Persistent-variable thermal compositional simulation of multiphase flow with phase separation in porous media

TL;DR

The paper develops a persistent-variable, enthalpy-based formulation for thermal compositional multiphase flow in porous media, incorporating phase separation via a local equilibrium subproblem without phase-stability tests. It embeds a local solver for the equilibrium calculations within a global fully implicit Newton solver, leveraging locality and modularity to reduce nonlinear iterations and enable high-enthalpy, narrow-boiling scenarios. Demonstrations with a high-enthalpy manufactured case and a Peng-Robinson EOS, including viscosity modeling via Lohrenz-Bray-Clark correlations, show significant improvements in convergence and robustness, with guidance on the cadence and precision of local equilibration. The approach offers a scalable, continuous, and thermodynamically consistent framework for challenging non-isothermal multiphase flow problems in porous media, suitable for high-enthalpy applications and complex phase behavior.

Abstract

Thermal compositional multiphase flow in porous media with phase transitions involves complex nonlinear interactions among flow, transport, and phase equilibrium. This paper presents a persistent-variable formulation for thermal compositional flow using enthalpy to formulate the energy balance and the local equilibrium problem. Equilibrium conditions are derived from a thermodynamically consistent minimization problem using a persistent set of variables, allowing for seamless integration of equilibrium calculations into a fully coupled flow and transport model. This formulation does not require phase stability tests and provides a continuous and full mathematical description of the multiphysics system, suitable for challenging non-isothermal scenarios. To tackle the nonlinearities arising from phase transitions, we embed a local solver for the thermodynamic subproblem within a global Newton solver for the fully implicit system. The local solver exploits the locality of the subproblem for parallelization and leverages the modularity of the persistent-variable formulation for both isothermal and isenthalpic equilibrium conditions locally. We demonstrate the capability of our approach to simulate complex high-enthalpy systems, including narrow-boiling phenomena. The impact of the embedded local solver is analyzed through numerical experiments, demonstrating a reduction in global nonlinear iterations of up to 23 \% with increased use of the local solver. The number of local iterations is controlled with a local solver tolerance and no significant impact on the global iteration number was observed for local residual tolerances as high as $1e-3$. The persistent-variable approach using enthalpy and the modularity of the embedded local solver advance the usage of equilibrium calculations in multiphase flow simulations and are suitable for high-enthalpy applications.

Figures (12)

• Figure 1: Flow chart for nonlinear solver. The Newton algorithm applied to each time step forms the basis. Within each iteration, the solution strategy is adapted to use the local equilibrium solver and feed the results back into the global system.
• Figure 2: Layout of 2D simulation setup. A rectangular domain with an injection and a production well. Mass is injected at rate $r_i$ and temperature $T_i$. The pressure at production is fixed to $p_p$. The production well is modeled as a perfect sink, extracting all mass and energy flowing into respective cell. The central bottom part of the boundary has a fixed temperature $T_h$. We impose no-flow conditions everywhere else. The domain has lower permeability $K_l$ in the middle section.
• Figure 3: Simulation after (a) 90, (b) 240 , (c) 300 and (d) 600 days. Rows from top to bottom: specific fluid enthalpy $h$ [kJ / mol], temperature $T$ [K], gas saturation $s_g$ [-] and local iterations summed over iterations per displayed time step.
• Figure 4: Top: Pressure $p$ [MPa] (left axis) and temperature $T$ [K] (right axis). Bottom: Gas saturation $s_g$ [-], fraction of carbon dioxide $z_{\text{CO$_2$\,}}$ [-] (left axis) and specific fluid enthalpy $h$ [kJ / mol] (right axis). The profiles correspond to \ref{['fig:simulation']} (a), horizontally through the domain intersecting wells. The injection and production point are positioned at 15 and 85 meters respectively.
• Figure 5: Progress in time (left axis) and time steps size (right axis) in days.

Theorems & Definitions (3)

• Remark 3.1
• Remark 3.2
• Remark 4.1