Geochemical Phenomena—Classification, Formulation, Modeling, and Software
Fluids and minerals in petroleum-bearing formations may undergo various interactive chemical reactions in response to the alteration of the in-situ conditions by various operations, including drilling, workover, and improved recovery. Geochemical models provide scientific guidance for controlling adverse reactions that may result from rock-fluid interactions. Excellent treaties of the geochemical reaction modeling are available from several sources, including Melchior and Bassett (1990), Ortoleva (1994), and Bethke (1996). This subject is extremely complex, therefore, only the fundamentals of the overall subject are outlined here. The readers are encouraged to resort to literature for details and to use ready-made software available from various sources.
Petroleum-bearing formations can be generally viewed as being geochemical systems in which fluids consisting of oil, gas, water, and immobile solid phases formed from an assemblage of minerals interact through various chemical reactions. Lichtner (1985) classified such reactions into four categories:
(1) aqueous ion complexing,
(2) oxidation and reduction,
(3) mineral precipitation and dissolution, and
(4) ion exchange and adsorption reactions.
As stated by Kharaka et al. (1988) and Amaefule et al. (1988), such reactions occur in response to changing temperature, pressure, and fluid composition by various factors, including the addition of incompatible fluids during drilling, workover and improved recovery processes and liberation of light gases, such as CH4, CO2, H2S, and NH3, during pressure-drawdown. Changes in temperature and pressure often cause the variation of the pH of the reservoir aqueous phase, which in turn induces adverse processes such as the precipitation of iron and silica gels (Kharaka et al., 1988; Rege and Fogler, 1989; Labrid and Bazin, 1993). Geochemical reactions can also be classified as homogeneous and heterogeneous depending on whether the reaction occurs inside a phase or with another phase, respectively. Geochemical reactions can also be classified as reversible and irreversible.
As explained by Lichtner (1985), the rates of reversible reactions are independent of the surface area. Reversible reactions can attain local equilibrium over a sufficiently long period of time, at which time, the reaction rates terms vanish in the transport equations. However, Lichtner (1985) adds that irreversible reactions require kinetic or rate expressions, in terms of the pertinent driving forces, that is chemical affinity, and/or the surface available for reactions. Detailed geochemical description is a very cumbersome task and often unnecessary and unjustified in view of the lack of the basic thermodynamic and kinetic data required for description.
Rather, geochemical models are constructed to emphasize the chemical reactions of the important aqueous species and minerals, which are essential for adequate description of the rock-water interactions, and neglect all other reactions. This is done to compromise between the quality of description and the effort necessary to gather basic thermodynamic and kinetic data and to carry out the numerical computations. Among the various alternatives, the kinetic and equilibrium models are extensively utilized. The kinetic models describe the rate of change of the amount of mineral and aqueous species in porous media in terms of the relevant driving forces and factors, such as deviation from equilibrium concentration and mineral-aqueous solution contact surface.
The proportionality constant is called the rate constant. The equations formed in this way are called the rate laws or kinetic equations. The equilibrium models assume geochemical equilibrium between the pore water and the minerals of porous formation. Since equilibrium can be reached over a sufficiently long time, equilibrium models represent the closed systems at steady-state conditions. Mathematically, the equilibrium models can be derived from kinetic models in the limit of infinitely large rate constants. Hence, rapid reactions reach equilibrium faster. Therefore, the equilibrium models represent the limiting conditions and yield conservative predictions (Schneider, 1997). Equilibrium models are particularly advantageous for determining the mineral stability and graphical representations of the mineral and aqueous species interactions (Bj0rkum and Gjelsvik, 1988; Stumm and Morgan, 1996; Schneider, 1997).
Because of the highly intensive numerical computations involved, the geochemical models of the rock-water interactions are usually implemented by computer-coded software. The geochemical computer software is constantly evolving and becoming more robust and accurate as a result of the advancement in computer technology, availability of accurate thermodynamic data, and development of efficient numerical solution methods. The engineers responsible for developing operational strategies and procedures for scale control in petroleum reservoirs should rely on such software. However, efficient use of the ready-made software requires some familiarity with the fundamental concepts, theories, and methods involved in the treatment and formulation of geochemical reactions. This information is usually provided with the user's guide and/or by relevant publications. In the following, a brief review of the description and graphical representation of aqueous and mineral species reactions and various approaches to geochemical modeling are presented.
Reactions in Porous Media
The various chemical reactions occurring in the pore space can be classified into the groups of homogeneous and heterogeneous reactions (Lichtner, 1992). The reactions occurring within the aqueous fluid phase are called the homogeneous or aqueous reactions. The reactions of the aqueous phase species with the solid minerals of porous formation, occurring at the pore surface, are called the heterogeneous or mineral reactions. A convenient treatment of the geochemical reactions can be achieved by grouping the various reacting solute species into the primary and secondary sets of species (Kandiner and Brinkley, 1950; Lichtner, 1992). The primary set of species is formed by selecting a minimum, critical number of reacting aqueous species, Sα, necessary for adequate description of the homogeneous and heterogeneous reactions. Thus, all other species form the set of the secondary species. The secondary species are derived from the primary species by means of the equations of the relevant chemical reactions.
Aqueous Phase Reactions
(a) ion pairing/exchange reactions,
(b) complexing reactions, and
(c) redox reactions.
The aqueous phase reactions are generally rapid relative to the mineral reactions (Liu et al., 1997). The rapid rates of aqueous phase reactions require kinetic descriptions with significantly large rate constants. Thus, for all practical purposes, these reactions can be assumed instantaneous and a transport controlled, local chemical equilibrium assumption is usually considered reasonable (Walsh et al., 1982, Lichtner, 1992; Liu et al., 1997; Liu and Ortaleva, 1996, 1996). Consider an aqueous phase undergoing a total of Nfr chemical aqueous reactions, r = 1,2,...,Nf denotes the index for the various aqueous reactions. Nr represents the total number of aqueous species involved in the rth aqueous or homogeneous reaction. Sa:а. = l,2,...,Nr denotes the various aqueous species involved in the r̩th aqueous reaction. Then, the aqueous reactions can be typically represented by (Walsh et al., 1982; Liu et al., 1996):
where νrfα denotes the stochiometric coefficient of species a involved in the rth aqueous reaction. Note v/a is negative for the reactants and positive for the products. Applying the mass action law of Prigogine and DeFay (1954), the chemical equilibria between the products and reactants of the rth reaction can be expressed as (Walsh et al., 1982; Liu et al., 1996):
Kfr denotes the thermodynamic equilibrium constant for the rth aqueous reaction given as the ratio of the rate constants kfrf and kfrb of the forward and backward reactions represented by Eq. 13-3:
αa is the chemical activity of the aqueous species a, which can be expressed in terms of the molal concentration, Cα, of species α as:
in which үa is the activity coefficient determined by the Debye-Huckel theory (Helgeson et al., 1970).
The reactions of the aqueous phase species with the solid mineral matter of the porous matrix are referred to as the mineral reactions. Most mineral reactions are typically hydrolysis reactions. The interactions of minerals and aqueous species are generally slow relative to the aqueous phase reactions (Lichtner, 1992). Their reaction kinetics are controlled by the external mineral surface area contacting the aqueous phase. The mineral surface area is determined by the sizes of the grains of the porous formation. The rates of mineral reactions are gradual and, therefore, require kinetic descriptions with finite reaction rate constants. Consider a porous formation containing a total of Ns different mineral species and undergoing a total of Nsr different chemical reactions between its minerals species and the aqueous phase species. s = 1,2,...,Ns denotes the index for the participating mineral species of the porous formation. Nr denotes the total number of aqueous species involved in the rth heterogeneous reaction. Msr denotes the sth mineral species undergoing the rth heterogeneous reaction. Sa:a = l,2,...,Nr denotes the various aqueous species involved in the rth mineral reaction. Then, the reactions between porous formation minerals and aqueous species can be typically represented by (Lichtner, 1992, Liu et al., 1996):
where νsra denotes the stochiometric coefficients associated with the aqueous phase species per one participating mineral species. Note that vs m is negative for the reactants and positive for the products. Applying the mass action law (Prigogine and DeFay, 1954), and assuming that the activity of the solid minerals is equal to one, the chemical equilibria between the minerals of the porous formation and the aqueous species of the rth reaction can be expressed in terms of the equilibrium saturation solubility product as (Walsh et al., 1982):
For a mineral s undergoing a dissolution/precipitation reaction r, the reaction driving force is given by (Liu et al., 1996):
ΔGrs > 0 for mineral dissolution and ΔGri < 0 for mineral precipitation. Κrs denotes the dissolution rate constant of the rth reaction of the sth mineral. Approximating the shape of the mineral grains by a sphere, and assuming that the mineral reactions are kinetic reactions, Liu et al. (1996) express the rate of dissolution or precipitation of the sth mineral by the rth reaction by:
as being proportional to the number of the Sth mineral grains per formation bulk volume, ns, the surface area of the sth mineral grain, 4ԉR2s, and the reaction driving force, ΔGrj. The grain mass density of the sth mineral, Ps, is inserted to express the mass rate of dissolution or precipitation of the ith mineral grain. Rs represents the radius of the Sth mineral grain. Thus, Liu et al. (1996) express the rate of change of the sth grain radius by dissolution and/or precipitation by various reactions as:
The mass conservation equation for species a undergoing transport through porous media by various mechanisms is given by Eq. 7-34 derived in this article. The rate of generation of the Oith aqueous species per bulk formation volume, required for Eq. 7-34, is given by Liu et al. (1996) as:
where Na is the total number of aqueous species involved; Nrf and Nrs denote the total number of aqueous and mineral reactions, respectively; Wrf and Wrs represent the rates of the rth aqueous and mineral reactions, respectively; Ѵfar. and Ѵsar. denote the stochiometric coefficients of the species a in the aqueous and mineral reactions, respectively, and qa represents the rate of species a addition per bulk formation volume by means of direct injection of fluids through wells completed in the reservoir.
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