Abstract
We present a novel nonlinear formulation for modeling reactive-compositional flow and transport in the presence of complex phase behavior due to a combination of thermodynamic and chemical equilibria in multi-phase systems. We apply this formulation to model precipitation/dissolution of minerals in reactive multiphase flow in subsurface reservoirs. The proposed formulation is based on the consistent element balance reduction of the molar (overall composition) formulation. To predict the complex phase behavior in such systems, we include the chemical equilibrium constraints to the multiphase multicomponent negative flash calculations and solve the thermodynamic phase and chemical phase equilibria simultaneously. In this solution, the phase equilibrium is represented by the partition coefficients, whereas the chemical equilibrium reaction is represented by the activity coefficients model. This provides a generic treatment of chemical and thermodynamic equilibrium within the successive substitution loop of mulmultiphase flash to accommodate chemical equilibrium reactions (precipitation and dissolution reactions). Equilibrium Rate Annihilation matrix allows us to reduce the governing component conservation equations to element conservation equations, while the coupling between chemical and thermodynamic equilibrium is captured by a simultaneous solution of modified multiphase flash equations. The element balance equation written in terms of overall component mole fractions is modified and defined in terms of element mole fractions. Therefore, the primary set of governing equations are the element balance equations and the kinetic equations. This element composition of the mixture serves as an input to the modified multiphase flash computations, whereas the output is fractions of components in each phase, including solids. The nonlinear element–based governing equations are solved with the modified version of the operator-based linearization (OBL) approach where the governing equations are formulated in terms of space- and state-dependent parameters constrained by the solution of the extended multiphase flash. The element balance molar formulation along with the modified multiphase flash has been tested in a simple transport model with dissolution and precipitation reactions. The simulation of multidimensional problems of practical interest has been performed using the adaptive OBL technique. This is the first time when a robust multiphase multicomponent flash based on element fractions is coupled with an element balance–based compositional formulation and tested for multidimensional problems of practical interest. The proposed technique improves both robustness and performance of complex chemical models.
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Acknowledgments
We thank the research group at the Department of Hydromechanics and Hydrosystem modeling, University of Stuttgart for helpful discussions and support during this project. We would also thank Mark Khait for his valuable contribution in OBL implementation and Alireza Iranshahr for his help with the negative flash.
Funding
The authors received financial support from the SRP-NUPUS consortium, University of Stuttgart.
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Appendices
Appendix A: Negative flash results
To predict phase behavior, we solve the multiphase multicomponent Rachford-Rice equation using the negative flash approach suggested by [7]. Equations 10 to 12 can be used to derive the Rachford-Rice formulation as given below
In this technique, the multi-stage flash is applied using the bisection method to solve the Rachford-Rice equations. Since the bisection method is robust, it is guaranteed to converge to a solution for a monotone function, but the convergence rate can be slow. Figure 8 above shows the solution for a three-phase three-component system. This technique provides a robust mechanism to resolve the thermodynamic equilibrium by solving the RR equation. To improve the convergence rate, we include a Newton loop inside the bisection. In the case when the solution based on Newton iterations fails to converge or moves out from the physical bounds, the solver switches from Newton method back to bisection.
The comparison between the original bisection method and the combination of bisection and Newton methods is given in Table 1. We parameterized the complete compositional space for a three-phase three-component system. The first column in Table 1 shows the total number of iterations required for 18997 flash computations with a pure bisection strategy. The second column uses a combination of Newton and bisection approaches for only one of the two RR equation. The last column shows the result when applying Newton for both the RR equations. Here, the residual tolerance was set to ε = 10− 12.
From the table, we can see that there are some waste Newton iterations in the case when Newton solver fails to converge and, hence, the system switches back to bisection. Even though some of the Newton iterations were wasted, the total number of iterations for the system with Newton and bisection is almost an order of magnitude less than the total iterations for the pure bisection method. This shows that including Newton method along with bisection for phase computations can significantly improve the computational efficiency of the thermodynamic flash solver and can be extended further to find more robust and effective methods to solve the RR equation. The compositional diagram generated using the flash calculations is shown as the third figure, where the yellow tie-triangle represents a three-phase region, the blue regions are the single-phase regions, and the green regions are the two-phase regions.
Appendix B: Solid phase treatment
This appendix describes the concept of fluid and reactive porosity which combine to give the total porosity of the system. We treat the volume, occupied by the mineral component, as a part of the pore volume. The classic porosity, which represents the volume occupied by fluids, is what we call a fluid porosity. For a system without chemical reactions, the porosity only varies with changes in pressure due to the compressibility of the rock. But in the case of chemical reactions when mineral precipitation and dissolution are present, we have continuous changes in the pore space depending on the concentration of minerals. Therefore, the reactive porosity varies with mineral mole fraction. The bulk volume of the model is defined here by three parameters: non-reactive volume (Vnr), reactive volume (Vr), and the pore volume (Vϕ). The non-reactive volume is the part of the rock which is not involved in any of the chemical reaction hence its volume is always constant. The reactive volume is the mineral part of the rock, and the pore volume is the volume occupied by the fluids in the rock; both of these volumes are changing depends on the amount of mineral present. Therefore, we can define the total volume as the sum of all the three components as shown below
From the above equation, it can be seen that the total porosity of the system is always constant irrespective of the concentration of mineral. If there is less mineral deposited, the pore volume will be higher; otherwise, the reactive volume will be higher. The mineral saturation is defined as
Using the definition of total porosity, the above equation can be written as
The relation for fluid porosity can be written as
where M stands for the number of mineral species. Therefore, knowing the initial total porosity of a control volume, we can calculate the porosity based on the saturation value of the mineral. As the reaction progresses, the dissolution/precipitation process occurs which alters the fluid porosity ϕp. This porosity value can be used to determine the permeability values using empirical relations and update the velocities in the governing equations.
Appendix C: Simulation parameters
In this section, we describe the fluid and rock properties as well as numerical parameters of OBL used in the simulation. Table 2 shows the rock and fluid properties. The rock is considered low compressible and the total mineral and fluid porosity are taken as 0.3.
Table 3 shows the initial pressure condition of the reservoir and the OBL parameters which are used for adaptive simulation.
Table 4 shows the initial and injection reservoir mole fractions.
Table 5 shows the thermodynamic partition coefficient and the chemical equilibrium constants for two equilibrium reactions. These values were slightly elevated in comparison with real physical values.
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Kala, K., Voskov, D. Element balance formulation in reactive compositional flow and transport with parameterization technique. Comput Geosci 24, 609–624 (2020). https://doi.org/10.1007/s10596-019-9828-y
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DOI: https://doi.org/10.1007/s10596-019-9828-y