Crystal Growth and Scale Formation in Porous Media
Civan (1996) describes that: Injection of fluids and chemicals for improved recovery, and liberation of dissolved gases, such as CO2 and light hydrocarbons from the reservoir fluids approaching the wellbore during production, and variation of fluid saturations can alter the temperature, pressure, and composition of the fluids in the near wellbore region and tubing. Consequently, the thermodynamic and chemical balance may change in favor of precipitate separation, aggregation of precipitates, crystal growth, and scale formation. Precipitates cause formation damage by changing the wettability and permeability of petroleum bearing rock and cause scale formation and clogging in tubing and pore throats.
Typical inorganic precipitates include anhydrate (CaCO3), gypsum (CaSO4'2H2O), hemihydrate (CaSO4']/2H2O\ barite (BaSO4), celestite (SrSO4), magnesium sulfide (MgSO4) originating from mixing sea water with brine, and rock and brine interactions (Oddo and Tomson, 1994; Atkinson and Mecik, 1997); ironhydroxide gel (Fe(OH)^ originating from the acid dissolution and precipitation of iron minerals such as pyrhotite (FeS), pyrite (FeS2), hematite (Fe2O3), magnetite (Fe3O4), and siderite (FeCO3) (Rege and Fogler, 1989); silicium tetra hydroxide gel (Si(OH)4] originating from the alkaline dissolution and precipitation of minerals in shaly sandstones such as quartz and kaolinite (Labrid, 1990); and polymeric substances produced by in-situ gelation (Todd et al., 1993), alcohol induced crystallization (Zhu and Tiab, 1993), separation of elemental sulfur (Roberts, 1997); and surfactant precipitation (Arshad and Harwell, 1985). Following Oddo and Tomson (1994), precipitation/dissolution reactions can be symbolically represented by:
Oddo and Tomson (1994) correlated the saturation solubility product, Ksp, empirically as a function of temperature, T, pressure, p, and ionic strength, 5"., for typical systems. Hence, the saturation ratio given by the following equation can be used to determine whether the conditions are favorable for precipitation (Oddo and Tomson, 1994):
Fs < 1 indicates an undersaturated solution, condition unfavorable for scaling, if Fs = 1, the solution is at equilibrium with the solid scale, and Fs > 1 indicates a supersaturated solution, condition favorable for scaling.
Typical organic precipitates encountered in petroleum production are paraffins and asphaltenes. Paraffins are inert and asphaltenes are reactive substances. They are both sticky, thick, and deformable precipitates (Chung, 1992; Ring et al., 1994). Therefore, they can seal the pore throats and reduce the permeability to zero without needing to reduce the porosity to zero and their deposition at the pore surface and tubing wall is irreversible unless a solvent treatment is applied (Leontaritis et al., 1992): The saturation ratio is given by:
where Fs < 1 for undersaturated solution, Fs = 1 for saturated solution, and Fs > 1 for supersaturated solution. XA is the mole fraction of the issolved organic in oil and (XA)S is the organic solubility at saturation conditions. (XA)S is predicted using the thermodynamic model by Chung (1992).
Majors (1999) explains that "Crystallization is the arrangement of atoms from a solution into an orderly solid phase." and "Growth is simply the deposition of material at growth sites on an existing crystal face." The process is called primary nucleation if there are no crystals present in the solution to start with and crystallization is occurring for the first time. Primary nucleation can be homogeneous or heterogeneous (Majors, 1999). Homogeneous nucleation occurs inside the solution without contact with any surface. Heterogeneous nucleation occurs over a solid surface. The process is called secondary nucleation if there are already some crystals present in the system over which further deposition can occur. The schematic chart given in this article by Majors (1999) describes the concentration-temperature relationship for nucleation. As can be seen, the primary nucleation process requires a sufficiently high concentration of
supersaturated solution. Whereas, secondary nucleation can occur at relatively lower concentrations above the saturation line. The metastable region represents the favorable conditions for crystal growth (Majors, 1999). The schematic chart given in this article by Majors (1999) describes the effect of the supersaturation ratio on the crystal growth and nucleation rates. Crystal growth rate is a low-order function of supersaturation and can be represented by a linear relationship, while nucleation rate is a highorder function of supersaturation and requires a more difficult nonlinear relationship (Majors, 1999). Majors (1999) explains that "Crystal growth is a dynamic process. While most of the crystals in the solution will grow, some may dissolve."
Grain Nucleation, Growth, and Dissolution
The formation of crystalline particulates from aqueous solutions of salts nvolves a four step phase change process (Dunning, 1969):
1. Alteration of chemical and/or physical conditions to lead to supsersaturation of the solution,
2. Initiation of the first small nuclei of the crystals,
3. Crystal growth, and
4. Relaxation leading to coagulation of crystalline particles.
The process is called homogeneous or heterogeneous crystal nucleation depending on the absence or presence, respectively, of some impurities, seed crystals, or contact surfaces, called substrates (see Leetaru, 1996). As stated by Schneider (1997), "Nucleation commonly occurs at sites of anomalous point defects on the grain surface, at structural distortions caused by edge or screw dislocations, or at irregular surface features produced by dissolution and etching." Because, Schneider (1997) adds, "When nucleation occurs at one of these sites, the free energy of the defect, dislocation, or surface irregularity can contribute to help overcome any energy barrier to nucleation." Also, the mineral grain surfaces serve as seed for nucleation if the mineral crystal lattice structure matches that of the precipitating substance (Schneider, 1997). The free energy change associated with heterogeneous nucleation at a surface is expressed by (Schneider, 1997):
Stumm and Morgan (1996) expressed the interface free energy change by
where cw, cs, and sw denote the deposit-water, deposit-substrate, and substrate-water interfaces, respectively. A denotes the surface area and o denotes the interfacial free energy. Gcs and ocw denote the surface energies per unit surface area of the deposited particle-substrate interface and the deposited particle-solution interface, respectively, e is the strain energy per unit volume. V is the volume of particle formed by precipitation, o is the surface energy per unit particle surface. AGV is the change of volume free energy from solution to solid phases per unit particle volume, given by (Stumm and Morgan, 1996):
where kb is the Boltzmann constant, T is absolute temperature, v is the molar volume, and a and a0 are the activity of the mineral dissolved in solution and its theoretical activity at saturation, respectively. Considering a semi-spherical deposition of radius r over a planar substrate surface as an approximation.
By combining the various efforts, Eq. 9-6 can be expressed as (Walton, 1969; Putnis and McConnell, 1980; Richardson and McSween, 1989; Schneider, 1997; and Stumm and Morgan, 1996):
The depositing substance and the substrate surface match well when acs<acH" and GCS is negligible and GSW=GCW for perfect matching (Schneider, 1997). Thus, the critical minimum radius necessary for formation of stable particles can be determined by equating the derivative of Eq. 9-11 with respect to the radius to zero as:
Then, substituting Eq. 9-12 into Eq. 9-11 yields the expression for the activation energy necessary for formation of stable particles as:
Approximating the shape of the deposit by a sphere of radius r such that
The minimum critical particle radius for a homogeneous nucleus to form a stable deposit at a given super saturation state can be estimated by equating the derivative of Eq. 9-17 to zero:
Thus, the activation energy necessary for starting homogeneous nucleation can be estimated by substituting Eq. 9-18 into Eq. 9-17 as:
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