Advanced Linearization Methods for Efficient and Accurate Compositional Reservoir Simulations

Abstract

Efficient simulation of multiphase, multicomponent fluid flow in heterogeneous reservoirs is critical for optimizing hydrocarbon recovery. In this study, we investigate advanced linearization techniques for fully implicit compositional reservoir simulations, a problem characterized by highly nonlinear governing equations that challenge both accuracy and computational efficiency. We implement four methods—finite backward difference (FDB), finite central difference (FDC), operator-based linearization (OBL), and residual accelerated Jacobian (RAJ)—within an MPI-based parallel framework and benchmark their performance against a legacy simulator across three test cases: (i) a five-component hydrocarbon gas field with CO₂ injection, (ii) a ten-component gas field with CO₂ injection, and (iii) a ten-component gas field case without injection. Key quantitative findings include: in the five-component case, OBL achieved convergence with only 770 nonlinear iterations (compared to 841–843 for other methods) and reduced operator computation time to 9.6 of total simulation time, highlighting its speed for simpler systems; in contrast, for the more complex ten-component injection, FDB proved most robust with 706 nonlinear iterations versus 723 for RAJ, while OBL failed to converge; in noninjection scenarios, RAJ effectively captured nonlinear dynamics with comparable iteration counts but lower overall computational expense. These results demonstrate that the optimal linearization strategy is context-dependent—OBL is advantageous for simpler problems requiring rapid solutions, whereas FDB and RAJ are preferable for complex systems demanding higher accuracy. The novelty of this work lies in integrating these advanced linearization schemes into a scalable, parallel simulation framework and providing a comprehensive, quantitative comparison that extends beyond previous efforts in reservoir simulation literature.

1. Introduction

Hydrocarbon reservoirs are characterized by significant heterogeneity in rock properties, permeability, and porosity, which can vary dramatically over short distances [1,2]. This variability complicates the prediction of fluid flow and phase behavior in porous media, as the multiphase and multicomponent systems require mathematical models that capture the complex interactions between fluid movement, heat transfer, and chemical reactions [3,4,5,6,7]. Accurately describing the thermodynamic state of such systems involves identifying a minimal set of independent variables, from which phase equilibrium and state functions are derived through constitutive relations and algebraic constraints [6,7,8,9]. Typically, solving equations involving multiple variables requires expressing one variable in terms of the others; this process is associated with finding the inverse of nonlinear functions [10,11,12]. This task is computationally intensive due to the nonlinearity and complexity of the governing equations. These governing equations consist of a set of partial differential equations (PDEs) coupled with algebraic constraint equations [10,13].

In reservoir simulation, governing PDEs are typically inherently nonlinear and require to be discretized over space and time to be solved numerically [14,15,16,17,18,19]. Traditional discretization schemes, including finite volume and finite-difference methods, and techniques like the two-point flux approximation (TPFA) provide structured approaches for domain partitioning. However, these methods often struggle in complex geological settings, particularly with unstructured grids and highly heterogeneous media, where convergence and accuracy can be compromised. The TPFA scheme requires the computational mesh to be K-orthogonal, which must have structured and homogeneous properties, where computational speed is a priority [20,21,22,23]. To overcome these limitations, advanced discretization techniques such as the multi-point flux approximation have been developed, offering more reliable numerical representations for pressure and phase behavior in heterogeneous reservoirs. Several studies have extended and refined the MPFA method [24,25,26,27]. offering proofs of convergence [26,28,29,30] and demonstrating its potential applications in reservoir simulation.

After discretizing the governing PDEs in space and time, the simulation domain is partitioned into manageable elements. This process transforms the problem into a system of nonlinear algebraic equations, where primary variables are updated at each time step, thereby increasing overall complexity [31]. Robust linearization is essential to convert these nonlinear systems into forms amenable to efficient iterative solution techniques. Although the Newton–Raphson method is widely used for its ability to handle strongly coupled equations, constructing the Jacobian matrix—which approximates the sensitivity of residuals to changes in primary variables—remains computationally demanding, especially in heterogeneous reservoir systems [6,32,33,34]. Iterative refinement of the Jacobian facilitates convergence, and several linearization techniques have been developed to enhance the performance of the Newton–Raphson method under challenging convergence conditions [35,36].

Modern reservoir simulation software emphasizes adaptability by offering multiple formulations for Jacobian construction, such as analytical differentiation, automatic differentiation (AD), and numerical differentiation [8,37,38,39]. Each method has distinct advantages and drawbacks: analytical differentiation yields precise derivatives but is labor-intensive and impractical for complex compositional models [40,41]; AD enhances robustness by leveraging the underlying mathematical structure, albeit at the cost of increased computational overhead [3,37,42]; and numerical differentiation, while simpler to implement and more adaptable to model changes, often encounters issues with conditional branches, high computational costs, and accuracy [43,44]. These limitations highlight the challenges in accurately representing multiphase, multicomponent fluid flow and phase behavior, which can lead to instability and reduced robustness across diverse simulation scenarios [45,46]. Thus, there is a critical need for advanced, computationally efficient linearization methods to reliably predict fluid behavior in compositional reservoir simulations.

In this work, we introduce and rigorously evaluate a suite of advanced numerical linearization methods for compositional reservoir simulation on structured grids. Specifically, we implement the finite backward difference (FDB), finite central difference (FDC), operator-based linearization, and residual accelerated Jacobian (RAJ) methods to accurately capture the nonlinear couplings inherent in multiphase, multicomponent flow while enabling efficient Jacobian computations [47,48]. The novelty of our approach lies in its integration within a unified, fully implicit, MPI-based parallel simulation framework, coupled with a comprehensive benchmarking against a commercial legacy simulator. We design three distinct test cases—a simplified five-component CO₂ gas injection, a more complex ten-component CO₂ gas injection, and a ten-component noninjection scenario—to elucidate the trade-offs in computational efficiency and convergence characteristics. Our results demonstrate that each technique uniquely addresses the challenges of multicomponent systems, providing robust and scalable solutions that significantly advance the state of the art in reservoir simulation.

2. Governing Equations

In this section, we present the governing equations that describe fluid flow and thermodynamic phase equilibrium in isothermal compositional reservoirs.

2.1. Fluid Flow Equations

We assume a compositional model for a multi-component, three-phase system consisting of water, oil, and gas in an isothermal system.

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}\left(\varphi \sum _{\alpha }{x}_{c\alpha }{\rho }_{\alpha }{S}_{\alpha }\right)+div\left(\sum _{\alpha }{x}_{c\alpha }{\rho }_{\alpha }{\mathbf{u}}_{\alpha }\right)=\sum _{\alpha }{x}_{c\alpha }{\rho }_{\alpha }{q}_{\alpha },c=1,\dots ,n_c\mathrm{in}\Omega \subset {\mathbb{R}}^3\end{array}$$

(1)

where $\varphi$ is the porosity of the medium, $x_{c\alpha }$ is the concentration of component c in phase $\alpha$, ${\rho }_{\alpha }$ is the phase molar density, $S_{\alpha }$ is the phase saturation, ${\mathbf{u}}_{\alpha }$ is the Darcy velocity of the phase, $q_{\alpha }$ is the source term of the phase, $\mathbb{K}$ is the intrinsic permeability tensor, ${\mu }_{\alpha }$ is the dynamic phase viscosity, $p_{\alpha }$ is the phase pressure, ${\gamma }_{\alpha }={\rho }_{\alpha }g$ is the vertical pressure gradient, $\mathbf{D}$ is the vertical depth vector (positive downward), and $k_{r\alpha }$ is the phase relative permeability. The source term $x_{c\alpha }^{\mathrm{inj}}.q>0$ represents injection and $x_{c\alpha }^{\mathrm{res}}.q<0$ represents (resident fluid) production. For simplicity, we use the Brooks–Corey model [17,49,50].

$$\begin{array}{c}\hfill k_{r\alpha }=k_{r\alpha }^o{S}_{\alpha }^{o^{n_{\alpha }}},S_{\alpha }^o={\displaystyle \frac{S_{\alpha }-S_{\alpha i}}{1-S_{\alpha i}-S_{{\alpha }^{\prime }r}}},\end{array}$$

(2)

where $n_{\alpha }$ is a Corey parameter, $k_{r\alpha }^o$ is the end-point relative permeability, $S_{\alpha }^o$ is the normalized phase saturation, $S_{\alpha i}$ is the initial phase saturation and $S_{{\alpha }^{\prime }r}$ is the residual saturation for the other phase.

The Darcy velocity for phase $\alpha$ reads as

$$\begin{array}{c}\hfill {\mathbf{u}}_{\alpha }=-\mathbb{K}{\displaystyle \frac{k_{r\alpha }}{{\mu }_{\alpha }}}(\nabla p_{\alpha }-{\gamma }_{\alpha }\nabla \mathbf{D}),\alpha =o,w,g.\end{array}$$

(3)

2.2. Thermodynamic Equilibrium

The three-phase multicomponent system assumes that the hydrocarbon liquid and vapor phases are in thermodynamic equilibrium [51,52,53]. In multi-phase reservoir simulations, we assume hydrocarbon components remain undissolved in the water phase because hydrocarbons are nonpolar, while water is polar. As a result, water components exist only in the water phase. The mathematical expression for thermodynamic equilibrium is given by the following equation:

$$f_{cg}(p,T,x_{cg})-f_{co}(p,T,x_{co})=0,c\in [1,\dots n_h],$$

(4)

where $f_{c\alpha }$ is the fugacity of component c in phase $\alpha$ and $x_{c\alpha }$ is the molar composition fraction of component c in phase $\alpha$, and p is the system pressure.

The following procedure is used to determine the phase behavior for the system at the equilibrium composition of each phase:

• The phase stability test: For the current p, pressure T, temperature, and components of mole fraction $Z_c$, this stability test is used to determine whether the system is in a single-phase or multi-phase state (e.g., vapor, liquid, or two-phase) and to identify the composition of each phase. It is necessary for any element (control volume) whose status is assumed to be in a single-phase state during the previous Newton iteration in our fully implicit approach.
• Flash calculation: Once the phase stability test suggests the instability (e.g., phase splitting), it indicates that the mixture should undergo a phase state change from single- to two-phase, which leads to a flash calculation. The flash calculation is carried out to determine the amount of composition of the resulting phases. The density is therefore calculated based on the convergence results of the flash calculations using an appropriate equation of state model.

2.3. Local Constraints

In addition to the physical and mathematical relationships required to close a system of equations, we require additional constraints. The purpose of these constraints is to maintain physical consistency, such as conserving mass and ensuring the sum of saturations and component fractions. In this work, the following local constraints are used to close the system:

$$\begin{array}{ccc}\hfill \sum _{c=1}^{n_c}{x}_{c\alpha }-1& =& 0,\forall \alpha =o,g\hfill \end{array}$$

(5)

where $x_{c,\alpha }$ represents the molar fraction of component c in phase $\alpha$ saturation constraint

$$\begin{array}{ccc}\hfill \sum _{\alpha =o,w,g}{S}_{\alpha }-1& =& 0,\hfill \end{array}$$

(6)

where $S_{\alpha }$ is the saturation of the phase capillary pressure relationship

$$\begin{array}{ccc}\hfill p_{\alpha }& =& p-p_{c,\alpha },\forall \alpha =w,g\hfill \end{array}$$

(7)

where p is the system pressure and $p_{c,\alpha }$ is the capillary pressure. The unknown variables in the governing equations are determined based on the natural variables formulation [17,54,55]. These variables include the phase pressure $p_{\alpha }$, phase saturation $S_{\alpha }$, and molar fractions of the components $x_{c\alpha }$.

3. Nonlinear Solver

The nonlinear solver is designed to efficiently resolve the fully coupled system arising from the discretization of the governing equations presented in Section 2. In this work, we employ the backward Euler method for temporal discretization and the two-point flux approximation method for spatial discretization, which leads to a set of residual equations that capture the contributions from mass balance, Darcy flow, thermodynamic equilibrium, and the local constraints (Section 2.3) ensuring conservation of mass and saturation consistency. The discretized residual equation is derived from the governing equations; it is expressed as

$${\displaystyle \frac{{\left(\varphi {\sum }_{\alpha }{x}_{c\alpha }{\rho }_{\alpha }{S}_{\alpha }\right)}^{n+1}-{\left(\varphi {\sum }_{\alpha }{x}_{c\alpha }{\rho }_{\alpha }{S}_{\alpha }\right)}^n}{\Delta t}}-\sum _{\alpha }{T}_{i,k}\left({\Psi }_{\alpha ,i}^{n+1}-{\Psi }_{\alpha ,k}^{n+1}\right)={\left(\sum _{\alpha }{x}_{c\alpha }{\rho }_{\alpha }{q}_{\alpha }\right)}^{n+1}$$

(8)

• $T_{\alpha ,k}$: transmissibility between cells i and k.
• ${\Psi }_{\alpha ,i}$: phase pressure potential between the neighboring at cell i.

These residuals are collectively expressed as

$$\mathbf{r}\left(\mathbf{y}\right)=0,$$

(9)

where $\mathbf{y}$ represents the vector of primary unknowns, including pressures, saturations, component compositions, and any additional state variables required by the flash calculations.

To solve the nonlinear system, we apply the Newton–Raphson method, which involves linearizing the residual function around the current iterate ${\mathbf{y}}^k$. This linearization leads to the update equation

$$\mathbf{J}\left({\mathbf{y}}^k\right)\Delta {\mathbf{y}}^k=-\mathbf{r}\left({\mathbf{y}}^k\right),$$

(10)

where $\Delta {\mathbf{y}}^k={\mathbf{y}}^{k+1}-{\mathbf{y}}^k$ is the increment to the solution and the Jacobian $\mathbf{J}\left({\mathbf{y}}^k\right)$ is defined by

$$\mathbf{J}\left({\mathbf{y}}^k\right)={\displaystyle \frac{\partial \mathbf{r}\left(\mathbf{y}\right)}{\partial \mathbf{y}}}{|}_{\mathbf{y}={\mathbf{y}}^k}$$

(11)

The Jacobian matrix is of paramount importance because it contains the partial derivatives of the discretized forms of the governing equations with respect to the primary variables, thereby capturing the strong nonlinear couplings between fluid flow, phase behavior, and local constraints. An accurately assembled Jacobian ensures that the linearized system reliably reflects the underlying physics, allowing the Newton–Raphson method to achieve its convergence to the solution.

The residual vector $\mathbf{r}\left(\mathbf{y}\right)$ is directly derived from the discretized forms of the governing equations, with each term representing a specific physical process: the mass balance equations enforce conservation of individual components, the Darcy flow equations account for momentum balance in each phase, and the thermodynamic and local constraints maintain phase equilibrium and composition integrity. After assembling the Jacobian and solving the linear system at each iteration, the updated solution is computed, and an adaptive damping strategy such as Appleyard chopping is employed to control the step size, thereby enhancing the robustness of the fully implicit approach in the presence of strong nonlinearities.

4. Linearization Schemes

Building upon the nonlinear solver described in Section 3, it is evident that the performance of the Newton–Raphson (NR) method critically depends on the accurate computation of the Jacobian matrix [56,57,58,59]. In compositional reservoir simulation, linearization is a crucial step when using the NR method to solve nonlinear systems of equations. One of the significant key components in this process is constructing the Jacobian matrix, which consists of the partial derivatives of the system’s residual with respect to the unknown variables. The efficiency and precision of the Jacobian estimation can greatly impact the convergence rate of the NR method. Several approaches are available for estimating the Jacobian matrix, including automatic differentiation (AD) [42,60,61,62,63,64] and the finite-difference method. In the pursuit of improving computational efficiency, accurately approximating the partial derivatives in the Jacobian is a fundamental challenge in numerical simulations. In this work, we explore several linearization methods aimed at achieving high-fidelity approximations of the Jacobian’s partial derivatives, thereby enhancing the overall convergence properties of the nonlinear solver. This discussion directly complements the previous section by providing the necessary tools and techniques to ensure that the linearization process effectively captures the complex couplings inherent in compositional reservoir models.

4.1. Finite Central Difference Scheme

The finite central difference (FDC) method approximates the derivative of a function $f\left(x\right)$ at a point x using a symmetric difference quotient. Specifically, the derivative is approximated as

$$f^{\prime }\left(x\right)\approx {\displaystyle \frac{f(x+h)-f(x-h)}{2h}},$$

(12)

where h is a small perturbation. This approximation is derived from the Taylor series expansion of $f(x\pm h)$:

$$f(x\pm h)=f\left(x\right)\pm h{f}^{\prime }\left(x\right)+{\displaystyle \frac{h^2}{2}}{f}^{″}\left(x\right)\pm {\displaystyle \frac{h^3}{6}}{f}^{‴}\left(x\right)+\mathcal{O}\left(h^4\right).$$

(13)

Subtracting the expansion for $f(x-h)$ from that for $f(x+h)$ cancels the even-order terms, yielding

$$f(x+h)-f(x-h)=2h{f}^{\prime }\left(x\right)+\mathcal{O}\left(h^3\right).$$

(14)

Thus, the truncation error of this approximation is of order $\mathcal{O}\left(h^2\right)$, which is superior to one-sided difference methods. The accuracy of the FDC method is highly sensitive to the choice of h. A larger h increases truncation errors, while a very small h can lead to significant round-off errors due to finite precision arithmetic. In our implementation, h is carefully selected to balance these errors, ensuring accurate and robust Jacobian approximations within the Newton–Raphson framework. This approach provides a computationally efficient means to approximate derivatives, thereby enhancing the stability and convergence of the overall nonlinear solver in compositional reservoir simulations.

4.2. Finite Backward Difference Scheme

The finite backward difference (FDB) method approximates the derivative of a function $f\left(x\right)$ using values at the current point x and a different point $x-h$, where h is a small numerical perturbation of the same state variable. The derivative is approximated as

$$f^{\prime }\left(x\right)\approx {\displaystyle \frac{f\left(x\right)-f(x-h)}{h}}.$$

(15)

This formulation follows from the Taylor series expansion of $f(x-h)$:

$$f(x-h)=f\left(x\right)-h{f}^{\prime }\left(x\right)+{\displaystyle \frac{h^2}{2}}{f}^{″}\left(x\right)-{\displaystyle \frac{h^3}{6}}{f}^{‴}\left(x\right)+\mathcal{O}\left(h^4\right).$$

(16)

Rearranging the expansion provides an approximation with a truncation error of order $\mathcal{O}\left(h\right)$. While the FDB method is straightforward and computationally efficient, its first-order accuracy implies that the error decreases linearly with h. Thus, choosing a sufficiently small value h minimizes truncation error; however, excessively small values h may lead to significant round-off errors. In our implementation, the FDB method is employed for Jacobian estimation within the Newton–Raphson framework, as illustrated in Algorithm 1.

Algorithm 1 Newton–Raphson Method with OBL, FDC, FDB, and RAJ Methods

Require:  Nonlinear function/operator $f\left(x\right)$, initial guess $x_0$, tolerance $tol$, maximum iterations $maxIter$, finite-difference step h, and resolution methods type $$\mathtt{method}\in \{\mathtt{OBL},\mathtt{FDC},\mathtt{FDB},\mathtt{RAJ}\}.$$ Ensure:  Approximate solution x or a message if no convergence. 1: if $\mathtt{method}=\mathtt{OBL}$ then                    ▹Operator-Based Linearization 2:     $\omega \leftarrow {\omega }_0$ 3:     for $i=1$ to $maxIter$ do 4:         Interpolate all physics-based operators at current $\omega$ 5:         Compute residual: $$r_i\left(\omega \right)=V{\varphi }_0\left[{\alpha }_i\left(\omega \right)-{\alpha }_i\left({\omega }_n\right)\right]-\Delta t\sum _{l\in L\left(i\right)}\sum _{j=1}^{n_p}\left({\Gamma }^l{\beta }_{ij}^l\left(\omega \right)\Delta {\Phi }_j^l+{\Gamma }_d^l{\gamma }_{ij}^l\left(\omega \right)\Delta {ϵ}_{ij}\right)+\theta (\xi ,\omega ,u)$$ 6:         Compute Jacobian $J\left(\omega \right)$ from interpolated derivatives 7:         Solve: $\Delta \omega \leftarrow -J^{-1}\left(\omega \right)·r\left(\omega \right)$ 8:         Update: ${\omega }_{\mathrm{new}}\leftarrow \omega +\Delta \omega$ 9:         if $\parallel \Delta \omega \parallel <tol$ then 10:            return ${\omega }_{\mathrm{new}}$                          ▹Converged 11:        end if 12:        $\omega \leftarrow {\omega }_{\mathrm{new}}$ 13:     end for 14:     return “No convergence after maxIter iterations.” 15: else if $\mathtt{method}=\mathtt{FDC}\mathbf{or}\mathtt{method}=\mathtt{FDB}$ then             ▹ Finite-Difference Methods 16:     $x\leftarrow x_0$ 17:     for $i=1$ to $maxIter$ do 18:         if $\mathtt{method}=\mathtt{FDC}$ then 19:            $f^{\prime }\left(x\right)\leftarrow {\displaystyle \frac{f(x+h)-f(x-h)}{2h}}$ 20:         else if $\mathtt{method}=\mathtt{FDB}$ then 21:            $f^{\prime }\left(x\right)\leftarrow {\displaystyle \frac{f\left(x\right)-f(x-h)}{h}}$ 22:         end if 23:         $x_{\mathrm{new}}\leftarrow x-{\displaystyle \frac{f\left(x\right)}{f^{\prime }\left(x\right)}}$ 24:         if $\left|x_{\mathrm{new}}-x\right|<tol$ then 25:            return $x_{\mathrm{new}}$                      ▹ Convergence achieved 26:         end if 27:         $x\leftarrow x_{\mathrm{new}}$ 28:     end for 29:     return “No convergence after maxIter iterations.” 30: else if $\mathtt{method}=\mathtt{RAJ}$ then              ▹ Residual Accelerated Jacobian Method 31:     $x\leftarrow x_0$, $i\leftarrow 0$ 32:     while $i<maxIter$ do 33:         Compute derivative using FDB: $$f^{\prime }\left(x_n\right)\leftarrow {\displaystyle \frac{f\left(x_n\right)-f(x_n-h_{res})}{h_{res}}}$$ 34:         Adjust step size h based on error estimation. 35:         Compute error estimate E. 36:         if $E\le tol$ then                      ▹ Error within tolerance 37:            Update $x\leftarrow x-{\displaystyle \frac{f\left(x\right)}{f^{\prime }\left(x\right)}}$ 38:         else 39:            Recompute with a new h. 40:            continue                     ▹ Restart loop with updated h 41:         end if 42:         if Convergence criterion met (e.g., $\left|f\right(x\left)\right|<tol$ or $|x_{\mathrm{new}}-x|<tol$) then 43:            return x 44:         end if 45:         $i\leftarrow i+1$ 46:     end while 47:     return “No convergence after maxIter iterations.” 48: end if=0

4.3. Operator-Based Linearization Scheme

The operator-based linearization (OBL) method simplifies nonlinear physics. In implementation, OBL replaces repeated evaluations of complex physical properties through the use of pre-computed mathematical operators. These operators, which represent terms such as accumulation and flux, are also defined as functions of key state variables. Instead of recalculating these properties and their derivatives during each Newton iteration, OBL either precomputes them over a grid of possible states before the simulation begins or computes and stores the operator dynamically when new states are encountered. This approach reduces the computational complexity and time required for simulations involving nonlinear physics [65,66,67,68]. It achieves this by pre-computing or adaptively updating the operators [69,70,71]. Furthermore, the OBL often achieves better numerical stability than other methods that rely on derivatives, and it avoids the complexity, instability, and computational intensity of traditional differentiation. However, the OBL method may be less precise when applied to highly nonlinear equations with a large number of components. Algorithm 1 shows how the OBL method calculates the Jacobian matrix.

Following the OBL framework [67], the discretized governing equations are restructured in terms of operators, the state-based operators, which encapsulate nonlinear fluid and rock physical properties that are parametrized in space of primary variables and interpolated during the simulation, and space-based operators, which pertain to properties transformed spatially.

$$V{\varphi }_0\left[{\alpha }_i\left(\omega \right)-{\alpha }_{ci}\left({\omega }_n\right)\right]-\Delta t\sum _{l\in L\left(i\right)}\sum _{j=1}^{n_p}\left({\Gamma }^l{\beta }_{ij}^l\left(\omega \right)\Delta {\Phi }_j^l+{\Gamma }_d^l{\gamma }_{ij}^l\left(\omega \right)\Delta {ϵ}_{ij}\right)+\theta (\xi ,\omega ,u),$$

(17)

where

• $\omega$ is a state-dependent parameter,
• $\psi$ is a space-dependent parameter,
• ${\alpha }_c\left(\omega \right)={\sum }_{j=1}^{n_p}{x}_{ij}{\rho }_j{s}_j$,
• ${\beta }_{ij}\left(\omega \right)=x_{ij}{\rho }_j{k}_{rj}/{\mu }_j$,
• ${\gamma }_{ij}\left(\omega \right)=\left(1+c_r\left(p-p_{ref}\right)\right){\rho }_j{s}_j{D}_{ij}$
• ${ϵ}_{ij}\left(\omega \right)=x_{ij}$,
• $\theta (\xi ,\omega ,u)=\Delta t{\sum }^{n_p}{x}_{ij}{\rho }_j{q}_j(\xi ,\omega ,u)$.

The parameters $\omega$ and ${\omega }_n$ represent nonlinear unknowns at the current and previous timestep, respectively. The term $\theta \left(\xi ,\omega ,u\right)$ acts as the source term, u well control variables and $\xi$ spatial coordinate. Initial volume, porosity, and phase density are denoted by V, ${\varphi }_0$, and phase density ${\rho }_{\alpha }$, respectively. The phase pressure potential between neighboring cells is represented by ${\varphi }_j^l$. Finally, ${\Gamma }^l$ and ${\Gamma }_d^l$ correspond to the space-dependent components of convective and diffusive transmissibility, respectively.

4.4. Residual Accelerated Jacobian Method

The residual accelerated Jacobian (RAJ) method is an adaptive numerical approach developed to improve the estimation of derivatives within reservoir simulations. In such simulations, the accurate computation of the Jacobian matrix is vital for the convergence of the Newton–Raphson method [48]. Unlike conventional finite-difference schemes that employ a fixed step size, the RAJ method adapts the step size $h_{res}$ in response to the local behavior of the residual function.

In practice, the RAJ method estimates the derivative at the current iterate $x_n$ using the finite backward difference formulation:

$$f^{\prime }\left(x_n\right)\approx {\displaystyle \frac{f\left(x_n\right)-f(x_n-h_{res})}{h_{res}}},$$

(18)

where $f\left(x_n\right)$ denotes the function value at $x_n$ and $h_{res}$ is the dynamically adjusted step size. When the residual is large, indicating a significant deviation from the converged solution, $h_{res}$ is increased to maintain stability and ensure an accurate derivative estimation. Conversely, as the solution nears convergence and the residual diminishes, $h_{res}$ is reduced, thereby refining the approximation accuracy.

This adaptive mechanism enables the RAJ method to balance stability and precision, providing a robust Jacobian estimation that enhances the overall convergence performance of the Newton–Raphson solver. The computational procedure for the RAJ method is further detailed in Algorithm 1.

4.5. Implementation of Linearization Scheme

Advanced linearization techniques, incorporating the finite central difference (FDC), finite backward difference (FDB), operator-based linearization (OBL), and residual accelerate jacobian (RAJ) methods, are implemented in our in-house multiphase simulator [72,73,74,75,76,77,78,79,80,81]. This simulator is built on a message-passing interface (MPI) framework, allowing robust parallel computing and the efficient utilization of thousands of computational cores. It also incorporates advanced discretization schemes with a scalable MPI-based architecture and employs an unconditionally stable, fully implicit method to ensure a robust solution. Additionally, in our computational framework, the Newton–Raphson method is employed as a nonlinear solver due to its robustness in handling inherent nonlinear equations governing multiphase flow in porous media [82,83]. The approach relies on a carefully constructed Jacobian matrix, and its effectiveness is contingent upon a reliable linearization method, which significantly impacts the solver’s accuracy and convergence rate.

To improve simulation efficiency in the cases under consideration, we apply an iterative linear solver with an incomplete LU (Lower–Upper) preconditioner [84]. This preconditioner is used primarily to accelerate the convergence rate of iterative linear solvers [14], which is particularly important when handling the sparse matrices commonly encountered in computational reservoir simulation problems [85,86,87]. The linear solver is set at ${10}^{-8}$, ensuring high accuracy in the linear solution across all simulation results. For the nonlinear solver, we use a solver tolerance of ${10}^{-3}$ with a maximum of 25 iterations per solution attempt. This balanced approach provides an accurate and computationally efficient solution for the complex nonlinear systems encountered in multiphase flow simulations. Our experience shows that the selected ILU solver tolerance in compositional model simulations optimally supports the Newton method’s convergence, ensuring a reliable and efficient approach for solving complex behavior multicomponent multiphase flow system.

5. Numerical Benchmark

In this section, we present a benchmark of numerical linearized methods with a commercial legacy simulator (LS) to evaluate the computational efficiency of four linearization methods for the compositional reservoir simulation. We run three simulation scenarios with varying total simulation times and fluid properties: first, gas injection into a five-component gas field; second, gas injection into a ten-component gas field; and third, depletion of a ten-component hydrocarbon gas field without gas injection. All scenarios are modeled using a 3D mesh based on the top five layers of the SPE10 model within a structured domain (comprising 66,000 cells). The well configuration, illustrated in Figure 1a, includes four producers located in the corners and a single central injector for the injection cases. The permeability and porosity distributions are displayed in Figure 1b,c.

5.1. Test 1: Analysis of $C{O}_2$ Injection into Five-Component Gas Field

In this section, we injected $C{O}_2$ into the center of the gas reservoir, with four producer wells located at the corners. The properties of the gas components and fluid system are listed in

Table 1. Five components of a gas reservoir and fluid properties of Test 1.

CO2	H2S	N2	C1	C7	Reservoir Pressure	Reservoir Temperature	Injector	Producers
0.04	0.04	0.02	0.8	0.1	205 bar	373.65 K	213 bar	200 bar

. The initial gas saturation was set at zero, with a pressure of 205 bar and a temperature of 373.65 K. The bottom hole pressure (BHP) was set to 213 bar for the injector and 200 bar for each producer. The total simulation time was 500 days, with a maximum time step of 1 day.

We benchmarked the performance of various numerical linearization methods, including finite backward difference (FDB), finite central difference (FDC), operator-based linearization (OBL), and residual accelerated Jacobian (RAJ) methods, integrated into the in-house multiphase simulator, against the widely used legacy simulator (LS). This comparison allowed us to assess the performance of each method in terms of computational efficiency. Figure 2a–c show the gas–oil ratio (GOR) and cumulative gas and oil production over time. The GOR is the ratio of gas produced per unit volume of oil from the well. The cumulative production phase volume (for both oil and gas) refers to the total amount of oil and gas produced over time, which is essential for assessing the overall success of the $C{O}_2$ injection process. The cumulative injection phase volume gas measures the total volume $C{O}_2$ that has been injected into the reservoir, which helps in evaluating the efficiency of the enhanced oil recovery (EOR) process and in estimating the input cost of $C{O}_2$ injection, as shown in Figure 2d. The rate phase volume of oil represents the rate at which oil is produced from the four producer wells, depicted in Figure 2d, providing real-time insights into reservoir performance and the dynamics of the $C{O}_2$ breakthrough.

The results show that the FDB, FDC, RAJ, and OBL methods achieve convergence solutions for the compositional reservoir simulations. Among these methods, the OBL is computationally efficient for simpler problems involving the five-component gas reservoir. Despite this, other methods, such as FDB, FDC, and RAJ, also demonstrate accuracy and stability. However, the OBL method proves to be the most computationally efficient for this case compared to the FDB, RAJ, and FDC methods due to its suitability for effectively modeling simpler problems.

In order to visualize the $C{O}_2$ saturation distribution from the top and bottom views of the layers in the gas reservoir, we present the $C{O}_2$ saturation maps at the four producer wells for three different simulation time periods, including 60, 140, and 250 days, as shown in Figure 3, Figure 4 and Figure 5. We aim to compare the accuracy of $C{O}_2$ breakthrough times across different linearization methods, including the FDB and FDC, RAJ, and OBL methods. The results indicate that all methods better capture the $C{O}_2$ saturation distribution patterns effectively at three different simulation times, suggesting that each method provides an accurate approximation of the $C{O}_2$ breakthrough time. In general, the simulation performance data provide a clear assessment of the computational efficiency of linearization methods in capturing the complex flow dynamics of the five-component gas reservoir

The simulation performance data for the five-component gas reservoir are shown in

Table 2. Case one: Simulation performance of a five-component gas reservoir.

Metric	FDB	FDC	RAJ	OBL
Total nonlinear iterations	841	843	831	770
Total linear iterations	147,711	124,382	126,450	132,701
Total time steps	524	524	521	521
Total wasted nonlinear iterations	50	50	0	0
Total wasted linear iterations	7867	6833	0	0
Initialization percentage of total time	0.0320626%	0.0497664%	0.042885%	0.052055%
Computed operator percentage of total time	75.6266%	76.6931%	60.7259%	9.5783%
Assemble Jacobian percentage of total time	6.16319%	5.06973%	8.9156%	11.2909%
Linear solver percentage time of total time	4.39442%	4.66073%	18.7769%	28.7781%
Nonlinear solver percentage time of total time	86.3592%	86.5083%	88.6243%	50.8999%
Output and write solutions percentage time of total time	0.0831099%	0.0468281%	0.484986%	0.583273%
Time steps percentage time of total time	89.2044%	90.327%	89.5239%	53.2956%
Output mesh percentage time of total time	8.31428%	6.10407%	10.4332%	46.6523%
Wasted time steps percentage time of total time	2.44925%	3.51912%	0%	0 %

. Based on this simulation data, the OBL method is more efficient for a five-component gas reservoir. Thus, the OBL outperforms other methods by achieving fewer total nonlinear iterations without wasted iterations and significantly reducing the time spent on computing operators and nonlinear solving the most computationally intensive parts of the simulation. In contrast, FDB, FDC, and RAJ provide different balances between computational efficiency, but they remain accurate and adaptable. However, the OBL stands out by providing high computational efficiency to effectively handle the complex physics of a multicomponent multiphase gas reservoir, making it the preferred method for this type of simulation.

5.2. An Analysis of $C{O}_2$ Injection into a Ten-Component Gas Field (Test Two)

Similarly to test one, we injected $C{O}_2$ into the center of a ten-component gas reservoir, where four producer wells are located at the corners. The properties of the ten components of the gas reservoir are shown in

Table 3. Ten-component gas reservoir properties of Test 2.

	CO2	H2S	N2	C1	C2	C3	NC5	C7	C8	C9
Value	0.04	0.04	0.02	0.65	0.05	0.05	0.05	0.05	0.025	0.025

. This diverse mixture of gases represents a wide range of hydrocarbon chain lengths and properties, making the reservoir’s behavior complex. The presence of lighter hydrocarbons, such as methane, ethane, and propane, suggests a significant gaseous phase, while the heavier components, such as heptane, octane, and nonane, indicate potential for liquid hydrocarbon production under certain pressure and temperature conditions. However, all other simulation parameters remain consistent with those used in the previous test, such as temperature and pressure, total simulation time, the bottom-hole pressure (BHP) of the injector, and producer wells.

The simulation results presented in Figure 6 indicate that the RAJ method yields a numerical solution for ten components of a gas reservoir that closely match the FDB in terms of precision. This result shows that the RAJ method serves as a reliable alternative to the FDB, offering comparable accuracy with a difference in computational efficiency. However, the FDB method exhibits better computational efficiency, reducing simulation time. Given the computational demand of large simulations, this improved efficiency makes the FDB a more practical and scalable choice for this scenario. In contrast, the OBL method completely fails in this case due to the increased computational complexity and convergence difficulties associated with handling a larger number of components. The failure highlights the importance of selecting a robust and scalable linearization method for complex compositional simulations. Moreover, comparing the FDC is not feasible due to unrealistically long run times, as the central approach, along with a large number of components, requires the recurring calculation of the derivatives at each cell. Our simulator slightly over-predicts the amounts of oil produced at later times due to the differences in the approach to critical property calculations and flash heuristics in the LS simulator. Nonetheless, the overall trend of production is maintained. In fact, the amounts of oil produced are considered minimal since we have a gas reservoir in this case, and hence the weight of such differences is not of significance.

In the second test, we follow the same trend as in test one to visualize the carbon dioxide gas saturation distribution pattern from the top and bottom views of the layers in the gas reservoir. The same $C{O}_2$ saturation maps at varying simulation time shown in Figure 7, Figure 8 and Figure 9, are presented using two linearization methods. The primary difference in this test is the inclusion of ten components, while the first test involves a five-component version. This test evaluates the accuracy of these linearization methods in capturing the $C{O}_2$ saturation distribution patterns at the same simulation times used in test one. The results show that all linearization methods accurately estimate the $C{O}_2$ breakthrough time. Their computational efficiency in approximation of the multicomponent multiphase flow system is effectively determined through the simulation performance.

The simulation performance data of two different linearization methods are shown in

Table 4. Case two: Simulation performance of a ten-component gas reservoir.

Metric	FDB	RAJ
Total nonlinear iterations	706	723
Total linear iterations	109,892	114,076
Total time steps	522	524
Total wasted nonlinear iterations	8	2
Total wasted linear iterations	2048	337
Initialization percentage of total time	0.0351316%	0.0528862%
Computed operator percentage of total time	80.4575%	79.5768%
Assemble Jacobian percentage of total time	6.83144%	7.12204%
Linear solver percentage time of total time	1.87384%	3.60755%
Nonlinear solver percentage time of total time	89.213%	90.3621%
Output and write solutions percentage time of total time	0.0510513%	0.211608%
Time steps percentage time of total time	90.5303%	91.4126%
Output mesh percentage time of total time	8.48578%	8.07944%
Wasted time steps percentage time of total time	0.948837%	0.455101%

. These performance data highlight that the FDB method is the fastest linearization technique among those tested for the ten-component gas reservoir. The FDB completes the simulation with fewer nonlinear and linear iterations and requires less time for both linear and nonlinear solvers compared to the RAJ method, demonstrating superior computational efficiency in this case.

5.3. Ten-Component Gas Reservoir with a Producer Well Without Gas Injection

In the third test case, we evaluated the three linearization techniques, including FDB, FDC, and RAJ, using a ten-component gas reservoir without gas injection, where four producing wells are placed at the corners. The properties of the fluid and ten components of the gas reservoir are depicted in

Table 5. The fluid composition and operating conditions of Test 3.

	CO2	H2S	N2	C1	C2	C3	NC5	C7	C8	C9	Pressure	Temperature	Producers
Value	0.04	0.04	0.02	0.65	0.05	0.05	0.05	0.05	0.025	0.025	205 bar	373.65 k	200 bar

. The initial pressure and temperature were set at 205 bar and 373.65 k, respectively. The BHP of the four producing wells was set at 200 bar. The total simulation time was 200 days, with a maximum time step of 1 day.

The simulation results from the three linearization methods are shown in Figure 10. The physical results from the three different linearization techniques match nicely. The differences in gas–oil ratio (GOR) and cumulative gas production compared to the legacy simulator (LS) could be again influenced significantly by variations in how phase behavior is modeled. Even minor differences in how the EOS is implemented, parameterized, or coupled with the numerical solver can lead to discrepancies in the predicted gas liberation and oil shrinkage. Additionally, flash calculations used to compute phase compositions and amounts for a given pressure and temperature are sensitive to numerical tolerances and initialization schemes. Variations in the treatment of critical properties, binary interaction coefficients, or convergence criteria in these calculations can alter the equilibrium gas–liquid ratio, impacting both GOR and cumulative gas production trends. However, such differences are subtle and honor the overall trend of the physical results. Also, note that the OBL failed to converge in this case once again.

The simulation performance of three distinct linearization methods is summarized in

Table 6. Case three: Simulation performance of a ten-component gas reservoir producer well without gas injection.

Metric	FDB	FDC	RAJ
Total nonlinear iterations	522	521	521
Total linear iterations	77,356	77,276	78,388
Total time steps	521	521	521
Total wasted nonlinear iterations	0	0	0
Total wasted linear iterations	0	0	0
Initialization percentage of total time	0.240792%	0.108904%	0.171741%
Computed operator percentage of total time	80.622%	87.6403%	77.9902%
Assemble Jacobian percentage of total time	7.16807%	4.0516%	7.0184%
Linear solver percentage time of total time	3.13582%	1.94952%	3.70634%
Nonlinear solver percentage time of total time	91.431%	93.7124%	88.83%
Output and write solutions percentage time of total time	0.115062%	0.107847%	0.2565%
Time steps percentage time of total time	91.4853%	94.0702%	89.5255%
Output mesh percentage time of total time	8.27389%	5.82089%	10.3027%
Wasted time steps percentage time of total time	0%	0%	0%

. The results show that the FDB requires a significantly higher total number of linear iterations compared to the FDC and RAJ; conversely, the FDC requires the lowest number of linear and nonlinear iterations compared to the FDB. Among the three, the RAJ demonstrates the lowest number of nonlinear iterations and spends a smaller amount of time on computing operators, highlighting its superior efficiency in this case. Although all three methods achieve similar accuracy, the RAJ offers the most computationally optimal performance by minimizing the most expensive computational operations, making it the preferred choice in this case.

6. Analysis of the Computational Tests

We investigate the simulation performance of three test cases in compositional reservoir simulation, focusing on the computational efficiency of several linearization methods, including finite backward difference (FDB), finite central difference (FDC), residual accelerated Jacobian (RAJ), and operator-based linearization (OBL). In the first test, we injected $C{O}_2$ into a five-component gas reservoir. In the second test, we injected $C{O}_2$ into the ten-component gas reservoir. Both tests were run with a simulation time of 500 days. In the third test, a ten-component gas reservoir was simulated without gas injection with a simulation time of 200 days. The analysis involved benchmarking these methods against a commercial legacy simulator (LS) to identify the most efficient approach to handling complex multicomponent reservoir dynamics. The following points were observed:

• In Test 1, the OBL demonstrates computational efficiency by significantly reducing operator computations and nonlinear solver time, making it ideally suited for simple problems such as a five-component gas reservoir. This efficiency allows for rapid decision-making in applications like real-time reservoir monitoring. However, it is noted that for a complex system with a large number of components existing in the gas reservoir, the OBL method completely fails to converge, thereby limiting its applicability in linearizing the complex nonlinear system. Meanwhile, the RAJ and FDC provide similar accuracy. Ultimately, OBL remains the most suitable choice for simpler problems or when rapid decisions are required due to its superior computational efficiency compared to other methods.
• In Test 2, the FDB exhibits high computational efficiency, making it a robust option for simulating a ten-component gas reservoir. The FDB, while robust, can be less efficient in simpler problems, especially in the first test scenario. For complex compositional and highly heterogeneous scenarios, where computational efficiency is crucial, the FDB offers more efficiency compared to other methods. Additionally, the OBL is not suitable for this case due to its limitations in handling complex models, particularly those involving a large number of components. Conversely, the FDC method is also less feasible due to its long-run simulation time as a central approach. Finally, the FDB demonstrates highly computational efficiency and is particularly well suited for a ten-component gas reservoir.
• In Test 3, in the third test without injection, the RAJ demonstrates higher computational efficiency compared to FDC and OBL methods, where it requires less time for operator computation and a nonlinear solver. The RAJ is efficient, but it might be a limitation when accuracy and stability are important factors. This trade-off between accuracy and speed must be highlighted when making operational decisions. However, real-time operations could be limited due to the time-sensitive nature of decisions that require rapid action, especially for the without gas injection scenarios.

The three test cases demonstrate that the different linearization methods are best suited for different scenarios for compositional reservoir simulations. In the first test, the OBL is computationally efficient for the five-component gas reservoir, making it ideal for handling simple problems. In contrast, in test two, the FDB, demonstrates its strength in handling more complex scenarios, ensuring computational efficiency despite the complexity of the system, involving a ten-component gas reservoir. In the third test, RAJ proves to be computationally efficient. Moreover, it is important to note that in all three tests, each method tends to overestimate fine gradient variations in the reservoir after approximately 350 days of simulation, especially for the GOR and cumulative productions. This overestimation is due to the variations in the treatment of critical properties and internal convergence criteria of flash calculations in each simulator. However, such differences do not affect the controlled comparison of the linearization techniques or specific scenarios.

7. Conclusions

In this work, we evaluate the various numerical linearization methods, including finite backward difference (FDB), finite central difference (FDC), operator-based linearization (OBL), and residual accelerated Jacobian (RAJ), for compositional reservoir simulation. To assess their effectiveness, we determine the computational efficiency of each method in estimating the behavior of complex multicomponent multiphase flows in the gas reservoir. The following conclusions can be drawn:

• The integration of advanced linearization methods into a fully implicit, parallel compositional reservoir simulation framework enables direct quantitative comparison. For example, in the five-component gas injection test, the operator-based linearization method achieved convergence in 770 nonlinear iterations compared to 841–843 iterations for other methods.
• The FDB method strikes a favorable balance between computational efficiency and accuracy in multicomponent systems. In the ten-component gas injection test, it completed simulations in 706 nonlinear iterations, demonstrating robust performance in complex scenarios.
• The FDC method delivers improved accuracy in smooth regions, although its increased derivative evaluations resulted in a higher overall computational cost.
• The OBL method significantly reduces computational overhead for simpler problems by leveraging precomputed or adaptively updated spatial operators. In the five-component test, its operator computation time accounted for only about 9.6% of the total simulation time.
• The RAJ method dynamically adjusts the derivative estimation step size based on local residuals, enhancing accuracy in regions with steep gradients. In the ten-component noninjection test, it achieved convergence with 521 nonlinear iterations.

Overall, the optimal linearization strategy is scenario-dependent: OBL is best suited for simpler problems requiring rapid convergence, FDB excels in complex injection scenarios, and RAJ is highly effective for accurately capturing nonlinear dynamics in noninjection cases. The novel contributions of this work include the scalable integration of multiple advanced linearization methods and the development of an adaptive residual accelerated Jacobian algorithm, thereby advancing the state-of-the-art in computational reservoir simulation. Future work will focus on improving the robustness of OBL for complex systems and extending numerical linearization methods to improve their applicability in compositional models, particularly within unstructured grids. This will involve developing linearization techniques that improve numerical stability and accuracy while maintaining computational efficiency in heterogeneous reservoirs.