Uncontrolled vertical propagation of hydraulic fractures (HFs) can compromise extraction efficiency in layered oil/gas reservoirs and potentially induce unintended aquifer contamination and fault reactivation. This study presents a two-dimensional semi-analytical criterion for a finite-length, toughness-dominated HF interacting with an orthogonal stratigraphic interface between dissimilar formations. Utilizing the Fourier transform technique and the Lobatto–Chebyshev collocation method, stress distribution induced by the finite-length, toughness-dominated HF is initially derived. By evaluating critical stress conditions at critical radius of two dissimilar formations, interaction behaviors (crossing, slippage, and opening) can be predicted. The model's predictions are validated through comparisons with existing analytical solutions and published experimental results. The induced horizontal stress components exhibit discontinuity at the interface, differing from that in the homogeneous formations. The elastic property contrast between dissimilar formations induces differing stress intensity factors (SIFs) at the upper and lower HF tips, leading to asymmetric vertical propagation. As for the interaction behaviors, crossing behavior is more probable with larger HF half-lengths, smaller ratios of elastic modulus contrast, and negative interlayer stress differences. The evolution of the critical vertical stress difference ratio \ (= (ₕ - ₇) /₇\) for crossing mode varies depending on HF initiation layer, underscoring the significant influence of elastic modulus contrast on the interaction behavior. When the interlayer stress difference approaches a critical value controlled by the critical radius of plastic zone, it can affect the interaction behavior. The proposed model offers a quantitative framework for predicting HF interaction behavior at stratigraphic interfaces, thereby facilitating the fracturing optimization design in layered formations.
Li et al. (2026) studied this question.