Rock Mechanics as a Multidisciplinary Science

Proceedings of the 32nd U.S. Symposium

Jean-Claude Roegiers

• Publisher: CRC Press
• Year: 2020
• Print ISBN:
  9789061911944
• eBook ISBN:
  9781000151404
• Topic: Tecnologia e ingegneria
• Subtopic: Ingegneria civile

Papers in the proceedings of the 32nd U.S. Symposium on Rock Mechanics were solicited to address the theme of 'Rock Mechanics as a Multidisciplinary Science'. The major goal was to assemble scientists and practitioners from various fields with interrelated interests in rock mechanics to share their common problems and approaches.

The proceedings include three papers related to a special session on 'Lunar Rock Mechanics', as well as 121 technical papers covering areas such as: field observations, in-situ stresses, instrumentation/measurement techniques, fracturing, rock properties, dynamics/seismicity, modelling, laboratory testing, discontinuities/fluid flow, design, wellbore stability, and analysis.

1. In-situ stresses

Hydraulic fracture stress measurement in rocks with stress-dependent Young’s Moduli

Yarlong Wang

Department of Earth Sciences, University of Waterloo, Ont., Canada (Presently: Esso Resources Limited of Canada, N.W., Calgary, Alb., Canada)

Maurice B.Dusseault

Department of Earth Sciences, University of Waterloo, Ont., Canada

Abstract

Stress measurement interpretation using the hydraulic fracture method explicitly assumes that the rocks are linear elastic materials. Because this is seldom an acceptable assumption in high porosity or fractured materials, we present a non-linear elastic model that more closely describes real rock behavior. The analytical model demonstrates that fracture initiation pressure (FIP) depends on the material stiffness, which is stress-dependent. Also, we will show that FIP depends on the Poisson’s ratio of the non-linear material. Focusing only on Poisson’s ratio, we demonstrate that, compared to a linear elastic model, a higher FIP is expected for rocks with a Poisson’s ratio greater than certain value, and a lower FIP is expected for rocks with a Poisson’s ratio less than the value. Our calculated results demonstrate that the conventional interpretation of FIP for stress measurements may be significantly in error, particularly for high porosity strata. The new equations developed may help in interpretation of fracturing data.

1 INTRODUCTION

Hubbert and Willis [1957] suggested that, by pressuring a drilled borehole, the equation

$$p_b=3{\sigma }_{hmin}-{\sigma }_{hmax}+T_0$$

(1)

can be used to estimate in situ stress (σ_hmax). Although this has been widely accepted in petroleum engineering and geomechanics, many unsatisfactory results are known in practice. Assuming hydrostatic loading, Wang and Dusseault [1991a,b] show that no information of in situ stress can be obtained from fracture breakdown analyses in a borehole surrounded by plastic yielded material. Such a conclusion has also been extended to a simple non-hydrostatic case [Wang, 1991]. For relatively hard rocks, only a trivial plastic zone may be induced during borehole excavation and drilling, but aspects of non-linear behavior may be nonetheless dominant [Santarelli et al., 1986]. Thus, in this paper, we investigate the effect of a stress-dependent Young’s modulus on the stress distributions around a wellbore, and analyze the consequences on in situ stress measurements.

Haimson and Fairhurst [1969] conducted experimental confining stress determination, interpreting results with the linear elastic model. They observed that predicted breakdown pressures (BP) were always higher than measured BP for impermeable media. Whereas part of this disparity may be the result of poor estimates for T₀, the larger part is likely the result of an inadequate constitutive model.

Although BP is related to the pressure which initiates tensile parting (fracture initiation pressure, FIP), the former is larger than the latter, and specifies the pressure at which a fracture suddenly propagates in a temporarily unstable manner through the stress concentration around a borehole. The FIP is the pressure at which the local T₀ is exceeded, and may be equal to or less than the BP. According to fracture mechanics concepts, a critical length required for unstable propagation has been invoked to explain discrepancies [Abou-sayed et al., 1978; Ishijima and Roegiers, 1983]. If we accept the concept of a stress concentration induced at the leading tip of a fracture, then we must also accept that the theoretical prediction from linear elastic approach will be lower than observed data. Boone [1989], however showed that numerical non-linear fracture analysis predictions may be 30% higher than for linear prediction.

Here, we present an alternative and analytical approach to analyze fracture data, and we will argue that at least part of the discrepancies are the result of a non-linear stiffness, which causes a redistribution of the tangential stresses, reducing the magnitude of the stress concentration and perhaps relocating it away from the borehole wall. This simple analysis may have limitations, but it clearly shows that hydraulic fracturing pressures are not independent of material properties, and that FIP may be much lower than BP because σ_θ]_max is located away from the borehole wall.

Santarelli et al. [1986], and Santarelli and Brown [1987] analyzed the stress distribution near a wellbore for borehole instability assessments by introducing a stress-dependent Young’s modulus (E = f(σ₃)). In their model, the radial stress is considered as the minimum principal stress on which the Young’s modulus depends. They concluded that a stress-dependent Young’s modulus will result in a smaller tangential stress concentration in comparison to that from a linear elastic model. Their model can be used to explain why borehole yield predictions are so inconsistent with experimental data. Biot [1974] qualitatively analyzed the fracture initiation problem with a nonlinear Young’s modulus. Motivated by Santarelli et al. [1986] and following Biot’s approach, we intend to analyze the effects of stress-dependent Young’s modulus on FIP.

It is known, particularly for relatively weak rock, that the strength and Young’s modulus are controlled by the confining stress. During hydraulic fracturing (Figure 1), the tangential stress must approach zero for tension-free rocks, and become negative for rocks with a tensile strength (we assume compression is positive). Accordingly, we introduce a non-constant Young’s modulus, related to the minimum stress, which in our case of injection, is

2 GENERAL FORMULATION

Starting from a 2-D plane strain condition, and assuming that Hooke’s law applies, but that Young’s modulus is a function of the minimum principal stress, then:

$$\begin{array}{l}\frac{u}{r}=...\end{array}$$

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Preface
• Acknowledgements
• Organization of the symposium
• U.S. National Committee for Rock Mechanics
• Previous symposia proceedings
• Table of Contents
• 1990 U.S. Rock Mechanics Awards
• 1. In-situ stresses
• 2. Instrumentation/Measurement techniques
• 3. Fracturing
• 4. Rock properties
• 5. Dynamics/Seismicity
• 6. Modelling
• 7. Laboratory testing
• 8. Analysis
• 9. Field observation(s)
• 10. Wellbore stability
• 11. Design
• 12. Discontinuities/Fluid flow
• 13. Lunar rock mechanics
• Author index