Emmanuel Detournay

Fundamentals of poroelasticity

Publisher Summary This chapter focuses on fundamentals of poroelasticity. The presence of a freely moving fluid in a porous rock modifies its mechanical response. Two mechanisms play a key role in the interaction between the interstitial fluid and the porous rock: (i) an increase of pore pressure induces a dilation of the rock; and (ii) compression of the rock causes a rise of pore pressure, if the fluid is prevented from escaping the pore network. These coupled mechanisms bestow an apparent time-dependent character to the mechanical properties of the rock. If excess pore pressure, induced by compression of the rock, is allowed to dissipate through diffusive fluid mass transport, further deformation of the rock progressively takes place. The rock is more compliant under drained conditions than undrained ones. The chapter discusses the formulation and analysis of coupled deformation–diffusion processes, within the framework of the Biot theory of poroelasticity. The Biot model of a fluid-filled porous material is constructed on the conceptual model of a coherent solid skeleton and a freely moving pore fluid.

THE CRACK TIP REGION IN HYDRAULIC FRACTURING

We present analytical tip region solutions for fracture width and pressure when a power law fluid drives a plane strain fracture in an impermeable linear elastic solid. Our main result is an intermediate asymptotic solution in which the tip region stress is dominated by a singularity which is particular to the hydraulic fracturing problem. Moreover this singularity is weaker than the inverse square root singularity of linear elastic fracture mechanics. We also show how the solution for a semi-infinite crack may be exploited to obtain a useful approximation for the finite case.

A phenomenological model for the drilling action of drag bits

Abstract This paper is concerned with an investigation of the drilling response of drag bits (or PDC bits, as they are often referred to), i.e. with the study of the relations between weighht-on-bit W , torque T , angular velocity ω and rate of penetration v . Following an early suggestion (Fairhurst and Lacabanne, Min. Quarry Engng, 157–161, 194–197, 1957) that the bit-rock interaction is characterized by the coexistence of rock cutting and frictional contact, the torque and weight-on-bit are each decomposed into two components associated with these basic processes. By postulating that the cutting component of T and W is proportional to the depth of cut per revolution, δ = 2πv/ω , and that a linear constraint exists between the frictional component of T and W , a linear relation is derived between the specific energy E and the drilling strength L two quantities with dimensions of stress, that are respectively proportional to T and W , and inversely proportional to δ. The original assumptions appear to be justified when the model is tested against published experimental results, as the data point cluster along a line in the E-L diagram. Interpretation of experimental data suggests that the contact friction coefficient actually reflects the internal frictional property of the rock. It is also proposed that the influence of the bit design on the drilling response is embodied in a single number γ, which depends on the shape of the cutting edge and on the distribution of cutters on the bit body.

Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions

This paper presents an analysis of the propagation of a penny-shaped hydraulic fracture in an impermeable elastic rock. The fracture is driven by an incompressible Newtonian fluid injected from a source at the center of the fracture. The fluid flow is modeled according to lubrication theory, while the elastic response is governed by a singular integral equation relating the crack opening and the fluid pressure. It is shown that the scaled equations contain only one parameter, a dimensionless toughness, which controls the regimes of fracture propagation. Asymptotic solutions for zero and large dimensionless toughness are constructed. Finally, the regimes of fracture propagation are analyzed by matching the asymptotic solutions with results of a numerical algorithm applicable to arbitrary toughness.

The Tip Region of a Fluid-Driven Fracture in an Elastic Medium

The focus of this paper is on constructing the solution for a semi-infinite hydraulic crack for arbitrary toughness, which accounts for the presence of a lag of a priori unknown length between the fluid front and the crack tip. First, we formulate the governing equations for a semi-infinite fluid-driven fracture propagating steadily in an impermeable linear elastic medium. Then, since the pressure in the lag zone is known, we suggest a new inversion of the integral equation from elasticity theory to express the opening in terms of the pressure. We then calculate explicitly the contribution to the opening from the loading in the lag zone, and reformulate the problem over the fluid-filled portion of the crack. The asymptotic forms of the solution near and away from the tip are then discussed, It is shown that the solution is not only consistent with the square root singularity of linear elastic fracture mechanics, but that its asymptotic behavior at infinity is actually given by the singular solution of a semi-infinite hydraulic fracture constructed on the assumption that the fluid flows to the tip of the fracture and that the solid has zero toughness. Further, the asymptotic solution for large dimensionless toughness is derived, including the explicit dependence of the solution on the toughness. The intermediate part of the solution (in the region where the solution evolves from the near tip to the far from the tip asymptote) of the problem in the general case is obtained numerically and relevant results are discussed, including the universal relation between the fluid lag and the toughness.

From mixture theory to biot’s approach for porous media

Abstract Two apparently different approaches are used in dealing with the mechanics of a deformable porous medium : mixture theories on the one hand, and purely macroscale theories, which are mainly associated with the work of Biot, on the other hand. In the mixture theories, the porous medium is represented by spatially superposed interacting media, while macroscale theories assume that standard concepts of continuum mechanics are still relevant at the macro-level. The aim of this paper is two-fold. First, it is shown that the macroscale field equations derived from mixture theories can be reformulated in terms of the measurable quantities involved in the macroscale theories. Second, it is demonstrated how these field equations, including the fundamental inequality obtained from the second law, entail the existence of a macroscale C-potential upon which a thermo- dynamically consistent formulation of the constitutive equations can be firmly founded.

Plane-Strain Propagation of a Fluid-Driven Fracture: Small Toughness Solution

The paper considers the problem of a plane-strain fluid-driven fracture propagating in an impermeable elastic solid, under condition of small (relative) solid toughness or high (relative) fracturing fluid viscosity. This condition typically applies in hydraulic fracturing treatments used to stimulate hydrocarbons-bearing rock layers, and in the transport of magma in the lithosphere. We show that for small values of a dimensionless toughness K, the solution outside of the immediate vicinity of the fracture tips is given to O1 by the zero-toughness solution, which, if extended to the tips, is characterized by an opening varying as the 2/3 power of the distance from the tip. This near tip behavior of the zero-toughness solution is incompatible with the Linear Elastic Fracture Mechanics (LEFM) tip asymptote characterized by an opening varying as the 1/2 power of the distance from the tip, for any nonzero toughness. This gives rise to a LEFM boundary layer at the fracture tips where the influence of material toughness is localized. We establish the boundary layer solution and the condition of matching of the latter with the outer zero-toughness solution over a lengthscale intermediate to the boundary layer thickness and the fracture length. This matching condition, expressed as a smallness condition on K, and the corresponding structure of the overall solution ensures that the fracture propagates in the viscosity-dominated regime, i.e., that the solution away from the tip is approximately independent of toughness. The solution involving the next order correction in K to the outer zero-toughness solution yields the range of problem parameters corresponding to the viscosity-dominated regime. DOI: 10.1115/1.2047596

Multiscale tip asymptotics in hydraulic fracture with leak-off

This paper is concerned with an analysis of the near-tip region of a fluid-driven fracture propagating in a permeable saturated rock. The analysis is carried out by considering the stationary problem of a semi-infinite fracture moving at constant speed V. Two basic dissipative processes are taken into account: fracturing of the rock and viscous flow in the fracture, and two fluid balance mechanisms are considered ― leak-off and storage of the fracturing fluid in the fracture. It is shown that the solution is characterized by a multiscale singular behaviour at the tip, and that the nature of the dominant singularity depends both on the relative importance of the dissipative processes and on the scale of reference. This solution provides a framework to understand the interaction of representative physical processes near the fracture tip, as well as to track the changing nature of the dominant tip process(es) with the tip velocity and its impact on the global fracture response. Furthermore, it gives a universal scaling of the near-tip processes on the scale of the entire fracture and sets the foundation for developing efficient numerical algorithms relying on accurate modelling of the tip region.

Drilling in extreme environments : penetration and sampling on Earth and other planets

1 Introduction 2 Principles of Drilling and Excavation 3 Ground Drilling and Excavation 4 Ice Drilling and Coring 5 Underwater Drilling 6 Extraterrestrial Drilling and Excavation 7 Planetary Sample Acquisition, Handling and Processing 8 Instruments for In-Situ Sample Analysis 9 Contamination and Planetary Protection 10 Conclusions

Plane strain analysis of a stationary hydraulic fracture in a poroelastic medium

Abstract This paper presents a plane strain analysis of a constant length hydraulic fracture embedded in an infinite poroelastic domain. The fracture is uniformly loaded by fluid pressurisation. For clarity of physical interpretation, this loading is decomposed into two modes, consisting respectively of a unit step for the normal stress and a unit step for the pore pressure along the fracture. For each loading mode, the transient fracture profile, the fracture volume, the leak-off volume, and the stress intensity factor are analyzed. First, short- and long-term asymptotic expressions are derived in closed form based on analytical arguments. The full transient behaviors are then formulated as a pair of coupled singular integral equations. The solutions are found via Laplace transform, and numerical discretization of the integral equations.