A three-dimensional solution of hydraulic fracture width for wellbore strengthening applications

Determining the width of an induced hydraulic fracture is the first step for applying wellbore strengthening and hydraulic fracturing techniques. However, current 2-D analytical solutions obtained from the plane strain assumption may have large uncertainties when the fracture height is small. To solve this problem, a 3-D finite element method (FEM) is used to model wellbore strengthening and calculate the fracture width. Comparisons show that the 2-D plane strain solution is the asymptote of the 3-D FEM solution. Therefore, the 2-D solution may overestimate the fracture width. This indicates that the 2-D solution may not be applicable in 3-D conditions. Based on the FEM modeling, a new 3-D semi-analytical solution for determining the fracture width is proposed, which accounts for the effects of 3-D fracture dimensions, stress anisotropy and borehole inclination. Compared to the 2-D solution, this new 3-D semi-analytical solution predicts a smaller fracture width. This implies that the 2-D-based old design for wellbore strengthening may overestimate the fracture width, which can be reduced using the proposed 3-D solution. It also allows an easy way to calculate the fracture width in complex geometrical and geological conditions. This solution has been verified against 3-D finite element calculations for field applications.