The shape factor is essential for characterizing matrix-fracture fluid transfer and scaling of spontaneous imbibition in fractured reservoirs. Most existing models assume isotropic permeability, while limited studies on anisotropic cases adopt constant shape factors that inadequately capture the transient behavior of counter-current imbibition. This study proposes a generalized time‑dependent shape factor for anisotropic matrix blocks. Two types of permeability tensors are considered: a diagonal permeability tensor aligned with matrix block boundaries, and a full permeability tensor representing general anisotropic configurations. Accordingly, anisotropic imbibition models were developed with approximate analytical solutions. Shape factors for 2D and 3D boundary conditions were determined using the scaling analysis method. Fine-grid simulations were conducted to extract imbibition recovery curves and validate the derived shape factor at the single-block scale. Field-scale dual-porosity simulations further demonstrate that the proposed model more accurately reproduces fine-grid results compared to conventional models. The results demonstrate that the shape factor is related to matrix block dimensions, boundary conditions, normalized recovery (or average water saturation), and permeability anisotropy. Crucially, the shape factor depends on both the magnitude and directions of principal permeabilities, significantly improving the accuracy of recovery predictions. This work enhances existing dual-porosity numerical models by integrating the proposed shape factor, enabling more reliable simulations of fluid exchange in anisotropic matrix blocks within fractured systems.

Spontaneous counter-current imbibition is a crucial recovery mechanism in water-wet fractured reservoirs, especially in unconventional formations like tight and shale reservoirs. The geometric characteristics of microscale fractures require further clarification regarding their impact on imbibition. In this paper, the numerical simulation method is used to study the influence of fracture aperture, length, density, and relative position between fracture and imbibition open face on the counter-current imbibition process of a matrix block. For fractures perpendicular to the imbibition surface and in contact with water, the embedded discrete fracture model is utilized to simulate the impact of varying fracture apertures on counter-current imbibition. For fractures parallel to the imbibition surface, considering the impact of fracture on the capillary discontinuity of the matrix, the effects of varying fracture lengths and densities on counter-current imbibition are simulated. The results show that when fractures are perpendicular to the imbibition surface and in contact with water, the imbibition rate can be increased, and as the fracture aperture decreases, the imbibition rate first increases and then decreases. On the other hand, fractures parallel to the imbibition surface inhibit the imbibition process, with the imbibition rate decreasing as fracture length or density increases. This paper proposes an empirical shape factor considering the geometric characteristics of fractures, which can effectively characterize the influence of microfractures on matrix block imbibition, thus improving the dual-medium numerical simulation model.

Tight sandstone gas reservoirs are characterized by low single-well productivity, rapid natural decline, and generally low recovery rates, which limit the effectiveness of conventional development strategies. The injection of CO₂ into tight gas reservoirs offers broad prospects for both enhanced gas recovery (EGR) and integrated CO₂ capture, utilization, and storage (CCUS). However, during the depletion and CO₂ injection processes, fracture conductivity, reservoir fluid distribution, and gas–water relative permeability evolve continuously, exhibiting pronounced time-varying behavior. Conventional models have certain limitations in characterizing these time-varying mechanisms, often resulting in significant discrepancies between predicted performance and actual conditions. To address this, we develop a robust and generic model that explicitly incorporates the time-varying mechanisms and stress sensitivity. This model also accounts for the non-equilibrium flash (NEF) behavior of the gas–liquid mixture during CO₂ injection into tight gas reservoirs. The successive substitution iteration method is adopted for non-equilibrium flash calculation, and the Newton–Raphson iteration (NRI) method is used for obtaining a fully implicit solution of the reservoir flow equations. The accuracy and reliability of the proposed model are verified through comparisons with commercial simulator results, and a quantitative comparison with traditional models highlights the key discrepancies caused by neglecting time-varying mechanisms. Parametric analyses further clarify the impacts of time-varying behavior and stress sensitivity on CO₂-EGR performance and CO₂ sequestration efficiency. The results show that the displacement process of CO₂ in tight gas reservoirs can be divided into four characteristic stages. The results also indicate that if the time-varying effects and the stress sensitivity of the reservoir permeability are ignored, the calculations tend to overestimate the true gas production capacity of the reservoir and underestimate the CO₂ sequestration performance.