The rate of scale formation at the pore surface can be expressed similarly to crystal growth. The pore volume relates to the diameter of an equivalent spherical shape pore space by (Civan, 1996):

Thus, the pore surface relates to the porosity according to:

C3 and C4 are some shape factors and C5=C4/C^3. The porosity of the solid porous matrix can be expressed as a sum of the instantaneous porosity, <|>, and the pore space occupied by the scales, <{>5, as:

Therefore, substituting Eqs. 9-32 and 9-34 into Eq. 9-23 yields (Civan, 1996):

and k'c is a scale formation rate constant incorporating the above mentioned shape factors and some constants. The minus sign in Eq. 9-35 is for the reduction of porosity by scale formation at the pore surface. Thus, assuming the rock porosity, (|)r, remains constant and substituting Eq. 9-33 into 9-35 leads to an equation similar to Ortoleva et al. (1987):

Assume that the surface area of crystal available for growth can be expressed empirically by:

in which /(({>) is the specific surface of the mineral-fluid contact area (surface area per unit mineral mass) expressed as a function of porosity. Civan (1996) approximated this function according to Eq. 9-32. Thus, substituting Eq. 9-40 into Eq. 9-23 leads to Holstad's (1995) equation:

Holstad (1995) expressed the temperature dependency of the crystallization rate constant by the Arrhenius equation:

where FM, AM, and EM denote an empirical mineral property factor, an Arrhenius pre-factor, and the activation energy. Liu et al. (1997) used a similar equation

where k° is the high-temperature (T —> °°) limit of the rate constant. The effects of various conditions on dissolution rates, including lithologic variation, hydrodynamics, ionic strength, saturation state, mixedkinetic control, and surface treatment, have been investigated by Raines and Dewers (1997, 1997), Hajash Jr. et al. (1998), and Merino and Dewers (1998).

Crystal Surface Displacement by Dissolution and Precipation

The dissolution and precipitation of a crystalline matter in contact with a solution can be studied by measuring the progress of the crystal surface as a function of time. Hunkeler and Bohni (1981) and Dunn et al. (1999) used this technique. Civan (2000) determined that the position of the progressing crystal surface could be correlated by:

for which x, x0 and xt are the instantaneous, initial, and final surface positions, respectively, k is a rate constant, and M is the amount of solute precipitated or dissolved, given by:

where t is time, c0 and c, are the solute concentrations of the solution at the beginning and equilibrium, respectively, and D is the diffusion coefficient of the solute. Civan (2000) verified this model using the Dunn et al. (1999) measurements of the pit depth during barite dissolution.


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