1. Introduction
Against the background of global warming, geothermal energy [
1], as a type of renewable energy, has been taken as an effective alternative for fossil fuels and utilized in power supply for decades [
2,
3]. During the development of geothermal reservoirs [
4], the injected cold water will be heated by in situ fluid/rock and the heat can be carried up to the surface through water reinjection and cycling. For high-enthalpy geothermal systems [
5], water can be present in the forms of vapor phase or vapor-liquid mixture under reservoir conditions. When developing the high-enthalpy geothermal reservoir with cold water injection, hot steam condensation happens after its contact with cold water. Therefore, multiphase flow and transport with phase changes appear in high-enthalpy geothermal systems.
Numerical simulation, as an efficient tool, can be used to model the mass and heat transfer processes happened in the subsurface reservoir. In geothermal simulation, the mass and energy conservation equations are usually tightly coupled since the fluid thermodynamics are functions of primary variables [
6]. A fully coupled, fully implicit strategy is generally utilized in solving the coupled formulations. In a high-enthalpy geothermal simulation, numerical simulators can experience great difficulties, one of which is commonly known as ‘negative compressibility’. This problem was first described by Coats [
7] with a single-cell numerical experiment, where a cell with saturated steam is invaded by cold water from the cell boundary with fixed pressure. Due to the invading of cold water, hot steam will condense, and the cell pressure will drop with steam shrinkage during phase transition. The cell pressure will constantly decrease until the steam is condensed and cell pressure then goes up to the injection pressure. To guarantee the convergence of a simulation, the timestep should be severely restricted which is often addressed as ‘stalling behavior’; see Pruess [
8] for an example.
Pruess et al. [
9] and Falta et al. [
10] discussed the ‘negative compressibility’ problem. They concluded that the ‘negative compressibility’ effect is because of the idealization of complete thermodynamic equilibrium within the computational grids. Spurious pressure variation could occur in control volumes with the two-phase front because of the instantaneous thermodynamic equilibrium assumption, which will enforce severe limitations to the nonlinear convergence. Wang [
11] made an analysis of the ‘negative compressibility’ problem in the fully implicit formulation. In that analysis, to ensure convergence of the fully implicit solution, a stability criterion for the timestep was developed and therefore, repeated timestep cuts were, to some extent, avoided. Nevertheless, the derived stability criterion is still quite restricted for simulations with larger timesteps. Wong et al. [
12] applied a nonlinear preconditioner to the fully coupled, fully implicit solution. The formulations were first solved with a sequential fully implicit approach (SFI) and then the solutions were taken as the initial guess for the fully implicit method (FIM) [
13]. Using this approach, the severe timestep restriction was reduced for some practical scenarios. However, there is still no robust strategy for converging a high-enthalpy nonlinear solution at a target timestep in the presence of the ‘negative compressibility’ phenomena. Therefore, it is of special interest to investigate an efficient way to tackle the issue of slow convergence appearing in the ‘negative compressibility’.
The partial differential equations included in a geothermal simulation are usually discretized within both spatial and temporal domains for approximate solutions. Generally, the governing system in discretized form has different degrees of nonlinearity and should be linearized. The discretized formulation is often linearized with a Newton-based procedure, where an assembly of the Jacobian matrix and residuals are needed. The values of fluid properties and their derivatives are involved in Jacobian assembly. With complex physical processes (i.e., multiphase compositional flow) involved in the setup, accurate physical properties should be evaluated through multiphase flash calculation [
14]. This process is required to solve highly nonlinear local constraints in each Newton iteration for molar formulation [
15]. Therefore, a heavy part of the overall simulation is occupied by the Jacobian assembly. Recently, the Operator-Based Linearization (OBL) approach was proposed by Voskov [
16] to facilitate this process and therefore accelerate the linearization. Similar to discretization in spatial and temporal domains, the physics will be discretized within the domain of nonlinear unknowns in OBL.
In OBL, the governing equations are written in the form of operators with two categories: state-dependent and space-dependent. The state-dependent operators can be parameterized in physical space constructed by primary unknowns with different resolutions. The tables consist of pre-computed supporting points. By interpolation, the values and derivatives of the operators are evaluated with supporting points in parameter space. To further accelerate the linearization process, the adaptive parameterization technique has been proposed. In adaptive parameterization, operator values at the supporting points are evaluated along the simulation and stored for later reuse, which is especially efficient for parameterization in high-dimensional parameter space. In the meantime, it makes the Jacobian assembly simple and flexible even with complex physical processes. The OBL approach has been successfully utilized in simulating many kinds of physical processes in the subsurface [
17]. The OBL technique also behaves as the underlying framework of the Delft Advanced Research Terra Simulator [
18], whose robustness and efficiency have been sufficiently verified [
19].
The OBL approach makes it possible to flexibly control the degree of physical nonlinearity by adjusting the resolution in parameter space. In other words, if fewer supporting points are chosen in parameter space, the nonlinear physics will become more linear, which makes it easier for the nonlinear solver to converge [
16]. In this work, we follow the hierarchy of physical approximation in parameter space using the OBL formalism and construct a continuous solution in physics to solve the ‘negative compressibility’ problem. We start with general formulations and numerical strategies used in thermal-compositional simulations and briefly introduce the OBL approach. Next, the ‘negative compressibility’ problem in a single cell is described from the Newton path, residual distribution and operator surface. Afterward, the continuous localization in physics is adopted to solve the ‘negative compressibility’ problem. Finally, an idealized one-dimensional test case and a heterogeneous two-dimensional test case are used to verify the feasibility of the proposed method.
2. Methodology
During the production of a high-enthalpy geothermal reservoir containing aqueous phase, the mass and energy conservation equations are utilized to describe the governing system with two-phase thermal fluid flow and transport.
where
is reservoir porosity,
refers to density of phase
,
refers to saturation of phase
,
refers to internal energy of phase
,
is rock internal energy,
is enthalpy of phase
, and
is thermal conduction.
It is assumed that the fluid flow in the reservoir follows Darcy’s law,
where
refers to Darcy velocity,
is permeability tensor,
refers to relative permeability of phase
,
refers to viscosity of phase
,
refers to pressure of phase
,
refers to specific weight of phase
and is defined as the product of the density of phase
and the gravity acceleration, and
is depth. In addition, to close the system, the summation of phase saturation should be equal to one,
Next, Darcy’s law is substituted into the governing equation and the resulting nonlinear equations are discretized with a finite-volume method in space on a general unstructured mesh and with backward Euler approximation in time:
where
is the volume of the grid block and
refers to the source and sink term of phase
.
is the phase pressure difference (including gravity and capillary pressure) between blocks connected via interface
, and
is a temperature difference between these neighboring blocks;
is cell transmissibility for phase
, and
is the geometrical part of transmissibility, which is evaluated by the permeability and the geometry of the control volume.
is the thermal transmissibility.
In the framework of molar formulation [
12,
20], pressure and enthalpy are selected as the primary variables for the nonlinear solution. In general, the Newton–Raphson method is used to solve a linearized system of equations in each nonlinear iteration in the following form:
where
refers to the Jacobian matrix constructed within the
nonlinear iteration. In the traditional method, the assembly of a Jacobian matrix requires accurate evaluation of the values and derivatives of physical properties. During this process, either different sets of interpolations (for properties such as relative permeabilities) or solution of a highly nonlinear system (e.g., multiphase flash) is needed. To meet the convergence requirement, the nonlinear solver has to undertake many iterations to resolve the minor variations in properties, which are sometimes unnecessary because of the numerical nature and uncertainties of physical properties. The Operator-Based Linearization approach, described below, is proposed to resolve this issue.
3. Operator-Based Linearization (OBL) Approach
The mass and energy conservation equations, in the OBL approach, are distinguished as different types of operators, the state-dependent and space-dependent operators. By name it can be recognized that the state-dependent operators are functions of a physical state
, whereas the space-dependent operators are correlated as functions of a physical state
and a spatial coordinate
[
16,
21]. Pressure and enthalpy are selected as the primary state variables in geothermal simulations.
The discretized governing equation of mass in operator form is expressed as:
where
is defined as the mass accumulation operator, and
is called the mass flux operator.
The discretized governing equation of energy in operator form is expressed as follows:
where
is defined as the energy accumulation operator, and
is called the energy flux operator. This representation in operator form will be used to significantly simplify the general-purpose simulation framework. Without conducting complicated estimations of the value and derivatives of each property during the simulation, we can parameterize the state-dependent operators in the space of unknowns with a limited number of supporting points and use multilinear interpolation to evaluate them [
16]. This not only makes the Jacobian assembly simpler but also improves the simulation performance since almost all expensive property evaluations are replaced by interpolations. In addition, due to the piece-wise multilinear approximation of physical operators, the system will become more linear and the performance of the nonlinear solver can be improved.
6. Conclusions
Since mass and energy conservation equations are tightly coupled through the fluid thermodynamics in high-enthalpy geothermal processes, they are usually solved in a fully implicit manner. The ‘negative compressibility’ phenomena can significantly impede the convergence of the nonlinear solver. Because of the large variation in thermodynamic properties between water and steam, the governing equations show high nonlinearity with phase transition. We analyze the problem in a single-cell setup with a cold water injection at fixed pressure. The analysis of the residual map demonstrates that two different minima can be present in the parameter space when simulation is performed with a sufficiently large timestep, which brings challenges for a gradient-based nonlinear solver. A suitable initial guess is essential for the Newton-based nonlinear strategy.
Applying the Operator-Base Linearization approach, we propose the continuous localization in physics method to solve the governing system of equations. With parametrization in physics changing from a coarse to fine resolution, the state-dependent operators and resulting residual changes from an almost linear and monotone behavior to a highly nonlinear and non-monotone shape. In the proposed nonlinear strategy, the solution at a coarser parametrization in physics is taken as an initial guess for the solution at a finer physical resolution. This continuous localization approach makes the nonlinear convergence process more robust in the presence of the ‘negative compressibility’ phenomena. To verify the feasibility of this approach, we prepare synthetic one- and two-dimensional test cases and make a comparison between the conventional and proposed approaches. The results demonstrate that the simulation of high-enthalpy geothermal applications can benefit from continuous localization by running the model at a sufficiently large timestep with a limited number of nonlinear iterations.