Fracture height is a critical input parameter for 2D hydraulic fracturing design models, and also an important output result of 3D models. While many factors may influence fracture height evolution in multi-layer formations, the consensus is that the so called “equilibrium height belonging to a certain treating pressure” provides an upper limit, at least for non-naturally fractured media. How to solve the “equilibrium height” problem has been known since the 1970s. However, because of the complexity of the algebra involved, published equations are overly simplified and do not provide reliable results.

We revisited the equilibrium height problem, and our theoretical and numerical investigations led to a new model that fully characterizes height evolution amid various formation properties (fracture toughness, in-situ stress, thickness, etc.). The new model, for the first time, rigorously solves the equilibrium height mathematically. With the help of computer algebra software, we applied a definition of fracture toughness, incorporated the effects of hydrostatic pressure, and considered non-symmetric variations of layered formation properties. Results were determined for the classic 3-layer problem and then were extended to 6 layers.

The effects of reservoir and fluid properties on the fracture height growth were investigated. Tip jump is caused by low in-situ stress; tip stability is imposed by large fracture toughness and/or large in-situ stress. If the fluid density is ignored, the result regarding to which tip will grow into infinity could be totally different. For any multi-layer formation problem, two, 3-layer pseudo problems were constructed to create an outer and inner height envelope, in order to assess the potential effects of reservoir parameter uncertainties on height profile. Second solution pair in the 3-layer problem was investigated numerically and analytically, to avoid misleading results in the 3D models.

This improved model can rapidly (in seconds) and reliably calculate the theoretical maximum equilibrium fracture height in layered formations with various rock mechanical properties. Then, the equilibrium height can be used to (1) provide input data for 2D model, (2) improve 3D model governing equations, (3) determine the net pressure needed to achieve a certain height growth, and (4) obtain the maximum net pressure assuring no fracture invasion into aquifers. This model may be incorporated into current hydraulic fracture simulation software to yield more accurate and cost-effective hydraulic fracturing designs.