File:Clay particle expansion and pore space reduction by swelling.png
Clay particle expansion and pore space reduction by swelling

Ladd 1960 explains that: "The exchangeable cations are attracted to the clay particles by the negative electric field arising from the negative charge on the particles. Hence, the electric field acts as a semi-permeable membrane in that it will allow water to enter the double layer but will not allow the exchangeable cations to leave the double layer." Thus, when the total ion cations plus anions concentration in the double-layer between the clay particles is higher than that in the aqueous pore fluid, the water in the pore fluid diffuses into the double-layer to dilute its ion concentration. This phenomenon creates an osmotic repulsive pressure between the clay particles. As a result, the interparticle distance increases causing the clay to expand and swell. Therefore, the driving force for osmotic pressure is the difference of the total ion concentrations between the clay double-layer, cc, and the surrounding pore fluid For only very dilute aqueous solutions, the van't Hoff equation given below can be used to estimate the osmotic pressure Ladd, 1960

Non-ideal models are required for concentrate solutions.

## Water Absorption Rate

File:Mechanism of osmotic pressure generation between two clay.png
Mechanism of osmotic pressure generation between two clay
File:Mechanism of formation swelling by water absorption.png
Mechanism of formation swelling by water absorption

Civan et al. 1989 assumed that water diffuses through the solid matrix according to Pick's second law over a short distance near the surface of the solid exposed to aqueous solution, because the coefficient of water diffusion in solid is small. Thus, the water absorption in the solid can be predicted by the one-dimensional transientstate diffusion equation:

File:Transient state diffusion equation.png
Transient state diffusion equation

where c0 and c are the initial and instantaneous water concentrations, respectively, in the solid, Cj is the water concentration of the aqueous solution, z is the distance from the pore surface, t is the actual contact time, k is the film mass transfer coefficient, and D is the diffusivity coefficient in the solid matrix. Eq. 2-4 expresses that the water diffusion into clay is hindered by the stagnant fluid film over the clay surface. Thus, similar to Civan (1997), the analytical solution of Eqs. 2-2 through 2-5 can be used to express the cumulative amount of water diffusing into the solid surface as given by Crank (1956):

File:Water absorption is given by differentation.png
Water absorption is given by differentiation

where h = kID. Civan et al. (1989) have resorted to a simplified approach by assuming that the film mass transfer coefficient k in Eq. 2-4 is sufficiently largeso that Eq. 2-4 becomes:

File:Analytical solution.png
Analytical solution

and, therefore, an analytical solution of Eqs. 2-2, 3, 8, and 5 according to Crank (1956) yields the expression for the cumulative and rate of water absorption, respectively, as:

File:Cumulative and rate of water.png
Cumulative and rate of water

The rate of formation damage by clay swelling also depends on the variation of the water concentration in the aqueous solution flowing through porous rock. Whereas, the analytical expressions given above assume constant water concentrations in the aqueous pore fluid. However, they can be corrected for variable water concentrations by an application of Duhamel's theorem. For example, if the time-dependent water concentration at the pore surface is given by:

File:Water concentration.png
Water concentration

where F(t) is a prescribed time-dependent function, the analytic solution can be obtained as illustrated, by Carslaw and Jaeger (1959). Then, using Eq. 2-10, the rate of water absorption can be expressed by:

File:Water absorptions.png
Water absorptions

However, in the applications presented here the water concentrations involved in the laboratory experiments are essentially constant. The preceding derivations assume a plane surface as supposed to a curved pore surface. From the practical point of view, it appears reasonable because of the very short depth of penetration of the water from the solid-fluid contact surface.

## Clay Swelling Coefficient

The rate of clayey formation swelling is derived from the definition of the isothermal swelling coefficient given by Collins, 1961:

File:Volumes of the solid and the water.png
Volumes of the solid and the water

V and Vw are the volumes of the solid and the water absorbed, respectively. Ohen and Civan 1991 used the expression given by Nayak and Christensen 1970 for the swelling coefficient:

File:Swelling coefficient.png
Swelling coefficient

in which c is the water concentration in the solid and CI is the plasticity index. <;, and qt are some empirical coefficients, m is an exponent. Chang and Civan 1997 used the expression given by Seed et al. 1962:

File:Clay content of porous rock as weight percent.png
Clay content of porous rock as weight percent

where Cc is the clay content of porous rock as weight percent, PI is the plasticity index, and k' is an empirical constant.

## Water Content During Clay Swelling

The rate of water retainment of clay minerals is assumed proportional with the water absorption rate, 5, and the deviation of the instantaneous water content from the saturation water content as:

File:Saturation water content.png
Saturation water content

where kw is a water retainment rate constant, w denotes the weight percent of water in clay and the subscripts o and t refer to the initial (t = 0) and terminal (t -» °o) conditions, respectively. An analytical solution of Eqs. 2-16 and 17 yields. Osisanya and Chenevert (1996) measured the variation of the water content of the Wellington shale exposed to deionized water.

File:Correlation of water pickup during swelling.png
Correlation of water pickup during swelling

shows the correlation of their data with Eq. 2-18 using Eq. 2-6. The best fits were obtained using w0= 2.7 wt.%, wt= 3.27 wt.%, A = kw(c{ - c0)/ h = 0.26 and h-Jo = \ for their Gage 1 data, w0 = 2.77 wt.%, wt = 3.28 wt.%, A = 0.06 and h^D = 0.8 for their Gage 2 data, and w0= 2.77 wt.%, wt = 3.28 wt.%, A = 0.035 and h-^j~D = 0.8 for their Gage 3 data. Brownell (1976) reports the data of the moisture content of a dried clay piece containing montmorillonite soaked in water.

## Time-Dependent Clay Expansion Coefficient

By contact with water the swelling clay particles absorb water and expand.The rate of volume increase is assumed proportional to the water absorption rate, 5, and the deviation of the instantaneous volume from the terminal swollen volume that will be achieved at saturation, (Vt - V). Therefore, the rate equation is written as:

File:Rate equation is written as.png
Rate equation is written as

where at is the terminal expansion coefficient at saturation. ka is the rate coefficient of expansion. Seed et al. (1962), Blomquist and Portigo (1962), Chenevert (1970), and Wild et al. (1996) measured the rates of expansion of the samples of compacted sandy clay, hydrogen soil, typical shales, and lime-stabilized kaolinite cylinders containing gypsum and ground granulated blast furnace slag, respectively.

File:Correlation of water pickup during swelling after Civan.png
Correlation of water pickup during swelling after Civan
File:Data Correlation of volume.png
Data Correlation of volume
File:Data Correlation of Wild.png
Data Correlation of Wild

Wild et al. (1996) tested lime-stabilized compacted kaolinite cylinders containing gypsum and ground granulated blast furnace slag. After oistcuring for certain periods, they soaked these samples in water and measured the linear expansion of the samples. The second set of data is for a 28-day moist-cured kaolinite containing 6% lime and 4% gypsum. The third set of data is for a 28-day moistcured kaolinite containing 2% lime, 4% gypsum and 8% ground granulated blast furnace slug. The best fits of Eq. 2-22 using Eq. 2-6 to the first, second, and third data sets were obtained with A = k(cl - c0}lh =1.1, W# = 1.0 and a, = 10.8 vol.%, A = 20, h^D = 0.2 and cc, = 1.48 vol.%, and A = 2.4, H-jD = 0.7 and oc, = 0.655 vol.%, respectively. Ladd (1960) measured the volume change and water content of the compacted Vicksburg Buckshot clay samples during swelling. For a linear plot of Ladd's data first, the S term is eliminated between Eqs. 2-18 and 22 to yield.

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