Permeability Relationships
The permeability relationships can be classified in two groups: static and dynamic. The static correlations have been derived using the properties of various porous materials that have not been subjected to formation damage processes. The dynamic correlations or models consider porous media undergoing alteration due to rock-fluid interactions during formation damage and, therefore, are preferred for formation damage prediction. In the following, selected models pertaining to formation damage are reviewed and presented with some modifications for consistency and applications in the formation damage prediction.
Contents
The Carman-Kozeny Hydraulic T\ibes Model
The hydraulic tubes model was derived based on the analogy between the flow of fluid through porous media and parallel flow through a bundle of tortuous capillary tubes (Carman-Kozeny, 1938). The number, diameter, and the tortuous length of the hydraulic tubes are denoted by n, Dh, and Lh, respectively. The porosity, specific pore or grain surface, and length of the porous media are (|>, Z, and L. Vp and Vb denote the pore and bulk volumes, respectively. The tortuosity of porous media is expressed as the ratio of the actual tortuous tube length to the length of porous media: where Dg is the grain diameter. Next consider that the laminar flows through porous media and the bundle of tortuous tubes can be described by the
Darcy and the Hagen- Poiseuille laws given, respectively, as:
A" is the intrinsic permeability of porous media. The cross-sectional area of porous media open for flow can be expressed by:
Therefore, equating Eqs. 5-9 and 10, and substituting Eqs. 5-1 and 10 results in the following relationship for the mean hydraulic tube diameter:
Bourbie et al. (1986) determined that n = 1 for ()><0.05 and n = 3 for 0.08 <(j)<0.25. In view of this evidence and Eq. 5-14, the Carman-Kozeny equation appears to be valid for the 0.08 < <)) < 0.25 fractional porosity range. Reis and Acock (1994) warn that these exponents may be low "because the permeabilities were not corrected for the Klinkenberg effect."
The Modified Carman-Kozeny Equation Incorporating the Flow Units Concept
Based on the Carman-Kozeny model, Eq. 5-14, Adin's (1978) correlation of experimental data leads to a permeability-porosity model as:
where oc and n are some empirical parameters. Arshad's equation accounts for the formation of the dead-end pores during deposition, which do not conduct fluids.
The Flow Efficiency Concept
Rajani (1988) concluded that permeability function can be separated into and expressed as a product of a function incorporating the pore geometry and a function of porosity as:
This approach is particularly useful in porous media undergoing alteration during formation damage. Frequently, the Carman-Kozeny equation fails to represent the cases where the pore throats are plugged without significant porosity reduction. This problem can be alleviated by introducing a flow efficiency factor, y, in view of Eq. 5-19 (Ohen and Civan, 1993; Chang and Civan, 1991, 1992, 1997). Hence, the permeability variation can be expressed by (Chang and Civan, 1997):
where a, b, and c are some empirically determined parameters and K0 and §0 denote the permeability and porosity at some initial or reference state. The flow efficiency factor, y, can be interpreted as a measure of the fraction of the open pore throats allowing fluid flow. Thus, when the pore throats are plugged, then y = 0, and therefore K = 0, even if <|) * 0. This phenomenon is referred to as the "gate or valve effect" of the pore throats (Chang and Civan, 1997; Ochi and Vernoux, 1998). In order to estimate the flow efficiency factor, Ohen and Civan (1993) assumed that, although the pore throat sizes vary with time, they always remain log-normally distributed:
in the range of dl<y<dh, where sd is the standard deviation and dt is the mean pore throat diameter. Then, assuming that the pore throats smaller than the size, dp, of the suspended particles will be plugged, the flow efficiency factor is estimated by the fraction of pores remaining open at a given time:
where Ep is the plugging efficiency factor. Particles that are sticky and deformable can mold into the shape of pore throats and seal them. Then, the plugging is highly efficient and Ep is close to unity. Particles that are rigid and nonsticky cannot seal the pore throats effectively and still allow for some fluid flow. Thus, E < 1 for such plugs.
The lower and upper bounds of the pore throat size range are estimated by a simultaneous solution of the non-linear integral equations given by:
for which the mean pore throat size is estimated by solving the following equation which relates the pore throat size variation to the rate of eposition:
where k6 is a rate constant and ep is the volume of deposition per unit bulk volume, subject to the initial mean pore throat diameter, either etermined from the initial pore throat size distribution using Eq. 5-28, or estimated as a fraction of the mean pore diameter using:
Note that r\ is not a fraction because it is a lumped coefficient including the mentioned fraction, some unit conversion factors, and the shape factor. Chang and Civan (1991, 1992, 1997) considered that the pore throat and particle diameters can be better represented by bimodal distribution functions over finite diameter ranges, given by Popplewell et al. (1989) as:
where w is an adjustable weighting factor in the range of 0 < w < 1, and ./i(y) and/2(j) denote the distribution functions for the fine and coarse fractions, each of which are described by:
with different values of the parameters a, ra, dt, and dh. Chang and Civan (1991, 1992, 1997) used the critical particle diameter, \dp] , necessary for pore throat jamming, determined according to the criteria. For applications with multiphase flow systems, Liu and Civan (1993, 1994, 1995, 1996) used a simplified empirical equation for permeability reduction in porous media as:
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