Mathematical Modeling of CO₂ Storage in a Geological Formation

Abstract

Chapter 2 discusses, in a descriptive manner, the processes occurring during the geological storage of CO₂. In this chapter, the mathematical models describing these processes are described. The chapter starts from the basic properties of the injected CO₂ and of the native brine, proceeding to the relevant models for multiphase flow of CO₂ and brine , the related chemical and reactive transport processes, the non-isothermal effects of CO₂ injection and the mechanical deformation . The concept of degrees of freedom , facilitating the selection of a smaller number of equations to be solved, in order to obtain a complete solution for this multifaceted problem, is also discussed. The numerical and analytical approaches for solving these mathematical models are presented in the following Chap. 4.

Appendix: Primary Variables and Degrees of Freedom

3.1.1 Extension of Gibbs Phase Rule

As shown throughout this chapter, a rather large number of variables are required in order to describe the complete behavior of a problem of transport that involves one or two fluid phases, a large number of chemical species, non-isothermal conditions, and a deformable solid matrix.

However, if we assume that locally, i.e., at every (macroscopic) point in the considered porous medium domain, all phases and chemical species are in thermodynamic equilibrium , or when the rate of transformation of the system from one state to another is sufficiently slow so that it can be assumed to be continuously close to equilibrium, this number can be significantly reduced, thus simplifying the task of solving the mathematical model .

The number of degrees of freedom is the smallest number of independent variables needed to fully define a system’s present and future behavior. We refer to these variables as primary variables. The values of the primary variables are obtained by solving partial differential balance equation. Values of all other system’s state variables can be obtained from the primary ones through the use of constitutive relationships and definitions.

Gibbs phase rule states that (at the microscopic level of description) the state of a system composed of NP phases and NC non-reacting components, under conditions of equilibrium, is fully defined by NF state variables, with NF, which is the number of degrees of freedom of the problem, determined by the relationship

$$ {\text{NF}} = {\text{NC}} - {\text{NP}} + 2 $$

(3.8.1)

Bear and Nitao (1995), on the basis of balance considerations and thermodynamic relationships, showed that under (exact of approximate) conditions of thermodynamic equilibrium among all phases and components present in a deformable porous medium domain under non-isothermal conditions, the number of degrees of freedom , NF, in a problem of non-isothermal mass transport, involving NP fluid phases and NC chemical species, is given by the relationship

$$ {\text{NF}} = {\text{NC}} + {\text{NP}} + 4 $$

(3.8.2)

Under conditions of non-equilibrium between the phases, this rule becomes

$$ {\text{NF}} = {\text{NC}} \times {\text{NP}} + 2 \times {\text{NP}} + {\text{NC}} + 4 $$

(3.8.3)

In both cases, when Darcy’s law is used to determine the velocities of the fluid phases, NF is reduced by NP. When the solid matrix is non-deformable, NF is reduced by 3, leading to the relationship

$$ {\text{NF}} = {\text{NC}} + 1 $$

(3.8.4)

Under isothermal conditions, the rule reduces to

$$ {\text{NF}} = {\text{NC}} $$

(3.8.5)

These rules are, thus, extensions of the Gibbs phase rule mentioned in Sect. 3.2 to phenomena of transport in porous media. The number of degrees of freedom for reactive transport problems was also discussed by Saaltink et al. (1998), and by Molins et al. (2004).

For a system with chemical reactions , let NS be the number of reacting species and ${\text{NR}}_{\text{eq}}$ denote the number of equilibrium reactions. Then, by expressing the reactions in the form a canonical set of equations, and using of the law of mass action, there are ${\text{NC}} = {\text{NS}} - {\text{NR}}_{\text{eq}}$ independent species concentrations . Thus, the number of degree of freedom, NF, in the above equations for a non-reacting system, still applies to a reacting system, as long as we use

$$ {\text{NC}} = {\text{NS}} - {\text{NR}}_{\text{eq}} $$

(3.8.6)

Once the number of degrees of freedom has been determined for a given problem, we select the most convenient variables to be declared as primary ones, and identify the (same number of) balance (partial differential) equations which have to be solved in order to determine the values of these variables. All other variables are, subsequently, determined by using the remaining equations—constitutive relations and definitions.

3.1.2 Degrees of Freedom for Phases in Motion

Essentially, in a porous medium, a system that undergoes changes in time due to motion of the phases can never be strictly in complete/exact thermodynamic equilibrium (Bear and Nitao 1995). Conditions of mechanical nonequilibrium prevail as a consequence of the transfer of momentum from the moving fluid to the solid by viscous forces. This gives rise to pressure gradients at the microscopic level within the REV. Temperature gradients may also occur because of viscous dissipation. If these pressure and temperature gradients are large, the system will be far from chemical and thermal equilibrium. In a multiphase REV, flow can cause some phases to be under nonequilibrium conditions, and some of the phases may be in nonequilibrium with each other. Table 3.4 summarized degrees of freedom for the non-equilibrium case.

As seen in the above table, the number of degrees of freedom depends on the type of problem that is being modeled and on the simplifications involved. e.g., whether or not the solid is assumed to be deformable, in an isothermal case and when Darcy’s law is employed (Table 3.4).

Table 3.4 Degrees of freedom for non-equilibrium case

When the phases are in thermal equilibrium with each other, the value of NF is reduced in all of the above cases by the amount NP. When the components absorbed on a solid are in chemical equilibrium with the fluid phases, the value of NF is reduced in all of the above cases by the amount of NC.

3.1.3 Degrees of Freedom Under Approximate Chemical and Thermal Equilibrium

Under the assumption that at every point within the domain and at every instant of time, the system is in approximate thermodynamic equilibrium between the averaged (over the REV) behavior of the phases and the components at that point, the number of degrees of freedom is as follow:

For a reactive system, the number of components, NC, is determined by reducing the number of the reactive species (NS) by the number of equilibrium equations (NE)—${\text{NC}} = {\rm NS} - {\rm NE}$ (Table 3.5).

Table 3.5 Degrees of freedom for equilibrium case