A Comparative Study of Solvers and Preconditioners for an SPE CO₂ Storage Benchmark Reservoir Simulation Model

Abstract

This study analyzes and evaluates the performance of various solvers and preconditioners for reservoir simulations of CO₂ injection and long-term storage using the model 11B of SPE CSP (Society of Petroleum Engineers, 11th Comparative Solution Project) and the MATLAB Reservoir Simulation Toolbox (MRST). The SPE CSP 11 model serves as a benchmark for testing numerical methods for solving partial differential equations (PDEs) in reservoir simulations. The research focuses on the Biconjugate Gradient Stabilized (BiCGSTAB) and Loose Generalized Minimum Residual (LGMRES) solver methods, as well as multiple preconditioning techniques designed to improve convergence rates and reduce computational effort for CO₂ storage. Extensive simulations were performed to compare the performance of different solver-preconditioner combinations under varying reservoir conditions, leveraging MRST’s flexible simulation capabilities. Key performance metrics, including iteration counts and computational time, were analyzed for the project. The results reveal trade-offs between computational efficiency and solution accuracy, providing valuable insights into the effectiveness of each approach. This study offers practical guidance for reservoir engineers and researchers seeking to analyze and optimize simulation workflows within MRST by identifying the strengths and limitations of specific solver-preconditioner combinations for complex linear systems.

1. Introduction

In this article, we use the model 11B in the 11th Comparative Solution Project (CSP) introduced by the Society of Petroleum Engineers (SPE) as a benchmark to compare the performance of certain combinations of solvers and preconditioners for CO₂ injection and long-term storage simulations. The CSP 11B model represents a meticulously designed reservoir simulation structure for analyzing the behavior of CO₂ once injected into the reservoir. By focusing on the dynamics of CO₂ storage and ensuring accurate modeling of key physical processes, our study provides valuable insights into the long-term viability of CO₂ storage projects and supports the success of carbon capture and storage (CCS) initiatives. We rely on the MATLAB Reservoir Simulation Toolbox (MRST) for reservoir simulations, iteration counts, and CPU time measurements.

Subsurface carbon dioxide (CO₂) storage, often called carbon sequestration, is the process of capturing CO₂ emissions and storing them underground to prevent them from entering the atmosphere. The captured CO₂ is typically injected into deep geological formations, where it is stored in porous rock layers under impermeable cap rocks, ensuring that the gas remains trapped for extended periods. This technique is a critical component in the global effort to mitigate climate change by reducing the concentration of greenhouse gases in the atmosphere. The primary goal of subsurface CO₂ storage is to achieve net-zero emissions by balancing the amount of CO₂ released into the atmosphere with the amount sequestered [1,2,3], though UN-set targets have continuously been revised, and other greenhouse gas emission targets are also often missed [4]. Moreover, the CO₂ removal industry had its own challenges in growing due to economic, political, and social barriers or constraints [4,5]. Even though our focus in this study is on geological/subsurface storage, we acknowledge other approaches to CCS including soil-based CO₂ sequestration and utilization through natural and safer organic and inorganic synergistic processes [4,5].

There are four principal trapping mechanisms for CO₂ storage within subsurface geological formations: structural trapping, capillary (residual) trapping, solubility trapping, and mineral trapping [3,6]. These mechanisms vary both in their CO₂ trapping capacity over time and the security they offer for long-term storage. Structural trapping is the most significant factor in the volume of CO₂ that can be stored, as it relies on geological formations such as anticlines and fault traps to contain the CO₂. Capillary trapping, on the other hand, occurs when CO₂ becomes immobilized in the pore spaces of the reservoir rock due to capillary forces, making it less mobile. Solubility trapping involves the dissolution of CO₂ into the formation brine, reducing the likelihood of leakage. In contrast, mineral trapping involves chemical reactions between CO₂ and the reservoir minerals, forming stable carbonate minerals that permanently sequester the CO₂ [6].

Some recent research also examines trapping mechanisms for containment risks associated with subsurface CO₂ storage, underscoring the critical importance of accurately evaluating caprock, faults, and overburden to assess the risks associated with CO₂ injection [7]. By proposing an interdisciplinary workflow in a 3D geomechanical model, [7] addresses the complex nature of rock behavior under stress. Their findings reveal significant differences between isotropic and anisotropic overburden models, demonstrating the importance of accurate overburden characterization for reliable risk assessment. Their proposed interdisciplinary workflow broadly applies to CO₂ storage projects and provides valuable insights for other subsurface injection projects globally.

A particularly notable aspect of the study by [7] is the analysis presented in Figure 1, which illustrates the vertical deformation of rock after 50 years of CO₂ injection within the Smeaheia reservoir in the Northern North Sea. The figure compares the results from isotropic and anisotropic overburden models, revealing significant spatial and vertical variations. This comparison underscores the issue of overestimating potential geomechanical risks when relying on an isotropic model, which could significantly influence decision-making processes in such projects. They also discuss the critical role of accurate modeling techniques and the impact of numerical stability criteria in simulations, particularly the Courant-Friedrichs-Lewy (CFL) condition. This condition is fundamental in the numerical study of (especially hyperbolic) PDEs, using, for example, the finite difference method (FDM). To maintain accuracy, the CFL condition mandates that the time step be shorter than the time required for a wave to travel between adjacent grid points, a constraint that explicit time integration methods (such as the Euler method) must strictly adhere to.

1.1. Mathematical and Computational Aspects

Reservoir simulations typically begin by constructing a mathematical model comprising a set of nonlinear, coupled partial differential equations (PDEs) that describe nonisothermal, multiphase, and multicomponent flow in porous media. These governing PDEs are typically discretized and transformed into nonlinear systems of algebraic equations, which are subsequently linearized into large, often nonsymmetric systems of equations using numerical methods such as the finite difference, finite volume, finite element methods, or their variants, in combination with Newton-like schemes.

The primary unknowns in the PDE system arising from reservoir simulations often include phase pressures, phase saturations, component molar fractions, and temperature. Phase pressures typically obey equations that are nearly elliptic, while molar fractions and saturation components follow equations that are almost hyperbolic. Temperature, in turn, exhibits elliptic or hyperbolic behavior depending on whether the diffusive or advective component dominates. Researchers have employed a variety of specialized numerical approaches to address the mixed elliptic/hyperbolic nature of the governing equations, including fully implicit (FI) methods and sequential techniques such as implicit pressure explicit saturation (IMPES) and sequentially fully implicit (SFI) schemes, among others. Below, we briefly describe some of the techniques used to tackle the governing equations in reservoir simulation.

The IMPES method generally applies to models with conservation equations. The IMPES pressure equation combines linearized conservation equations, allowing the method to simulate black oil reservoirs, as first demonstrated by [8]. The method can then be generalized to encompass simulations with multiple conservation equations and stability conditions (e.g., see [9] and the references there). In the context of reservoir simulations, the fundamental idea of IMPES is to isolate pressure by removing non-pressure variables from the system of equations, leading to a single pressure equation.

The Finite Difference Simultaneous Solution (SS) (FDSS) method addresses all coupled nonlinear equations simultaneously and implicitly. Known for its stability, it can manage large time steps, making it applicable to oil and thermal models with few components. However, the computational burden of solving large linear systems limits its suitability for complex compositional models [10].

Sequential (SEQ) methods offer software development advantages by decoupling flow and mechanics problems in sequence while retaining implicit time discretization for each problem. SEQ is particularly effective for compositional and chemical flow problems, offering a practical balance between stability and efficiency. By incorporating finite difference adaptive implicit methods, one can dynamically switch between implicit and explicit treatment of the saturation equation. This allows costly SS to be applied selectively only where necessary while using IMPES elsewhere, thus optimizing computation.

The classical finite element method (FEM) approximates solutions to governing PDEs by constructing certain finite dimensional subspaces of the original infinite-dimensional function space in which the exact solution resides and seeking approximate solutions in those finite-dimensional subspaces. The key steps in FEM include generating a mesh that divides the problem domain into simple geometric elements (such as triangles or rectangles in 2D and tetrahedra or cubes in 3D), transforming the governing PDEs into systems of algebraic equations for the nodal values of the unknowns using shape functions, and solving the resulting linear systems. Originally developed for analyzing aircraft structures, FEM has since become an indispensable tool in engineering and physics. In FDM, a uniform grid is used, while in FEM, there is flexibility to work with non-uniform meshes.

The control volume FDM introduces control volumes around nodes to manage fluxes and rates of change, providing a physical connection between flux and volume. However, it lacks the geometric flexibility of FEM. Winslow’s control volume FEM and other versions of the finite volume method (FVM) added local adaptability, allowing control volumes around nodes on unstructured meshes especially in fluid dynamics and reservoir simulations [10,11]. The FVM involves discretizing the computational domain into small control volumes (or cells) and applying the integral form of the governing equations to each control volume. This method ensures local conservation of quantities like mass and energy.

The FVM is the underlying method used in MATLAB Reservoir Simulation Toolbox (MRST) simulations. It is coupled with specific discretization techniques to handle differential operators in MRST. This converts the continuous model into a set of linear algebraic equations that can be solved computationally. The software uses various approaches to approximate the fluxes at the boundaries of each control volume and applies numerical methods to solve the resulting system of equations efficiently. Some other details of these methods are in the MRST manual developed by a research group at SINTEF Digital.

1.2. How CO₂ Storage Works: Recent Technological Advances

CO₂ storage involves steps that begin with capturing CO₂ from industrial sources, such as power plants, cement factories, and steel mills. The captured CO₂ is then compressed into a supercritical fluid, allowing it to be injected into deep geological formations through wells. Once injected, the CO₂ migrates through the porous rock and is eventually trapped by various mechanisms, including structural trapping, residual trapping, solubility trapping, and mineral trapping [3,7,12]. Moreover, geological repositories like saline aquifers and depleted gas fields have different storage efficiency factors, challenges, and leakage risks [3,13].

Recent advancements in CO₂ capture technology have significantly improved the efficiency, cost-effectiveness, and security of the process. For instance, developing advanced solvent-based capture systems and solid sorbents has enhanced CO₂ capture rates while reducing energy consumption. Additionally, novel monitoring techniques, such as fiber-optic sensing and 4D seismic imaging, have improved the ability to track CO₂ movement underground, ensuring that it remains securely stored. The reviewed monitoring techniques for CO₂ storage sites to detect potential leaks include seismic, borehole, geophysical, and atmospheric monitoring methods, which are critical for assessing the effectiveness of CO₂ storage and ensuring environmental safety [6]. A notable example of successful CO₂ storage is the Northern Lights project in Norway, which aims to store millions of tons of CO₂ annually in the North Sea’s subsea formations, contributing to the region’s decarbonization efforts. Figure 2 illustrates the various CO₂ trapping mechanisms within subsurface formations (redrawn from [6]). It highlights the distinct structural, capillary, solubility, and mineral trapping stages and their contributions to long-term CO₂ storage security. By showing how CO₂ is immobilized over time, it provides a clear understanding of the processes that contribute to the effective sequestration of CO₂.

1.3. Challenges in Reservoir Simulations and the Comparative Solutions Project

Simulating CO₂ storage reservoirs presents unique challenges not found in traditional oil and gas reservoirs. Key issues include modeling diverse CO₂ trapping mechanisms, ensuring the long-term stability of stored CO₂, and examining interactions between CO₂ and reservoir rocks. Additionally, the greater scale and complexity of CO₂ storage models require substantially more computational power [14].

Solvers and Preconditioners. The efficiency of numerical reservoir simulations depends on the choice of solvers and preconditioners. Solvers in this context refer to the numerical techniques used to approximate solutions to large linear systems obtained after discretizing the governing PDEs. Preconditioners typically improve the convergence rate of iterative solvers. Each method has certain advantages and disadvantages, depending on the specific characteristics of the reservoir being simulated. The choice of methods impacts CPU time, iteration count, and memory usage, which are critical factors in large-scale simulations [10,15,16,17]. Our focus in this study is on CPU time and iteration count as the primary metrics for evaluating the performance of various numerical solvers. CPU time measures the computational resources required to run a reservoir simulation. It is influenced by grid resolution, the physical models’ complexity, and the numerical methods’ efficiency. A detailed breakdown of CPU time involves analyzing the time spent on various simulation components, such as matrix assembly, solver execution, and data input/output operations. Understanding how CPU time is allocated allows researchers to optimize simulation performance by selecting appropriate solvers and preconditioners, reducing grid resolution, and parallelizing computations across multiple processors. Table 1 in Section 3 summarizes advantages, disadvantages, and CPU time impact for a list of major MRST solvers. More details about these and some other solvers/ preconditioners and references are given in Section 3.

1.4. Outline of the Paper

In Section 2, we provide a brief overview of the 11th SPE CSP project, with a focus on the problem described in CSP 11B, which serves as the basis for our analysis. Then, we summarize the governing equations adopted from the CSP 11 project description as well as the leading numerical methods for solving these governing equations. Additionally, we present an overview of key solvers and preconditioners used in their numerical solution, along with general remarks on the advantages and disadvantages of different approaches. It also contains a detailed description of the specific problem under consideration in this paper, following the constraints and conditions outlined in the CSP 11B project. In Section 3, we present the results of our numerical simulations and compare the performance of various solvers and preconditioners employed in our study. Finally, in Section 4, we summarize our findings and provide concluding remarks.

2. Materials and Methods

The geometry and context of the 11th SPE CSP (https://spe.org/csp) is a synthetic geological cross-section with water-filled porous media, similar to the structures found in the Norwegian Continental Shelf. The model contains seven facies (rock types) to describe three versions of the CSP: Version 11A is a 2D experiment at laboratory scale and surface conditions. Version 11B is a 2D transect at field scale and conditions. Version C is a full synthetic field study as a 3D version of 11B. Their common 2D transect geometry with reasonable geological and operational realism is designed to keep problem statement simple and contain key computational challenges associated with numerical simulation of CO₂ injection, migration, and long-term storage [18]. Some other considerations about the geology, fluid complexity, and petrophysical properties include the structural homogeneity for facies and faults, two-phase, two-component flow with thermal effects (pure water in the reservoir, with pure CO₂ injection) and the capillary forces and dispersion (for details, see [18] and the CSP site mentioned above).

A sketch of the benchmark geometry of CSP 11B is given in Figure 3 below. It is based on the description of the eclipse problem on the project site at https://github.com/sintefmath/spe11-decks (accessed on 10 April 2025) (for other sketches, one can check [18]). The coordinate system is oriented with the vertical direction pointing up and the origin in the lower-left corner. The dimensions of the cross-section are 8.4 km (horizontal) and 1.2 km (vertical). Moreover, a nominal depth of 1 m is assigned (to the y-direction) to be able to utilize volumetric expressions and units. The geometry includes seven internally homogeneous facies (six permeable and one impermeable), two injection wells (for pure CO₂), and three (almost vertical) fault-like structures with different permeabilities. The domain consists of the full porous medium except for the two injection wells. The boundaries are assumed impermeable. The figure contains two injection wells, which are specified in terms of their bottom left and top right corners, stated as (x, z) coordinates (measured relative to the lower left corner of the domain) with the coordinates (2700, 300) and (5100, 700).

2.1. Governing Equations and Constitutive Laws

The governing equations are adopted from the CSP 11 project description [18], which is based on an isothermal two-phase, two-component version of Darcy’s law where constant model parameters and constitutive functions are used within each facies. For details, one can refer to [18] and the references there, including [19].

For the phases α = n (CO₂-rich non-wetting gas phase) and α = w (H₂O-rich wetting liquid phase), the main equations are summarized below:

The volumetric flux u_α of each phase α is governed by the multi-phase Darcy’s law:

$$u_{\mathsf{\alpha }}=-{\displaystyle \frac{{\mathrm{k}}_{r,\alpha }\mathrm{k}}{{\mathsf{\mu }}_{\mathsf{\alpha }}}}(\nabla {\mathrm{p}}_{\mathsf{\alpha }}-{\mathsf{\rho }}_{\mathsf{\alpha }}\mathrm{g})$$

(1)

where ${\mathrm{p}}_{\mathsf{\alpha }}$ is the phase pressure, ${\mathrm{k}}_{r,\alpha }$ and k are the relative and intrinsic permeabilities, respectively, ${\mathsf{\mu }}_{\mathsf{\alpha }}$ is the phase viscosity, ${\mathsf{\rho }}_{\mathsf{\alpha }}$ is the phase density, and g is the gravitational force pointing “down” with the approximate magnitude |g| = 9.81 ms⁻².

Component mass conservation for components i = CO₂ and i = H₂O is as follows:

$${\sum }_{\mathsf{\alpha }=\mathrm{w},\mathrm{n}}\left[{\displaystyle \frac{\partial \left({\rho }_{\alpha }\varphi s_{\alpha }{\chi }_{\alpha }^{i}J\right)}{\partial t}}+\mathbf{\nabla }\left({\rho }_{\alpha }{\mathit{u}}_{\alpha }{\chi }_{\alpha }^i{+{\rho }_{\alpha }\mathit{j}}_{\alpha }^i\right)\right]=0$$

(2)

where ϕ is the porosity, ${\chi }_{\alpha }^i$ is the component mass fraction in phase α, t is the time variable, $s_{\alpha }$ is the saturation, J is a volumetric density term (i.e., reservoir volume per domain volume) for adding volume to boundaries, and ${\mathit{j}}_{\alpha }^i$ is the sum of diffusive and dispersive fluxes for component i in phase α given in Equation (3) below:

$${\mathit{j}}_{\alpha }^i=-(s_{\alpha }\varphi D_{\alpha }+E|{\mathit{u}}_{\alpha }\left|\right)\nabla {\chi }_{\alpha }^i$$

(3)

where D_α is the mutual diffusivity in phase α, while the term E|u_α| is due to the assumption of linear dispersion in the model.

Capillary pressure: The phase pressures are related by saturation as follows:

$${\mathrm{p}}_{\mathrm{n}}-{\mathrm{p}}_{\mathrm{w}}={\mathrm{p}}_{\mathrm{c}\mathrm{a}\mathrm{p}}\left({\mathrm{s}}_{\mathrm{w}}\right)$$

(4)

Completeness of model: For phases α = n, w and components i = CO₂, H₂O,

$${\sum }_{\mathsf{\alpha }=\mathrm{w},\mathrm{n}}{s}_{\alpha }=1\mathrm{a}\mathrm{n}\mathrm{d}{\sum }_{\mathrm{i}={\mathrm{C}\mathrm{O}}_2,{\mathrm{H}}_2\mathrm{O}}{\chi }_{\alpha }^i=1$$

(5)

Thermodynamics: The thermal equation for multi-phase flows in porous media (with the omission of kinetic energy) with the sum being over $\mathsf{\alpha }=\mathrm{w},\mathrm{n},\mathrm{s}$ (as in [19]) is as follows:

$$\sum _{\mathsf{\alpha }}\left[{\displaystyle \frac{\partial }{\partial t}}\left({\rho }_{\alpha }\varphi s_{\alpha }\left(e_{\alpha }-gz\right)J\right)+\nabla ({\mathit{u}}_{\alpha }\left({\rho }_{\alpha }{e}_{\alpha }+p-{\rho }_{\alpha }gz\right)-\varphi s_{\alpha }{\kappa }_{\alpha }\nabla T)\right]=0$$

(6)

where the solid phase is designated by α = s, and its saturation is set to ϕ s_s = 1 − ϕ along with zero Darcy flux: ${\mathit{u}}_s$ = 0. Moreover, $e_{\alpha }$ represents the internal energy per mass, and ${\kappa }_{\alpha }$ is the thermal conductivity in Equation (6).

In addition to the governing equations, the following constitutive laws are considered.

Brooks–Corey type relative permeability and capillary pressure: For primary drainage (initial period of injected gas displacing water), saturations are normalized:

$$s_{w,n}=max\left({\displaystyle \frac{s_w-s_{w,imm}}{1-s_{w,imm}}}\right)\mathrm{and}{s}_{w,n}=max\left({\displaystyle \frac{s_w-s_{w,imm}}{1-s_{w,imm}}}\right)$$

(7)

where $s_{w,imm}$ is the saturation below which the phase is immobile. The relative permeability is modeled as a nonlinear function of saturation: $k_{r,\alpha }$($s_w$) =${{(s}_w)}^{c_{\alpha }}$. If ${\mathrm{p}}_{\mathrm{c}\mathrm{a}\mathrm{p}}\left({\mathrm{s}}_{\mathrm{w}}\right)$ is the basic Brooks–Corey capillary pressure, then an extended capillary pressure expression that is valid for all saturations with smooth transition to the maximum value, $p_{cap,max}$, can be obtained using Equation (8) below that also involves the error function, erf:

$$p_{cap}\left(s_w\right)=p_{cap,max}·erf\left({\displaystyle \frac{{\sqrt{\pi }\tilde{p}}_{cap}\left(s_w\right)}{{2p}_{cap,max}}}\right)$$

(8)

Thermodynamics: The thermodynamic properties for phase partitioning, pure-phase characteristics, and mixture properties follow those outlined in [18]:

(i)

The solubility limit of each phase (CO₂ solubility limit in the water-rich wetting phase and H₂O solubility limit in the CO₂-rich non-wetting phase) is calculated based on the pressure of the same phase. Mass transfer between phases occurs instantaneously at each point, immediately dissolving or vaporizing available CO₂ and H₂O until the solubility limit is reached.

(ii)

The pure-phase properties of CO₂ and H₂O and phase pressures are defined according to the NIST database [20], available at https://webbook.nist.gov/chemistry/fluid/, accessed on 10 April 2025. For the solid phase, the internal energy is a constant heat capacity, while for other phases, it is a function of both pressure and temperature.

(iii)

Due to the very small mutual solubilities in this setup, all mixture properties are assumed to be equal to those of the pure phases, except for water density, which follows the form described in [21]. Moreover, the rock is considered incompressible, and thus ϕ does not vary over time.

Initial conditions: The CSP 11B model is initialized at t = −3.1536 × 10¹⁰ s (1000 years before injection begins). The initial conditions are given by considering an initially stagnant water-filled reservoir, the geothermal gradient, and a consistent pressure specification at the center of Well 1.

Boundary conditions: All the boundaries are impermeable (no-flow boundary conditions for the fluid): u_α ⋅ n = 0 and jⁱ_α ⋅ n = 0, where n is the (outward) normal vector to the boundary. For the energy equation, an insulating boundary condition, n ⋅ ∇T = 0, is applied to the left and right boundaries, while a constant temperature is maintained at the top and bottom boundaries (based on the geothermal gradient and the depth z).

To prevent an unphysical increase in reservoir pressure, additional volume is introduced at the horizontal boundaries using a variant of pore volume multipliers. This approach typically involves assigning elevated volume content to cells at the domain boundary. The rigorous mathematical treatment is provided in [18], utilizing a Dirac-type distribution on the boundary and describing volume per area at each boundary and within the corresponding facies. Additionally, the two injection wells (j = 1, 2) are equipped with fixed temperature boundary conditions during injection and zero heat transfer.

The material properties of the six permeable and one impermeable facies and model parameters (permeability, porosity, saturation, heat conductivity, diffusion constants, pressure, dispersivity, rock density etc.) closely follow the values given in Tables 4 and 5 of [18]. However, the actual parameters used in the simulations are as in the problem sets of the CSP 11 research team of SINTEF: https://github.com/sintefmath/spe11-decks, accessed on 10 April 2025. The full facies permeability is defined based on a 10:1 horizontal to vertical anisotropy ratio from the horizontal permeability. In addition, the capillary entry pressure is defined based on the Leverett J-scaling and a constant term reported by [22] and also used in [18].

2.2. Numerical and Computational Methods

Equations (1)–(6) form a system of coupled nonlinear partial differential equations (PDEs) of multi-phase and multi-component flow with thermal effects. Suitable numerical methods and discretizations including finite volume, finite difference, or finite element methods are employed to solve them approximately. These techniques ultimately transform the governing system of PDEs into linear algebraic systems that usually involve very large, sparse (populated with many zero entries), non-symmetric, indefinite, and ill-conditioned coefficient matrices. Such a large sparse linear system may not be solved via direct methods like Gaussian elimination or LU decomposition. However, it is possible to find robust and efficient approximate solutions via iterative linear solvers based on Krylov subspace methods, especially if the iterative solver is paired with a suitable preconditioner to address the ill-conditioned nature of the system [10,23]. A significant proportion of the overall computational time of reservoir simulations is used in the linear solver/preconditioner stage. Thus, selection of a computationally efficient method that produces accurate and stable results is of crucial importance in reservoir simulations. A list of linear solvers and their key properties are summarized in Table 1. More details about these and other solvers are given below.

The classical conjugate gradient (CG) method and minimum residual (MINRES) algorithm apply only to symmetric matrices, but they have several extensions for more general linear systems including conjugate gradient squared (CGS), generalized conjugate residual, biconjugate gradient (BiCG), biconjugate gradient stabilized (BiCGSTAB), orthogonal minimum residual (ORTHOMIN), and generalized minimum residual (GMRES) methods, along with their variations such as restarted GMRES and loose GMRES (LGMRES), among others. Due to their flexibility, computational efficiency, and empirical convergence in reservoir simulations, GMRES, BiCGSTAB, and ORTHOMIN are preferred as linear solvers, often being used in conjunction with appropriate preconditioning and relaxation methods [10,23,24,25,26]. There is no fixed combination of a linear solver and preconditioner pair that works best in terms of both computational efficiency and accuracy/convergence properties for all types of nonlinear problems. Moreover, there is no general convergence theory for a number of Krylov subspace methods that are designed for non-symmetric indefinite systems, including BiCGSTAB and restarted GMRES. So, an effective choice for a solver/preconditioner pair depends on the type and complexity of the problem, constraints, and method of discretization [10,15]. It is noteworthy that another robust and efficient method for approximating solutions to some non-symmetric large linear systems is induced dimension reduction (IDR), though it has not yet become a popular choice in reservoir simulations.

As previously mentioned, the most commonly used linear solvers in reservoir simulations are GMRES and BiCGSTAB. These methods, along with their various variants, offer distinct advantages depending on the specific characteristics of the simulation. Each solver has scenarios where it can outperform the other, as demonstrated by examples in the literature [10]. A thorough mathematical analysis of both GMRES and BiCGSTAB, including their underlying principles and performance considerations, can be found in works such as [24,25], providing a deeper understanding of their applications in reservoirs simulations. GMRES is an extension of MINRES (which is only applicable to symmetric matrices) to non-symmetric matrices, and the iterations are based on minimizing the 2-norm of the residual vector in a corresponding Krylov subspace. Unfortunately, unlike the CG method, GMRES is based on long-term recurrence. Thus, it becomes impractical beyond a certain number of iterations in terms of memory storage. Restarted GMRES was introduced in part to circumvent this disadvantage and to keep the growth of the computational cost under control. In the restarted version of GMRES, one selects a threshold l, and all the generated Krylov basis vectors after l iterations are discarded. After l iterations, a new sequence of Krylov subspaces is built using the residual, $r_l=b-A{x}_l$. The resulting method is denoted as GMRES(l). General theoretical results for the convergence rate of GMRES(l) are not available. Restarting may slow the convergence, but when it works, it reduces storage [10]. The iterative method LGMRES was introduced in part to potentially enhance the convergence rate of restarted GMRES. The residual vectors at the end of each restart cycle often shift direction in a repeating pattern. LGMRES applies a technique to improve the convergence of restarted GMRES by breaking this repeating pattern and acting as an accelerator for GMRES [27].

BiCGSTAB is an extension of the BiCG method, specifically designed to address the often irregular convergence patterns observed in BiCG and CGS. The method BiCGSTAB combines elements of both BiCG and GMRES, with each BiCG step followed by a GMRES(l) iteration. Because BiCGSTAB employs short-term recurrences at each iteration, similar to CG, it generally requires less memory than GMRES. However, unlike GMRES, which minimizes the norm of the residual vector, BiCGSTAB does not guarantee any optimal property for the $m^{th}$ approximate solution vector. Consequently, as observed in our experiments, its convergence behavior can be more erratic compared to GMRES. The method BiCGSTAB(l) refers to a version where a GMRES(l) step follows each BiCG step. Although the computational cost of BiCGSTAB(l) increases with larger l, numerical experiments suggest that in some cases, the faster convergence may justify the added expense.

For nonsymmetric indefinite systems, such as those encountered in reservoir simulations, the choice of solver is delicate and problem-dependent. GMRES (without restarting) generally performs well if it converges within a few iterations, as is often the case with matrices that have a clustered spectrum [28]. However, if GMRES requires additional iterations, it may be more effective to use LGMRES(l) or BiCGSTAB(l). Of course, when BiCGSTAB works, it works exceptionally well. Considering these factors, we decided to use LGMRES(l) and BiCGSTAB(l) as our solvers for the current study and to compare their performance for various values of l. As previously mentioned, since the coefficient matrices arising from the discretization of the governing PDEs are typically ill-conditioned, it is important to use a preconditioner for fast convergence of a Krylov subspace method. Roughly speaking, a preconditioner is a matrix that transforms the linear system into a system that has the same solution, but the transformed coefficient matrix has a better condition number. For example, consider an (easily) invertible n-by-n matrix P that approximates A. Then the matrix product $P^{-1}A$ approximates the n-by-n identity matrix I in the sense that the eigenvalues of $P^{-1}A$ are expected to be clustered around 1. Thus, the transformed system, $P^{-1}Ax=P^{-1}b$, is simpler than the original system, as it should be possible to do matrix-vector products of the form $P^{-1}y$ at a lower cost.

There are a multitude of preconditioners that can be paired with Krylov subspace methods including Jacobi, Gauss–Seidel, algebraic multigrid (AMG), and incomplete LU (ILU) methods and their extensions/variations such as ILU(k), ILUP(k), ILUT(τ), and nested factorization. Moreover, there are a variety of factors that can impact the choice of a preconditioner, including suitability of the algorithms for parallel computing, physics-based implementation, and the use of structured versus unstructured grids [29]. Many commercial solvers like Eclipse 100, IMEX, STARS, and OPM use serial or parallel versions of such ILU factorizations [26].

Table 1. Solver methods for MRST reservoir simulations.

Method	Advantages	Shortcomings	CPU Time	Source
BiCGSTAB	Stable and efficient for large-scale simulations	May require more iterations	Longer with more iterations	[10,23,26]
GMRES	Robust with good convergence	Memory-intensive	Longer due to memory needs	[23,26,27,28]
BiCGSTAB (l)	Enhanced stability, fewer iterations	Complexity increases per iteration	May increase with complexity	[23,26,28]
LGMRES	Improved convergence for certain problems	Increased memory usage	Longer due to memory needs	[27,28]
FGMRES	Flexible with preconditioning	Requires careful tuning of preconditioners	Shorter if well-tuned	[10,27,28]

Table 1. Solver methods for MRST reservoir simulations.

Method	Advantages	Shortcomings	CPU Time	Source
BiCGSTAB	Stable and efficient for large-scale simulations	May require more iterations	Longer with more iterations	[10,23,26]
GMRES	Robust with good convergence	Memory-intensive	Longer due to memory needs	[23,26,27,28]
BiCGSTAB (l)	Enhanced stability, fewer iterations	Complexity increases per iteration	May increase with complexity	[23,26,28]
LGMRES	Improved convergence for certain problems	Increased memory usage	Longer due to memory needs	[27,28]
FGMRES	Flexible with preconditioning	Requires careful tuning of preconditioners	Shorter if well-tuned	[10,27,28]

In most practical reservoir simulation scenarios, Gauss–Seidel and Jacobi preconditioners may offer some benefits, but they should not be expected to have a significant impact [10]. On the other hand, if the problem can theoretically be stably factored into LU, the incomplete LU preconditioner is likely to improve the convergence rate. Indeed, ILU is commonly used as the default preconditioner in commercial software like COMSOL. In this study, we have employed a variant of ILU factorization, ILU(k), as our preconditioner, due to its simplicity, low setup cost, and robust performance in the presence of strong coupling and heterogeneity, which are typical features of reservoir simulations. It should be noted that although multigrid methods like AMD can be highly efficient for solving a wide range of symmetric positive definite (SPD) problems, achieving similar performance for non-SPD matrices or for large scale heterogeneous problems remains an open challenge [29,30].

A generic ILU algorithm can be derived from the exact LU decomposition by introducing rules that replace some aspects with zeros in the factors L and U. In ILU(0), the zero pattern for the triangular factors L and U matches exactly the zero pattern of the matrix A. However, the accuracy of ILU(0) may be insufficient to ensure a satisfactory convergence rate. More accurate incomplete LU factorizations tend to be both more efficient and reliable. For example, ILU(1) is obtained by extending the zero pattern for the triangular factors to the zero pattern of the product $L_0{U}_0$, where $L_0$ and $U_0$ are derived from ILU(0). Incomplete factorizations that rely on sparsity patterns are insensitive to numerical values because the elements that are dropped depend solely on the zero structure of A. In contrast, ILUT(τ), the preconditioner chosen for this study, drops elements during the Gaussian elimination process based on their magnitude rather than their position. An element is replaced by zero if it is smaller than the relative tolerance ${\mathsf{\tau }}_{\mathrm{i}}$, which is calculated by multiplying τ by the original 2-norm of the $i^{th}$ row.

It is noteworthy that the system of equations originating in reservoir simulations typically exhibits a mixed elliptic/hyperbolic character, with the pressure block being nearly elliptic and the saturation/concentration part being almost hyperbolic. There are indeed matrix structuring approaches that leverage the relative homogeneity of blocks within the coefficient matrix, based on their elliptic behavior (e.g., pressure variables), hyperbolic behavior (e.g., saturations), or hybrid character (e.g., temperature). For example, constrained pressure residual (CPR) framework can be used to take advantage of the partitioning of the coefficient matrix into pressure and saturation components. One can conduct first-stage pre-conditioning for pressure variables before a full-system preconditioning and even combine CPR with other preconditioners like AMG [26,29,30,31].

In this simulation study, which is essentially a 2D field-scale project from the 11th SPE CSP, our focus has been on LGMRES and BiCGSTAB solvers along with ILU(k) preconditioners which improve their efficiency and convergence properties. Moreover, they can be easily implemented via the MRST package using the AMGCL [31] solver library. Our simulation procedure and results are described in the next section.

2.3. Simulation Deck Description and Computations

The CSP11B is designed as a two-dimensional cross-sectional representation with grid dimensions of 130 × 1 × 62, comprising 130 cells in the X direction, a single cell in the Y direction, and 62 cells in the Z direction. The overall dimensions consist of a vertical cross-section measuring 8.4 km horizontally and 1.2 km vertically. The geometry includes three fault-like structures with varying permeability: two have high permeability, while one has low permeability. This configuration allows for detailed vertical and horizontal resolution, making it particularly effective for studying vertical CO₂ migration and trapping mechanisms within stratified formations.

Permeability and porosity are critical for accurately modeling the reservoir. The geometry includes seven facies, six of which are permeable and one that is impermeable. These facies define the reservoir porosity (Figure 4) and permeability (Figure 5), which are synthetic and represent the typical characteristics of storage reservoirs on the Norwegian Continental Shelf. Permeability in the Y and Z directions is derived from the X direction, ensuring uniformity and simplifying data input. The model defines initial equilibrium conditions with a reservoir pressure of 196.2 bars and a reference temperature of 60 °C. These parameters provide a realistic basis, closely mirroring actual reservoir conditions.

Since the properties of brine-CO₂ mixtures are not well-characterized in literature, salts and minerals are excluded to simplify the problem and to avoid complications from salt precipitation and other geochemical processes. Thermal effects are also excluded to account for the fact that CO₂ is often injected at temperatures lower than the reservoir’s. Detailed rock and fluid property data, such as relative permeability, capillary pressure, and fluid phase behavior, are utilized to capture the complex interactions within the reservoir.

The simulation includes a plan for gas injection through two wells, INJ0 and INJ1 (in Figure 4 and Figure 5), each set to inject gas at a rate of 1618.47 Sm³/day with a maximum injection pressure of 400 bars, which is selected to prevent any phase transition from water to vapor with given reservoir conditions [18]. The simulation operates with recurring time steps: it repeats annually for the first 44 steps, then shifts to centennial intervals for the final 10 steps to facilitate long-term analysis of the reservoir’s response to gas injection. This approach allows for monitoring key performance indicators like field pressure and gas production rates.

The simulations take detailed control parameters and tuning settings into account to maintain model stability and accuracy throughout the simulation period. This ensures that the model produces reliable and precise results, supporting informed decision-making in reservoir management and optimization.

3. Results and Discussion

With focus on CPU time and iteration count, all simulations are run using the MRST package in MATLAB version 2023b on a laptop computer with Intel^® Core™ i7-13700H, 2400 Mhz, 14 Core processor, and 32 GB installed memory. CPU and wall times are calculated separately by the intrinsic reporting functions of the MRST. As noted in Section 1, CPU time provides a practical measure of the actual runtime of an algorithm on a given problem, directly impacting user experience. The iteration count reflects the mathematical efficiency of a solver and allows for comparisons that are largely independent of hardware specifications or implementation details. Together, these two metrics offer a transparent assessment of solver performance in our setting. In contrast, memory usage was not a critical factor in our experiments, as the simulations remained well within available system memory. Moreover, memory consumption can be heavily influenced by programming language choices and data structures, which can obscure the evaluation of the core algorithmic behavior.

3.1. Simulation Runs and Results

MRST seeks to achieve the specified solver tolerance by adaptively dividing the defined time step into smaller substeps. Within each substep, the LGMRES(l) or BiCGSTAB(l) algorithm is called to solve the linear systems appearing within the substep. LGMRES(l) (or BiCGSTAB(l)) further introduces its own iterations. In the following sections and graphs except Figure 6, “iteration” count specifically refers to the number of LGMRES(l) (or BiCGSTAB(l)) calls—excluding the inner workings of LGMRES’s or BiCGSTAB’s iteration cycles. In Figure 6, it is important to note that each call to the preconditioned LGMRES (or BiCGSTAB) method includes both outer and inner iterations, owing to the use of an inexact preconditioner. While running the experiments, maximum LGMRES(l) and BiCGSTAB(l) iterations are set to 200, and maximum iterations are set to 50 to reach solver tolerance of 1 × 10⁻⁶. The maximum number of time-step cuts to divide the current time step to smaller substeps is set to 100.

In order to illustrate the general solver behavior, residuals for LGMRES(l) and BiCGSTAB(l) are given versus outer iteration numbers for the last iteration applied to solve the last substep of the (44th) time step in Figure 6. For a fair comparison, both solvers are fed with the identical problem and the same ILU(0) preconditioner. Both solvers perform multiple inner iterations in each outer iteration, yet only the residuals for the outer iterations are shown in the figure. A comparison of these methods reveals that the BiCGSTAB(l) results in a rather unstable residual pattern. Moreover, the LGMRES(l) algorithm is observed to behave very similarly for l = 10 and l = 20, while using larger l (like l = 30) improves the convergence behavior significantly, avoiding the local minima better than the other two.

3.2. CPU Time vs. Method

The CPU times for each time step are stacked separately for each solver configuration in Figure 7. The total number of substeps and the total number of iterations for each method are shown above respective bars using black and red text, respectively. The first centennial time step, in which large changes in the system are present, impacted solution times significantly for all solver configurations and can be seen as the largest item in each stacked bar. The remaining centennial time steps are subject to more gradual changes, and their time consumption is comparable to that of the annual time steps. As shown in the diagrams below, LGMRES(l) generally enables the software to achieve the prescribed tolerance more quickly and with fewer substeps compared to BiCGSTAB(l). Additionally, higher-order ILUs appear to have minimal impact on the overall convergence behavior of the software. The LGMRES(l) method is observed to behave nearly identically for l = 10 and l = 20 for ILU(0), which can be explained by the convergence behavior given in Figure 6. The highest CPU time is observed for BiCGSTAB(l = 5) and ILU(3), suggesting potential inefficiencies in this specific configuration.

3.3. Iteration Count vs. Method

In Figure 8, given below, iteration counts for each time step are stacked for each solver configuration. Total CPU time for each method is given above respective bars with black text. Similar to Figure 7, the first centennial time step caused a significant increase in the number of iterations needed to attain an acceptable solution, and the iteration counts are similar for both LGMRES(l) and BiCGSTAB(l) solvers, while CPU times are much better for LGMRES(l).

4. Conclusions

This study investigates the performance of various solver-preconditioner pairs applied to an SPE CO₂ storage benchmark model using the MRST toolbox and the AMGCL solver library. The primary focus is on the BiCGSTAB and LGMRES solvers, coupled with different levels of ILU(k) preconditioners. Results indicate that both solvers effectively reduced residuals to near machine precision within a reasonable time, demonstrating their robustness for large-scale reservoir simulations.

LGMRES(l) consistently exhibited rapid and stable convergence across various preconditioner levels, while BiCGSTAB(l) was also effective, albeit with greater variability in convergence behavior. CPU time analysis revealed that the choice of solver-preconditioner combination significantly influences computational efficiency, with certain configurations offering an optimal balance between performance and resource utilization.

Interestingly, higher-order ILU preconditioners (ILU(1) and above) with LGMRES(l) increased CPU time, while ILU(0) provided sufficient stabilization. In contrast, BiCGSTAB(l) performed best with ILU(1), as it provided a more accurate matrix inverse approximation, enhancing convergence stability and reducing the number of iterations. These findings highlight that LGMRES(l) benefits from simpler preconditioning, while BiCGSTAB(l) performs optimally with more robust preconditioners.

Furthermore, higher-level ILU preconditioners (e.g., ILU(2) or higher) did not significantly affect convergence rates or reduce the iteration count. The increased per-iteration cost outweighed the small improvements in preconditioning accuracy, making simpler preconditioners such as ILU(0) or ILU(1) more suitable for the simulations.

The first centennial time step was solved most efficiently using LGMRES(l = 30) with ILU(0), showing notably improved performance. This suggests that switching solver configurations during the simulation may enhance performance, rather than using a fixed combination throughout. This observation warrants further investigation.

Overall, the study underscores the importance of selecting appropriate solver-preconditioner pairs based on the specific characteristics of the simulation model and available computational resources. The findings provide valuable insights for optimizing reservoir simulation workflows in MRST, contributing to more efficient and reliable numerical reservoir simulation practices, particularly in complex scenarios such as those modeled in the SPE CSP 11 project. In particular, their extensions along with sensitivity analysis methods can be applied to industrial-scale CCS and utility projects like the Sleipner CCS project. We will address such applications in a future article.