respectively, by (Wennberg and Sharma, 1997):

subject to the initial and boundary conditions given by (Pang and Sharma, 1994):

The analytical solution of Eqs. 19-24 through 27 used by Pang and Sharma (1994) implies some simplifications. It applies for the injection of dilute suspension of particles. Therefore, the effect of small amount of particle deposition, a, compared to the initial porosity can be neglected, the deposition rate coefficient is assumed constant, and a constant rate injection is considered. Thus, the analytic solution for constant ɸ~ɸ0, X,~X0 , and u = u0 can be adopted from Rhee et al. (1986) as:

where the term inside the square brackets expresses that cf is a function of (t-ɸx/u). Considering that cf(t) = cf is constant and c0(x) = Q and a0(x) = 0 in the laboratory core flow tests, Pang and Sharma (1994) simplify Eqs. 19-29 and 30, respectively, as:

Pang and Sharma (1995) assumed that the permeability reduction primarily occurs by pore throat plugging. Therefore, they estimated the permeability of the porous formation as a harmonic average permeability of the combined plugged and unplugged regions as:

where fp represents the volume fraction of the deposited particles contributing to pore throat plugging, Kp denotes the permeability of the plugged region near the pore throats, and Km is the permeability of the formation matrix, assumed to remain constant, which is equal to the initial nondamaged permeability, K0 (i.e., Km =K0}. Therefore, Pang and Sharma (1995) rearranged Eq. 19-35 for the relative or fractional retained permeability of the porous formation undergoing particle deposition from dilute suspensions as, inferred by Payatakes et al. (1974):

Thus, they calculate the harmonic average permeability of the damaged portion of the core by:

Substituting Eqs. 19-33, 34, and 36 into Eq. 19-38, and then integrating, they derive the following expression:

Note that Eq. 19-39 was previously derived by Wojtanowicz et al. The injectivity ratio for the linear cases, applicable to laboratory core plugs without external cake formation (/?c = 0), is given by:

Pang and Sharma (1994) simplify Eq. 19-41 by considering that the injection front reaches the outlet end of the core rapidly. Therefore, neglecting the damage during the short period of time until the front reaches the core outlet, Eqs. 19-39 and 41 yield for xf = L the following equation indicating that the reciprocal injectivity ratio is a linear function of time:

## External Filtration Models

Considering the formation of an incompressible external cake without any particle invasion into the core plug, Pang and Sharma (1994) expressed the harmonic average permeability of the cake and the core system as:

Sharma et al. (1997) determine the thickness of the external filter cake by means of a volumetric balance of the particles in the cake as:

where A is the cross-sectional area of the core plug and cf is the volume fraction of the fine particles in the water injected at a flow rate of q. For constant cf and q, Eq. 19-47 simplifies as:

Thus, substituting Eq. 19-48 into 45 and considering that the filter cake thickness is much smaller than the length of the core plug (i.e., hc « L), they obtained the following expression indicating that the reciprocal injectivity ratio is a linear function of time:

## Filtration Coefficient

Wennberg and Sharma (1997) point out that the filtration coefficient varies by particle deposition according to:

Their review of the various expressions available for prediction of the filtration coefficient is summarized and presented in the following. Ives (1967) proposed a general expression as:

in which x, v, z, and p are some empirical parameters and aM is the maximum of the volume fraction of the deposited particles necessary to make the filtration coefficient of porous media zero. This equation indicates that the filtration coefficient is equal to one when there is no deposited particles in porous media, and the filtration coefficient becomes zero when the volume fraction of deposited particles reaches a certain characteristic value of maximum aM. Chiang and Tien (1985) developed an empirical correlation as:

in which the dimensionless groups are defined as following. As is Happel's dimensionless geometric parameter. A^ is the London parameter given by

v is velocity, ILL is viscosity, and pp and pe denote the particle and fluid densities, respectively. (3 is a packing parameter given by

where ɸ is the porosity in fraction. D is the coefficient of diffusion for the Brownian motion of particles. Wennberg and Sharma (1997) analyzed the measurements of the filtration coefficient reported by various investigators and determined that these data mostly indicate power law-type relationships to the volumetric flux, the suspended particle size, and the porous media grain size as:

## Diagnostic-Type Curves for Water Injectivity Tests

Pang and Sharma (1994, 1997) identified four distinct type curves that can be used for interpretation of the water-quality tests. They justified these type curves with experimental data obtained from the literature as shown in Figure 19-3. Type curve 1 is a straight line indicating the formation of an incompressible external filter cake or a thin internal cake near the injection face of the core plug according to Eq. 19-49. The slope remains constant. Type curve 2 is for the similar case, but applies for compressible cakes. In this case, the porosity and permeability of the cake decrease by increasing filtration pressures. As a result, the slope of the curve increases with the filtration time or pore volume injected. Type curve 3 refers to a deep particle invasion and pore filling in the core plug, leading to a slower gradual permeability decrease. As a result, the slope of the curve decreases with the filtration time. Type curve 4 may be an S-shaped or other types of curves indicating a shift of the dominance of the different damage mechanisms during the filtration process.

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