In this study, we revisit a semi-analytical fracture height growth model and propose a fast-marching procedure to obtain the full height map of a given fracture growing in a multi-layered formation. A full height map offers substantial information about fracture growth, which can help infer fracture tip locations, estimate required pressure buildup, and evaluate operational parameters in a stimulation practice. We provide a detailed derivation of the underlying complex mathematical formulations and develop a procedural solution to construct the equilibrium height map. Our fast-marching solution algorithm can consider numerous interbedded laminations with a negligible increase in computational expenses, and it has been tested in various conceptual cases. Subsequently, the impacts of stress and toughness barriers, lateral landing depth of a horizontal well, and fracturing fluid density on fracture height growth and fracture-mouth pressure are studied to illustrate their associated potential consequences on an effective hydraulic-fracture design. Our results show that the stress barriers contribute more to the containment of fracture height than the toughness barriers. A higher minimum horizontal stress leads to more evident fracture aperture loss of that specific layer as the fracture-mouth pressure drops. Breaking through a relatively tough layer (e.g., ash bed), regardless of the layer thickness, requires substantial pressure buildup. This pressure buildup potentially leads to fracture arrest at the bedding plane and fracture growth diversion toward the bedding plane. Such fracture diversion probably creates a step-over that is detrimental to proppant transport and further fracture-height growth. The farther the horizontal well from tough layers, the lower the required pressure buildup for breaking through these layers, thus minimizing step-overs in the fracture height growth. Also, a lower-density fracturing fluid gives more symmetric fracture geometry. In contrast, a high-density fracturing fluid helps avoid undesirable upward fracture growth toward a possibly overlying shallow aquifer.
Improved hydrocarbon production from usually laminated unconventional resources owes to effective hydraulic fracture propagation. A production-effective hydraulic fracture growing normal to a horizontal well needs to propagate through several bedding planes overlying or underlying the horizontal-drilling landing zone. This is due to the fact that shale layers hold sub-micro scale intrinsic permeability values, and pore fluid is almost locked against diffusion through the interfaces of these layers toward the horizontal lateral. However, the bedding planes between these laminated layers can perform as areal conduits to drain hydrocarbons to a transverse high-permeability pathway; in our case, a cross-cutting hydraulic fracture generated during a stimulation practice. The effectiveness of this drainage depends on fracture height growth; the larger the fracture height, the larger the number of laminated layers penetrated by a transverse hydraulic fracture and the higher the probability of tapping into a production sweet spot.