A Rigorous Hydraulic-Fracture Equilibrium-Height Model for Multilayer Formations
Authors: Liu Songxia , Valkó Peter P. 
Published: August 2017

Summary
Fracture height is a critical input parameter for 2D hydraulic-fracturing-design models, and also an important output result of 3D models. Although many factors may influence fracture-height evolution in multilayer formations, the consensus is that the so-called “equilibrium height belonging to a certain treating pressure” provides an upper limit. However, because of the complexity of the algebra involved, published height models are overly simplified and do not provide reliable results.
We revisited the equilibrium-height problem, started from the definition of the fracture stress-intensity factor (SIF), considered variation of layered formation properties and effects of hydrostatic pressure, and developed a multilayer fracture-equilibrium-height (MFEH) model by use of the programming software Mathematica (2017). The detailed derivation of SIF and work flow of MFEH model are provided.
The model is compared with existing models and software, under the same ideal geology condition. Generally, MShale (2013) calculated smaller height, and FracPro (2015) larger height, than the MFEH model. Most of the difference is attributable to the different interpretation of the “net pressure,” and the solving of the nonlinear equations of SIF as well. In the normally stressed case, they are both acceptable, although MShale is more reliable. The discrepancy is much larger when there is abnormally high or low stress in the adjacent layers of the perforated interval. The effects of formation rock and fluid properties on the fracture-height growth were investigated. Tip jump is caused by low in-situ stress, whereas tip stability is imposed by large fracture toughness and/or large in-situ stress. If the fluid density is ignored, the result regarding which tip will grow into infinity could be totally different. Second and even third and fourth solutions for a three-layer problem were found by Excel experiments and this model, and proved unrealistic; however, they can be avoided in our MFEH model. The full-height map with very-large top- and bottom-formation thicknesses shows the ultimate trend of height-growth map (i.e., when the fracture tip will grow to infinity) and suggests the maximum pressure to be used. To assess the potential effects of reservoir-parameter uncertainties on the height map, two three-layer pseudoproblems were constructed by use of a multilayer formation to create an outer- and inner-height envelope.
The improved MFEH model fully characterizes height evolution amid various formation and fluid properties (fracture toughness, in-situ stress, thickness, and fluid density), and for the first time, rigorously and rapidly solves the equilibrium height. The equilibrium height can be used to provide input data for the 2D model, improve the 3D-model governing equations, determine the net pressure needed to achieve a certain height growth, and suggest the maximum net pressure ensuring no fracture propagation into aquifers. This model may be incorporated into current hydraulic-fracture-propagation simulators to yield more-accurate and -cost-effective hydraulic-fracturing designs.