Porosity Reduction by Swelling

File:Swelling of porous matrix.png
Swelling of porous matrix

Based on the definition of the swelling coefficient, Civan and Knapp (1987) expressed the rate of porosity change by swelling of porous matrix as: where § is porosity, t is time, S is the rate of water absorbed per unit bulk volume of porous medium. Civan 1996 developed an improved equation assuming that the rate of porosity variation by swelling is proportional to the rate of water absorption and the difference between the instantaneous and the terminal or saturation porosities:

from which the porosity variation by swelling can be expressed by:

File:Terminal or saturation porosities.png
Terminal or saturation porosities

where ksw is the formation swelling rate constant, t is the actual time of contact with water. Therefore, the swelling rate constant can be determined by fitting Eq. 2-27. However, due to the lack of experimental data, the application of Eq. 2-28 could not be demonstrated. It is difficult to measure porosity during swelling. Permeability can be measured more conveniently. Ohen and Civan (1993) used a permeability-porosity relationship to express porosity reduction in terms of permeability reduction.

File:Porosity variation.png
Porosity variation

Permeability Reduction by Swelling

File:Instantaneous permeability and terminal permeability.png
Instantaneous permeability and terminal permeability

Civan and Knapp (1987) assumed that the rate of permeability reduction by swelling of formation depends on the rate of the water absorption and the difference between the instantaneous permeability and terminal permeability that will be attained at saturation as: where asw is the rate constant for permeability reduction by swelling, from which the permeability variation by swelling is obtained as:

File:Permeability variation.png
Permeability variation

Civan and Knapp (1987) and Civan et al. (1989) have confirmed the validity of Eq. 2-31 using the Hart et al. (1960) data for permeability reduction in the outlet region of a core subjected to the injection of a suspension of bacteria. Because bacteria is essentially retained in the inlet side of the core, the permeability reduction in the near-effluent port of the core can be attributed to formation swelling by water absorption. The best linear, least-squares fit of Eq. 2-31 to Hart et al. (1960) data using Eq. 2-9 for S yields (Civan et al., 1989): [[File:Best linear, least-squares ormula.png|thumb|400px|Best linear, least-squares formula]] with (Kt/K0) = 0.57 and B = 2asw(cl -c0)Jo/n = QMhr~{/2 with a correlation coefficient of R2 = 0.93 the Hart et al. (1960) data can also be correlated using Eq. 2-6 for S. In this case, the best fit is obtained using the parameter values of A = asw (q - c0)/h = 0.95, h^D = 1, and Kt/K0 = 0.57. Ngwenya et al. (1995) conducted core flood experiments by injecting an artificial seawater into the Hopeman (Clashach) sandstone. They report that the core samples used in their experiments contained trace amounts of clays. Therefore, they concluded that the effect of clay swelling, and entrainment and deposition of clay particles to permeability impairment would be negligible.

File:Correlation of Hart et al data function.png
Correlation of Hart et al data function
File:Correlation of Ngw enya et al data function.png
Correlation of Ngw enya et al data function

Consequently, it can be concluded that the swelling of some constituents of the sandstone formation should be contributing to permeability reduction. The best least-squares linear fit of Eq. 2-31 was obtained using Eq. 2-9 for 5 with the parameter values of B = asw (c, - c0)/h = 0.038hr~1/2 and (KtIK0} = 0.087 with a correlation coefficient of R2 = 0.89. The best fit of Eq. 2-31 using Eq. 2-6 for 5 was obtained using the parameter values of A = asw (cl - c0)/h = 0.035, hjD = 1, and KtIK0 = 0.087. Hart et al. (1960) and the Ngwenya et al. (1995) experimental data does not permit etermining whether Eqs. 2-6 or 9 with Eq. 2-31 better represents the data. Because Eq. 2-6 led to successful representation of the other data correlated in the preceding sections, it is reasonable to assume that Eq. 2-6 should also represent the permeability reduction data equally well. Therefore, Eq. 2-6 may be preferred over Eq. 2-9.

Graphical Representation of Clay Content

File:A ternary clay distribution chart.png
A ternary clay distribution chart

The distribution of clays can be conveniently depicted by Lynn and Nasr-El-Din (1998). Lynn and Nasr-El-Din (1998) classified reservoir formations having less than 1 wt. % total clay and permeability higher than one Darcy as the high quality, and the low quality vice versa. Amaefule et al. (1993) measured the formation quality by the reservoir quality index defined as:

Hayatdavoudi Hydration Index (HHI)

The Hayatdavoudi hydration index (HHI=[O/OH]) is defined as the ratio of the oxygen atoms to hydroxyl groups in clays and it controls the enthalpy or free energy of the clays available for the work of swelling by hydration (Hayatdavoudi, 1999, 1999). Hence, higher hydration index is indicative of more clay swelling, according to (Hayatdavoudi, 1999, 1999):

File:Hayatdavoudi clay hydration charts.png
Hayatdavoudi clay hydration charts
File:Indicative of more clay swelling.png
Indicative of more clay swelling

where G, H, S, T, and R denote the free energy, enthalpy, entropy, absolute temperature, and the universal gas constant, respectively. Hayatdavoudi (1999) shows that all clays possess some degree of the work of swelling and therefore classification of different clays as swelling or non-swelling has no significance.


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