Pore-Scale Thermo-Hydro-Mechanical Simulation of Heavy Oil Displacement in 3D Anisotropic Porous Media

Abstract

Three-dimensional synthetic porous models were generated using an improved Quartet Structure Generation Set (QSGS) method to examine heavy oil displacement. Flow and heat transfer were coupled, with a one-way thermoelastic response on a fixed geometry. Between-axis anisotropy was distinguished from the response to inlet–outlet reversal. In the main model, reversal changed absolute permeability by no more than 0.062%, whereas water saturation indicators at 0.5 s differed by 7.2%, 1.2%, and 31.3% along X, Y, and Z. An additional ensemble of 15 realizations was compared using saturation-based incremental recovery referenced to each initialized oil volume. Within the common supported range up to 2.90 injected pore volumes (PV), the one-sided growth design showed negative reverse-minus-forward gaps in all five realizations at every evaluated checkpoint from 0.5 PV. At 1 PV, the mean gap was −17.08 percentage points with a sample standard deviation of 6.20. The isotropic and symmetric designs showed smaller means with mixed individual signs. These results support directional sensitivity within the tested structures while distinguishing fixed-time throughput effects from matched-PV comparisons. The growth-probability trend remains an association for the selected geometries, and the finite-time results do not establish ultimate recovery.

1. Introduction

Heavy oil reservoirs are rich in reserves and have enormous exploitation potential [1,2,3]. With technological breakthroughs and the growth of energy demand, heavy oil will play an important role in future energy supply. At present, studies on micro-porous media reservoirs mainly focus on two-dimensional anisotropic and 3D isotropic models [4,5]. However, studies on 3D anisotropic porous media are relatively limited. Anisotropy widely exists in oil reservoir formations, where parameters such as permeability vary with spatial direction due to sedimentation, diagenesis, and other geological processes [6]. Therefore, it is urgent to use a thermo-hydro-mechanical multi-physics coupling model for 3D anisotropic porous media to study fluid flow and heat transfer problems in heavy oil reservoirs.

In recent years, significant progress has been achieved in the theories and methods of numerical simulation for porous media [7,8]. Since the proposal of Darcy’s law, scholars have gradually developed flow simulation methods for porous media ranging from macroscopic scales to microscopic scales [9,10]. These include macroscopic simulations based on the governing equations of traditional porous media, as well as micro-scale simulations using technologies such as the Lattice Boltzmann method (LBM) and pore network model (PNM) at pore scale [11,12,13,14]. With the improvement of computing power, it has become possible to construct 3D porous media models using micro-CT scanning and 3D reconstruction methods, especially for two-phase flow in complex pore structures [15]. However, most existing studies are limited to 2D or simplified 3D structures, which are difficult to fully reflect the complex geometric features of real porous media. On the other hand, although image-based 3D digital core construction techniques can improve model accuracy, the balance between computational cost and data processing efficiency still needs to be addressed [16].

Early scholars preliminarily explored the influence of anisotropic permeability on fluid flow characteristics by modifying Darcy’s law and the oil film Reynolds equation [17]. At the theoretical level, scholars have systematically revealed the influence mechanisms of microscopic characteristics on macroscopic fluid transport behavior by establishing turbulent diffusion coefficient models for anisotropic porous media, analyzing elastic wave propagation characteristics, and applying the effective stress law. These theoretical frameworks have elucidated how microscale features like crack orientation and pore structure anisotropy govern macroscopic fluid flow and transport processes in porous media [18,19,20]. In terms of experimental research, techniques such as X-ray micro-tomography have been used to reveal that grain arrangement and aggregation patterns significantly govern porosity and anisotropic parameters, while the heterogeneity of pore size distribution and anisotropic factors collectively determine the dynamic response of permeability. Experimental investigations employing high-resolution imaging methods have demonstrated that microstructural features (e.g., particle packing orientation and clustering) exert critical control over macroscopic transport properties. Specifically, the coupling effect between pore size distribution heterogeneity and anisotropic coefficients has been found to dictate permeability variations under dynamic flow conditions, highlighting the importance of integrating microscale geometric characteristics into transport modeling [21,22]. The phase field method has demonstrated outstanding performance in simulating complex interfaces and multi-physics coupling. Compared with the pore network model, the lattice Boltzmann method, the level set method, and the volume of fluid method, it features superior interface handling capability, physical consistency, higher stability in dealing with dynamic interfaces, a balance between computational efficiency, and suitability for numerical calculations of multi-physics coupling [23,24,25,26].

Thermo-hydro-mechanical multi-physics coupling studies have achieved remarkable progress in the field of numerical simulation in recent years. This interdisciplinary approach integrates heat transfer, fluid flow, and solid mechanics to characterize the complex interactions within porous media, which is particularly critical for understanding energy extraction, geological storage, and environmental remediation processes [27,28]. Through integrating the multi-field interaction mechanisms of fluid dynamics, heat transfer, and structural mechanics, collaborative analysis of temperature fields, stress fields, and flow fields has been realized. Early studies mainly focused on the establishment of bidirectional coupling models [29,30]. By the late 20th century, with the rapid development of computer technology, numerical calculation methods such as the finite element method (FEM) and finite difference method (FDM) were introduced into thermo-hydro-mechanical coupling research, making it possible to solve complex coupling problems. The research scope has also expanded from simple models to practical engineering structures [31,32]. Subsequently, research has expanded from single-discipline applications to interdisciplinary integration, accompanied by remarkable progress in multiscale modeling and the development of multi-physics coupling solvers [33].

This study uses improved QSGS reconstruction to examine displacement in irregular three-dimensional porous models. Axial anisotropy denotes contrasts among X, Y, and Z, whereas a forward–reverse difference denotes the response to exchanging the inlet and outlet along one axis. End-to-end structural asymmetry describes unequal pore openings and connectivity at the opposing faces of a finite realization. Section 3.1 and Section 3.2 retain the original fixed-time comparisons; Section 3.3 adds a stochastic ensemble and comparisons at matched injected pore volumes and states the limits of these interpretations. The geometries are controlled synthetic structures rather than digital rocks calibrated to a particular reservoir. Recent work on intelligent digital rock physics [34] provides context for stochastic reconstruction and pore-scale multiphase simulation, while studies of coupled non-isothermal seepage [35] motivate the flow-heat component of the present model.

2. Methods

2.1. Physical Model and Hypothesis

The Quartet Structure Generation Set (QSGS) is a probabilistic control-based algorithm specifically designed to generate 2D and 3D porous media models with complex pore structures. The core principle dynamically simulates the formation of the pore-solid structure using the following parameters: distribution probability (determining the spatial distribution of initial solid-phase nuclei), growth probability (controlling the stochastic expansion of solid voxels into unoccupied voxels in the 26 neighborhood directions, with neighborhood direction No. i ranging from 1 to 26), porosity (setting the target pore volume ratio), and probability density (the multiphase material interaction parameter). Solid-phase nuclei were randomly distributed in the voxel domain. The solid phase subsequently grew into unoccupied neighboring voxels, while the remaining voxels constituted the pore phase. Growth continued until the prescribed solid fraction corresponding to the target porosity was reached. Compared with 2D models, 3D QSGS can more accurately represent pore tortuosity, connectivity, and anisotropy by assigning solid-phase growth probabilities to face-, edge-, and vertex-connected neighbors within the 26-direction neighborhood. The schematic diagram of the 26 neighborhood directions in the 3D QSGS is shown in Figure 1.

Figure 1. The 26 neighborhood directions of growth probability in a 3D porous media model generated by QSGS.

The QSGS code was used to control the model size, porosity, solid-phase seed probability, number of solid-phase seeds, and growth probabilities in all directions. The number of solid-phase seeds was set to less than 1500, and the solid-phase seed probability was 0.015.

In this work, the 3D microporous reservoir models were generated on a 50 × 50 × 50 μm voxel grid with a target porosity of 0.4. Solid-phase nuclei (seeds) were first randomly distributed in the grid with the prescribed seed probability. The solid phase then grew iteratively from the seeds: each solid voxel attempted to occupy empty neighboring voxels in the 26 directions shown in Figure 1, and occupation in direction i occurred with the corresponding growth probability di (i = 1, 2, …, 26). Growth ended when the number of solid voxels reached (1 − φ) × N, where φ is the target porosity and N is the total number of voxels. A 3 × 3 × 3 median filter was subsequently applied to smooth the pore-solid interface, and the porosity was recalculated. Finally, the pore phase was extracted by 26-connectivity analysis: only pore domains connected to both the inlet face (z = 1) and the outlet face (z = 50) were retained as the fluid domain, whereas isolated pore domains were converted to a solid. Axial anisotropy was imposed by assigning different growth probabilities to different coordinate directions; for example, d5 = 0.05 was assigned to the positive Z direction and identical prescribed growth probabilities of 0.01 to all remaining directions, including the transverse X- and Y-axis directions. The model possesses both axial anisotropy and directional structural asymmetry because only the growth probability in the positive Z direction was increased. Because QSGS is stochastic and the computational sample is finite, the two opposing end faces can still have different pore openings and connectivity. This end-to-end structural spatial asymmetry is evaluated separately from the between-axis anisotropy. Such process-based stochastic reconstruction offers a controllable complement to image-based digital-rock workflows, as reviewed in recent intelligent digital rock physics studies [34].

The randomly generated reservoir model will be automatically divided into two parts: the fluid domain and the solid domain. During the process of generating the model, isolated pore channels can be automatically screened and eliminated. The schematic diagram of the 3D anisotropic micro-porous media reservoir model produced is shown in Figure 2. Anisotropy is widely present in actual oil reservoirs. Moreover, deep-sea oil reservoirs have characteristics such as complex pore structure, high temperature, and high pressure. Therefore, in order to facilitate research, it is necessary to simplify the established anisotropic porous media reservoir model, which should follow the following conditions:

Figure 2. Schematic diagram of 3D anisotropic micro-porous media reservoir model.

(1)

The reservoir is composed of rock layers and fluids, with pore channels containing only water and heavy oil;

(2)

The fluid in the reservoir is uniform and incompressible;

(3)

The fluid flow in the pore channels of the reservoir is unidirectional;

(4)

The fluid flow in the reservoir follows Darcy’s law;

(5)

The solid phase is homogeneous and isotropic in its material properties and capable of deformation.

The boundary conditions and parameters of 3D anisotropic micro-porous media reservoirs are shown in Table 1. The physical properties of heavy oil and rock formations change with temperature. The physical property change functions of heavy oil and rock layers related to temperature are shown in Table 2.

Table 1. Boundary conditions and parameters of reservoirs [36,37].

Property	Symbol	Value	Unit
Cube reservoir side length	L	50	μm
Target porosity	φ	0.4
Initial temperature of reservoir	Te	360	K
Initial pressure of reservoir	Pe	30	MPa
Reservoir contact angle	θ	90	°
Water injection temperature	Tw	468	K
Water density	ρw	1000	kg/m3
Water viscosity	μw	0.001	Pa·s
Water heat capacity	Cpw	4200	J/(kg·K)
Water thermal conductivity	Kw	0.59	W/(m·K)
Density of heavy oil	ρo	860	kg/m3

Table 2. Temperature-dependent material-property inputs and their literature sources.

Property	Formula	Source
Rock heat capacity (J·kg−1·K−1)	Cps = log10T × 100 + 550	[38]
Rock thermal conductivity (W·m−1·K−1)	ks = −4 × 10−7 × (T + 500)2 + 6.5 × 10−5 × (T + 500) + 2.65	[38,39]
Rock thermal expansion coefficient (K−1)	αs = 5.82 × 10−6 + 4.19 × 10−9 × T	[40]
Heavy oil viscosity (Pa·s)	μo = ${10}^{{10}^{-3.6337\times \left(\log 10\right(T\left)\right)+9.6888}}$	[41,42]
Heavy oil heat capacity (J·kg−1·K−1)	Cpo = (10−5 × (T − 100)2 − 5 × 10−4 × (T − 100) + 1.55) × 103	[43,42]
Heavy oil thermal conductivity (W·m−1·K−1)	ko = 5 × 10−8 × (T − 500)2−10−4 × (T − 500) + 0.15	[42]

2.2. Governing Equation

The flow and temperature fields interact through advection, heat exchange, and temperature-dependent material properties. The temperature field additionally drives the thermoelastic response of the solid. The coupling scope is specified below so that this model is distinguished from a fully coupled poroelastic formulation. Recent non-isothermal seepage analyses [35] provide context for the coupling between temperature and seepage.

The Navier–Stokes equations describe incompressible viscous flow, as shown in Equation (1) [44].

$$\left\{\begin{array}{l}\rho \cdot {\displaystyle \frac{\partial u}{\partial t}}+\rho \cdot (u\cdot \nabla )\cdot u=\nabla \cdot [-P\cdot I+\mu \cdot (\nabla u+{(\nabla u)}^{\mathrm{T}})]+F_{\mathrm{st}}\\ u\cdot \nabla =0\end{array}\right.$$

(1)

where ρ denotes the fluid density (kg·m⁻³), P denotes the pressure (Pa), I denotes the unit vector, μ denotes the fluid viscosity (Pa·s), and F denotes the interfacial tension (N·m⁻³), t denotes the displacement time, and u denotes the fluid velocity (m·s⁻¹).

The Cahn–Hilliard formulation is given in Equation (2) [44].

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial \varphi }{\partial t}}+u\cdot \nabla \varphi =\nabla \cdot {\displaystyle \frac{\mathit{\gamma \lambda }}{{\epsilon }^2}}\cdot \nabla \psi \hfill \\ \psi =-\nabla \cdot {\epsilon }^2\cdot \nabla \varphi +({\varphi }^2-1)\cdot \varphi \hfill \\ \lambda ={\displaystyle \frac{3\mathit{\epsilon \sigma }}{\sqrt{8}}}\hfill \end{array}\right.$$

(2)

where ϕ denotes the phase field variable; ϕ = −1 and ϕ = 1 are, respectively, for heavy oil and water; λ represents mixed energy density (N); ε and ψ separately represent interfacial heaviness (m) and the phase field auxiliary variable; and γ expresses mobility (m³).

The coupling equation of the two-phase field is shown in Equation (3) [44].

$$\left\{\begin{array}{l}{F}_{\mathrm{st}}=({\displaystyle \frac{\lambda }{{\epsilon }^2}}\psi -{\displaystyle \frac{\partial f}{\partial \varphi }})\cdot \nabla \varphi \hfill \\ \rho ={\rho }_{\mathrm{o}}{\displaystyle \frac{1-\varphi }{2}}+{\rho }_{\mathrm{w}}{\displaystyle \frac{1+\varphi }{2}}\hfill \\ \mu ={\mu }_{\mathrm{o}}{\displaystyle \frac{1-\varphi }{2}}+{\mu }_{\mathrm{w}}{\displaystyle \frac{1+\varphi }{2}}\hfill \end{array}\right.$$

(3)

where F_st indicates the surface tension of the fluid.

The seepage equations of Darcy’s law are shown in Equation (4) [45].

$$u=-{\displaystyle \frac{k_a}{\mu }}\nabla P$$

(4)

The heat transfer between solids and fluids in the reservoir is shown in Equation (5) [46]. The heat transfer between fluids follows Equation (6) [47].

$$\left\{\begin{array}{l}\rho C_{\mathrm{p}}{\displaystyle \frac{\partial T}{\partial t}}+\rho C_{\mathrm{p}}u\cdot \nabla T+\nabla q=Q\\ q=-k\nabla T\end{array}\right.$$

(5)

$$\left\{\begin{array}{l}-n\cdot q=\rho \Delta \mathit{Hu}\cdot n\\ \Delta H={\displaystyle \int \begin{array}{l}T\\ T_{\mathrm{ustr}}\end{array}}{C}_p\mathit{dT}+{\int }_{P_{\mathrm{ustr}}}^{P_{\mathrm{A}}}{\displaystyle \frac{1}{\rho }}(1-{\alpha }_{\mathrm{p}})\mathit{dP}\end{array}\right.$$

(6)

where the symbol n is the total static energy rate (W); T_ustr and P_ustr, respectively, indicate upstream temperature and upstream absolute pressure (Pa); P_A is initial rock pressure (Pa); and α_p is the isobaric compression coefficient.

The solid momentum balance and the temperature-dependent thermal strain used in the thermoelastic calculation are given by Equations (7) and (8), respectively [48].

$$\rho {\displaystyle \frac{{\partial }^2{u}_{\mathrm{solid}}}{\partial t^2}}=\nabla \cdot S+F_{\mathrm{V}}$$

(7)

ε_th = α(T) ⋅ (T − T_ref)

(8)

Here, usolid is the solid displacement (m), initially zero; S denotes the stress measure used in the solid balance (Pa); Fv is the body-force density (N·m⁻³); and εth is thermal strain. The coefficient α(T) has units K⁻¹ and Tref is the reference temperature (K).

Absolute permeability is defined from the single-phase volumetric flow rate, bulk cross-sectional area, and imposed pressure drop as follows [49]:

$$\left\{\begin{array}{l}{q}_{\mathit{out}}={\displaystyle \frac{M_{\mathit{out}}}{{\rho }_w}}\hfill \\ u_{\mathit{out}}={\displaystyle \frac{q_{\mathit{out}}}{A}}\hfill \\ \Delta P=P_{\mathit{in}}-P_{\mathit{out}}\hfill \\ k_a={\displaystyle \frac{{\mu }_w{u}_{\mathit{out}}L}{\Delta P}}\hfill \end{array}\right.$$

(9)

Here, Mout is the outlet mass flow rate (kg·s⁻¹), ρw is water density (kg·m⁻³), qout is the volumetric flow rate (m³·s⁻¹), A = L² is the bulk cross-sectional area normal to flow (m²), uout = qout/A is the superficial Darcy velocity (m·s⁻¹), μw is water dynamic viscosity (Pa·s), and ΔP = P_in−P_out is the pressure drop (Pa) over sample length L (m). The bulk area A used for permeability differs from the open pore area used for inlet throughput.

The flow and heat-transfer equations are solved jointly as a non-isothermal two-phase system. The solid calculation is a one-way, temperature-driven thermoelastic response on a fixed mesh: Equations (7) and (8) do not include pore-pressure loading through an effective-stress/Biot term, and solid deformation does not update pore geometry or flow. The solid material is homogeneous and isotropic; structural anisotropy arises from the pore geometry. The model therefore does not represent full two-way poroelastic thermo-hydro-mechanical coupling.

This coupling choice provides a controlled comparison of pore-structure effects and their associated thermal-stress patterns. Its adequacy for pressure-sensitive deformation has not been independently established. The reported stresses are conditional thermoelastic responses, and neglect of pressure loading and deformation feedback is a model limitation rather than evidence that these effects are negligible in real reservoirs. The displacement output is the pore-domain volume average Sw(t) of the COMSOL water-volume-fraction variable pf.Vf2. Section 3.1 and Section 3.2 retain the original Sw indicators at 0.5 s. For the additional ensemble, saturation-based incremental recovery is Rf(t) = [Sw(t) − Sw(0)]/[1 − Sw(0)], referenced to the initialized oil volume. This follows from Vo(t) = Vp[1 − Sw(t)] for fixed pore volume and incompressible phases. Rf is inferred from domain oil-volume decrease; it is not an independently integrated cumulative outlet production measurement. Each direction uses its own initial state, and the correction does not make the initialized phase distributions identical or establish a closed boundary-flux mass balance. The synthetic structures have not been calibrated to the pore-throat distribution, specific surface area, connectivity, or anisotropy of a particular reservoir. Independent thermal and mechanical validation remains a limitation. The material-property sources in Table 2 identify the adopted inputs, rather than validating each coefficient or establishing reservoir-specific calibration.

2.3. Grid Division and Model Validation

The generated voxel model was stored in a binary raw file, in which the values 1 and 0 denote the solid and pore phases, respectively. To import the voxel geometry into COMSOL5.6, a MATLAB R2026a script was used to convert the raw file into a NASTRAN/BDF mesh: for each pore voxel, one hexahedral element (CHEXA) was constructed from its eight corner nodes (GRID). In the conversion code, scale = 1 specified only a unit increment between nodal coordinates. The code did not assign a physical length unit. The BDF mesh was then imported into COMSOL through File > Import > Mesh, and the geometry length unit was set to μm. Consequently, the coordinate range of 0–50 represented a reservoir side length of 50 μm. After the BDF fluid-domain mesh was imported, a cube with the same side length as the voxel model was created in COMSOL. The solid-rock domain was then obtained by a Boolean difference operation that subtracted the imported fluid domain from the cube. The resulting complementary fluid and solid domains were used in the coupled flow, heat-transfer, and solid-mechanics calculations. During this preprocessing sequence, the imported BDF hexahedral mesh was first converted into geometry, and inspection confirmed that there were no damaged or open surfaces requiring repair. After construction of the complementary fluid and solid domains, both domains were remeshed with free tetrahedral elements for the final coupled simulations.

The two-phase-flow implementation was assessed by reproducing the benchmark case reported by Feng et al. [50], including its key geometric, fluid-property, and boundary-condition settings. This reference employs the same class of Navier–Stokes/Cahn–Hilliard formulation and includes a theoretical assessment of capillary filling at different wetting angles. Figure 3 compares the recovery histories at three water-injection velocities of 1 × 10⁻⁴, 3 × 10⁻⁴, and 5 × 10⁻⁴ m·s⁻¹. The recovery trends agree with the reference results, and the reported endpoint differences are below 1%. These matched-case comparisons provide relevant numerical evidence supporting the two-phase displacement implementation under the tested conditions.

Figure 3. Numerical benchmark of oil-recovery histories against the reference results of Feng et al. [50] at three water-injection velocities.

The imported fluid and complementary solid geometries were remeshed with free tetrahedral elements. For the fluid domain, the maximum element growth rate was 1.3, the curvature factor 0.8, and the narrow-region resolution was 0.5. For the solid domain, these settings were 1.3, 0.7, and 0.5, respectively. The transient calculation used relative and absolute tolerances of 0.005 and 0.05. GMRES was used for flow and phase field, and PARDISO for heat transfer and solid mechanics. The mesh study used positive-Z injection at 0.0001 m·s⁻¹ and 468 K, with d5 = 0.05 and all other directional growth probabilities equal to 0.01. Figure 4 shows a substantial change on the coarse meshes, followed by a stable range of the reported 0.5 s saturation-based displacement indicator. Two additional finer meshes, containing 734,556 and 803,465 elements, give values of 0.50548 and 0.50562. Relative to the finest mesh, the selected 510,772-element mesh gives 0.50571, a difference of 0.018%; the largest deviation among the examined plateau meshes is 0.028%. The 57.3% increase in element count from 510,772 to 803,465 therefore provides additional evidence of mesh insensitivity for this global quantity under the tested conditions. The selected count is the combined fluid-and-solid tetrahedral element count.

Figure 4. Mesh refinement of the original 0.5 s water saturation indicator Sw. The ordinate-labeled recovery degree denotes Sw in this comparison.

3. Results and Discussion

Water at 468 K displaces heavy oil in the connected pore domain, with a contact angle of pi/2 and a prescribed inlet velocity of 0.0001 m·s⁻¹. The original comparisons in Section 3.1 and Section 3.2 use the 0.5 s reporting time and stored outputs at 0.01 s intervals. The additional stochastic ensemble in Section 3.3 extends to 1.0 s with 101 stored time levels. Its initialized phase distribution is quantified by the nonzero Sw(0) values in the ensemble analysis, rather than being assumed to have exactly zero water volume.

3.1. Axial Permeability and Fixed-Time Inlet-Reversal Responses

The growth probability in neighborhood direction No. 5 (positive Z) was set to 0.05, while the probabilities in the other directions were set to 0.01. Six displacement configurations were simulated, forming one forward–reverse pair for each coordinate axis. Within each pair, the geometric model and all operating conditions were unchanged; only the inlet and outlet were exchanged between the two opposing faces. Comparisons among the X-, Y-, and Z-axis pairs characterize axial anisotropy, whereas comparisons within a pair characterize the forward–reverse displacement difference. Any within-pair difference is discussed in relation to structural spatial asymmetry between the two end faces rather than being labeled as axial anisotropy. For consistency throughout Section 3.1 and Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, the six configurations are numbered as Case 1 (+X), Case 2 (−X), Case 3 (+Y), Case 4 (−Y), Case 5 (+Z), and Case 6 (−Z). The six boundary configurations are illustrated in Figure 5.

Figure 5. Opposing inlet–outlet directions. Cases 1–6 denote +X, −X, +Y, −Y, +Z, and −Z inlet configurations, respectively; the outlet is on the opposite face.

Figure 6. Absolute permeability for Cases 1–6 (+X, −X, +Y, −Y, +Z, and −Z).

Figure 7. Original time histories for Cases 1–6. The ordinate-labeled recovery degree denotes uncorrected pore-domain water saturation Sw.

Figure 8. Original water saturation indicators Sw at 0.5 s for Cases 1–6 (+X, −X, +Y, −Y, +Z, −Z).

Figure 9. Oil–water distributions for Case 5 (+Z) and Case 6 (−Z) at 0.5 s.

Figure 10. Temperature distributions for Case 5 (+Z) and Case 6 (−Z) at 0.5 s.

Figure 11. Thermoelastic stress distributions for Case 5 (+Z) and Case 6 (−Z) at 0.5 s.

The absolute permeability of the six configurations was calculated using Equation (9) with water as the only pore fluid and a steady-state solver. As defined in Figure 6, Figure 7 and Figure 8, Cases 1 and 2 are the positive- and negative-X configurations, Cases 3 and 4 are the positive- and negative-Y configurations, and Cases 5 and 6 are the positive- and negative-Z configurations. The corresponding permeabilities are 1.2184 × 10⁻¹⁴, 1.2183 × 10⁻¹⁴, 1.0063 × 10⁻¹⁴, 1.0061 × 10⁻¹⁴, 1.6027 × 10⁻¹⁴, and 1.6017 × 10⁻¹⁴ m². Reversing the pressure-driven flow along the same axis changes the permeability by no more than 0.062%, so these small within-axis differences do not constitute axial anisotropy. In contrast, the Z-axis permeability differs from the X- and Y-axis permeabilities by 23.98% and 37.22%, respectively. These between-axis contrasts quantify the axial anisotropy of the model.

Figure 7 and Figure 8 compare the original saturation-based displacement indicators at the same prescribed inlet velocity. At 0.5 s, the positive- and negative-X values are 0.57311 and 0.61742; the corresponding Y values are 0.65185 and 0.64405, and the Z values are 0.50571 and 0.73659. The pairwise relative contrasts are 7.2%, 1.2%, and 31.3%, calculated as the absolute difference divided by the larger value in each pair. These are fixed-time water saturation contrasts, not differences in ultimate recovery or direct measures of axial anisotropy. Because the opposing pore openings differ, equal local inlet velocity can produce unequal volumetric injection. The 31.3% contrast therefore combines throughput and displacement history effects and is not treated as an equal-PV recovery difference.

Figure 9 compares the oil–water distributions for positive-Z and negative-Z injection at 0.5 s. The larger and more concentrated inlet pore openings in Case 6 admit a larger volumetric throughput at fixed local velocity and can also alter the activated pathways. The higher fixed-time saturation indicator in this case cannot separate these contributions. Section 3.3 therefore compares additional realizations at matched injected pore volume; its recovery estimates use the initialized oil volume and are not numerically interchangeable with the fixed-time Sw values reported here.

Figure 10 shows the temperature distributions at 0.5 s for Cases 5 (+Z) and 6 (−Z), with average temperatures of 410.88 and 419.51 K. Figure 11 shows the associated thermoelastic stress distributions. The reported mean and maximum stress magnitudes are 192,260 and 898,413.8 Pa for Case 5, and 205,052 and 1,673,625.3 Pa for Case 6. These statistics describe the temperature-driven solid response. Differences in throughput, pathway activation, and thermal exposure may contribute jointly; this comparison does not isolate their individual effects or include pore-pressure loading.

3.2. Effect of Directional Growth Probability on Displacement

Three synthetic geometries were generated with positive-Z growth probabilities of 0.05, 0.15, and 0.25, while the other 25 directional probabilities remained 0.01. Water was injected in the positive-Z direction in each case. Changing d5 can alter pore morphology, connected volume, pore-throat geometry, and inlet opening. This series therefore examines the response to the imposed growth-probability setting; it does not hold all other geometric descriptors fixed or isolate a pure anisotropy-strength effect.

Figure 12 shows the original water saturation displacement indicators for the three growth-probability settings. At 0.5 s, the values are 0.50571, 0.67742, and 0.72434 for d5 = 0.05, 0.15, and 0.25, respectively. The initially different ordering and subsequent curve crossings demonstrate that the ranking depends on the reporting time. The distributions in Figure 13 are consistent with changes in the swept pathways. The higher 0.5 s value at the larger d5 is consequently reported as an association for these selected structures.

Figure 12. Original histories for the three positive-Z growth-probability settings. The ordinate-labeled recovery degree denotes water saturation Sw.

Figure 13. Oil–water distributions at 0.5 s for d5 = 0.05, 0.15, and 0.25 under positive-Z injection.

Figure 14 and Figure 15 show the temperature and thermoelastic stress distributions at 0.5 s for the selected growth-probability settings. Across d5 = 0.05 to 0.25, the reported average temperature increases from 410.88 to 414.17 K, while average solid stress decreases from 192,260 to 135,575 Pa. These finite-time associations include the combined influence of the generated morphology, end-face openings, and displacement history. They are not a forward–reverse comparison and do not establish a pressure-coupled geomechanical response.

Figure 14. Temperature distributions at 0.5 s for the three positive-Z growth-probability settings.

Figure 15. Thermoelastic stress distributions at 0.5 s for the three positive-Z growth-probability settings.

3.3. Stochastic Ensemble and Comparison at Matched Injected Pore Volume

An additional ensemble comprises five QSGS realizations with seeds 101, 202, 303, 404, and 505 for each of the three designs. Iso sets all 26 growth probabilities to 0.01; SymAniso sets d5 = d6 = 0.05 and the remaining 24 to 0.01; and AsymAniso sets d5 = 0.05, d6 = 0.01, and the remaining 24 to 0.01. Each design uses a 50 × 50 × 50 voxel grid with 1-micrometer voxels, target porosity 0.40, seed probability 0.015, at most 1500 solid seeds, one 3 × 3 × 3 median filter, and the same 26-connectivity screening. Each retained pore domain links the opposing Z faces.

Table 3 summarizes the final connected geometry, permeability variability, and forward Rf variability at 1.0 s for five realizations per design. Final porosity ranges from 0.3142 to 0.3328; group means are 0.3195, 0.3276, and 0.3265 for Iso, SymAniso, and AsymAniso. Mean end-face open-area imbalance is 2.65%, 4.84%, and 16.50% of the total face area. The small porosity spread characterizes this ensemble but does not exclude differences in throat distributions or connectivity. Within a forward–reverse pair, the geometry, total connected porosity, and pore volume are unchanged. Here, φc denotes final connected porosity; Vp is the connected pore volume; |ΔA| is the absolute difference in end-face pore openings divided by the 2500 μm² face area; k+ CV and Rf+ CV are the sample coefficients of variation in forward permeability and Rf at 1.0 s; and reversal is |k+ − k−|/k+ in percent (group median and maximum).

Table 3. Connected geometry, permeability and forward Rf variability at 1.0 s for 15 QSGS realizations (five per design; sample SD; ranges or maxima as indicated).

Group (n = 5)	φc, Mean ± SD	Vp Range (×103 μm3)	|ΔA| (%), Mean (Range)	k+ CV (%)	Reversal (%), Median (Max)	Rf + CV (%)
Iso	0.3195 ± 0.0035	39.28–40.46	2.65 (0.40–8.36)	20.9	0.13 (4.67)	13.5
SymAniso	0.3276 ± 0.0044	40.29–41.60	4.84 (1.16–10.36)	13.9	0.03 (0.44)	6.6
AsymAniso	0.3265 ± 0.0031	40.42–41.45	16.50 (12.76–19.24)	34.4	0.02 (0.13)	8.1

The median relative forward–reverse permeability discrepancy is 0.033%; 14 of 15 realizations are below 1%, while Iso seed 101 has a 4.67% discrepancy. The latter is retained explicitly as a numerical outlier and limits any claim of uniformly reciprocal numerical results. The predominance of small reversal discrepancies supports the comparison of displacement responses in the same geometry.

Injected pore volume is PV(t). Mean Q−/Q+ ratios are 1.039, 1.088, and 1.311 for Iso, SymAniso, and AsymAniso. The limiting history, forward AsymAniso seed 303, reaches 2.949932 PV at 1.0 s (Vp = 40,645 μm³; Qin = 119,900 μm³·s⁻¹). All common comparisons end at 2.90 PV and use interpolation within the available histories; the former 3-PV endpoint retention is excluded. Raw Sw(0) ranges from 0.061533 to 0.115500, so the comparisons use the initial-oil-volume-referenced Rf defined in Section 2.2. At 0.05 PV, seven of 30 directional curves still exceed the necessary ΔSw ≤ PV bound by 0.24–1.60 percentage points in saturation; the maximum discrepancy at the first stored post-zero time is 2.47 points. The stored histories do not establish the cause. Figure 16 retains the early points; 0–0.10 PV is treated as initialization-sensitive, and quantitative directional interpretation is confined to 0.5–2.90 PV. The mean recovery gaps and throughput ratios (Q−/Q+) at matched injected pore volume arepresented in Table 4.

Figure 16. Group-mean saturation-based incremental recovery Rf versus injected pore volume for forward and reverse injection (five realizations per design). The initial value is subtracted, and recovery is referenced to the initialized oil volume. Early points are retained; 0–0.10 PV is initialization-sensitive. All curves are supported through 2.90 PV; quantitative directional comparisons use 0.5–2.90 PV.

Figure 16. Group-mean saturation-based incremental recovery Rf versus injected pore volume for forward and reverse injection (five realizations per design). The initial value is subtracted, and recovery is referenced to the initialized oil volume. Early points are retained; 0–0.10 PV is initialization-sensitive. All curves are supported through 2.90 PV; quantitative directional comparisons use 0.5–2.90 PV.

Table 4. Mean reverse-minus-forward saturation-based incremental-recovery gaps at matched injected pore volume (percentage points), and mean throughput ratio Q−/Q+. Five paired realizations contribute to every entry, with no extrapolation.

Group	Mean Q−/Q+	0.5 PV	1.0 PV	1.5 PV	2.0 PV	2.5 PV	2.90 PV
Iso	1.039	−4.88	−5.31	−5.14	−2.76	−1.86	−1.97
SymAniso	1.088	−6.65	−1.67	−2.10	−1.74	−0.52	−1.32
AsymAniso	1.311	−25.05	−17.08	−15.83	−17.34	−15.76	−14.86

Table 4. Mean reverse-minus-forward saturation-based incremental-recovery gaps at matched injected pore volume (percentage points), and mean throughput ratio Q−/Q+. Five paired realizations contribute to every entry, with no extrapolation.

Group	Mean Q−/Q+	0.5 PV	1.0 PV	1.5 PV	2.0 PV	2.5 PV	2.90 PV
Iso	1.039	−4.88	−5.31	−5.14	−2.76	−1.86	−1.97
SymAniso	1.088	−6.65	−1.67	−2.10	−1.74	−0.52	−1.32
AsymAniso	1.311	−25.05	−17.08	−15.83	−17.34	−15.76	−14.86

At 1 PV, the mean reverse-minus-forward Rf gaps are −5.31, −1.67, and −17.08 percentage points for Iso, SymAniso, and AsymAniso. The AsymAniso mean has a sample SD of 6.20 percentage points and an individual range of −27.33 to −10.49 percentage points. Its gap is negative in all five realizations at every evaluated 0.05-PV increment from 0.5 through 2.90 PV. The Iso and SymAniso results have mixed individual signs at 1 PV. At 2.90 PV, their mean gaps are −1.97 and −1.32 percentage points, compared with −14.86 for AsymAniso. Figure 16 shows the group-mean Rf histories; Table 3 quantifies between-realization variability. In the five AsymAniso realizations, the reverse inlet has the larger open area but lower Rf at matched PV. Pathway selection, breakthrough, and the different elapsed times required to reach a given PV may contribute. The end-face imbalance and the 1-PV gap have Pearson r = −0.814 within the five AsymAniso samples (Figure 17). This is a descriptive association in a small ensemble, not an independent causal test. Because equal PV can correspond to different exposure times in this non-isothermal system, throughput normalization does not equalize the thermal histories. The extended histories are finite-time results rather than ultimate-recovery estimates. Between 0.9 and 1.0 s, the individual Rf increments range from 0.49 to 6.28 percentage points, so a strict plateau is not established for all cases. The consistent equal-PV sign in the tested AsymAniso samples supports the directional comparison over 0.5–2.90 PV.

Figure 17. End-face open-area imbalance versus the reverse-minus-forward Rf gap at 1 PV for all 15 realizations. The dashed fit and Pearson correlation apply only to the five AsymAniso samples (r = −0.814); the other designs are shown for context.

4. Conclusions

The simulations distinguish between axis-based permeability contrasts, fixed-time inlet-reversal responses, and saturation-based incremental recovery at matched injected pore volume. The conclusions are restricted to the tested synthetic structures and operating windows as follows:

(1)

The main geometry has Z-axis permeability contrasts of 23.98% and 37.22% relative to the X and Y values under the reported comparison convention. Reversing flow along a given axis changes the computed permeability by no more than 0.062%. The between-axis contrast and the sign of flow direction therefore describe different properties.

(2)

At 0.5 s, the main model has relative forward–reverse water saturation contrasts of 7.2%, 1.2%, and 31.3% along X, Y, and Z. These fixed-time indicators include unequal injected volume when the open inlet areas differ. Within the tested model, they therefore indicate directional sensitivity to the inlet-facing pore structure rather than a universal recovery gap.

(3)

Increasing d5 from 0.05 to 0.25 is associated with an increase in the selected positive-Z 0.5 s saturation indicator from 0.50571 to 0.72434. This is a combined morphology-and-displacement association in the chosen geometries. Therefore, the stronger directional growth setting, which produces wider and less tortuous Z-oriented channels, accompanies a higher 0.5 s saturation indicator under the fixed orientation.

(4)

In the additional ensemble, saturation-based recovery referenced to the initialized oil volume gives a mean reverse-minus-forward gap of −17.08 percentage points at 1 PV for AsymAniso, with negative gaps in all five realizations over the evaluated 0.5–2.90 PV interval. Iso and SymAniso have smaller means and mixed signs at 1 PV. The observation supports sensitivity to one-sided structural design within this ensemble.