Papers by Emmanuel Detournay
Journal of Sound and Vibration, 2009
Engineering Geology, 2012
This paper analyses the problem of a hydraulically driven fracture, propagating in an impermeable... more This paper analyses the problem of a hydraulically driven fracture, propagating in an impermeable, linear elastic medium. The fracture is driven by injection of an incompressible, viscous fluid with power-law rheology and behaviour index n50: The opening of the fracture and the internal fluid pressure are related through the elastic singular integral equation, and the flow of fluid inside the crack is modelled using the lubrication theory. Under the additional assumptions of negligible toughness and no lag between the fluid front and the crack tip, the problem is reduced to self-similar form. A solution that describes the crack length evolution, the fracture opening, the net fluid pressure and the fluid flow rate inside the crack is presented. This self-similar solution is obtained by expanding the fracture opening in a series of Gegenbauer polynomials, with the series coefficients calculated using a numerical minimization procedure. The influence of the fluid index n in the crack propagation is also analysed.
IEEE Transactions on Control Systems Technology, 2015
To enable access to unconventional reservoirs of oil and (shale-) gas, geothermal energy and mine... more To enable access to unconventional reservoirs of oil and (shale-) gas, geothermal energy and minerals, complex curved boreholes need to be drilled in the Earth's crust. Directional drilling techniques, incorporating down-hole robotic actuation systems called rotary steerable systems, are used to generate these curved boreholes. In practice, however, boreholes drilled with such systems often show instability-induced borehole spiraling, which negatively affects the borehole quality and increases drag losses while drilling. As a basis for controller synthesis, we present a directional drilling model in terms of delay differential equations. Next, the problem of curved well-bore generation is formulated as a tracking problem and a model-based robust control strategy is developed, solving this tracking problem while guaranteeing the prevention of borehole spiraling. The effectiveness of the proposed approach is illustrated by representative case studies for the generation of curved boreholes.
The classic Mandel solution associated with the well-known Mandel–Cryer effect is re-examined. Us... more The classic Mandel solution associated with the well-known Mandel–Cryer effect is re-examined. Using Biot theory of poroelasticity, the solution of the Mandel problem is extended to include material transverse isotropy, as well as the compressibility of the pore fluid and the solid constituents of the soil–rock skeleton. Le résultat classique de Mandel, associé à l'effet bien comnnu de Mandel—Cryer, est réexamineé. En utilisant la théorie poro-é1astique de Blot, la solution du probléme de Mandel est étendue afin d'introduire l'anisotropie transverse du matériau, ainsi que la compressibilité des fluides présents dans les pores et celle des constituants solides du squelette du sol ou de la roche.
The focus of this paper is on constructing the solution for a semi-infinite hydraulic crack for a... more 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.
Journal of Computational and Nonlinear Dynamics
This work describes an alternative formulation of a system of nonlinear state-dependent delay dif... more This work describes an alternative formulation of a system of nonlinear state-dependent delay differential equations (SDDDEs) that governs the coupled axial and torsional vibrations of a 2 DOF drillstring model considering a Polycrystalline Diamond Compact (PDC) bit with realistic cutter layout. Such considerations result in up to 100 state-dependent delays due to the regenerative effect of the drilling process, which renders the computational efficiency of conventional solution strategies unacceptable. The regeneration of the rock surface, associated with the bit motion history, can be described using the bit trajectory function, the evolution of which is governed by a partial differential equation (PDE). Thus the original system of SDDDEs can be replaced by a nonlinear coupled system of a PDE and ordinary differential equations (ODEs). Via the application of the Galerkin method, this system of PDE-ODEs is transformed into a system of coupled ODEs, which can be readily solved. The ...
This paper reviews recent results of a research program aimed at developing a theoretical framewo... more This paper reviews recent results of a research program aimed at developing a theoretical framework to understand and predict the different modes of propagation of a fluid-driven fracture. The research effort involves constructing detailed solutions of the crack tip region, developing global models of hydraulic fractures for plane strain and radial geometry, and identifying the parameters controlling the fracture growth. The paper focuses on the propagation of hydraulic fractures in impermeable rocks. The controlling parameters are identified from scaling laws that recognize the existence of two dissipative processes: fracturing of the rock (toughness) and dissipation in the fracturing fluid (viscosity). It is shown that the two limit solutions (corresponding to zero toughness and zero viscosity) are characterized by a power law dependence on time and that the transition between these two asymptotic solutions depends on a single number, which can be chosen to be either a dimensionless toughness or a dimen...
This paper describes a KGD-type model of a hydraulic fracture created by injecting fluid in weak,... more This paper describes a KGD-type model of a hydraulic fracture created by injecting fluid in weak, poorly consolidated rocks. The model is based on the key assumption that the hydraulic fracture is propagating within a domain where the hydraulic fields are quasi-stationary. By further assuming a negligible toughness, the fracture is shown to grow as a square root of time. The asymptotic fracture propagation regimes at small and large time are constructed and the transient solution is computed by solving a nonlinear system of algebraic equations formulated in terms of the fracture aperture. At early time the fracture is hydraulically invisible and the injection pressure increases with time t as log t, while at late time leak-off from the borehole is negligible and the injection pressure decreases as t−1/4. According to this model, the peak injection pressure observed when injecting fluid in weak, poorly consolidated rocks should not be interpreted as indicating a breakdown of the form...
This paper proposes an efficient formulation for the simulation of waves propagating in the down-... more This paper proposes an efficient formulation for the simulation of waves propagating in the down-the-hole percussive drilling systems typically used in mining applications. In particular, it treats the problem of elastic wave propagation in 1-dimensional colliding bodies. The presented algorithm performs the time integration of the equations of motion arising from the space semi-discretisation of the continuous problem by the Galerkin method, using a dissipative midpoint rule as to the time dimension, and builds its efficiency on the innovating handling of the contact opening/closure events. Specific to our approach is the use of an event-driven scheme to perform the time integration of the discrete equations of motion, combined with a penalty-based definition of the contact forces. This formulation, most adapted to systems presenting a low number of contact interfaces, takes its efficacy in the exploitation of the linearity of the governing equations between events, and is based on...
These MATLAB codes demonstrate the method proposed by the paper of the same name. HeMo evaluates ... more These MATLAB codes demonstrate the method proposed by the paper of the same name. HeMo evaluates the dynamic response of a cantilever beam against two symmetrical stops ((end)-point and/or horizontal wall). The impact motion is simulated by two methods: a) Moreau's midpoint method and b) Moreau-Jean's theta method. The second code, Rayleigh, measures the response of a Rayleigh beam for unit impulse load with beta=0.1 (Fourier series - 6 modes).
International Journal for Numerical and Analytical Methods in Geomechanics, 2015
The site contains a Mathematica code for computing the displacement and stress induced by injecti... more The site contains a Mathematica code for computing the displacement and stress induced by injection of fluid in a poroelastic layer bounded by impermeable elastic half-spaces
IFAC-PapersOnLine, 2015
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
Journal of Fluid Mechanics
This paper investigates the motion of a liquid blister, trapped between an elastic sheet and a ri... more This paper investigates the motion of a liquid blister, trapped between an elastic sheet and a rigid substrate. The blister is driven by a frictionless blade moving at a constant velocity, forcing a constant gap that causes fluid to bleed from the blister. The sheet adheres to the substrate ahead of the blister. The main goal of the study is to assess the magnitude and orientation of the force applied by the blade on the moving blister. The solution is constructed for the asymptotic case of a long blister. Thanks to a separation of scales, the asymptotic solution is obtained by matching the boundary layers at the front end and at the back end of the blister to an outer solution characterised by a uniform pressure in the bulk. Both boundary layers are formulated as travelling-wave equations for the gap between the sheet and the substrate. The formulation accounts for a moving fluid front, distinct from the separation edge, and for a tail with a gap tending to an a priori unknown value far behind the blister. Scaling of the governing equations indicates that the solution depends on two numbers: a dimensionless toughness K and a scaled gap W imposed by the moving blade. The key result concerns the dependence of the scaled force on the two numbers controlling the solution of the moving liquid blister. There are two asymptotic solutions: for small gaps at the blade, the force on the blade is dominated by viscous dissipation at the back end and only depends on aperture W; for large gaps, the horizontal force H only depends on toughness K, a function of both fluid viscosity and energy of separation at the front end, whereas the vertical force V depends on both K and W.
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Papers by Emmanuel Detournay