Study on Hydrogen Seepage Laws in Tree-Shaped Reservoir Fractures of the Storage Formation of Underground Hydrogen Storage in Depleted Oil and Gas Reservoirs Considering Slip Effects

Abstract

Underground hydrogen storage (UHS) in depleted oil and gas reservoirs is regarded as a highly promising subsurface option due to its large storage capacity. In such reservoirs, the pore structure provides the primary space for hydrogen storage and governs matrix flow and diffusion. Tree-shaped fracture networks generated by hydraulic fracturing or cycling injection–production typically exhibit much higher transmissivity and serve as the dominant pathways. In this study, the geometry of multilevel branching fractures was parameterized, and two classes of tree-shaped fracture configurations were constructed, including point–line-type (PLTSF) and disc-shaped (DSTSF) networks. Analytical models were developed to evaluate the equivalent permeability of tree-shaped fracture networks with either elliptical or rectangular cross-sections. The Klinkenberg slip correction and a gas-type factor associated with molecular kinetic diameter were incorporated. The apparent equivalent permeability of hydrogen (k_app,H2) was quantified and compared with those of nitrogen and methane under identical conditions. The main findings were as follows: (1) the fracture width ratio (β) was identified as the primary factor controlling network conductivity, while the height ratio (α) amplified or attenuated this effect at a given β; (2) as the main-fracture aspect ratio, the branching order (n) or branching angle (θ) increased, the rectangular cross-sections were more favorable for maintaining higher permeability compared to the elliptical cross-section; (3) under typical operating pressures of 5–30 MPa, the apparent permeability of hydrogen was approximately 2–9% higher than that of methane and nitrogen; and (4) by introducing the fracture volume fraction, the REV-scale equivalent-permeability expression was derived for fractured rock masses containing tree-shaped fracture networks. The proposed framework provides a theoretical basis and parametric support for quantifying fracture flow capacity for UHS in depleted reservoirs.

1. Introduction

To achieve global carbon neutrality, hydrogen—an energy carrier with high specific energy and low carbon emissions [1]—has been increasingly regarded as a pivotal vector for deep decarbonization of energy systems [2,3,4]. With the large-scale deployment of hydrogen for power peak shaving, seasonal energy storage, and industrial decarbonization, the demand for storage solutions that are efficient, safe, and scalable has continued to increase [5,6,7]. Among the available options, UHS has emerged as a key enabler of large-scale hydrogen utilization because it offers large capacity, long storage duration, low cost, and high safety [8,9]. Compared with above-ground storage systems, UHS generally exhibits superior economic performance and scalability [10,11]. Although salt-cavern hydrogen storage is relatively mature, its applicability is constrained by geological conditions and the spatial distribution of salt resources [12]. By contrast, depleted oil and gas reservoirs—featuring wide availability, substantial storage potential, intact caprock, and well-established surface and wellbore infrastructure—have been considered attractive candidates for constructing large-scale UHS facilities [3,13]. Notably, long-term reservoir development commonly involves hydraulic fracturing and repeated injection–production cycles, which can promote the formation of hierarchical tree-shaped fracture networks (TSFNs). These TSFNs typically comprise a main-fracture framework with progressively developed branching fractures [14,15]. Previous studies have indicated that the fracture conductivity of TSFNs is often much higher than that of the porous matrix and can strongly control gas injectivity, deliverability, and pressure propagation [5,16]. Therefore, accurate characterization of the flow parameters of TSFNs is essential for evaluating the injection–production capacity of UHS in depleted oil and gas reservoirs.

Focusing on injectivity and deliverability for UHS in depleted oil and gas reservoirs, existing studies have generally agreed on one point. Hydrogen permeability in porous media is a fundamental input parameter [17,18]. It governs injectivity, deliverability, and cyclic efficiency [19,20,21]. Rezaei et al. [22] conducted unsteady-state drainage experiments on two sandstones and one carbonate and systematically examined the effects of pressure, brine salinity, and lithology on H₂–brine relative permeability. Their results were benchmarked against CH₄ and N₂ cases, and they emphasized that H₂-specific two-phase flow functions should be established under reservoir conditions to reduce uncertainties in injection–production forecasting. Under cyclic operation, Lysyy et al. [23] investigated hysteresis in hydrogen relative permeability and pointed out that directly adopting empirical curves derived from other gases or systems may introduce systematic bias. Boon et al. [24] used X-ray CT to visualize core-scale hydrogen distributions in heterogeneous sandstones and inversely derived relative permeability and capillary pressure curves. Al-Yaseri et al. [25] performed pressurized NMR coreflooding experiments and compared the initial and residual gas saturations and trapping behaviors of H₂ and N₂, showing that distinct trapping responses can still occur even under similar capillary-number conditions. At the mechanistic-parameter level, Hashemi et al. [26] quantified the contact angle in the H₂–brine–sandstone system using a high-pressure captive-bubble method, indicating an overall water-wet condition and providing key boundary conditions for numerical simulations. In addition, steady-state co-injection coreflooding tests of H₂ and brine have been conducted in carbonates and other materials, and the Klinkenberg correction has been applied to gas-flow measurements to obtain slip-corrected permeability parameters for engineering-scale simulations [27,28,29]. On the numerical side, Huang et al. [30] developed a compositional simulation model for UHS in depleted reservoirs and simulated injection–shut-in–production cycles on heterogeneous grids. Their results demonstrated that permeability and two-phase functions strongly control hydrogen recoverability and loss assessment. Moreover, reservoir-scale simulations based on experimentally measured hysteretic relative permeability have indicated that neglecting hysteresis, or adopting curves from other systems, can substantially amplify uncertainties in injection–production predictions [31,32]. More broadly, a central challenge is to convert complex fracture architectures into permeability representations that are practical for engineering forecasts and injection–production simulations. [33,34,35].

In depleted oil and gas reservoirs used for UHS, long-term reservoir development—particularly hydraulic fracturing stimulation and repeated injection–production perturbations—often causes the fracture system to deviate from an idealized single planar conduit [36,37,38]. Instead, it tends to evolve into a complex fractured system comprising a main-fracture backbone, multilevel branching fractures, and interconnected units. Such a system can be conceptualized as a TSFN and commonly serves as the primary flow pathway for injection and production in engineering practice [19,20]. Based on microseismic fracture imaging in the Barnett Shale, Fisher et al. [39] showed that post-fracturing propagation is markedly complex, often involving multi-azimuth growth and the activation of natural fractures, thereby forming a networked flow-conducting structure rather than a simple biwing fracture. Warpinski et al. [40] further emphasized that the extent of connectivity and the effective conductivity of the fracture network directly control well deliverability and pressure-depletion propagation in ultra-low-permeability reservoirs. The stimulated reservoir volume (SRV) concept proposed by Mayerhofer et al. [41] similarly suggests that the effective stimulated volume created by hydraulic fracturing corresponds to a composite fracture network, in which network geometry and connectivity are key determinants of fluid mobilization. At the mechanistic level, Cipolla et al. [42] reported that interactions among hydraulic fractures, natural fractures, bedding, and reservoir heterogeneities can induce branching, turning, and coordinated multi-fracture growth, thereby promoting the formation and evolution of a trunk–branch network architecture.

For UHS, the dominant role of fracture conduits during injection and production has been directly demonstrated in numerical studies of depleted fractured gas reservoirs. Zamehrian et al. [43] showed, through UHS simulations of naturally fractured gas reservoirs, that fractures can markedly accelerate hydrogen production and improve both recovery and produced-gas purity. Using a full-cycle UHS numerical model for a depleted fractured gas reservoir, Wang et al. [44] similarly found that fracture properties exert critical control over injection–production responses. In addition, Li et al. [45] indicated, based on a mixed-kinetics analysis of hydrogen storage in depleted gas reservoirs, that fractures may enhance sweep efficiency and flow connectivity at early times, but can intensify gas mixing at later times. Such mixing can, in turn, affect produced-gas purity and operational strategies. Overall, the tree-shaped fracture network system that forms and continuously evolves during the long-term development of depleted oil and gas reservoirs constitutes a dominant pathway for flow conduction and pressure transmission during hydrogen storage. Therefore, it should be explicitly accounted for in subsequent injectivity and deliverability modeling and engineering design [33,34,35].

Characterizing complex fracture-network architectures and their flow-conducting capacity is a critical step in translating geological fracture information into flow parameters that are usable for injection–production forecasting [46,47,48]. In general, structural attributes—including strike, dip, length, spacing, aperture, and connectivity—are first extracted from multi-source observations (e.g., outcrops, cores, image logs, and seismic data). Discrete fracture network (DFN) models are then constructed to represent the geometry of the fractured system [16,49]. Dershowitz et al. [50] summarized advances in statistical modeling and flow analysis, emphasizing that representing network structure through statistical geometry forms the basis for parametric modeling of complex fractured media. For permeability representation, when fractures are sufficiently dense and a representative elementary volume (REV) exists, the equivalent porous medium (EPM) concept can be adopted to upscale the fracture network into an anisotropic permeability tensor. Long et al. [51] provided criteria for when a fractured medium can be treated as a continuum and for the existence of an REV, noting that connectivity and scale separation are essential prerequisites. The fracture-tensor approach proposed by Oda et al. [52] further links fracture orientation, density, and geometric parameters to the permeability tensor, thereby establishing an analytical bridge between fracture-network structure and permeability.

When fracture networks are highly heterogeneous and flow is strongly controlled by connectivity, a representative elementary volume (REV) is often difficult to define robustly. In such cases, discrete representations are generally more reliable. Berkowitz et al. [53] pointed out that flow and transport in fractured media are highly sensitive to network topology and connectivity, and that a single equivalent parameter may mask critical channeling effects. Cacas et al. [54] demonstrated, through calibration and validation of stochastic DFN models, how small-scale hydraulic and tracer data can be used to invert and constrain large-scale parameters such as network equivalent permeability. During upscaling from network-scale descriptions to continuum-equivalent parameters, Ref. [55] proposed an approach to extract continuum-equivalent parameters directly from DFN flow and tracer responses. Hadgu et al. [56] compared DFN and equivalent-continuum models for flow and transport prediction and noted that applicability limits and error sources should be assessed cautiously in a scenario-dependent manner. For engineering simulation, the permeability representation of fracture networks also depends on the governing law for single-fracture conductivity and the numerical discretization strategy. Witherspoon et al. [57] validated the applicability of the parallel-plate cubic law under variable stress-induced closure, and Zimmerman et al. [58] further discussed the validity range of cubic-law parameterization and possible correction directions in view of realistic geometric deviations such as roughness. In terms of reservoir-simulation implementation, the discrete fracture model proposed by Karimi-Fard et al. [59] enabled explicit treatment of complex fracture geometries within general-purpose simulators. Moinfar et al. [60] adopted an embedded representation of fractures to reduce meshing difficulty and improve computational efficiency for large-scale fractured systems, which has become a mainstream option for engineering-scale evaluation of equivalent fracture-network conductivity. Overall, the approaches above provide a comprehensive framework that links fracture-network geometry to permeability representation. However, most existing permeability formulations still implicitly assume classical continuum boundary conditions and single-phase viscous flow, and additional corrections may be required for micro- to nano-scale fractures or gas-specific transport regimes. In particular, the no-slip boundary condition at fluid–wall interfaces is widely adopted, making it difficult to explain the pronounced deviations in near-wall velocity observed in microscale fractures [61]. For low-density, small-molecule gases such as hydrogen, the molecular mean free path in nano- to microscale fractures can become comparable to the characteristic fracture size. The resulting rarefaction effects, including slip and transition flow and Knudsen diffusion, can enhance hydrogen transport capacity well beyond that predicted by classical no-slip models [31,32].

However, previous studies neglected the slip-flow property of hydrogen in the pores and its effect on the fluid flow in the fracture. This study proposed a parameterized method to evaluate the injection and production performance for UHS in depleted oil and gas reservoirs. The multilevel geometric models have been constructed for the tree-shaped fracture networks that control injection and production. Then, the equivalent-permeability expressions have been derived for representative cross-sections considering the Klinkenberg correction together with gas-property effects. Finally, the method was extended to the representative elementary volume (REV) scale, aiming to improve the injection–production operating conditions of UHS in depleted oil and gas reservoirs.

2. Equivalent-Permeability Model for Hydrogen Flow in a Tree-Shaped Fracture Network

2.1. Conceptual Model and Fundamental Assumptions for Tree-Shaped Fractures in UHS in Depleted Oil and Gas Reservoirs

In depleted oil and gas reservoirs, long-term production and hydraulic fracturing commonly generate a high-conductivity fracture system. This system consists of a main-fracture backbone and multilevel branching fractures. As illustrated in Figure 1, after the reservoir enters the UHS stage, the main fracture near the wellbore and its branches serve as the dominant pathways for hydrogen injection and withdrawal. Flow through the low-permeability matrix is comparatively minor. Accordingly, the primary flow-conducting system is idealized as a tree-shaped fracture network embedded in a low-permeability matrix. Based on this conceptual model, an equivalent-permeability model is established for single-phase hydrogen flow within the fracture network.

To facilitate the development of the equivalent-permeability model for the tree-shaped fracture network and the single-phase flow simulations, the following assumptions are adopted:

• The gas is treated as a compressible Newtonian fluid. During one injection–production operation, pressure variations are assumed to be small compared with the mean pore pressure. An equivalent mean pressure is therefore used. Steady-state Darcy flow is assumed. The flow is laminar at the fracture scale.
• Under the in situ stress conditions, the fracture geometry is assumed to remain stable. Pronounced aperture closure and shear slip induced by pressure fluctuations are neglected. Flow–solid coupling effects are not considered.
• Matrix permeability is assumed to be much lower than fracture permeability. The representative elementary volume scale equivalent permeability is therefore controlled primarily by the tree-shaped fracture network.
• The tree-shaped fracture network is assumed to include one main fracture and multiple levels of branching fractures. In the plane, branching is represented by pairwise symmetric bifurcations at a prescribed branching angle. Geometric parameters are scaled proportionally across branching levels. The scaling is characterized by the length ratio, width ratio, and height ratio.
• Consistent with classical hydraulic-fracturing models, the main-fracture cross-section is approximated as either elliptical or rectangular.

Accordingly, this study is positioned as a conceptual and parametric analysis.

2.2. Geometric Parameterization of the Tree-Shaped Fracture Network

Under these assumptions, the geometry of the tree-shaped fractures is parameterized to support the subsequent derivation of the equivalent-permeability model. As shown in Figure 2, a representative tree-shaped fracture network in a depleted oil and gas reservoir is considered. The length, height, and aperture (width) of the main fracture are denoted by l₀, h₀, and w₀, respectively. The branching angle between two adjacent fracture levels is denoted by θ. The branching order is denoted by n, where n = 0 indicates that only the main fracture is present. The number of branches generated at each level is denoted by m. For the j-th level fracture (j = 1, 2, …, n), the length, height, and aperture are denoted by l_j, h_j, and w_j, respectively.

To describe the geometric scaling relationship between two adjacent fracture levels, three dimensionless ratios are defined, including the length ratio γ, height ratio α, and width ratio β. For the j-th level fracture (j = 1, 2, …, n), these ratios are given by

$$\gamma ={\displaystyle \frac{l_j}{l_{j-1}}},\alpha ={\displaystyle \frac{h_j}{h_{j-1}}},\beta ={\displaystyle \frac{w_j}{w_{j-1}}},j=1,2,\dots ,n$$

(1)

The width-to-height ratio of the main fracture, λ, is defined as

$$\lambda ={\displaystyle \frac{w_0}{h_0}}$$

(2)

2.2.1. Total Fracture Volume and Total Fracture Length

For an elliptical cross-section, the volume of the j-th level fracture is given by

$$V_{j,e}=m^j\cdot {\displaystyle \frac{\pi }{4}}{w}_j{h}_j{l}_j$$

(3)

Using Equation (1), Equation (3) can be rewritten as

$$w_j={\beta }^j{w}_0,h_j={\alpha }^j{h}_0,l_j={\gamma }^j{l}_0$$

(4)

The total volume of the tree-shaped fracture network with an elliptical cross-section is obtained by summing over all branching levels:

$$V_e=\sum _{j=0}^n{V}_{j,e}={\displaystyle \frac{\pi }{4}}{w}_0{h}_0{l}_0{\displaystyle \frac{1-{(m\gamma \alpha \beta )}^{n+1}}{1-m\gamma \alpha \beta }}$$

(5)

If the cross-section is approximated as rectangular, the volume of the j-th level fracture is given by

$$V_{j,s}=w_j{h}_j{l}_j$$

(6)

Accordingly, the total volume for a rectangular cross-section is obtained by summing over all branching levels:

$$V_s=w_0{h}_0{l}_0{\displaystyle \frac{1-{(m\gamma \alpha \beta )}^{n+1}}{1-m\gamma \alpha \beta }}$$

(7)

The total geometric length of the tree-shaped fracture network is obtained by summing the fracture lengths over all levels:

$$L_f=\sum _{j=0}^n{l}_j=l_0{\displaystyle \frac{1-{\gamma }^{n+1}}{1-\gamma }}$$

(8)

2.2.2. Equivalent Cross-Sectional Area and Characteristic Length

If the tree-shaped fracture network is represented as a single tortuous fracture conduit, an equivalent cross-sectional area is defined as the ratio of the total fracture volume to the total fracture length.

For an elliptical cross-section, the equivalent area is written as

$$A_e={\displaystyle \frac{V_e}{L_f}}={\displaystyle \frac{\pi }{4}}{w}_0{h}_0{\displaystyle \frac{1-\gamma }{1-{\gamma }^{n+1}}}{\displaystyle \frac{1-{(m\gamma \alpha \beta )}^{n+1}}{1-m\gamma \alpha \beta }}$$

(9)

For a rectangular cross-section, the equivalent area is written as

$$A_s={\displaystyle \frac{V_s}{L_f}}=w_0{h}_0{\displaystyle \frac{1-\gamma }{1-{\gamma }^{n+1}}}{\displaystyle \frac{1-{(m\gamma \alpha \beta )}^{n+1}}{1-m\gamma \alpha \beta }}$$

(10)

To distinguish the point–line tree-shaped fracture (PLTSF) and the disc-shaped tree-shaped fracture (DSTSF), as shown in Figure 3, a characteristic flow-path length, L_0n, is introduced. For a PLTSF that develops approximately along a straight line, the characteristic length is given by

$$L_{0n}^{(L)}=l_0\left[1+{\displaystyle \frac{\gamma (1-{\gamma }^n)}{1-\gamma }}\cos \theta \right]$$

(11)

For a DSTSF whose terminal branches are distributed along an arc, the azimuth of the j-th level fracture relative to the main fracture is denoted by φ_j. The disc-shaped fractures are assumed to be uniformly distributed within a total spreading angle Φ. The fracture length at level j is given by l_j = γ^jl₀. The characteristic length along the main flow direction is defined as the sum of the projected fracture lengths over all levels:

$$L_{0n}^{(\mathrm{C})}=l_0+\sum _{j=1}^n{l}_j\cos {\phi }_j=l_0+l_0\sum _{j=1}^n{\gamma }^j\cos {\phi }_j$$

(12)

Here, φ_j denotes the included angle between the j-th level fracture and the main fracture. If the fractures are uniformly distributed within the spreading angle Φ, then

$${\phi }_j={\phi }_0+j\Delta \phi ,\Delta \phi ={\displaystyle \frac{\mathsf{\Phi }}{n}}$$

(13)

By substituting Equation (13) into Equation (12), the characteristic length is obtained in a summation form. With γ, α, β, m, n, θ, and Φ specified, the tree-shaped fracture network is represented as an equivalent single fracture conduit with the characteristic length $L_{0n}^{(C)}$ (for PLTSF or DSTSF) and the equivalent cross-sectional area A_e or A_s. These geometric quantities are then used to support the development of the equivalent-permeability model for single-phase hydrogen flow in the fracture network.

2.3. Equivalent Permeability of Tree-Shaped Fractures Without Slip

2.3.1. Governing Equation for Hydrogen Seepage in a Single-Level Fracture

Considering the local Reynolds number was typically small under conventional operating pressures for UHS in depleted oil and gas reservoirs (approximately 5–30 MPa), the flow was treated as steady and laminar. Weak compressibility effects were assumed. For the j-th level fracture, the volumetric flow rate was expressed in a Hagen–Poiseuille-type form when an elliptical cross-section was considered. Although hydrogen was compressible, the pressure drop across one fracture level during a single injection and production operation was assumed to be small compared with the mean pore pressure. Thus, the equivalent mean pressure, p_m, was introduced. Hydrogen is then treated as quasi-incompressible with viscosity μ_g(P_m, T). This allows a Hagen–Poiseuille-type expression to be used [62,63]:

$$Q_{j,e}={\displaystyle \frac{\pi w_j^3{h}_j^3}{64{\mu }_g(w_j^2+h_j^2)}}{\displaystyle \frac{\Delta p_{j,e}}{l_j}}$$

(14)

where μ_g denotes the hydrogen viscosity, and Δp_j,e denotes the pressure difference between the two ends of the j-th level fracture.

If a rectangular cross-section is considered, the volumetric flow rate is approximated as

$$Q_{j,s}={\displaystyle \frac{w_j^3{h}_j}{12{\mu }_g}}{\displaystyle \frac{\Delta p_{j,s}}{l_j}}$$

(15)

In the tree-shaped network, all branching fractures at the same level are subject to the same pressure drop. Their flow rates are additive. Therefore, for the j-th level fractures, the following relation is obtained:

$$Q_g=m^j{Q}_{j,e}=m^j{Q}_{j,s}$$

(16)

where Q_g denotes the total flow rate passing through the cross-section of the main fracture.

Accordingly, the pressure drop across the j-th level fracture is expressed as follows.

$$\Delta p_{j,e}={\displaystyle \frac{64{\mu }_g{Q}_g}{\pi }}{\displaystyle \frac{(w_j^2+h_j^2)l_j}{m^j{w}_j^3{h}_j^3}}$$

(17)

For the elliptical cross-section,

$$\Delta p_{j,s}={\displaystyle \frac{12{\mu }_g{Q}_g{l}_j}{m^j{w}_j^3{h}_j}}$$

(18)

By substituting the geometric scaling relationships in Equation (1) and rearranging, the following forms are obtained.

$$\Delta p_{j,e}={\displaystyle \frac{64{\mu }_g{Q}_g{l}_0}{\pi w_0^3{h}_0}}{\displaystyle \frac{{\gamma }^j\left({\lambda }^2{\beta }^{2j}+{\alpha }^{2j}\right)}{m^j{(\alpha \beta )}^{3j}}}$$

(19)

For the rectangular cross-section case,

$$\Delta p_{j,s}={\displaystyle \frac{12{\mu }_g{Q}_g{l}_0}{w_0^3{h}_0}}{\displaystyle \frac{{\gamma }^j}{m^j{\beta }^{3j}{\alpha }^j}}$$

(20)

If additional local losses at branching junctions are neglected, the total pressure drop across the tree-shaped fracture network is approximated as the sum of the pressure drops over all levels:

$$\Delta p_e=\sum _{j=0}^n\Delta p_{j,e},\Delta p_s=\sum _{j=0}^n\Delta p_{j,s}$$

(21)

A dimensionless geometric coefficient is defined as

$${\mathsf{\Psi }}_e(\gamma ,\alpha ,\beta ,m,n,\lambda )=\sum _{j=0}^n{\displaystyle \frac{{\gamma }^j\left({\lambda }^2{\beta }^{2j}+{\alpha }^{2j}\right)}{m^j{(\alpha \beta )}^{3j}}}$$

(22)

and

$${\mathsf{\Psi }}_s(\gamma ,\alpha ,\beta ,m,n)=\sum _{j=0}^n{\displaystyle \frac{{\gamma }^j}{m^j{\beta }^{3j}{\alpha }^j}}$$

(23)

So that Equations (19) and (20) are written in a more compact form:

$$\Delta p_e={\displaystyle \frac{64{\mu }_g{Q}_g{l}_0}{\pi w_0^3{h}_0}}{\mathsf{\Psi }}_e(\gamma ,\alpha ,\beta ,m,n,\lambda )$$

(24)

$$\Delta p_s={\displaystyle \frac{12{\mu }_g{Q}_g{l}_0}{w_0^3{h}_0}}{\mathsf{\Psi }}_s(\gamma ,\alpha ,\beta ,m,n)$$

(25)

2.3.2. Equivalent Permeability of the Tree-Shaped Fracture Network

Taking a representative case with fracture apertures of 1–3 mm, temperatures of 40–60 °C, and pressures of 5–30 MPa, the Knudsen number of hydrogen reaches 10⁻³–10⁻². This range indicates a regime approaching slip flow, and slip effects may therefore be non-negligible. The Knudsen number is defined as K_n = λ/L_c, where λ is the molecular mean free path and L_c is the characteristic length scale of the fracture aperture [64]. To isolate the influence of fracture geometry on flow capacity, an intrinsic equivalent permeability, K_eq, is derived under the continuum-flow assumption without slip in this section. The slip correction is incorporated in Section 2.4. The tree-shaped fracture network is represented as an equivalent “total fracture conduit” with a characteristic length L_0n and an equivalent cross-sectional area A (either A_e or A_s). Darcy’s law is assumed to hold

$$Q_g={\displaystyle \frac{K_{eq}A}{{\mu }_g}}{\displaystyle \frac{\Delta p}{L_{on}}}$$

(26)

where K_eq is the equivalent permeability without gas slippage, A is the equivalent cross-sectional area, and Δp is the total pressure drop. From Equations (22) and (23), the total pressure drop is expressed in terms of the dimensionless geometric functions Ψe and Ψs. By substituting Equation (22) or Equation (23) into Equation (26) and setting Δp = Δpe or Δps, the general forms of equivalent permeability are obtained.

For a rectangular cross-section, the equivalent permeability is given by

$$K_e={\displaystyle \frac{\pi w_0^3{h}_0}{64A_e}}{\displaystyle \frac{L_{0n}}{l_0}}{\displaystyle \frac{1}{{\mathsf{\Psi }}_e(\gamma ,\alpha ,\beta ,m,n,\lambda )}}$$

(27)

The general equivalent permeability for the tree-shaped fracture network with a rectangular cross-section is given by

$$K_s={\displaystyle \frac{w_0^3{h}_0}{12A_s}}{\displaystyle \frac{L_{0n}}{l_0}}{\displaystyle \frac{1}{{\mathsf{\Psi }}_s(\gamma ,\alpha ,\beta ,m,n)}}$$

(28)

The required expressions for the equivalent area and the characteristic length are provided in Section 2.2. For different tree-shaped fracture configurations, the corresponding equivalent permeability is obtained by using the appropriate L_0n in Equations (27) and (28).

For a point–line tree-shaped fracture, the characteristic length is given by Equation (11):

$$L_{0n}^{(L)}=l_0\left[1+{\displaystyle \frac{\gamma (1-{\gamma }^n)}{1-\gamma }}\cos \theta \right]$$

(29)

By substituting Equation (29) into Equations (27) and (28), the equivalent permeabilities of the point–line configuration are obtained for the elliptical and rectangular cross-sections, denoted by $K_e^{(L)}$ and $K_s^{(L)}$, respectively.

For a disc-shaped tree-shaped fracture, the characteristic length L_0n is evaluated using a recursive relation. First, a dimensionless function is defined as

$$L_{0n}^{(C)}=l_0\mathsf{\Omega }(\gamma ,n,\mathsf{\Phi })$$

(30)

where Ω depends only on the geometric parameters. By substituting L_0n into Equations (27) and (28), the equivalent permeabilities of the disc-shaped configuration are obtained for the elliptical and rectangular cross-sections, denoted by $K_e^{(L)}$ and $K_s^{(L)}$, respectively.

For subsequent comparisons, the intrinsic permeability of a single main fracture without branches is defined. For an elliptical cross-section,

$$K_{e0}={\displaystyle \frac{w_0^2}{16(1+{\lambda }^2)}}$$

(31)

For a rectangular cross-section,

$$K_{s0}={\displaystyle \frac{w_0^2}{12}}$$

(32)

Accordingly, a dimensionless equivalent permeability is defined as

$$K_e^{+}={\displaystyle \frac{K_e}{K_{e0}}},K_s^{+}={\displaystyle \frac{K_s}{K_{s0}}}$$

(33)

When K_e and K_s are taken as the equivalent permeabilities of the point–line or disc-shaped configuration, the amplification or attenuation of flow capacity relative to a single main fracture is quantified.

2.4. Hydrogen Slip Effects and Gas-Type Correction

Knudsen number is defined as K_n = λ/L_c, where λ is the molecular mean free path and Lc is a characteristic aperture length scale of the fracture. As K_n increases, the flow departs from the classical no-slip continuum regime and slip effects become progressively more important. In this study, slip effects are represented using a first-order Klinkenberg-type correction. This correction provides a practical approximation over the pressure range of interest.

Under UHS in depleted oil and gas reservoirs, fractures are typically at the millimetre scale, whereas the kinetic diameter of hydrogen is much smaller than that of conventional subsurface gases such as methane. In addition, the mean free path within the operating pressure range can be relatively large. As a result, pronounced slips can occur near fracture walls [65,66,67]. Therefore, a continuum Darcy model alone can underestimate the equivalent permeability of hydrogen, and the Klinkenberg slip correction is introduced [29,68,69].

For a given tree-shaped fracture geometry, the intrinsic equivalent permeability without slip is denoted by K_eq (for example, the intrinsic values derived in Section 2.3). Here, p_m is the arithmetic mean of the pressures at the two ends of the corresponding fracture level. At the mean pore pressure p_m, the apparent equivalent permeability of hydrogen is written as

$$k_{g,H_2}=K_{\mathrm{eq}}\left(1+{\displaystyle \frac{b_{H_2}}{p_m}}\right)$$

(34)

where b_H2 denotes the Klinkenberg coefficient for hydrogen. It accounts for the combined effects of gas type, fracture characteristic length scale, and temperature. For a fixed fracture geometry, previous empirical and theoretical results indicate that b is approximately inversely proportional to the kinetic diameter d_g and increases with temperature T. Accordingly, a gas-type correction factor is defined as

$$b_g=b_{\mathrm{ref}}{\eta }_g,{\eta }_g={\displaystyle \frac{d_{\mathrm{ref}}}{d_g}}{\left({\displaystyle \frac{T_{\mathrm{ref}}}{T}}\right)}^{1/2}$$

(35)

At ambient conditions, the kinetic diameter of hydrogen is approximately 0.29 nm, whereas that of methane is approximately 0.38 nm. This yields η_H2 ≈ 1.3. Therefore, at the same pressure, hydrogen exhibits stronger slip effects and a larger increase in apparent permeability.

By combining the relations above, the equivalent permeability of hydrogen in the tree-shaped fractures is written in a unified form as

$$k_{g,H_2}(\gamma ,\alpha ,\beta ,m,n,\lambda ,p_m,T)=K_{\mathrm{eq}}(\gamma ,\alpha ,\beta ,m,n,\lambda )\left[1+{\displaystyle \frac{b_{\mathrm{ref}}{\eta }_{H_2}(T)}{p_m}}\right]$$

(36)

In the numerical examples, a representative value of b_ref is selected based on typical measurements from depleted gas reservoirs. Sensitivity analyses with respect to p_m and geometric parameters are then performed to quantify the slip-induced increase in hydrogen permeability for different fracture morphologies.

2.5. Upscaling to the REV Scale for Depleted Oil and Gas Reservoirs Containing Tree-Shaped Fractures

In practical parameter evaluation for UHS, the quantity of interest is the representative elementary volume scale equivalent permeability, k_REV, rather than the fracture-scale permeability, k_g,H2 [70]. Within a given REV, the tree-shaped fractures and the low-permeability matrix are assumed to conduct flow in parallel. Serial pressure drops are neglected [71,72]. Under these assumptions, the block-scale equivalent permeability is approximated as

$$k_{REV}={\displaystyle \frac{V_f{k}_{g,H_2}+\left(V_{REV}-V_f\right)k_m}{V_{REV}}}$$

(37)

where φ_f denotes the fracture volume fraction and k_m denotes the matrix permeability (typically much smaller than k_g,H2). When k_g,H2 ≫ k_m, Equation (37) is simplified to

$$k_{\mathrm{REV}}={\varphi }_f{k}_{g,H_2}+(1-{\varphi }_f)k_m$$

(38)

where φ_f is calculated from the total fracture volume V_f and the REV V_REV:

$${\varphi }_f={\displaystyle \frac{V_f}{V_{\mathrm{REV}}}}$$

(39)

Under UHS conditions in depleted oil and gas reservoirs, k_g,H2 ≫ k_m is commonly satisfied. In this case, Equation (38) is further simplified to

$$k_{\mathrm{REV}}\approx {\varphi }_f{k}_{g,H_2}$$

(40)

3. Results and Discussion

3.1. Validation of the Analytical Equivalent-Permeability Model Against 3D Numerical Simulations

Three-dimensional simulations were performed in COMSOL (Multiphysics 6.3) Multiphysics. Steady, laminar, single-phase flow in the fracture network was solved using the Navier–Stokes equations under the assumptions adopted in Section 2.3. Hydrogen was treated as a Newtonian fluid, and a quasi-incompressible approximation was applied using the equivalent mean pressure, p_m. The gas viscosity was set to μ_g(p_m, T). A pressure-drop boundary condition was imposed between the inlet and outlet. After convergence, the volumetric flow rate Q_g was obtained. The numerical equivalent permeability was then inferred from Darcy’s law using the same definition as Equation (26). Slip effects were incorporated later using the analytical Klinkenberg correction in Section 2.4. The unstructured mesh models were generated, with local refinement near fracture walls and branching junctions. Grid independence was checked using progressively refined meshes. The solution was regarded as converged when COMSOL’s default convergence criteria were satisfied, and the monitored Q_g became stable.

To evaluate the applicability and accuracy of the analytical equivalent-permeability model developed in Section 2, a strictly corresponding three-dimensional numerical model was constructed under the same assumptions. Analytical predictions were compared with numerically inferred results. The baseline geometry was defined by l₀ = 10 mm, h₀ = 5 mm, and w₀ = 1 mm, together with n = 3, m = 2, and θ = 30°. Based on this baseline, four representative case groups were designed by varying the length ratio γ, width ratio β, height ratio α, cross-sectional shape, and spatial deployment pattern. Groups I and II corresponded to disc-shaped tree-shaped fractures with elliptical and rectangular cross-sections, respectively. Groups III and IV were used to compare point–line and disc-shaped configurations under rectangular cross-sections. Typical combinations of γ, β, and α (0.6, 0.7, and 0.8) were adopted. In Groups III and IV, the height ratio was fixed at α = 1.0 to highlight morphological differences between the point–line and disc-shaped configurations. The detailed geometries and parameter settings are summarized in

Table 1. Geometries and parameter configurations for the comparison cases.

Group	Geometry Type	Cross-Section	Main-Fracture Length l0 (mm)	Main-Fracture Width w0 (mm)	Main-Fracture Height h0 (mm)	Length Ratio γ	Width Ratio β	Height Ratio α	Branching Order n	Branches per Level m	Branching Angle θ (°)
I	DSTSF	Elliptical	10	1.0	5.0	0.6, 0.7, 0.8	0.6, 0.7, 0.8	0.6, 0.7, 0.8	3	2	30
II	DSTSF	Rectangular	10	1.0	5.0	0.6, 0.7, 0.8	0.6, 0.7, 0.8	0.6, 0.7, 0.8	3	2	30
III	PLTSF	Rectangular	10	1.0	5.0	0.6, 0.7, 0.8	0.6, 0.7, 0.8	1.0	3	2	30
IV	DSTSF	Rectangular	10	1.0	5.0	0.6, 0.7, 0.8	0.6, 0.7, 0.8	1.0	3	2	30

. Under the prescribed pressure drop and boundary conditions, the steady-state volumetric flow rate was obtained from the numerical model, and the numerical equivalent permeability was inferred via Darcy’s law. In parallel, the theoretical equivalent permeability was calculated using the expressions derived in Section 2. The relative error was defined as

$$\delta ={\displaystyle \frac{|K_{\mathrm{eq}}^{\mathrm{num}}-K_{\mathrm{eq}}^{\mathrm{th}}|}{K_{\mathrm{eq}}^{\mathrm{th}}}}\times 100\%$$

(41)

As shown in Figure 4, a representative three-dimensional velocity field was obtained for the disc-shaped configuration with an elliptical cross-section. As shown in Figure 5, the corresponding result was obtained for the disc-shaped configuration with a rectangular cross-section under typical conditions. Along the main-fracture direction, the velocity exhibited an approximately parabolic profile. High-velocity regions were concentrated near the main-fracture centerline. A pronounced contraction–expansion effect was observed at branching junctions. As the branching level increased, velocities in the branch fractures decreased progressively, while streamlines remained well connected. Compared with the elliptical case, a more uniform velocity distribution across the width direction was observed for the rectangular case. This behavior indicated that, under the same volume and length constraints, a rectangular cross-section provided a larger effective flow area. This trend was consistent with the analytical prediction regarding cross-sectional shape effects on equivalent permeability.

As shown in Figure 6, the numerical and analytical equivalent permeabilities agreed closely for all 12 cases. Within each group, the equivalent permeability increased monotonically with increasing γ, β, and α. Across groups, higher equivalent permeabilities were obtained for rectangular cross-sections than for elliptical cross-sections. In addition, higher flow-conducting capacity was generally obtained for disc-shaped configurations than for point–line configurations. These trends were consistent with the geometric amplification mechanisms embedded in the analytical expressions in Section 2. The relative error δ remained within 3–5% for all cases, without systematic overestimation or underestimation. For the disc-shaped groups with elliptical and rectangular cross-sections, the relative error was typically around 3%. For the point–line and disc-shaped comparison groups, slightly larger fluctuations were observed, but the error remained within an engineering-acceptable range.

Taken together, these results demonstrated that, under steady-state laminar flow with a no-slip boundary condition, the proposed analytical model predicted the flow-conducting capacity of tree-shaped fracture networks with good accuracy across different cross-sectional shapes and spatial configurations. The three-dimensional Computational Fluid Dynamics simulations verified the equivalent-permeability formulation and indicated that the macroscopic permeability of complex tree-shaped fracture networks can be characterized using a limited set of geometric parameters (γ, β, α, n, m, and θ). This validation provides a basis for the subsequent sensitivity analyses of geometric parameters and the hydrogen slip correction in Section 3.2, Section 3.3 and Section 3.4.

$$\delta ={\displaystyle \frac{|K_{\mathrm{eq}}^{\mathrm{num}}-K_{\mathrm{eq}}^{\mathrm{th}}|}{K_{\mathrm{eq}}^{\mathrm{th}}}}\times 100\%$$

(42)

3.2. Effects of Geometric Parameters on the Equivalent Permeability of Tree-Shaped Fractures

3.2.1. Coupled Effects of the Width Ratio β and Height Ratio α

Based on Equations (27) and (28) and the baseline cases in Table 1, the main-fracture geometry was set to l₀ = 10 mm, h₀ = 5 mm, w₀ = 1.0 mm. The branching order was set to n = 3, the number of branches per level was set to m = 2, and the branching angle was set to θ = 30°. The length ratio was varied within γ = 0.3–0.7. The width ratio was set to β = 0.4, 0.6, and 0.8. The height ratio α was varied in the subsequent dimensionless analysis to examine coupled cross-sectional effects on the equivalent permeability of tree-shaped fractures.

As shown in Figure 7, for rectangular cross-sections, K_eq decreased monotonically with increasing γ for both the point–line configuration and the disc-shaped configuration. This trend was consistent with increased viscous resistance along long flow paths [73,74]. At a fixed γ, increasing β markedly increased K_eq. For example, for the point–line configuration at γ = 0.3, increasing β from 0.4 to 0.8 increased K_eq from approximately 1.4 × 10⁻⁸ m² to 2.7 × 10⁻⁸ m². Over the same range, K_eq for the disc-shaped configuration increased from approximately 6.0 × 10⁻⁹ m² to 2.5 × 10⁻⁸ m², indicating a stronger enhancement. As γ increased, the separation between curves for different β values decreased. This behavior suggested that the sensitivity to β was strongest when fractures were relatively short. Overall, the results indicated that β controlled the flow-conducting capacity for rectangular tree-shaped fractures, and the disc-shaped configuration responded more strongly to β than the point–line configuration.

To remove the dimensional influence of the main-fracture permeability, the dimensionless equivalent permeability defined in Equation (33) was used to examine the coupled effects of β and α, as shown in Figure 8 and Figure 9. As shown in Figure 8, for elliptical cross-sections with γ = 0.6 and fixed branching parameters (n = 3, m = 2, θ = 30°), the dimensionless equivalent permeability increased monotonically with both β and α. A threshold-like behavior was observed. When β ≤ 0.5 or α ≤ 0.3, the dimensionless equivalent permeability remained close to zero, indicating weak connectivity and a limited effective flow area. When β increased to 0.6–0.9 and α exceeded approximately 0.4, the dimensionless equivalent permeability increased rapidly. At β = 0.9, values on the order of 0.3–0.45 were obtained for the elliptical-cross-section cases. Curves for λ = 0.1, 0.3, and 0.5 were nearly parallel. Smaller λ yielded consistently higher values, indicating that λ primarily shifted the overall magnitude of the coupled β–α response.

As shown in Figure 9, similar coupled behaviors were observed for rectangular cross-sections. In Figure 9a, with α fixed at 0.6, 0.7, and 0.8, the dimensionless equivalent permeability remained close to zero when β < 0.5. A pronounced nonlinear increase occurred as β increased from 0.6 to 0.9. Curves for different α values were nearly parallel and shifted upward with increasing α, indicating that β remained the dominant control. In Figure 9b, with β fixed at 0.6, 0.7, and 0.8, the dimensionless equivalent permeability increased approximately linearly with α. The effect of α became more apparent at larger β, but the gain remained smaller than that produced by increasing β. Across both cross-sectional shapes, β controlled the primary flow area, whereas α provided a secondary modulation by strengthening fracture continuity in the height direction. Only when both parameters exceeded a threshold (approximately β ≥ 0.6 and α ≥ 0.4) did the system transition from a low-permeability regime to a high-permeability regime.

Overall, β controlled the magnitude of the equivalent permeability in tree-shaped fracture networks, whereas α amplified or attenuated conductivity at a given β. Cross-sectional shape (elliptical compared with rectangular) and the main-fracture width-to-height ratio λ shifted the overall permeability level but did not change the fundamental trend of the coupled β–α response. These findings provided geometric constraints for hydraulic-fracturing design and indicated that the flow capacity of tree-shaped fractures under depleted-reservoir UHS conditions could be improved by increasing fracture width and selecting an appropriate cross-sectional shape.

3.2.2. Effect of the Main-Fracture Aspect Ratio λ on the Equivalent Permeability

To quantify the influence of the main-fracture aspect ratio on flow capacity, the baseline geometric parameters were kept consistent with Section 3.2.1, and only λ was varied. Specifically, λ was varied from 0.1 to 1.0. The width ratio was fixed at β = 0.7, and the height ratio was set to α = 0.6, 0.7, and 0.8. For the elliptical-cross-section case, the dimensionless equivalent permeability $K_e^{+}$ was calculated.

As shown in Figure 10, $K_e^{+}$ decreased monotonically with increasing λ for all three height ratios. When λ increased from 0.1 to 1.0, the curves decreased by approximately 40–60%. This trend indicated that a larger λ increased the overall viscous resistance of the network and reduced the equivalent permeability. At a given λ, larger α yielded higher $K_e^{+}$. For example, at λ ≈ 0.5, $K_e^{+}$ for α = 0.8 was higher than that for α = 0.6. The curves for different α values were approximately parallel, suggesting that α mainly shifted the permeability level by changing the effective flow area, whereas λ primarily controlled the along-path viscous resistance.

Overall, the main-fracture aspect ratio λ mainly governed the equivalent permeability level by changing the characteristic flow-path length of the tree-shaped fracture network. In addition, λ, together with the cross-sectional shape (elliptical compared with rectangular), controlled the conductivity contrast between the two cross-section models. From an engineering perspective, to increase the equivalent permeability while maintaining a prescribed branch-distribution extent, a smaller λ was preferred. In this case, higher-order branches were shortened to a moderate extent, and viscous losses along the flow path were reduced. Excessively large values of λ were not recommended because overly slender main-fracture cross-sections increased flow resistance and reduced conductive performance.

3.2.3. Effects of the Branching Order n and Branching Angle θ

With the parameters used in the preceding analysis kept at their baseline values, only the branching order n and branching angle θ were varied. Their effects on the dimensionless equivalent permeability $K_e^{+}$ of rectangular-cross-section tree-shaped fractures were examined, as shown in Figure 11.

As shown in Figure 11a, $K_e^{+}$ decreased monotonically with increasing n for β = 0.6, 0.7, and 0.8. When n increased from 1 to 5, the overall reduction was approximately 70–80% for all three cases. This trend indicated that, when local cross-sectional parameters were fixed, a higher branching order increased the total network length and the complexity of flow paths. As a result, viscous losses along the flow paths were amplified. Additional losses associated with flow partitioning at branching junctions were also increased. Consequently, the equivalent permeability was reduced. At a given n, a larger β produced higher $K_e^{+}$. For example, at n = 3, the value for β = 0.8 was higher than that for β = 0.6. This result suggested that increasing fracture width partially offset the permeability reduction caused by a higher branching order, although the overall decreasing trend with n remained unchanged.

As shown in Figure 11b, $K_e^{+}$ decreased monotonically with increasing θ for α = 0.6, 0.7, and 0.8. When θ increased from 10° to 80°, a more pronounced decline was observed at larger angles. For α = 0.8, $K_e^{+}$ decreased from around 0.7 to approximately 0.25, corresponding to a reduction of more than one half. At a given θ, larger α produced higher $K_e^{+}$. A slightly steeper decrease with θ was also observed at larger α. This result indicated higher sensitivity to the branching angle when fracture height increased. The physical interpretation was consistent across cases. Smaller branching angles kept branch fractures more aligned with the main-fracture direction. Streamline bending was limited, and local expansion–contraction losses were smaller. As θ increased, branch directions deviated further from the main flow direction. Flow paths became more tortuous, and the effective hydraulic length increased. As a result, the macroscopic permeability decreased rapidly. Taken together, Figure 11a,b indicate that increasing n and θ weakened flow-conducting capacity primarily by increasing flow-path length and tortuosity.

3.3. Comparison of Equivalent Permeability Between Different Cross-Sections

3.3.1. Effect of the Main-Fracture Width-to-Height Ratio λ

As shown in Figure 12, the equivalent-permeability ratio K_e/K_s decreased approximately linearly with increasing main-fracture width-to-height ratio λ when elliptical and rectangular cross-sections were compared. Similar trends were obtained for γ = 0.6, 0.7, and 0.8. When λ increased from 0.1 to 1.0, K_e/K_s decreased from about 0.70 to about 0.30. This corresponded to an overall reduction of approximately 55–60%. The result indicated that the conductivity disadvantage of the elliptical cross-section became more pronounced as the main fracture became slender. At a given λ, a slightly lower K_e/K_s was obtained at larger γ. This behavior suggested that longer, higher-order branches strengthened the contrast between the two cross-sections. However, the differences among the three λ curves remained small, typically within about 0.01–0.02. Therefore, λ dominated the variation in K_e/K_s, whereas γ played a secondary role.

3.3.2. Effects of the Width Ratio β and Height Ratio α

As shown in Figure 13a, K_e/K_s depended strongly on the width ratio β for λ = 0.1, 0.3, and 0.5. When β ≤ 0.4, the three curves nearly overlapped and remained at K_e/K_s ≈ 0.75. This result indicated a small difference between elliptical and rectangular cross-sections under relatively narrow-fracture conditions. When β > 0.4, K_e/K_s decreased markedly. A stronger decrease was observed at larger λ. For example, at λ = 0.3, K_e/K_s decreased from approximately 0.75 to approximately 0.63. At λ = 0.5, it decreased from approximately 0.75 to below 0.50. By contrast, the curve for λ = 0.1 remained nearly flat, with only a slight decline. These results indicated that, for a slender main fracture and a larger fracture width, the rectangular cross-section maintained higher equivalent permeability than the elliptical cross-section. In addition, the elliptical cross-section showed higher sensitivity to width increase. As shown in Figure 13b, K_e/K_s increased monotonically with the height ratio α for all λ values. An S-shaped trend was observed, with a gentle increase at small α, a rapid rise at intermediate α, and a plateau at large α. When α < 0.3, K_e/K_s remained within 0.05–0.15, indicating a much lower conductivity for the elliptical cross-section. When α increased from 0.3 to approximately 0.6, K_e/K_s increased rapidly to 0.4–0.6. With further increase to α ≈ 0.9, K_e/K_s approached 0.7–0.8, and the difference between the two cross-sections became small. Curves for different λ values were approximately parallel, and a slight downward shift was observed at larger λ. This behavior indicated that a slender main fracture increased the overall conductivity disadvantage of the elliptical cross-section across the full α range, while the qualitative response to α remained unchanged. Taken together, Figure 13a,b indicate that the equivalent permeabilities of elliptical and rectangular cross-sections were similar when the fracture width was small or the fracture height was large. In contrast, when fractures were wide and the main fracture was slender (large β and large λ), a clearer conductivity advantage was obtained for the rectangular cross-section, and a stronger reduction in K_e/K_s was observed for the elliptical cross-section.

3.3.3. Effect of the Branching Order n

As shown in Figure 14, the equivalent-permeability ratio K_e/K_s generally decreased with increasing branching order n for different main-fracture width-to-height ratios λ. A more pronounced decrease was observed at larger λ. When λ = 0.1, K_e/K_s remained nearly constant at approximately 0.74 and showed little sensitivity to n. In contrast, when λ = 0.5, K_e/K_s decreased from approximately 0.57 to 0.39 as n increased. This result indicated that the conductivity disadvantage of the elliptical cross-section was amplified when the main fracture was slender and the branching order was higher. From an engineering perspective, when the main fracture was slender and a high branching order was involved, a rectangular cross-section was preferred to maintain a higher equivalent permeability.

3.4. Gas Type and Slip Effects

To quantify the influence of gas type on the equivalent permeability of tree-shaped fractures and its pressure sensitivity, a comparative analysis was performed using the representative PLTSF geometry and typical operating conditions. The fracture geometry was kept identical for all gases. The intrinsic equivalent permeability was fixed at K_eq = 2.0 × 10⁻⁸ m². This value lies within the commonly reported range of 10⁻⁸–10⁻⁷ m² for millimetre-scale conductive fractures in depleted gas reservoirs.

When gas slippage was considered, the apparent equivalent permeability of gas g, k_app,g, was expressed using a Klinkenberg-type relation as

$$k_{\mathrm{app},g}=K_{\mathrm{eq}}\left(1+{\displaystyle \frac{b_g}{p_m}}\right)$$

(43)

where p_m was the mean pore pressure. Values of p_m = 5, 10, 15, 20, 25, and 30 MPa were used to cover typical operating pressures for depleted oil and gas reservoirs. Here, bg denoted the Klinkenberg coefficient of gas g and represented the magnitude of slip-induced amplification.

To account for differences among gas types, methane was used as the reference gas. A gas-type correction factor η_g based on the molecular kinetic diameter d_g was introduced as

$${\eta }_g={\displaystyle \frac{d_{{\mathrm{CH}}_4}}{d_g}},b_g={\eta }_g{b}_{\mathrm{ref}}$$

(44)

where b_ref = b_CH4 = 2.0 MPa was used as the reference Klinkenberg coefficient for methane. Using kinetic diameters reported in the literature, d_H2 = 0.289 nm, d_CH4 = 0.380 nm, and d_N2 = 0.364 nm, the correction factors were obtained as η_H2 ≈ 1.31, η_CH4 = 1.00, and η_N2 ≈ 1.04. The corresponding Klinkenberg coefficients were b_H2 ≈ 2.63 MPa, b_CH4 = 2.00 MPa, and b_N2 ≈ 2.09 MPa.

Based on these parameters, the apparent equivalent permeabilities were calculated, as shown in Figure 15a. As p_m increased from 5 MPa to 30 MPa, k_app,g decreased monotonically for all three gases. This trend reflected the progressive weakening of slip effects at higher pressures. The apparent permeability of hydrogen decreased from 3.05 × 10⁻⁸ m² to 2.18 × 10⁻⁸ m². The apparent permeability of methane decreased from 2.80 × 10⁻⁸ m² to 2.13 × 10⁻⁸ m². The apparent permeability of nitrogen decreased from 2.84 × 10⁻⁸ m² to 2.14 × 10⁻⁸ m². Over the full pressure range, the ordering $k_{\mathrm{app},{\mathrm{H}}_2}>k_{\mathrm{app},{\mathrm{N}}_2}>k_{\mathrm{app},{\mathrm{CH}}_4}$ was obtained. This result indicated that hydrogen exhibited the strongest slip-induced amplification under the same fracture geometry. Quantitatively, at p_m = 5 MPa, hydrogen was approximately 9% higher than methane and about 8% higher than nitrogen. At p_m = 30 MPa, this advantage decreased to approximately 2%, indicating that inter-gas differences became much smaller at higher pressures. The methane and nitrogen curves remained very close, with differences around 1%, consistent with their similar molecular scales.

To further quantify the permeability amplification attributable to gas type, a gas-type correction factor was evaluated, as shown in Figure 15b:

$$R_{{\mathrm{H}}_2/{\mathrm{CH}}_4}={\displaystyle \frac{k_{\mathrm{app},{\mathrm{H}}_2}}{k_{\mathrm{app},{\mathrm{CH}}_4}}},R_{{\mathrm{N}}_2/{\mathrm{CH}}_4}={\displaystyle \frac{k_{\mathrm{app},{\mathrm{N}}_2}}{k_{\mathrm{app},{\mathrm{CH}}_4}}}$$

(45)

The ratio R_H2/CH4 decreased from approximately 1.09 at 5 MPa to approximately 1.02 at 30 MPa. This trend indicated that the permeability advantage of hydrogen weakened with increasing pressure. Over the pressure range considered, the amplification remained within a modest band. In contrast, R_N2/CH4remained close to unity, varying only between about 1.01 and 1.00. This result indicated nearly equivalent slip-induced amplification for nitrogen and methane.

Overall, for a fixed tree-shaped fracture geometry, gas-type effects on equivalent permeability were smaller than the changes introduced by geometric parameters. Hydrogen still showed a modest permeability amplification relative to methane and nitrogen. Because R_H2/CH4 varied weakly with pressure, an approximately constant correction factor was considered adequate for engineering use. This approach allowed methane or nitrogen seepage-test results at different pore pressures to be extrapolated to hydrogen conditions with reasonable accuracy, providing a simplified gas-type correction strategy for UHS evaluation and design in depleted oil and gas reservoirs.

3.5. Limitations and Implications for Depleted-Reservoir UHS

To support proper interpretation and practical use of the results, the main assumptions and limitations of this study are summarized below.

(1)

This study focused on the fracture-dominated, pressure-driven flow, in which case, the matrix advective flow is typically much smaller than fracture flow and treated as a low-permeability boundary. However, hydrogen diffusion into the matrix can be non-negligible in some reservoirs. This behavior depends on pore structure and residence time. Capturing this effect requires coupling the present model with a fracture–matrix mass-transfer description.

(2)

Steady-state, single-phase gas flow is assumed to enable the analytical derivations and to quantify the flow capacity of tree-shaped fracture networks. Field injection and production are transient. Local two-phase effects (for example, hydrogen–brine) can also occur during pressure cycling. Therefore, the present results are best interpreted as a parametric baseline for fracture-network flow capacity. Extensions to transient and multiphase flow remain needed.

(3)

The comparison among hydrogen, nitrogen, and methane is made using apparent equivalent permeability. In practice, injection and production also depend on viscosity, density, and real-gas behavior, near-wellbore conditions, and operational constraints. These factors should be considered when permeability-based trends are translated into field-scale performance metrics.

(4)

Elliptical and rectangular cross-sections are used to derive permeability relations. Natural and induced fractures often exhibit roughness and variable aperture. These features can cause deviations from idealized conductivity relations and can modify near-wall transport. In engineering applications, such effects may be handled by correction factors or by calibration against laboratory and field data.

(5)

Fracture apertures and network geometry are treated as prescribed and time-invariant. Under cyclic injection and withdrawal, changes in effective stress can induce stress-sensitive aperture evolution and alter conductivity. Coupling the framework with stress-dependent aperture models would improve long-term applicability.

(6)

The REV-scale extension relies on statistical descriptors inferred from field observations (e.g., fracture volume fraction and network parameters). Sensitivity and uncertainty analyses, together with calibration using laboratory tests and field monitoring data, are recommended for engineering use.

(7)

Branch fractures can have sub-millimetre apertures. The Knudsen number can then increase, and slip-to-transition behavior can occur. In that regime, a first-order Klinkenberg correction can be insufficient. More comprehensive models may be required, such as the Beskok–Karniadakis–Civan model or dusty-gas formulations. The present framework adopts the first-order correction as an engineering approximation over the operating pressure range considered.

4. Conclusions

This paper proposed an equivalent-permeability model for the H₂ in the tree-shaped fracture networks, including the point–line type (PLTSF) and the disc-shaped type (DSTSF).

Slip effect of hydrogen was accounted for using a Klinkenberg-type correction. A REV-scale extension was then established by introducing the fracture volume fraction. The following conclusions can be achieved:

• The analytical predictions agreed well with three-dimensional numerical simulations under the sameconditions. The relative errors were generally within 3–5%, indicating the reliability of the proposed formulation.
• The width ratio β was the primary parameter controlling the flow capacity of tree-shaped fracture networks. The height ratio α modulated this effect at a given β. In contrast, increasing γ, λ₀, n, and θ reduced the equivalent permeability, because longer and more tortuous flow paths were produced.
• Under identical geometric parameters, the flow capacity of DSTSF was higher than that of PLTSF. Rectangular cross-sections consistently yielded higher equivalent permeability than elliptical cross-sections. The conductivity disadvantage of the elliptical cross-section became more pronounced for main fractures with a more extreme aspect ratio (λ), together with higher branching order, larger width ratios, and smaller height ratios.
• Under the same conditions, the apparent permeability of hydrogen was approximately 2–9% higher than that of methane and nitrogen. The difference was weakened as pressure increased, and inter-gas differences remained modest over the investigated range. Therefore, when direct hydrogen measurements were limited, a near-constant correction factor could be used as a practical approximation for converting methane- or nitrogen-based permeability estimates to hydrogen conditions.
• At the REV scale, the bulk permeability was governed mainly by fracture-network geometry, scale, and connectivity, followed by the influence of gas type under depleted-reservoir underground hydrogen storage conditions.

Future work should incorporate stress-dependent aperture evolution and hydrogen–brine two-phase flow during cyclic operation. Calibration of the fracture volume fraction, network parameters, and gas-type correction factors is also needed, using core-scale experiments and field monitoring data, together with uncertainty analysis. These developments will support integration of the proposed framework into reservoir simulation and site-selection workflows and improve its engineering applicability.