A Multiscale Compositional Numerical Study in Tight Oil Reservoir: Incorporating Capillary Forces in Phase Behavior Calculation

Abstract

Tight oil reservoirs offer significant development potential. Due to the pronounced capillary forces in their nanopores, phase behavior differs markedly from that in conventional reservoirs, challenging traditional equations of state and numerical simulation methods. This paper presents a multiscale compositional numerical simulation method that incorporates capillary forces, leveraging the parallel advantages of the multiscale finite volume method. The approach decouples the compositional model using a sequential format to derive pressure and transport equations, then solves the pressure equation iteratively in a multiscale format to enhance computational efficiency. Results show that the proposed method significantly improves simulation speed while maintaining accuracy. By considering capillary forces in phase equilibrium calculations, this model effectively characterizes phase behavior in tight oil reservoir development, making it highly relevant for Pressure Volume Temperature (PVT) simulation, development simulation, and forecasting development strategies.

1. Introduction

As oil and gas exploration transitions from conventional to unconventional fields, tight oil and gas have emerged as key resources for boosting China’s oil and gas production. China’s tight oil geological resources are estimated at approximately 200 × 10⁸ t, highlighting their abundance and potential, which have garnered increasing attention in recent years [1,2,3]. A defining characteristic of tight oil reservoirs is their extremely small pore size [4]. While conventional oil reservoirs typically have pore sizes exceeding 2 μm [5,6], tight oil reservoir pores range from 5 to 750 nm, with 20–40% measuring below 10 nm [7,8]. This nanometer-scale pore size amplifies capillary forces, significantly altering fluid phase behavior compared to conventional reservoirs [9,10,11].

Phase behavior in conventional reservoirs is typically described by traditional equations of state (EOS), such as Redlich-Kwong equation of state (RK-EOS), Soave-Redlich-Kwong equation of state (SRK-EOS), and Peng-Robinson equation of state (PR-EOS) [12,13,14]. These EOS assume that phase behavior is independent of pore size and provide sufficient accuracy for conventional oil and gas calculation [15]. However, in tight reservoirs, the nanometer-scale throats and strong interactions between pore walls and fluids significantly influence fluid phase behavior, deviating from the phase behavior observed in conventional reservoirs. Consequently, the phase state theory for fluids in large conventional pores becomes inadequate for tight reservoirs [16].

Numerous studies have confirmed that the phase behavior of fluids in nanometer pores cannot be accurately described by traditional equations of state [17,18]. Ribeiro [19] and Qiao [20] observed through adsorption-desorption experiments that the saturation pressure of fluids in nanometer pores is lower than in larger pores. Morishige [21], Russo [22], and Yun [23] further demonstrated that the critical temperature of fluids in nanometer pores is reduced compared to larger pores. Similarly, Zeigermann [24] found via diffusion methods that the critical temperature of n-pentane in 6 nm and 15 nm pores is lower than in larger pores. To directly observe and measure these phenomena, Ally [25] employed high-pressure nanofluidic chips and showed significant deviations in the dew point pressure of propane and carbon dioxide within 17–33 nm porous media compared to calculated values. Alfi [26] used microfluidic chips to simulate shale matrices and found that decreasing pore size significantly impacts the bubble point properties of fluids. Zhong [27], using precision-fabricated 8 nm silicon nanochannels, demonstrated that nanoscale pore and throat interactions influence fluid dew point and bubble point pressures. These experimental findings collectively indicate that pore size significantly affects the phase behavior of tight oil reservoirs, rendering traditional EOS inadequate for describing phase behavior in nanometer pores.

To accurately calculate the fluid phase behavior in tight reservoirs, improvements to traditional equations of state are essential. Brusilovsky [28] developed a new EOS for predicting the critical pressure of multi-component hydrocarbon systems and used numerical simulations to study the impact of capillary forces in porous media on dew and bubble points. Pang [29] analyzed the influence of capillary action on fluid saturation pressure in shale gas and oil reservoirs, while Teklu [30] incorporated critical temperature and pressure variations, along with capillary force effects, into the Pressure Volume Temperature (PVT) phase equilibrium of micropore fluids. Teklu also utilized an improved EOS and multi-component mixing algorithm (MMC) to calculate the minimum miscibility pressure for Bakken crude oil and carbon dioxide mixtures. Theoretical analyses by Qi [31], Nojabaei [32], and Zhang Yuan [33] concluded that bubble-point pressure decreases as pore size reduces, with the effect being more pronounced under conditions further from the critical point. Nojabaei [32] coupled capillary pressure with phase equilibrium equations, solving nonlinear fugacity equations and showing that micropores reduce fluid saturation pressure. Li et al. [34] proposed a phase equilibrium calculation method that incorporates pore throat-fluid molecule interactions and capillary forces, verifying through methane, butane, and octane mixtures that pore throat diameters below 10 nm significantly impact phase behavior. Zhang [35] further corrected the PR-EOS for Bakken shale oil by accounting for capillary effects on two-phase systems. These improvements to the traditional PR-EOS, considering capillary pressure and nanopore confinement effects, have been applied in numerical simulations of tight oil reservoirs [36,37,38,39]. However, the iterative computation required for component simulation and EOS adjustments increases computational complexity, limiting industrial adoption. As a result, these approaches are primarily used as alternatives when the accuracy of black oil simulations is insufficient.

To address convergence issues caused by the strong nonlinearity in compositional simulation, many researchers have employed multiscale techniques to optimize the method. These techniques construct multiscale basis functions for coarse grids using fine-scale detailed information, enabling the upscaling and downscaling of data across scales and achieving effective multiscale coupling. The mathematical foundation of the multiscale finite element method (MsFEM) originated from the pioneering work of Babuska and Osborn [40,41]. Hou and Wu [40] extended this concept to elliptic equation problems, introducing the innovative MsFEM. Building on this, Jenny [42] proposed the multiscale finite volume method (MsFVM), inspired by MsFEM, and applied it to incompressible flow in porous media. Moyner and Li [43], along with Zhou and Tchelepi [44], further extended the approach to black oil simulations. To address the challenges posed by reservoir models with complex geological structures, Lunati and Moyner et al. [45,46,47] incorporated the algebraic multigrid concept, developing multiscale formulations based on operator algebraic representations. These advancements have significantly enhanced the efficiency and applicability of multiscale methods in reservoir simulations.

A review of previous studies reveals that no existing multiscale compositional numerical simulation method for tight oil reservoirs incorporates the effects of capillary forces in phase behavior. Building on prior research, this study first modifies phase behavior calculations to account for capillary forces, proposing a phase calculation method that integrates these effects. Subsequently, the traditional multiscale finite volume method is enhanced to develop a multiscale compositional numerical model that considers capillary forces. Finally, the proposed model is validated, and a parameter sensitivity analysis is performed.

2. Materials and Methods

This section details the construction of a multiscale compositional numerical simulator incorporating capillary forces. First, it outlines the development of an equation of state that accounts for capillary forces. Next, it describes the sequential formulation and solution of the compositional numerical simulation, which is the foundation for multiscale construction. Following this, an improved iterative multiscale finite volume method is introduced. Finally, these components are integrated into a multiscale compositional numerical simulation method that fully considers capillary forces.

2.1. The Phase Equilibrium Calculation Theory Considers the Capillary Force Effect

The PR-EOS is modified to incorporate capillary pressure, which is influenced by pore geometry and the wettability of the rock surface. In the calculation of capillary pressure between oil and gas, the oil phase is regarded as the wetting phase relative to the gas phase. Under this assumption, the gas phase pressure can be expressed as:

$$p_v=p_l+p_c$$

(1)

where $p_v$ is the gas phase pressure (Pa), $p_l$ is the liquid phase pressure (Pa), $p_c$ is the capillary pressure (Pa), which is determined by the geometry of the pores and the wettability of the rock surface.

Mercury intrusion experiments quantify the relationship between capillary pressure and mercury saturation (i.e., the pore volume occupied by mercury). Volume averaging of pore radius and interfacial tension allows derivation of this relationship from the Young-Laplace equation, a result validated in multiphase flow simulations [48]. This study investigates the impact of nanopore capillary pressure on phase equilibrium. Consequently, phase behavior calculations incorporate these effects as described by Zhang et al. [35] the calculation of capillary force in Equation (1) refers to the Young-Laplace equation

$$p_c={\displaystyle \frac{2\sigma \cos \theta }{r}}$$

(2)

where $\sigma$ represents the interfacial tension (N/m); θ represents the contact angle (For gas-liquid phase equilibrium in reservoirs, the gas phase is regarded as non-wetting, while the liquid phase is considered wetting, assuming a contact angle of 45° [49] in this study; $r$ represents the pore radius (nm).

The method for calculating the interfacial tension $\sigma$ in Equation (2) refers to the work of MacLeod and Pedersen [50,51]

$$\begin{array}{l}\sigma ={({\mathsf{\Omega }}_l{\overline{\rho }}_l-{\mathsf{\Omega }}_v{\overline{\rho }}_v)}^v\\ {\mathsf{\Omega }}_l={\displaystyle {\displaystyle \sum }_{i=1}^{N_c}}{x}_i{\mathsf{\Omega }}_i\\ {\mathsf{\Omega }}_v={\displaystyle {\displaystyle \sum }_{i=1}^{N_c}}{y}_i{\mathsf{\Omega }}_i\end{array}$$

(3)

where $v$ is the scaling exponent, $v$ = 3.6; ${\overline{\rho }}_{l/v}$ is the average molar densities of the liquid and gas phases in the pore (mol/cm³); $x_i$ and $y_i$ are the molar fractions of component i in the liquid and gas phases, respectively; ${\mathsf{\Omega }}_i$ is the isobaric molar volumes of the liquid and gas phases, respectively [50,51]

$${\mathsf{\Omega }}_i=(8.21307+1.97473{\omega }_i)T_{ci}^{1.03406}{P}_{ci}^{-0.82636}$$

(4)

where ${\omega }_i$ is the acentric factor; $T_{ci}$ is the critical temperature of component i (K); $P_{ci}$ is the critical pressure of component i (bar).

Equations (1)–(3) provide a method for integrating microscopic mechanisms into phase behavior calculations. Since this study focuses on the impact of capillary pressure in nanopores on phase behavior, capillary pressure is computed using an average pore radius in subsequent calculations. However, the option to incorporate a pore size distribution remains. The proposed model can be extended to account for the actual pore radius distribution of a reservoir, based on geological pore size data.

In the PR-EOS, capillary pressure is not accounted for in the gas and liquid phases. Instead, Equations (1) and (2) compute capillary pressure and phase pressures for different pore sizes, subsequently affecting the compressibility factor and fugacity coefficient. As noted above, the original PR-EOS neglects capillary forces, resulting in no distinction between phase pressures. Consequently, the formulations for fugacity and compressibility factor must be modified to incorporate capillary effects and differentiate phase pressures. Referring to PR-EOS [12], the compressibility factor equation incorporating capillary forces is given as:

$$Z_{\alpha }^3-(1-B_{\alpha })Z_{\alpha }^2+(A_{\alpha }-3B_{\alpha }^2-2B_{\alpha })Z_{\alpha }-(A_{\alpha }{B}_{\alpha }-B_{\alpha }^2-B_{\alpha }^3)=0$$

(5)

where $\alpha$ denotes the phase, with l representing the gas phase and v the liquid phase. $Z_{\alpha }$ is the phase compressibility factors; A_α and B_α are the PR-EOS parameters for the liquid and gas phases, with the specific expressions given as:

$$A_{\alpha }={\displaystyle \frac{a{p}_{\alpha }}{R^2{T}^2}} B_{\alpha }={\displaystyle \frac{b{p}_{\alpha }}{RT}}$$

(6)

According to Gibbs free energy principles [12], the smallest real root of $Z_l$ represents the liquid-phase compressibility factor, while the largest real root of $Z_v$ corresponds to the gas-phase compressibility factor. Using these factors, fugacity and fugacity coefficients can be determined. At phase equilibrium, the fugacity of the gas phase equals that of the liquid phase:

$${f_l}^i(T,P_l,x_i)={f_v}^i(T,P_v,y_i)$$

(7)

where ${f_{l/v}}^i$ are the fugacities of component i in the gas and liquid phases, respectively.

For a gas-liquid two-phase system, fugacity is defined as:

$$f=\phi p$$

(8)

where $\phi$ is the fugacity coefficient, which is expressed as follows [12]:

$$ln\phi =Z-1-\ln Z+{\displaystyle \frac{1}{RT}}\underset{\infty }{\overset{V}{\int }}\left({\displaystyle \frac{RT}{V}}-p\right)dV$$

(9)

Based on Chen’s research [52], the modified fugacity coefficients for the gas and liquid phases, incorporating capillary forces, are expressed as:

$$\begin{array}{l}\ln {\phi }_v^i={\displaystyle \frac{b_i}{b}}(Z_v-1)-\ln (Z_v-B_v)-{\displaystyle \frac{Z_v}{2\sqrt{2}{B}_v}}\left({\displaystyle \frac{2{\displaystyle \sum _{j=1}^N{y}_j}{a}_{ij}}{a}}-{\displaystyle \frac{b_i}{b}}\right)\ln \left[{\displaystyle \frac{Z_v+(1+\sqrt{2})B_v}{Z_v+(1-\sqrt{2})B_v}}\right]\\ \ln {\phi }_l^i={\displaystyle \frac{b_i}{b}}(Z_l-1)-\ln (Z_l-B_l)-{\displaystyle \frac{Z_l}{2\sqrt{2}{B}_l}}\left({\displaystyle \frac{2{\displaystyle \sum _{j=1}^N{x}_j}{a}_{ij}}{a}}-{\displaystyle \frac{b_i}{b}}\right)\ln \left[{\displaystyle \frac{Z_l+(1+\sqrt{2})B_l}{Z_l+(1-\sqrt{2})B_l}}\right]\end{array}$$

(10)

where a and b represent the PR-EOS parameters for the mixture, while a_i and b_i denote those for pure component i. The calculation methods for these parameters are detailed in Reference [12].

2.2. Multiscale Compositional Model

2.2.1. Basis Model

Based on the compositional model, the following assumptions are made:

• The reservoir consists of three phases: oil (denoted by subscript l), gas (subscript v), and water (subscript w).
• The component model considers these three flowing phases. The water phase contains only the water component and does not achieve phase equilibrium with the other phases, whereas phase equilibrium is established among m components in the oil and gas phases.
• Fluid flow follows Darcy’s law.
• The reservoir is isothermal.
• Phase equilibrium is achieved instantaneously.

Under these assumptions, the compositional control equations are formulated, beginning with the continuity equations for components in the oil and gas phases [52].

$${\displaystyle \frac{\partial }{\partial t}}\left[\varphi ({\rho }_l{S}_l{X}_i+{\rho }_v{S}_v{Y}_i)\right]+\nabla \cdot ({\rho }_l{X}_i{\overrightarrow{v}}_l+{\rho }_v{Y}_i{\overrightarrow{v}}_v)-q_i=0$$

(11)

where $\varphi$ represents porosity, ${\rho }_{\alpha }$ and $S_{\alpha }$ denote the density and saturation of phase $\alpha$, respectively, l/v corresponds to the oil and gas phases, in accordance with phase equilibrium calculation theory.

The continuity equation for the water component is formulated as follows:

$${\displaystyle \frac{\partial }{\partial t}}(\varphi {\rho }_w{S}_w)+\nabla \cdot ({\rho }_w{\overrightarrow{v}}_w)-q_w=0$$

(12)

Phase velocities are determined according to multiphase Darcy’s law.

$$v_{\alpha }=-K{\lambda }_{\alpha }(\nabla p_{\alpha }-{\rho }_{\alpha }g\Delta z),\hspace{1em}{\lambda }_{\alpha }={\displaystyle \frac{k{r}_{\alpha }}{{\mu }_{\alpha }}}$$

(13)

where K denotes the absolute positive-definite permeability tensor, while kr_α, μ_α, λ_α represent the relative permeability, viscosity, and mobility of phase α, respectively.

For each component, the source and sink terms in the continuity equation, along with the well control equations, follow the standard Peaceman model [53]. Additionally, the compositional model accounts for component transfer between the oil and gas phases, incorporating phase equilibrium equations to ensure accurate phase behavior representation

$$f_l^i-f_v^i=0$$

(14)

In addition, the constraints must also be considered:

$$S_w+S_v+S_l=1,\hspace{1em}{S}_{\alpha }\in [0,1]\forall \alpha$$

(15)

$${\displaystyle \sum _{i=1}^m{x}_i}=1,\hspace{1em}{\displaystyle \sum _{i=1}^m{y}_i}=1,\hspace{1em}{x}_i,y_i\in [0,1]\forall i\in \{1,\dots ,m\}$$

(16)

By combining Equations (11) to (16), the governing equations are reformulated into residual forms. The Newton-Raphson iteration method is then applied to solve these equations efficiently.

$$-J_{\epsilon }\Delta \epsilon =r_{\epsilon },\hspace{1em}{J}_{\epsilon }={\displaystyle \frac{\partial r_{\epsilon }}{\partial \epsilon }},\hspace{1em}{r}_{\epsilon }=\left[\begin{array}{c}{R}_{\epsilon }\\ W_{\epsilon }\\ F_{\epsilon }\\ C_{\epsilon }\end{array}\right]$$

(17)

where $J_{\epsilon }$ represents the Jacobian matrix of the system, and $r_{\epsilon }$ denotes the residual term, consisting of the residuals from the continuity equations ($R_{\epsilon }$), the well control equations ($W_{\epsilon }$)—with $W_{\epsilon }$ constructed based on the Peaceman model [53], fugacity equations ($F_{\epsilon }$), and constraints ($C_{\epsilon }$), as defined in Equations (15) and (16).

While the above equations can be solved directly in compositional numerical simulation, the computational efficiency remains low. To enhance efficiency, the Schur complement reduction method [54] is employed to decrease the degrees of freedom of the matrix. By partitioning the variables ($\epsilon$) into primary variables (u) and eliminable variables (v), and reordering the Jacobian matrix accordingly, the system can be expressed as:

$$\left[\begin{array}{cc}{\displaystyle \frac{\partial U}{\partial u}}& {\displaystyle \frac{\partial U}{\partial v}}\\ {\displaystyle \frac{\partial V}{\partial u}}& {\displaystyle \frac{\partial V}{\partial v}}\end{array}\right]\left[\begin{array}{c}\Delta u\\ \Delta v\end{array}\right]=\left[\begin{array}{c}{r}_u\\ r_v\end{array}\right]$$

(18)

By reordering the system in Equation (17), the primary variable u is placed first, along with its corresponding continuity and well control equations U. Similarly, v represents the secondary variable, while V includes the fugacity equations and constraints associated with v. By applying block Gaussian elimination, a reduced system containing only the primary variables can be derived, effectively reducing the computational complexity

$$-{\tilde{J}}_u\Delta u={\tilde{r}}_u,\hspace{1em}{\tilde{J}}_u={\displaystyle \frac{\partial U}{\partial u}}-{\displaystyle \frac{\partial U}{\partial v}}{\left({\displaystyle \frac{\partial V}{\partial v}}\right)}^{-1}{\displaystyle \frac{\partial V}{\partial u}},\hspace{1em}{\tilde{r}}_u=r_u-{\displaystyle \frac{\partial U}{\partial v}}{\left({\displaystyle \frac{\partial V}{\partial v}}\right)}^{-1}{r}_v$$

(19)

The increments of the secondary variables are updated using the increments of the primary variables, ensuring consistency between the two sets of variables.

$$\Delta v={\left({\displaystyle \frac{\partial V}{\partial v}}\right)}^{-1}\left(r_v-{\displaystyle \frac{\partial V}{\partial u}}\Delta u\right)$$

(20)

2.2.2. Sequential Formulation

To facilitate the use of multiscale formulations later, the sequential formulation is employed to split the entire system of equations into pressure equations and transport equations.

Various methods exist to construct the sequential formulation pressure equation based on reservoir numerical models. Using the linear algebraic approach by Coats [54], in Equation (19), considering the physical significance of different elements in the system, the term U corresponding to u represents the accumulation term containing the time derivative, while V corresponding to v includes the remaining constraint terms, excluding the accumulation term. Consequently, the Jacobian matrix $J_{\epsilon }$ and its Schur complement reduced matrix ${\tilde{J}}_u$ can be reformulated to facilitate the explanation of subsequent transformations

$$J_{\epsilon }=\left[\begin{array}{cc}{\displaystyle \frac{\partial AC}{\partial u}}& {\displaystyle \frac{\partial AC}{\partial v}}\\ {\displaystyle \frac{\partial \vartheta }{\partial u}}& {\displaystyle \frac{\partial \vartheta }{\partial v}}\end{array}\right],\hspace{1em}{\tilde{J}}_u={\displaystyle \frac{\partial AC}{\partial u}}-{\displaystyle \frac{\partial AC}{\partial v}}{\left({\displaystyle \frac{\partial \vartheta }{\partial v}}\right)}^{-1}{\displaystyle \frac{\partial \vartheta }{\partial u}}.$$

(21)

where $AC$ represents the accumulation terms in the system of equations, $\vartheta$ represents the constraint equation block, and u and v represent the primary and secondary unknowns, respectively.

The purpose of the sequential formulation is to decouple the pressure variable from other variables. Referring to the derivation process of the sequential formulation in the black oil model, it achieves this by using a linear combination of the accumulated phases, which has zero derivatives with respect to non-pressure variables. Thus, the pressure variable can be decoupled from the non-pressure variables. Drawing on this concept, from a linear algebra perspective, we construct a linear combination coefficient vector—the weight factor w, which is formulated as follows:

$${({{\tilde{J}}_u}^T)}^{-1}w=e_p,\hspace{1em}{(e_p)}_i=\left\{\begin{array}{ll}1\hfill & \mathrm{if} u_i \mathrm{is} \mathrm{a} \mathrm{pressure} \mathrm{variable}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(22)

Mathematically, the weight factor w aligns with the derivation of the sequential formulation in the black oil model, as both eliminate non-pressure variables through weighted summation. Thus, by applying w, a linear algebraic transformation can be performed on Equation (19) to derive the pressure equation. The pressure equation is derived based on Equations (19) and (22):

$$-{\tilde{J}}_p\Delta u_p={\tilde{r}}_p,\hspace{1em}{\tilde{J}}_p\in R^{n_c\times n_c},\hspace{1em}{\tilde{r}}_p\in R^{n_c}.$$

(23)

The coefficients and the right-hand side of Equation (23) are constructed through the following expressions:

$${({\tilde{r}}_p)}_j={\displaystyle \sum _{i=0}^{m-1}{w}_{j+i{n}_c}}{({\tilde{r}}_u)}_{j+i{n}_c},\hspace{1em}{({\tilde{J}}_p)}_{j,k}={\displaystyle \sum _{i=0}^{m-1}{w}_{j+i{n}_c}}{({\tilde{J}}_u)}_{j+i{n}_c,k}+{\displaystyle \frac{\partial w_{j+i{n}_c}}{\partial p_k}}{({\tilde{r}}_u)}_{j+i{n}_c}.$$

(24)

The pressure increment is obtained by solving the above equations. Subsequently, the transport equations are solved using the computed pressure increment.

Once the pressure Equation (23) has converged, the Darcy velocities for different phases are determined using the multiphase Darcy equation (Equation (13)). The total volumetric Darcy velocity is defined as:

$${\overrightarrow{v}}_t={\overrightarrow{v}}_w+{\overrightarrow{v}}_l+{\overrightarrow{v}}_v$$

(25)

Based on Equation (25), the integrated fractional flow is as follows [55]:

$${\overrightarrow{v}}_{\alpha }={\displaystyle \frac{{\lambda }_{\alpha }}{{\displaystyle \sum _{\beta \in \{l,v,w\}}{\lambda }_{\beta }}}}\left[{\overrightarrow{v}}_t+K{\displaystyle \sum _{\beta \in \{l,v,w\}}{\lambda }_{\beta }}(G_{\alpha }-G_{\beta })\right],\hspace{1em}{G}_{\alpha }=g\Delta z{\rho }_{\alpha }+\nabla p_c^{\alpha }.$$

(26)

where K represents the absolute positive-definite permeability tensor, while λ_α denotes the mobility of phase α, $G$ accounting for the effects of gravity and capillary forces.

By substituting Equation (26) into the conservation Equations (11) and (12), the transport equations are derived, allowing the transport variables (components and saturations) to be solved.

2.2.3. Improved Multiscale Formulation

The multiscale formulation primarily targets the pressure Equation (23) and is written in the following form:

$${\tilde{J}}_p\mathsf{\Delta }p={\tilde{r}}_p$$

(27)

Since Equation (27) exhibits significant coefficient variations and strong coupling between adjacent cells, direct solution is inefficient. Instead, a multiscale expansion is employed to approximate pressure, enabling efficient computation on a coarser grid.

Following the algebraic multigrid approach, prolongation operator P and restriction operator R are defined to relate the fine-scale and coarse-scale pressure increments [44]. Fine-scale information is mapped to the coarse scale using a restriction operator. Solving on the coarse scale reduces the number of iterations due to low-frequency errors on the fine scale. The coarse-scale pressure increment is then mapped back to the fine-scale using a prolongation operator, approximating the fine-scale exact pressure increment:

$$\mathsf{\Delta }p\approx \mathsf{\Delta }{p}_f=P\mathsf{\Delta }{p}_c$$

(28)

Using Equations (28) and (27), the solution of the fine-scale pressure increment is reformulated as solving the coarse-scale pressure increment, enhancing computational efficiency.

$$(R{\tilde{J}}_{p}P)\mathsf{\Delta }{p}_c={\tilde{J}}_c\mathsf{\Delta }{p}_c=R{\tilde{r}}_p={\tilde{r}}_c$$

(29)

The construction of the operator format is key to achieving a multiscale solution. The restriction operator maps physical quantities from the fine-scale grid to the coarse-scale grid, with a common approach being the control volume restriction operator.

$${\left(R_{c\nu }\right)}_{ji}=\left\{\begin{array}{ll}1,\hfill & \mathrm{if} x_i\in {\mathsf{\Omega }}_j^b\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(30)

Zhou and Lunati et al. [44,45] employed this restriction operator in the multiscale finite volume method, demonstrating its ability to ensure mass conservation. Wang et al. [56] further proposed a Galerkin restriction operator to improve convergence rates. Wang et al. have combined these approaches, applying the Galerkin restriction operator in the early stages for faster convergence and the control volume restriction operator later to maintain mass conservation [56]. This hybrid strategy is also applicable to the multiscale solution method proposed in this study. Møyner and Li et al. conducted numerical experiments on various restriction operators in multiscale formulations using different test models. Their findings indicate that in the original multiscale formulation, the choice of restriction operator influences the convergence of the multiscale method. However, when an iterative scheme is used to construct the prolongation operator, both types of restriction operators demonstrate identical convergence performance [57]. Therefore, this study adopts the control volume restriction operator.

The prolongation operator extends physical quantities from the coarse-scale grid to the fine-scale grid. A standard approach for constructing the multiscale prolongation operator P involves solving a series of local flow problems numerically. Unlike the classical MSFV method, this study follows Møyner and Li [44,46] by constructing P using an iterative scheme, eliminating the need for complex boundary conditions when setting up flow problems. This approach addresses the challenge of constructing high-quality basis functions on unstructured grids. The iterative format is as follows:

$$P^{n+1}=P^n-\mathcal{M}(\omega {D_s}^{-1}J{P}^n)$$

(31)

In Equation (31) $D_s$ represents the constructed smoother, a diagonal matrix composed of the diagonal elements of the matrix ${\tilde{J}}_p$. The smoother is used to compute increments and progressively refine them. $\mathcal{M}$ serves as a restrictor, confining the smoothing of basis functions to a support region centered on the coarse-scale grid. The boundary of this region is defined by the eight surrounding grid center points, ensuring the accuracy of the prolongation operator.

2.3. Implementation of the Multiscale Compositional Simulator

The model consists of Equations (11)–(16), with phase equilibrium calculations governed by Equations (1), (5), (7), and (10), and the multiscale formulation described by Equation (29). The implementation is carried out using the MATLAB Reservoir Simulation Toolbox (MRST) [58]. This study refines the MRST-based compositional model by modifying the phase equation model class functions in accordance with the equations in Section 2.1. Additionally, the tool functions in the sequential formulation module are enhanced. A compositional pressure equation class and a transport equation class, accounting for capillary effects, are developed based on the sequential model base class. Finally, in line with the multiscale formulation, the local solution of the prolongation operator in the multiscale volume method is revised to an iterative solution. Within each nonlinear iteration, flash calculations incorporating capillary effects are first performed. The resulting component compositions are then integrated into the pressure and transport equations. The pressure equation is solved using a multiscale formulation for linear iterations. Upon convergence of the iteration error, pressure increments and secondary variables are transferred to the transport equation for subsequent calculations. The implementation process is illustrated in Figure 1.

3. Results and Discussion

3.1. Simulation Validation

Common commercial simulators still employ the traditional PR-EOS or SRK-EOS, without considering the effects of nanopore constraints on phase behavior [49]. For example, the CMG-GEM simulator (Version 2022) uses a hysteretic relative permeability approximation to simulate changes in flow caused by compositional variations. Therefore, this section adopts a model degradation approach, comparing the multiscale compositional model with the ECLIPSE E300 simulator (Version 2022) to validate the accuracy of the multiscale calculations in the current model. The validation employed a one-dimensional, two-phase model (1000 × 1 × 0.1 m), with its 3D view depicted in Figure 2. The parameters for the validation reservoir model are presented in

Table 1. Validation reservoir model parameters.

Model Grid	Grid Size (m)	Initial Reservoir Temperature (°C)	Initial Reservoir Pressure (MPa)	Porosity (%)	Permeability (mD)	Injection Pressure (MPa)	Production Pressure (MPa)	Simulation Time (Year)
1000 × 1 × 1 (coarse grid 100 × 1 × 1)	1 × 1 × 0.1	150	7.5	25	10	10	5	20

.

Carbon dioxide was injected into a mixture of CO₂, methane, and decane (10% CO₂ + 30% C₁ + 60% C₁₀). The model comprises 1000 grid cells, with an initial reservoir temperature of 150 °C and an initial pressure of 7.5 MPa. The initial component properties are summarized in

Table 2. Component properties of the fluid mixture in the one-dimensional validation model.

Component	Pc (Bar)	Tc (K)	Vc (L/mol)	Mol. Weight	Acentric Fact	Initial Total Composition Values
C1	45.922	190.654	0.09863	16.04	0.01142	0.3
C10	21.03	617.7	0.6098	142.28	0.4884	0.6
CO2	73.733	304.128	0.09412	44	0.22394	0.1

. The relative permeability curves are shown in Figure 3.

To ensure consistent time step selection across simulators, the problem is discretized into a large number of time steps. Given the relatively low pressure, phase behavior exhibits strong pressure dependence, with all components present in both phases. Since the well operates under bottom-hole pressure control, the injected fluid volume relies on precise calculations of fluid properties and densities within the reservoir. To validate the simulator (MS)’s accuracy, the predicted cumulative oil production curve over 20 years (Figure 4) and the total molar fractions of different components at various stages were compared with results from ECLIPSE E300 (Figure 5). The minimal discrepancies between the two sets of results confirm the reliability of the proposed simulator.

Additionally, due to the large number of time steps in the model calculations, the runtime logs for the first 50-time steps were extracted to compare the computational efficiency of the traditional fully implicit (FI) method with the multiscale method proposed in this study. As shown in Figure 6, the multiscale method (MS) demonstrates a clear advantage over the traditional fully implicit method (FI) in terms of iterative solution speed, significantly improving the model’s computational efficiency. The total runtime for FI and MS was measured, revealing that the runtime for MS was approximately 254 s, compared to about 408 s for FI. This indicates a 40% improvement in the computational efficiency of MS.

3.2. Sensitivity Analysis

The accuracy and efficiency of the multiscale compositional numerical simulator were validated in the previous section. This section focuses on the impact of capillary forces, emphasizing the proposed simulator’s importance in tight oil reservoir development. It is divided into two main parts: the first part examines the influence of capillary forces on phase equilibrium calculations, while the second part utilizes the previously established multiscale numerical simulator, incorporating capillary forces, to analyze a typical tight oil reservoir model. By referencing actual tight oil reservoir parameters, this part explores the effect of capillary forces on oil recovery.

3.2.1. The Influence of Capillary Forces on Phase Equilibrium Calculations

Referring to the mixture component properties of the Fifth SPE Comparative Solution Project [59], the PR-EOS was modified to incorporate capillary forces. The study compared the modified PR-EOS with capillary constraints (Equation (5) for pore radii of 15 nm, 30 nm, and 50 nm against the conventional PR-EOS without capillary constraints. Figure 7 and Figure 8 present the phase diagram and the iteration count, respectively.

As shown in Figure 7, both the bubble point and dew point lines decrease with the reduction of pore radius. The smaller the pore radius, the more pronounced the downward trend of both lines. During the pressure drop process, the capillary effect becomes more significant with decreasing pore radius, leading to a reduction in both the bubble point and dew point. As a result, the liquid phase fraction increases while the gas phase fraction decreases. Near the critical point, the calculation results for different pore radii converge, which is attributed to the interfacial tension approaching zero as the gas-liquid interface disappears, thus nullifying the capillary force. Therefore, capillary forces do not affect fluid phase behavior near the critical point. For larger pore radii, such as 50 nm and above, the capillary effect is negligible, and the phase diagram closely resembles that without considering capillary forces.

As shown in Figure 8, the number of iterations for phase equilibrium calculations increases with decreasing pore radius, particularly near the dew point line. This indicates that the PR-EOS incorporating capillary forces introduces greater computational overhead, highlighting the need for a multiscale fast solution. As described in Section 2.1, the conventional PR-EOS does not consider capillary pressure effects, leading to consistent phase pressures across different phases. However, when capillary pressure effects are incorporated, phase pressures differ, with smaller pores showing more significant disparities. This physical phenomenon increases the number of iterations required during the phase equilibrium solution process.

3.2.2. Key Findings on the Impact of Capillary Forces on Tight Oil Production

This section uses fluid data from B. Mallison et al. [60], including mixture composition, basic properties, and binary interaction parameters. A simple closed-external-boundary cubic reservoir model, incorporating 5 vertical wells and a hydraulic fracture (as detailed in

Table 3. Reservoir model parameters.

Model Grid	Grid Size (m)	Initial Reservoir Temperature (°C)	Initial Reservoir Pressure (MPa)	Porosity (%)	Matrix Permeability (mD)	Fracture Permeability (mD)	Initial Saturation (Water, Oil, and Gas) (f)	The Bottom-Hole Pressure of the Producing Well (MPa)	The Daily Gas Injection Volume of an Injection Well (m3/day)
25 × 25 × 8	40 × 40 × 5	114	15	8	10	1000	0.38, 0.62, 0	10	300

), was created with 5000 grid cells. The grid size settings are based on Ma’s research [49].

The initial component properties are summarized in

Table 4. Component properties of the fluid mixture.

Component	Pc (Bar)	Tc (K)	Vc (L/mol)	Mol. Weight	Acentric Fact	Initial Total Composition Values
N2/CH4	45.8	189.5	0.0997	16.1594	0.01142	0.463
CO2	73.733	304.128	0.09412	44	0.22394	0.0164
C2–5	41.0	387.6	0.02171	45.5725	0.167	0.2052
C6–13	33.5	597.5	0.03812	11.774	0.386	0.19108
C14–24	17.7	698.5	0.07214	24.8827	0.808	0.08113
C25–80	11.7	875.0	0.01136	48.125	1.231	0.04319

.

As shown in Figure 9, the five vertical wells form a five-spot pattern, with four injection wells at the reservoir edges and one production well at the center. A black line indicates the hydraulic fracture, and the target grid is marked with a red star. The production well operates at a constant pressure of 10 MPa, while the injection wells inject CO₂ at a constant rate of 300 m³/day. The injection and production period lasts 10 years. The relative permeability curves used in the simulation refer to the core displacement test report of a certain block in the Changqing Oilfield, China. The specific data can be found in Figure 10.

This section presents four scenarios:

Base Case: In gas-phase equilibrium calculations, existing numerical simulators neglect the impact of confinement on fluid phase behavior in nanopores. Consequently, capillary pressure effects and shifts in critical properties are excluded.

Case 1: Capillary forces are calculated using the Young-Laplace equation with a pore radius of 20 nm.

Case 2: Capillary forces are calculated using the Young-Laplace equation with a pore radius of 35 nm.

Case 3: Capillary forces are calculated using the Young-Laplace equation with a pore radius of 50 nm.

The proposed multiscale compositional simulator is used to analyze the influence of capillary forces on cumulative production, component distribution, and reservoir saturation.

Based on the results of the four scenarios, the impact of capillary forces on cumulative production was analyzed. As shown in Figure 11, including capillary forces (Case 1–3) generally leads to higher cumulative oil production compared to the scenario without capillary effects (Base Case). The smaller the pore radius, the more pronounced the capillary effect, resulting in a greater increase in cumulative oil production. This underscores the importance of considering capillary forces in tight oil reservoir development. It is important to note that this section primarily addresses how capillary pressure influences phase behavior and, consequently, oil recovery efficiency. However, in the actual development of tight oil reservoirs, various complex mechanisms and operating conditions must be considered. For instance, capillary trapping effects play a role during multiphase flow in tight oil reservoirs. Therefore, capillary pressure should not be viewed as the sole key mechanism for improving oil recovery.

Figure 12 shows that capillary forces (Case 1–3) inhibit the increase in gas-phase components. This is due to a decrease in the bubble point and dew point lines caused by capillary effects. Smaller pore radii result in a more significant decrease, which leads to an increase in the liquid phase and a relative reduction in the gas phase.

The impact of capillary forces on component distribution was analyzed, focusing on the cell marked by a red star in Figure 9. The molar fractions of components in this cell under the Base Case and Case 1 were compared. As shown in Figure 13, the model categorizes components into light (N₂/CH₄, CO₂) and heavy (C_2–5, C_6–13, C_14–24, C_25–80) groups. Compared to the model incorporating capillary forces (Case 1), the base-case model (excluding capillary forces) yields a higher proportion of light components and retains more heavy components in the reservoir. These findings suggest that accounting for capillary forces in nanopores may enhance recovery efficiency by modifying phase behavior.

The oil saturation at different time steps for various scenarios is compared in Figure 14. The bubble point pressure is higher in the scenario without capillary effects, leading to greater gas stripping. As a result, this scenario (Base Case) exhibits lower oil saturation and higher gas saturation compared to the scenario with capillary forces (Case 1–3).

Additionally, oil saturation varies with pore radius and capillary force magnitude. The analysis shifts to the characteristic scale to further investigate, focusing on the cell marked by a red star in Figure 9. Figure 15 shows that gas saturation is higher in the scenario without capillary effects (Base Case), and this trend becomes more pronounced with a stronger capillary effect. Eventually, saturation levels converge as CO₂ breakthrough occurs.

4. Conclusions

This paper presents a multiscale compositional numerical simulation method that incorporates capillary forces in phase equilibrium calculations. A one-dimensional model is used for validation, demonstrating the method’s accuracy and advancement in analyzing the impact of capillary forces on phase equilibrium and tight oil recovery.

• Phase behavior in tight oil reservoirs is influenced by pore size and capillary forces, which traditional equations of state fail to capture in nanopores. This paper introduces capillary forces into traditional equations of state and compositional numerical simulations, improving phase equilibrium calculations using MRST.
• To enhance iterative solution efficiency, a sequential solution format is employed, leveraging the parallel advantages of the improved multiscale finite volume method for pressure equation resolution. Numerical validation with a one-dimensional model confirms that this method enhances solution efficiency without compromising accuracy.
• Using the proposed multiscale compositional simulation method, the impact of capillary forces on phase equilibrium and tight oil recovery is analyzed. Sensitivity analysis shows that smaller pore radii lead to more pronounced capillary effects, with a decrease in bubble point and dew point lines, and an increase in calculation iterations. For tight oil recovery, capillary forces generally result in higher cumulative oil production and improved recovery rates for heavy components. Capillary effects lower bubble point pressure, suppress gas stripping and increase oil saturation while decreasing gas saturation.
• Compared to traditional numerical simulators, the proposed multiscale compositional method, which accounts for capillary forces, better characterizes phase behavior in tight oil reservoirs. This approach is significant for PVT simulation, development of numerical simulations, and forecasting development strategies in tight oil reservoirs.

In summary, the proposed model advances numerical simulation research for tight oil reservoirs, though several challenges remain. First, this study examines the reservoir-scale impact of capillary forces on phase behavior under an instantaneous equilibrium assumption, without addressing nanopore kinetic limitations and complex flow mechanisms. Second, large-scale physical experiments could help overcome the validation constraints of commercial simulators. Finally, although MATLAB Reservoir Simulation Toolbox (MRST 2024b) offer significant community and ecosystem potential, incorporating additional open-source high-performance components may further promote the development of large-scale 3D tight oil reservoir simulators.