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Emmanuel Detournay

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Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 55th U.S. Rock Mechanics/Geomechanics Symposium, June 18–25, 2021

Paper Number: ARMA-2021-1605

Abstract

ABSTRACT: There is an long-standing debate in the rock mechanics community about the dependence of fracture toughness K Ic on the confining stress. One of the tests used to assess this dependence relies on injecting fluid into a slotted borehole drilled along the axis of a cylindrical sample and interpreting the toughness from the breakdown pressure (Abou-Sayed et al. 1978). However, the interpretation of the observed breakdown pressure relies critically on assuming that (i) the fluid pressure in the slots and the cracks ahead is uniform and (ii) the peak (breakdown) pressure is the fracture initiation pressure. The model described in this paper challenges these assumptions by considering a fluid lag at the tip of the hydraulic fracture growing from the pre-existing notches and by incorporating the hydraulic compliance of the injection system. This model indicates that the peak pressure generally differs from the fracture initiation pressure due to an episode of stable hydraulic fracture propagation following initiation. The difference between the peak pressure and the fracture initiation pressure increases with the fluid viscosity and the confining pressure, which leads to an artificial dependence of the toughness on the confining pressure if the peak pressure is interpreted as the initiation pressure. Comparison between the predicted and experimental peak pressure for a series of tests on cement samples indicates similar trends between experiments and numerical simulations. However, the predicted peak pressure generally underestimates the experimental one. 1. Introduction There is a long-standing debate in the geomechanics community as to whether the fracture toughness depends on the confining stress in rocks (Schmidt and Huddle, 1977; Schmidt, 1980; Ko and Kemeny, 2007; Garagash, 2019; Yue et al., 2020; Gehne et al., 2020). Although fracture toughness is treated as a constant material parameter in the theory of linear elastic fracture mechanics (LEFM) (Rice, 1968), a dependence of the resistance to fracture propagation on the confining stress is expected in quasi-brittle materials. Indeed, these materials are characterized by the presence of a process zone ahead of the crack tip (Barenblatt, 1962; Ballarini et al., 1984; Garagash, 2019; Bazant and Le, 2017). The confining stress acting across this cohesive zone contributes to the energy dissipation associated with the creation of new surfaces. The combined energy dissipation can be captured via an apparent toughness K Ic . Here the term "toughness" is used to encompass both the intrinsic material property K Ic and the stress-related energy dissipation, in accordance with the terminology used in the literature.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 54th U.S. Rock Mechanics/Geomechanics Symposium, June 28–July 1, 2020

Paper Number: ARMA-2020-2004

Abstract

Water injection laboratory experiments in weak, poorly consolidated sandstones show evidence that the peak injection pressure is much larger than the one predicted by the Haimson-Fairhurst criterion. A model based on poroelasticity, fracture mechanics, and lubrication theory is constructed to simulate the laboratory experiments. It aims at computing the propagation of a bi-wing hydraulic fracture from a borehole with increasing injection rate, until the crack reaches the boundary of the sample. The model is applicable to situations for which the pore pressure field reaches steady-state quasi-instantaneously when changing the injection rate, on account of the large permeability of these rocks. Two asymptotic regimes of solution are found: (i) a rock-flow regime where the induced fracture is hydraulically invisible, and (ii) a fracture-flow regime where the fluid penetrates the rock via the crack. In the rock-flow regime, fracture propagation is stable, i.e., the borehole pressure increases with the injection rate, while in the fracture-flow regime, the reverse is true. It is therefore concluded that the peak injection pressure reflects a transition between two flow regimes, rather than breakdown. 1. INTRODUCTION Water injection laboratory tests have been conducted to investigate the abnormal breakdown pressure observed in waterflooding operations in oil reservoirs located in weak, poorly consolidated sandstones. In these experiments, the injection rate is increased by step until the induced hydraulic fracture propagates unstably and causes the sample to break apart. Counterintuitively, experimental evidence shows that the peak pressure achieved in these weak rock injection experiments is much larger than the one predicted by the Haimson-Fairhurst breakdown criterion (Haimson and Fairhurst, 1967). A large fracturing pressure has often been attributed to large apparent fracture toughness, linked to the existence of a plastic zone at the crack tip (Papanastasiou and Thiercelin, 1993; Papanastasiou, 1997; van Dam et al., 2002; Agarwal and Sharma, 2011). Here we suggest a different mechanism behind the large observed peak pressure, which involves a changing flow pattern with increasing injection rate, rather than strength considerations.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 54th U.S. Rock Mechanics/Geomechanics Symposium, June 28–July 1, 2020

Paper Number: ARMA-2020-1835

Abstract

This paper describes a generic lumped-parameter model with multiple degrees-of-freedom (referred to as the MDOF model) to study the coupled axial and torsional vibrations of the drilling system with a drag bit consisting of full and partial blades. The interaction between such a bit with the rock is composed of a cutting process and a frictional contact process. The cutting process introduces a feedback with state-dependent delays into the equations of motion. The feedback is related to the depth of cut whose value depends on the current and a prior axial positions of the bit—the axial regenerative effect. The friction contact process is associated with nonlinear boundary conditions due to the occurrence of stick events in both axial and torsional vibrations. In this work, the axial regenerative effect is captured by a bit trajectory function, which makes it more efficient to calculate the depth of cut of a drag bit with full and partial blades than the approaches used in previous studies. The evolution of the bit trajectory function is governed by a partial differential equation (PDE), which is inherently coupled with the ordinary differential equations (ODEs) that govern the axial and torsional dynamics of the MDOF model. Through the application of Galerkin method, the coupled system of PDE-ODEs is further approximated by a finite set of ODEs, which is integrated to obtain the dynamic responses of the MDOF model. The characteristics of axial and torsional vibrations along the drillstring are captured with the MDOF model in conjunction with drag bits equipped with full and partial blades. The dynamic responses between a bit with full and partial blades and a bit with identical full blades are also compared in this paper. 1. INTRODUCTION Rotary drilling systems equipped with drag bits, which are used to create deep boreholes for exploration and production of oil and gas, often exhibit self-excited torsional vibrations. These vibrations often manifest themselves in the severe form of stick-slip oscillations characterized by a periodic succession of stick phase with zero bit angular velocity for a period of time and slip phase, during which the maximum bit angular velocity could be multiple times larger than the constant rotary table speed at the surface. It is documented that the rotary drilling systems experience stick-slip oscillations more than half of the drilling time (Henneuse, 1992; Kriesels et al., 1999). Stick-slip oscillations have significantly negative impacts on the drilling process, including reducing drilling efficiency, accelerating the wear of drag bits, and causing downhole tool failures with associated non-productive time. It is thus essential to understand the mechanism behind these detrimental oscillations in order to avoid and mitigate the adverse effects.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 53rd U.S. Rock Mechanics/Geomechanics Symposium, June 23–26, 2019

Paper Number: ARMA-2019-2041

Abstract

ABSTRACT: One approach to analyze the stick-slip vibrations of a rotary drilling system with a PDC bit is based on the RGD model (Richard et al., 2007), a discrete model that assumes a rate-independent bit/rock interaction law. This model considers both axial and torsional vibration modes, which are coupled through the depth of cut, a state variable determined by the current bit positions and its motion history. The bit geometry is further simplified as n continuous blades that are evenly distributed around the axis of rotation. The model is governed by a set of delay differential equations, with one priori unknown time delay corresponding to the constant angle between two blades. According to the RGD model, there exists a critical rotational speed separating two regimes of instability (Depouhon and Detournay, 2014), a function of the number of blades and a characteristic time of the system. In this work, we extend the RGD model by considering a class of symmetric bits with the same number of full or partial blades. This extension results in two angular delays and a corresponding set of coefficients which captures the contribution of each angular delay to the cutting component of weight-on-bit and torque-on-bit. A linear stability analysis is conducted that leads to explicit expressions for the critical rotational speed(s), which is a function of the number of blades and the relative length of the partial blades. The stability maps indicate that the bits can have a better stability compared to their symmetric counterparts. Compared to the original RGD model, it is also discovered that for certain bit geometries, there may exist multiple critical rotation speeds. Time simulation is performed to verify the results from the linear stability analyses.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 48th U.S. Rock Mechanics/Geomechanics Symposium, June 1–4, 2014

Paper Number: ARMA-2014-7479

Abstract

Abstract The sudden change in drilling conditions taking place at an interface between two rock layers creates parasitic forces and moments at the bit, which locally perturb the borehole trajectory. This perturbation can be determined by coupling a bit/rock interaction law with a model for the bottom-hole assembly (BHA). In this paper a linear bit/rock interaction that accounts for the difference of rock properties across the interface is developed and coupled with a BHA model, simplified on account that the positions of the stabilizers are assumed fixed in position when computing the influence of the interface at the length scale of the bit. The BHA is then modeled by springs that prevent the bit from moving freely. The theoretical predictions for a cylindrically-shaped bit are consistent with experimental results reported in the literature. A preliminary study on the influence of the main parameters influencing the perturbation is also presented.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 48th U.S. Rock Mechanics/Geomechanics Symposium, June 1–4, 2014

Paper Number: ARMA-2014-7351

Abstract

Abstract Drilling with drag bits (PDC bits) simultaneously involves fragmentation of rock by the cutters and frictional contact on the cutter wear flats. While there is reasonable understanding of the forces arising from the cutting process, knowledge of the factors affecting the contact forces on the wear flats is still fragmentary. This paper focuses on determining the parameters that influence the mean normal stress s on the cutter wear flats, and on mapping the dependence of s on these parameters, by analyzing the idealized problem of a slightly inclined rigid slider moving on the surface of a Mohr-Coulomb elastoplastic half-plane.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Alaska Rocks 2005, The 40th U.S. Symposium on Rock Mechanics (USRMS), June 25–29, 2005

Paper Number: ARMA-05-818

Abstract

ABSTRACT: Design of hydraulic fracturing laboratory experiments that capture similar phenomena to those expected at the field scale requires consideration of the scaling laws intrinsic to the mathematical model. Analysis of the model for a radial, Newtonian-fluid-driven fracture in an infinite elastic homogeneous medium indicates that fractures evolve relative to three timescales associated with transitions between regimes characterized by large/small fluid lag, large/small effective fluid viscosity, and large/small °uid leak-o®. The three invariants of the model are given by the ratio of the treatment time with these timescales, hence they provide the key for experimental design and interpretation that properly accounts for the deference between the field and laboratory scales. This paper presents a practical experimental design method based on these considerations. Experimental results are presented for which the invariant associated with °uid viscosity takes on deferent values. The results are in close agreement with a published solution that is based on modelling the crack tip using the classical Linear Elastic Fracture Mechanics when the viscosity invariant is small. However, when the viscosity invariant is O(1) the experimental results are in agreement with a published solution that utilizes a unique crack tip singularity associated with fluid-solid coupling in the tip region. INTRODUCTION Hydraulic fracturing has drawn hundreds of contributions over the last ¯fty years, many of which have involved laboratory investigation. However, the scale of laboratory fractures is always vastly different than the field applications to which they are supposed to bear relevance. The fundamental desire is to make some observations on laboratory- scale fractures, which permit measurements that are unavailable in the field, and then make some inference regarding the nature of field-scale fractures. However, it seems few authors give formal consideration to the difference of scale when de- signing or interpreting their experiments. The result is contradictions in the literature, particularly with regards to the relative importance of certain parameters. One example of such a contradiction concerns the viscosity of the fluid which is driving hydraulic fractures. Spence and Sharpe [1] and Barenblatt [2] have argued from the mathematical model that viscosity is expected to play a crucial role at the field-scale. However, laboratory verification of fracture behavior consistent with the so- called viscosity-dominated regime of propagation has proven elusive, and in fact, some laboratory studies are seemingly interpreted to downplay the role of viscosity altogether (e.g. [3]). Hence the hydraulic fracturing community remains divided on some important basic issues of parametric analysis. The way forward requires formal application of scaling principles to not only numerical and theoretical analysis of hydraulic fracturing, but also to design and interpretation of experimental investigations. This assertion was previously made by de Pater et al. [4], who derived invariants for essentially the same mathematical model considered here.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the DC Rocks 2001, The 38th U.S. Symposium on Rock Mechanics (USRMS), July 7–10, 2001

Paper Number: ARMA-01-0243

Abstract

ABSTRACT: This paper describes a numerical model to simulate the propagation of a plane-strain (KGD) hydraulic fracture in an elastic, impermeable medium with zero toughness. The fracture is driven by injection of an incompressible fluid with power-law rheology. The numerical model, which is formulated in terms of a moving coordinates system, is based on the displacement discontinuity method and on an explicit finite difference scheme. The accuracy of the algorithm is validated against the available self-similar solution for a Newtonian fluid. INTRODUCTION Hydraulic fracturing (HF) is a technique widely used to enhance the flow of oil or natural gas from the reservoir formations towards the extracting wells. The uncertainty about the in situ conditions, the complexity of the mechanisms taking place and the difficulty in obtaining precise measurements of the fracture geometry, make necessary the use of idealized models (e.g., the KGD or "plane-strain" model, the "pennyshaped" or radial model and the PKN model) for studying this process. Even with these simplified models, the mathematical formulation for the propagation of hydraulic fractures is given by a relatively complicated system of integral and non-linear differential equations. Some analytical solutions of these mathematical models are already available (Spence & Sharp, 1985; Savitski & Detournay, 1999; Garagash, 2000; Savitski, 2000; Adachi, 2001; Adachi et al., 2001). However, these solutions are constructed on the basis of various restrictive assumptions (e.g., constant injection rate; very small or very large material toughness, Newtonian theology for the fracturing fluid, no fluid leakoff, etc.). In order to extend the applicability of these models, it is necessary to release some of these assumptions and consequently, the solution of the governing equations demands the use of numerical techniques. Commonly, the numerical solution of non-linear problems is restricted to the use of implicit schemes. Explicit schemes are not often applied in this type of problems due to the difficulty in reaching numerical stability, even though the latter are simpler to implement. In this paper, we introduce an explicit finite difference scheme with a moving mesh that can be used to simulate the propagation of a KGD hydraulic fracture. This scheme is shown to be numerically stable, accurate and "flexible", in the sense that additional features (e.g., fluid leak-off, poroelastic effects, etc.) can be easily incorporated into the model (Detournay et al., 1990).

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the DC Rocks 2001, The 38th U.S. Symposium on Rock Mechanics (USRMS), July 7–10, 2001

Paper Number: ARMA-01-0281

Abstract

ABSTRACT: In this paper, the problem of a penny-shaped hydraulic fracture propagating parallel to the freesurface of an elastic half-space is studied. The fracture is driven by an incompressible Newtonian fluid injected at a constant rate. The flow of viscous fluid in the fracture is governed by the lubrication equation, while the crack opening and the fluid pressure are related by singular integral equations. We construct two asymptotic solutions based on the assumption that the energy expended in the creation of new fracture surfaces is either small or large compared to the energy dissipated in viscous flow. One important outcome of the analysis is to show that the asymptotic solutions, when properly scaled, depend only on the dimensionless parameter ?, the ratio of the fracture radius over the distance from the fracture to the free-surface. The scaled solutions can thus be tabulated and the dependence of the solution on time can be retrieved for specific parameters, through simple scaling and by solving an implicit equation. INTRODUCTION Hydraulic fracturing is the most common method used to stimulate production from gas and oil wells. The fractures are propagated from the well to stimulate reservoirs that are typically located 500 to several thousand meters below the surface. Hydraulic fracture growth at these depths is not affected by the surface of the earth, although surface deformation (tilt) is sometimes measured to infer fracture orientation and size. Although most fractures arising from hydraulic fracture treatments can be conceptualized as propagating within an infinite space, there are specific cases where the influence of a free-surface on the fracture growth becomesignificant or even dominant. Hydraulic fracturing has recently been used in mining to induce and control the timing of rock caving event, see Jeffrey & Mills (2000). The work contained in this paper is motivated by hydraulic fracturig used underground at Moonee Colliery to control the timing of roof-rock caving events. At Moonee, a massive conglomerate roof rock does not cave behind the longwall face in a predictable way and, when it does cave, produces a strong windblast in the nearby mine workings. Moonee has adopted hydraulic fracturing as a way to induce the conglomerate to cave in a controlled time period. The hydraulic fractures are formed at the end of 8 m long vertical holes drilled up into the .conglomerate, as shown in Fig. 1. The treatments produce more or less axisymmetric horizontal fractures that grow parallel to and strongly affected by the free surface. Fracture behaviors are characterized by a ratio of the fracture radius over the distance from the free surface which can reach order I or more. (See, also, Pollard & Hozlhausen (1979) for a granite quarry example.) The problem of predicting the fluid pressure, opening and size of the fracture given the injection rate, fluid rheology, and rock properties has attracted an extraordinary number of contributions since the 1950s. This intense research activity has contributed to the formulation of a variety of models that emphasize either the design of a hydraulic fracturing treatment, or the exact solution of the coupled fluid-solid problem with simple fracture geometry (Savitski & Detournay 2000). However, the influence of a free-surface on the propagation of a fluid-driven fracture has not yet been addressed, except approximately (Jeffrey & Settaft, 2000) and by a few contributions dealing with the problem of a uniformly pressurized fracture(Wang et al. 1994).

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the DC Rocks 2001, The 38th U.S. Symposium on Rock Mechanics (USRMS), July 7–10, 2001

Paper Number: ARMA-01-0927

Abstract

ABSTRACT : This paper reports the results of an investigation aimed at understanding and quantifying the influence of the cutter geometry on the forces acting on a rock cutter. The original cutting model introduced by Detournay and Defourny (1992) is modified to account for an additional force component related to the groove geometry. The validity of this new model is confirmed by the results of an extensive series of experiments using the Rock Strength Device with rectangular cutters.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Vail Rocks 1999, The 37th U.S. Symposium on Rock Mechanics (USRMS), June 7–9, 1999

Paper Number: ARMA-99-0851

Abstract

ABSTRACT: This paper deals with the self-similar solution of a penny-shape hydraulic fracture propa- gating in an impermeable elastic rock. Growth of the fracture is driven by injection of an incompressible Newtonian fluid at the center of the fracture, at a flow rate varying according to a power law of time (which includes the practically important case of a constant injection rate). The solution is restricted to the so-called viscosity-dominated regime where it can be assumed that the rock has zero toughness. In this regime, the fracture tip is characterized by a singularity which is weaker than the classical square root singularity of linear elastic fracture mechanics. The paper describes the construction of a semi- analytical similarity solution, which incorporates the known singularity of the fluid pressure at the center of the fracture and at the tip and which is based on series expansions of the fracture opening and fluid pressure in terms of Jacobi polynomials. It is shown that very few terms in the expansions are needed to capture the solution accurately. INTRODUCTION Mathematical modeling of hydraulic fractures has attracted numerous contributions since the 1950's. While early efforts dealt mainly with (approxi- mate) analytical solutions (see e.g. Khristianovic and Zheltov, 1955; Barenblatt, 1962; Perkins and Kern, 1961; Nordgren, 1972; Geerstma and Haaf- kens, 1979), the focus of researchas shifted in re- cent years towards the development of numerical algorithms to model the three-dimensional propa- gation of hydraulic fractures in layered strata char- acterized by different mechanical properties and/or in-situ stresses (e.g. Clifton and Abou-Sayed, 1979; Advani et al., 1990, Sousa et al., 1993; Shah et al., Despite this trend towards the development of realistic models of hydraulic fractures, there is still interest in obtaining "exact" solutions for simpler models with rigorous consideration given to both the flow of fluid in the fracture (generally modeled according to the lubrication theory) and the elas- tic deformation and propagation of the fracture. Such solutions can be used not only to benchmark numerical codes but also to explore the depen- dance of the solution to various parameters and to establish the existence of different regimes of propagation. These solutions are notoriously dif- ficult to construct, however, because of the strong non-linear coupling between the lubrication and elasticity equations and the non-local character of the elastic response of the fracture. Within the realm of "simple" models for hy- draulic fracturing, the penny-shape fracture is guably the most relevant one. Yet it has been treated by an handful of authors only (e.g. Baren- blatt, 1959; Abe et al., 1976; Abe et al., 1979; Cleaxy and Wong, 1985; Nilson et al., 1985; Ad- vani et al., 1987; Barr, 1991; de Pater et al., 1996; Yuan, 1997). Furthermore, no rigorous analytical (or semi-analytical) solution of this problem ex- ists to our knowledge, as published analytical so- lutions are based on approximations involving ad hoc forms of the fluid pressure or the crac? aper- ture. This paper deals with the construction of a semi-analytical solution for the problem of a penny- shape crack propagating in an unbounded imper- meable elastic medium, see Fig. 1. The fracture is driven by an incompressible Newtonlan fluid in- jected at the center of the fracture. As discussed in the paper, the injection flow rate is restricted to a power law of time, which includes, however, the practically important case of a constant flow rate.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Vail Rocks 1999, The 37th U.S. Symposium on Rock Mechanics (USRMS), June 7–9, 1999

Paper Number: ARMA-99-0877

Abstract

ABSTRACT: This paper is concerned with an in situ thermo-hydraulic experiment carried out at the Underground Research Laboratory of Atomic Energy of Canada Limited. The thermo-hydraulic experiment was designed to determine a hydro-thermal coupling parameter as well as the permeability and the thermal and hydraulic diffusivities of the Lac du Bonnet granite. Several water injection and heater tests were conducted during this experiment. In an injection test, a given volume of water is pumped quasi instantaneously in a sealed-off interval of the borehole, while in a heater test, heat is produced at a constant rate over a certain period (ranging from about one day to several weeks). The experimental set-up involves a heater installed in a sub-horizontal borehole drilled from an underground gallery, and piezometers and thermistors located at different distances from the heater, in auxiliary boreholes drilled from an adjacent gallery. Determination of the material parameters relies on matching the measured pore pressure and temperature responses with the theoretical predictions based on singular solutions of thermoporoelasticity. INTRODUCTION Various geological structures, including thick clay layers, salt domes and hard rock, are being con- sidered in various countries to host nuclear waste repositories. The research described in this paper is associated with the Canadian used fuel program, which is focusing on developing technical capa- bility for the siting, design and safety assessment of a repository in a saturated granitic rock mass. One facility developed in the Canadian Nuclear Fuel Waste Management Program is Atomic En- ergy of Canada Limited's (AECL's) Underground Research Laboratory (URL), in southeastern Man- itoba, Canada. The geological setting of the URL is unique compared to other underground labora- tories in the world because the rock mass of the Lac du Bonnet batholith is essentially ,manufactured below 250 to 300 m depth ?. The Thermal-Hydranlic Experiment (abbrevi- ated to TI?) conducted at the URL included a series of water injection and heater tests. The main objective of TI? is to determine the in situ value of a hydro-thermal coupling parameter (7), characterizing the magnitude of the pore pressure induced by thermal loading, as well as the perme- ability (n) and the thermal (c.) and hydraulic (c) diffusivities of the Lac du Bonnet granite. In this paper, the experimental set-up of THE as well as some results from a back-analysis of the experiments are presented. These results are ob- tained by matching the time and the amplitude of the peak of the pore pressure and tempera- ture responses with theoretical predictions based on the singular thermoporoelastic solutions of a fluid source and a heat source. The analysis pre- sented in this paper is a follow-up of the prelimi- nary results discussed by Berchenko et al. (1998).

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Vail Rocks 1999, The 37th U.S. Symposium on Rock Mechanics (USRMS), June 7–9, 1999

Paper Number: ARMA-99-0123

Abstract

INTRODUCTION ABSTRACT: The paper deals with a numerical analysis of rock cutting experiments using the discrete element method. The main objective of this research is to establish if the occurence of the two failure modes observed in rock cutting experiments (ductile at small depth of cut, brittle at large depth) can be duplicated in numerical simulation. The numerical analysis is carried out with the discrete element code PFC ?v which modelsolids as a collection of bonded disks. Scaling laws are first established between the micro-properties at the particle scale (such as the mean particle radius, and bond strengths) and the apparent material properties at the macroscopic scale (such as the compressive strength ac and the toughness Kxc). Cutting tests are then performed with a particle assembly of rock-like properties. The paper deaJs with a numerical analysis of rock cutting experiments in which rock is scratched by a cutter at a constant velocity and at a prescribed depth of cut. The objective of this preliminary study is to establish whether or not results obtained in laboratory rock experiments can be duplicated in numerical simulations. Cutting experiments carried out with the Rock Strength Device developed at the University of Minnesota have shown that two failure modes can occur depending on the depth of cut: (i) a ductile mode associated with plastic flow of failed rock ahead of the cutting face at small depth of cut (larger than the grain size and typically less than i mm in sandstones) and (ii) a brittle mode associated with fracture propagation and chipping of the rock at the depth of cut above a certain threshold (Richard et al., 1998). The transition depth of cut between the two failure modes appears to be related to the length scale (KIC/óc) ?, where KIC is the rock toughness and ac the uniaxial compressive strength. Furthermore, there is a large body of evidence to suggest that the average cutting force in the duct fie mode is proportional to the crosssectional area of the cut (i.e. to the depth of cut for a rectangular cutter) and that the coefficient of proportionality (referred to as the specific energy e) is itself proportional to óc In this study, the rock cutting process is investigated using the discrete element method, based on the approach by Cundall & Strack (1979) and Cundall & Hart (1993). The discrete element code PFC 2D (Itasca Consulting Group, 1996), is employed in the analysis. This code models solids as a collection of distinct and arbitrarily sized circular particles. The particles are treated as rigid bodies and allowed to overlap one another at the contact points. The contacts between particles are characterized through the stiffness, slip condition, and bonding models. The constitutive behavior of the particles enables the simulation of both plasticity and fracture at the macroscale. As a prerequisite to realistically represent rock-like materials by a particle assembly, scaling laws are first established between the micro-properties at the particle scale and the apparent material properties at the macroscopic scale. Numerical simulations on rock cutting experiments with a sharp cutter (no wear fiat) are then carried out to establish the existence of two failure modes in relation to the depth of cut, and to investigate the influence of material parameters on the transition depth of cut and on the magnitude of the cutting force.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 35th U.S. Symposium on Rock Mechanics (USRMS), June 5–7, 1995

Paper Number: ARMA-95-0349

Abstract

ABSTRACT: This paper deals with the numerical modeling of the problem of wedge indentation into an elasto-plastic half-plane. It focuses on the initial regime of the indentation process (prior to the initiation of a tensile fracture), in which energy is solely dissipated in a diffuse mode. The main issues that need to be addressed in modeling such a problem are first discussed: preservation of self-similarity, accuracy, and computational efficiency. Then, a comparison between numerical and analytical results based on a cavity expansion model is presented. The numerical results are shown to support a major prediction of the cavity expansion model, namely that the indentation process is predominantly controlled by a single number ?, which is a function of the wedge angle, the unconfined compressive strength of the rock, and its elastic modulus. INTRODUCTION This paper deals with the generic problem of wedge indentation into an elasto-plastic half-plane, and is motivated by the need to improve our understanding of the process of rock breakage by a mechanical tool. During indentation of a rock by a tool, two modes of energy dissipation occur: (i) diffuse with plastic deformation of the rock, and (ii) localized with tensile fracture propagation. These two modes correspond, respectively, to a ductile and brittle regime of the indentation process. Although the most efficient mechanism of rock breakage is by propagation of tensile cracks, the development of a zone of crushed material beneath the cutting tool prior to fracture initiation (if any) appears unavoidable, see Figure 1. Indeed, it can be expected that high stress develops under the tip of the indentor, leading to the formation of a zone of damaged rock. Furthermore, it can be shown that the formation of this plastic zone provides a mechanism by which the Ml-compressive stress field immediately beneath the tool transforms into a stress field containing at least one tensile stress component (Detournay, Fairhurst, Labuz, 1995). In the absence of a far-field stress a0 parallel to the free-surface, the maximum tensile stress occurs at the elasto-plastic interface, along the indentor axis; this stress is parallel to the interface. In the work presented here, we axe focusing on the initial regime of the indentation process, prior to the initiation of a tensile fracture. This regime, in which energy is solely dissipated in a diffuse mode, is characterized by the growth of a plastic zone with movement of the indentor. Modeling of this early phase of the indentation process is cru

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 35th U.S. Symposium on Rock Mechanics (USRMS), June 5–7, 1995

Paper Number: ARMA-95-0465

Abstract

ABSTRACT: This paper presents an approach to analyze the breakdown process of a pressurized circular cavity. Such a process is associated with the initiation and propagation of two radial symmetric tensile fractures. This problem is solved semi-analytically by making use of the exact elastic solution of a finite dislocation, which satisfies homogeneous boundary conditions at the circular cavity and at infinity. This approach leads to the formulation of a singular integral equation which is solved numerically using Erdogan and Gupta (1972) method. Application of this method to the analysis of the breakdown pressure in a hydraulic fracturing stress test is then described and the influence of the far-field stress, pressurization rate, and size of the borehole on the wellbore pressure response is discussed. INTRODUCTION The breakdown process of pressurized circular cavities in rock has received special attention for many years. This is due to its practical importance, in a variety of geomechanical applications such as hydrofracture of oil wells, determination of the farfield stress from the pressure-time record obtained during a hydraulic fracturing experiment, interpretation of leak-off test, and others. In particular, interpretation of the breakdown pressure pb, the critical pressure at which "breakdown" takes place during pressurization of the cavity, has been the subject of much debate in the last years, and has been the primary motivation behind many recent research efforts (for example, Schmitt and Zoback, 1992 and 1993; Haimson and Huang, 1989; Haimson and Zhao, 1991; Ito and Hayashi, 1991; Detournay and Cheng, 1992; Thiercelin, 1992; Guo, Morgenstern and Scott,1993; Detournay and Carbonell, 1994). Only a few studies have actually relied on the application of fracture mechanics to the breakdown process (Hardy, 1973; Abou-Sayed et al. 1978; Rummel, 1987; Atkinson and Thiercelin, 1993; Detournay and Carbonell, 1994). The criterion of crack propagation in such an approach (in which the cracks are explicitly modelled) relies on the calculation of the stress intensity factor Kl. Various solutions of the stress intensity factor for the problem of a circular cavity with two symmetric radial cracks have been published for particular loading configurations. The first solution is apparently due to Bowie (1956), who used a complex mapping method. Other researchers have approached the problem numerically and semianalytically (i.e. Kutter, 1970; Newman, 1971; Tweed and Rooke, 1973). In this analysis, the cracks at the edge of the circular cavity are modeled by a continuous distribution of edge dislocations. This method, which has been applied extensively to model crack related problems in Fracture Mechanics (e.g. Bueckner, 1960; Sheng, 1987) leads to the formulation of a singular integral equation in terms of the unknown dislocation density f(x). Both the stress intensity factor Kl and the crack opening displacement c? (x) are directly related to the dislocation density function, which is the slope of the crack opening.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 34th U.S. Symposium on Rock Mechanics (USRMS), June 28–30, 1993

Paper Number: ARMA-93-0523

Abstract

ABSTRACT INTRODUCTION Virtually all Petroleum geomechanical processes deal with rocks that are infiltrated by fluids. In view of the strong influence that pore pressure exerts on the deformation and failure of rocks (as it known since the 1923 seminal work of Terzaghi), it should therefore not be of any surprise that the Petroleum Industry has been one of the driving force behind basic and applied research in the mechanics of fluid-infiltrated solids. In the last decade, there has been a strong research emphasis on evaluating the overall implication of the presence of pore fluid on the geomechanical process, through an analysis of initial/boundary value problems. This is also the main topic of the paper, which reviews the consequence of the mechanical interaction between the pore fluid and the rock on some petroleum engineering processes such as drilling, borehole stability, and hydraulic fracturing. First, however, some fundamental features of the response of fluid-infiltrated solids are outlined. Drained and Undrained ResponseRate Effects There are two basic mechanisms that cause pore pressure to evolve with time: diffusive mass transport of the pore fluid driven by non-equilibrated pore pressure perturbations on the one hand, and "mechanical" pore pressure variation associated with change in pore volume, on the other hand. Of fundamental importance, is the intrinsic rate-sensitivity (or time-dependency) introduced by the diffusion mechanism, as it is embodied in the dimension [L]2/[T] of the diffusivity coefficient. In contrast, the time-dependency associated with the "mechanical" pore pressure change is external, in the sense that it is forced by the boundary conditions. It is important to note that none of these two mechanisms relies on any particular constitutive description of the rock. A related issue is the existence of two limiting deformation states, drained and undrained, which is one of the key features of the response of fluidinfiltrated materials. In undrained deformation, there is no variation of fluid content in a material element, and the pore pressure change with respecto some initial conditions is exclusively related to the variation of pore volume. Depending on the constitutive model of the rock (and the loading history), the pore volume change can be either elastic or elastoplastic; in the former case, Ap is proportional to change in the mean stress. (Note that the undrained pore pressure change is sometimes neglected, although this can strictly be justified only if the fluid is much more compressible than the solid.) In drained deformation, the pore pressure is completely determined by the current fluid boundary conditions; this is a situation where the flow of pore fluid has either vanished (if the pore pressure is hydrostatic) or has reached steady state conditions.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 33rd U.S. Symposium on Rock Mechanics (USRMS), June 3–5, 1992

Paper Number: ARMA-92-0325

Abstract

ABSTRACT: A mathematical model is proposed to account for rate and size effects on the magnitude of the breakdown pressure during a hydraulic fracturing experiment. This model recognizes the existence of two length scales: a diffusion length ä(a lengthscale representative of the distance of propagation of the pore pressure perturbation from the boundary) and a microstructural length ë (which underpins the failure process). In this context, rate effects are seen as a consequence of the interaction of these two lengthscales. An expression for the breakdown pressure pt,, which depends explicitly on the pressur- ization rate, is derived. It is demonstrated that the Haimson-Fairhurst (H-F) and the Hubbert-Willis (H-W) expression for the breakdown pressure correspond respectively to the asymptotically slow and fast pressurisation regimes. However, the H-F limit is shown to be the appropriate expression for "permeable" rocks, as hydraulic fracturing experi- ments in these rocks are practically always in the slow regime. It is also shown that in low permeability/low porosity rocks, rate effects are potentially significant and that both the H-F and the H-W expressions are acceptable limits, depending on the pressurization rate.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 32nd U.S. Symposium on Rock Mechanics (USRMS), July 10–12, 1991

Paper Number: ARMA-91-539

Abstract

ABSTRACT: In this paper, the dependence of the drilling specific energy on the virgin pore pressure p o is examined by analyzing the mechanics of a PDC cutter. On the basis of a simple failure mechanism involving a single shear plane, it is first established that the specific energy depends linearly on the difference between the bottom-hole pressure p h and the average pore pressure p b on the shear plane. The relationship between p b and p o is then determined by considering fluid mass balance across the shear plane. It is shown that the drilling conditions for a shale is in the high-speed regime, while for a permeable sandstone it is typically in the low speed regime. 1 INTRODUCTION It is generally accepted in the Drilling Industry that the specific energy (the amount of energy required to drill a unit volume of rock), depends on the differential pressure, i.e. the difference between the bottom-hole pressure p h and the virgin pore pressure p o . This commonly held assertion is based on the results of drilling experiments on permeable rocks that were undertaken in the late fifties (e.g. Eckel 1958, Cunningham and Eenink 1959). These results have been rationalized on the ground that (i) most of the pressure drop between the mud pressure and the far-field pore pressure takes place across the filter cake, (ii) the pore pressure in the failed rock regions is close to the virgin pore pressure due to the high permeability of the rock, and (iii) the strength of rock depends on the difference between confining (bottom-hole) and pore pressure. However, it has also been recognized early on (Handin 1959, Robinson and Holland 1969) that this conceptual model is probably not applicable to low permeability rocks because fluid cannot be supplied rapidly enough to the dilatant failed regions, causing the pore pressure in the failed zones to drop or even vanish. This view is supported by results of cutting experiments with a single PDC cutter on Mancos shale (Zijsling 1987) and drilling experiments with a milled tooth bit on an over consolidated Jurassic II shale (Gray Stephens et al 1991). These experiments indeed indicate that the specific energy depends on the bottom-hole pressure and not on the virgin pore pressure, thus implying that cavitation takes place in the failed regions. This paper addresses the issue of the dependence of the drilling response on the virgin pore pressure within the restricted context of a drag (PDC) bit and on the assumption that the rock dilates during failure. 2 MERCHANT FAILURE MECHANISM The factors influencing the mechanical response of PDC bits can be analyzed by focusing on the mechanics of failure induced by the motion of a single cutter. Figure 1 shows a cutter removing material over a constant depth of cut £, at a constant velocity v. A mud pressure ph is applied on the free surface of the rock. Also illustrated in this figure is the simple flow mechanism considered by Merchant (1944, 1945) for the machining of metals.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 30th U.S. Symposium on Rock Mechanics (USRMS), June 19–22, 1989

Paper Number: ARMA-89-0377

Abstract

ABSTRACT ABSTRACT: The solution for a pressurized hollow poroelastic cylinder subjected to a ramping internal pressure, p i , is employed to interpret laboratory burst tests on 'thin- walled' cylinders of fluid saturated rocks. It is shown in the paper how a testing program based on thin-walled cylinders could be used to critically assess the validity of the 'Strength of Materials' interpretation of the breakdown and examine whether or not tensile failure is controlled by a Terzaghi effective stress. This in turn will help validate existing breakdown equations used in hydraulic fracturing stress measurements. 1 INTRODUCTION The role of pore pressure and fluid infiltration on the 'breakdown' in a hydraulic fracturing experiment is still a controversial issue. Three breakdown equations are currently used, which, at great depth, predict significantly different values for the maximum horizontal stress 1 . The 'Haimson-Fairhurst' breakdown equation (Haimson and Fairhurst, 1967), which is consistent with Biot's theory of poroelasticity (1941), takes into account the stress concentration induced by pore pressure changes in the vicinity of the wellbore. One of the basic hypotheses built into this equation is the assumption that, at breakdown, the effective Terzaghi stress at the borehole wall is equal to the 'tensile strength' of the rock. The validity of the assumptions, on which the 'Haimson-Falrhurst' breakdown equation is based, could be tested by a carefully designed laboratory experimental program, that would involve internal pressurization (up to bursting) of hollow rock cylinders. So far, laboratory experiments aimed at validating this theory have been performed on cylindrical samples characterized by a very small ratio, ß, of inner to outer radius (typically less than 0.05). These tests can be interpreted with the 'Haimson-Fairhurst' equation (which applies for a circular hole in an infinite medium); however, they suffer from the fact that the breakdown pressure can only be changed by modifying the confining pressure, thus leading to limited verification of the theory. The solution for a poroelastic hollow cylinder, subject to a ramping internal pressure P i and a constant external stress, has recently been obtained (Carvalho and Detournay, 1989). This solution can be used to interpret the results of laboratory burst tests on 'thin- walled' cylinders of fluid saturated rocks (note that the short- and long-term response of the pressurized hollow cylinder was previously derived by Rice and Cleary (1976) who also stressed the importance of rate effects in burst experiments). This paper presents the interpretation and application of this solution for a poroelastic cylinder subjected to an internal ramping pressure and a stress free, drained outer boundary. Then, a discussion is given of how a testing program (which involves changing the rate of pressurization, and the ratio ß can be devised to critically examine the validity of the breakdown equation. 2 PROBLEM DESCRIPTION The geometry of the problem under consideration is shown in Figure 1. A hollow cylinder of permeable rock, with an inner radius r i , and an outer radius r o , is internally pressurized by a fluid. The pressure increases linearly with time, i.e. P i = At, while the outer surface is maintained at zero confining pressure and zero pore pressure.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 28th U.S. Symposium on Rock Mechanics (USRMS), June 29–July 1, 1987

Paper Number: ARMA-87-0575

Abstract

ABSTRACT ABSTRACT Two problems, a borehole subject to a non-hydrostatic far-field stress and a pressurized crack, are examined within the framework of Biot's theory of poroelasticity. The significance of the coupled diffusion-deformation theory is demonstrated, and the implications of these effects in rock mechanics applications are discussed. IINTRODUCTION Following the pioneering work of Terzaghi (1923, 1936, 1943) on the role of pore fluid in the consolidation and failure of soil masses, Biot (1941) presented the first consistent theory, now referred to as 'poroelasticity,' of the coupled diffusion-deformation processes in fluid-infiltrated solids. This theory, of which various versions exist (e.g. Biot 1955, 1956, 1973, Verruijt 1968, Rice and Cleary 1976), has enjoyed many applications in soil mechanics and earthquake mechanics (see Rudnicki 1985, for an up-to-date review). Although Biot's theory received an early introduction in the discipline of rock mechanics (Geertsma 1957, 1966), not much progress has been made in applications pertaining to mining, and petroleum engineering. Reasons for such a slow pace include (i) the lack of physical character in the original formulation of Biot (1941, 1956), and (ii) the difficulty of solving Biot's equations under a general geometry. However, Rice and Cleary (1976) have recently reformulated Biot's equations, using a set of material constants that allows a straightforward interpretation of the limiting behaviors of a poroelastic system. In addition, several fundamental poroelastic solutions have been obtained (Cleary 1977, Rudnicki 1981, 1987, Cheng and Liggett 1984, Detournay and Cheng 1987a, Cheng and Predeleanu 1987), thus clearing the way for efficiently solving complex problems using the integral equation technique (Cheng and Liggett 1984, Cheng and Liu 1986, Cheng and Detournay 1987). The purpose of this paper is to review two relevant rock mechanics problems, that have been solved within the framework of Biot's theory, and to discuss their important implications. The first problem concerns a borehole drilled in a porous formation subject to a non-hydrostatic far-field stress. Solution to this problem can be obtained analytically (Detournay and Cheng 1987c), by generalizing the solution of Carter and Booker (1982), based on the assumption of incompressible constituents. Next, the problem of a pressurized Griffith crack is solved numerically with the poroelastic displacement discontinuity method (Detournay and Cheng 1987b). 2 LINEAR POROELASTICITY As noted earlier, there are many different ways to formulate the theory of poroelasticity.Following the continuum mechanical formulation of Rice and Cleary (1976), the total stress tensor s ij and the pore pressure p are chosen as basic dynamnic variables. The solid matrix strain e ij , defined on the average solid displacement vector u l , and the change of fluid volume per unit reference volume, ¿, are the corresponding conjugate kinematic quantities. An isotropic rock-fluid system is then described by a set of five measurable material constants: the shear modulus G, the drained and undrained Poisson's ratio ¿, and ¿u, Skempton's pore pressure coefficient B (ratio of the induced pore pressure over the variation of confining pressure under undrained conditions), and the permeability ¿ which can be expressed as k/µ, where k is the intrinsic permeability (dimension of length squared) and µ the fluid viscosity.