## An Improved Model for Predicting Hydraulic-Fracture-Height Migration

Topics: Hydraulic fracturing
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While 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, at least for nonnaturally fractured media. The authors have revisited the “equilibrium-height problem,” and their theoretical and numerical investigations led to a new model that fully characterizes height evolution amid various formation properties.

## Introduction

Solutions for the equilibrium-height problem have been known since the 1970s, and several models have been developed for calculating hydraulic-fracture height. However, because of the complexity of the algebra involved, the equations used in these early models were overly simplified and gave unreliable results.

The authors developed an improved, mathematically rigorous model that, for the first time, solves the equilibrium height under various formation-property conditions and fluid properties. The authors started from the definition of fracture toughness, incorporated the effects of hydrostatic pressure, and considered nonsymmetric variations of layered formation properties. The model was applied to the classic three-layer problem and then extended up to six layers.

With the new model, the effects of fracture toughness, in-situ stress, fluid density, and their interactive effects on fracture-height growth were investigated. The full height profile with very large top and bottom formation thicknesses showed the ultimate trend of fracture-height migration. Two three-layer pseudoproblems were constructed to create an outer and inner height envelope for any multilayer-formation problem, to assess the potential effects of reservoir-parameter uncertainties on height profile. The occurrence of a second solution pair and its analytical solutions were presented, to avoid misleading results in the 3D models.

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 173335, “An Improved Equilibrium-Height Model for Predicting Hydraulic-Fracture-Height Migration in Multilayer Formations,” by Songxia Liu, SPE, and Peter P. Valkó,SPE, Texas A&M University, prepared for the 2015 SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, USA, 3–5 February. The paper has not been peer reviewed.
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