Khristianovich-Geertsma-de Klerk problem with stress contrast

TL;DR

This paper extends the Khristianovich-Geertsma-de Klerk model to account for variable confining pressures during hydraulic fracture propagation, providing insights into fracture behavior across different stress layers.

Contribution

It introduces a numerical method for simulating fracture propagation through layers with varying stresses, enhancing the existing P3D model with a new stress contrast parameter.

Findings

Fractures stop in high-stress layers with increased pressure.

Fractures accelerate in low-stress layers with decreased pressure.

A simple dimensionless parameter characterizes stress boundary transitions.

Abstract

The paper contains an extension of the Khristianovich-Geertsma-de Klerk (KGD) model to the case when the confining rock pressure, which closes a hydraulic fracture, varies in the direction of its propagation. The extension is impelled by the need to simulate fracture hampering (acceleration) when it penetrates into a layer with increased (decreased) rock pressure. The paper presents the problem formulation, an efficient numerical method for its solving, examples of fractures propagating through layers with various stresses and general conclusions. It is established that when the fracture enters a layer with increased rock stresses (positive stress contrast), it actually stops. In this case, the fluid particle velocity drops practically to zero and the velocity near the tip oscillates about a small value until the fluid pressure increases to the level of the increased rock pressure. In the opposite case, when the fracture enters a layer with decreased stresses (negative stress contrast), the fracture front accelerates until the fluid pressure drops to the level of the decreased rock pressure. Physical considerations and numerical results imply that the transition through a boundary of layers with different stresses may be characterized by a simple dimensionless parameter. The latter is defined as the ratio of the average difference between the fluid and rock pressures (at the moment of reaching the contact) to the stress jump at the contact. The parameter is applicable to contacts with positive as well as negative contrasts. The model and the method developed are to serve for improvement of the popular pseudo three-dimensional (P3D) model.