Dmitry I. Garagash

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.

The near-tip region of a fluid-driven fracture propagating in a permeable elastic solid

This paper is concerned with an analysis of the near-tip region of a fluid-driven fracture propagating in a permeable saturated rock. It focuses on the calculation of the pore fluid pressure in the tip cavity, the region corresponding to the lag between the front of the fracturing fluid and the fracture tip. In contrast to impermeable rocks where the tip cavity can be considered to be at zero pressure, the fluid pressure in the tip cavity is here unknown and not uniform as exchange of pore fluid between the cavity and the porous medium and flow of pore fluid within the cavity is taking place. Solution of the fluid pressure in the tip region requires therefore simultaneous consideration of fracture mechanics (for the aperture of the tip cavity), diffusion theory for the movement of fluid within the porous medium, and viscous flow along the crack. Construction of such a solution within the framework of some simplifying assumptions is the main objective of this paper. It is shown that the problem depends, in general, upon two numbers with the meaning of a permeability and a propagation velocity. For the asymptotic case of large propagation speed, these two numbers merge into a single parameter, while the solution becomes independent of the propagation velocity in the limit of small velocity. The particular case of large velocity is solved analytically, while both the general and the small velocity cases are computed numerically but with different techniques. The paper concludes with a comprehensive analysis of numerical results.

Confined flow of suspensions modelled by a frictional rheology

We investigate in detail the problem of confined pressure-driven laminar flow of neutrally buoyant non-Brownian suspensions using a frictional rheology based on the recent proposal of Boyer et al., 2011. The friction coefficient and solid volume fraction are taken as functions of the dimensionless viscous number I defined as the ratio between the fluid shear stress and the particle normal stress. We clarify the contributions of the contact and hydrodynamic interactions on the evolution of the friction coefficient between the dilute and dense regimes reducing the phenomenological constitutive description to three physical parameters. We also propose an extension of this constitutive law from the flowing regime to the fully jammed state. We obtain an analytical solution of the fully-developed flow in channel and pipe for the frictional suspension rheology. The result can be transposed to dry granular flow upon appropriate redefinition of the dimensionless number I. The predictions are in excellent agreement with available experimental results, when using the values of the constitutive parameters obtained independently from stress-controlled rheological measurements. In particular, the frictional rheology correctly predicts the transition from Poiseuille to plug flow and the associated particles migration with the increase of the entrance solid volume fraction. We numerically solve for the axial development of the flow from the inlet of the channel/pipe toward the fully-developed state. The available experimental data are in good agreement with our predictions. The solution of the axial development of the flow provides a quantitative estimation of the entrance length effect in pipe for suspensions. A analytical expression for development length is shown to encapsulate the numerical solution in the entire range of flow conditions from dilute to dense.

The Impact of the Near-Tip Logic on the Accuracy and Convergence Rate of Hydraulic Fracture Simulators Compared to Reference Solutions

We benchmark a series of simulators against available reference solutions for propagating plane-strain and radial hydraulic fractures. In particular, we focus on the accuracy and convergence of the numerical solutions in the important practical case of viscosity dominated propagation. The simulators are based on different propagation criteria: linear elastic fracture mechanics (LEFM), cohesive zone models/tensile strength criteria, and algorithms accounting for the multi-scale nature of hydraulic fracture propagation in the near-tip region. All the simulators tested here are able to capture the analytical solutions of the different configurations tested, but at vastly different computational costs. Algorithms based on the classical LEFM propagation condition require a fine mesh in order to capture viscosity dominated hydraulic fracture evolution. Cohesive zone models, which model the fracture process zone, require even finer meshes to obtain the same accuracy. By contrast, when the algorithms use the appropriate multi-scale hydraulic fracture asymptote in the near-tip region, the exact solution can be matched accurately with a very coarse mesh. The different analytical reference solutions used in this paper provide a crucial series of benchmark tests that any successful hydraulic fracturing simulator should pass.

Scaling of Physical Processes in Fluid-Driven Fracture: Perspective from the Tip

A particular class of fractures driven in a solid by pressurized viscous fluids is considered. These fractures could be either tens or hundreds meters long man-made hydraulic fractures in oil and gas reservoirs, or natural fractures, such as kilometers-long volcanic dikes driven by magma coming from upper mantle beneath the Earth’s crust. Different physical mechanisms governing propagation of a fluid-driven fracture include (i) dissipation in the viscous fluid flow along the fracture, (ii) dissipation in the solid due to fracturing, (iii) lagging of the fluid front behind the fracture front, (iv) fluid leak-off (into the permeable solid), and others. Dissipation in the viscous fluid flow is often considered to be the dominant mechanism on fracture length and time scales of practical interest. Universal scaling of the non-dominant mechanisms (dissipation in the solid, fluid lag, etc.) in the global solution of fluid-driven fracture is derived in this paper based on the analysis of the boundary layer structure near the fracture leading edge. This scaling may be particularly important in guiding numerical solution of fractures when non-trivial fracture geometry or/and spatially varying properties of the solid prevent analytical investigation of the global solution.

Steadily propagating slip pulses driven by thermal decomposition

Geophysical observations suggest that mature faults weaken significantly at seismic slip rates. Thermal pressurization and thermal decomposition are two mechanisms commonly used to explain this dynamic weakening. Both rely on pore fluid pressurization with thermal pressurization achieving this through thermal expansion of native solids and pore fluid and thermal decomposition by releasing additional pore fluid during a reaction. Several recent papers have looked at the role thermal pressurization plays during a dynamically propagating earthquake, but no previous models have studied the role of thermal decomposition. In this paper we present the first solutions accounting for thermal decomposition during dynamic rupture, solving for steady state self-healing slip pulses propagating at a constant rupture velocity. First, we show that thermal decomposition leads to longer slip durations, larger total slips, and a distinctive along–fault slip rate profile. Next, we show that accounting for more than one weakening mechanism allows multiple steady slip pulses to exist at a given background stress, with some solutions corresponding to different balances between thermal pressurization and thermal decomposition, and others corresponding to activating a single reaction multiple times. Finally, we study how the rupture properties depend on the fault properties and show that the impact of thermal decomposition is largely controlled by the ratio of the hydraulic and thermal diffusivities χ = αhy/αth and the ratio of pore pressure generated to temperature rise buffered by the reaction Pr/Er.

Numerical Simulation of Hydraulic Fracturing in the Viscosity-Dominated Regime

Most hydraulic fracturing treatments are in the viscositydominated regime. Hence, fracture growth does not depend on the rock toughness and it can be shown that the fracture aperture w near the fracture front, when viewed at the scale of the whole fracture, is not characterized by the classical square root behavior predicted by linear elastic fracture mechanics 12 ~ ws / , where s is the distance from the tip. Instead, the asymptotic tip aperture that reflects the predominance of viscous dissipation is of the form 23 ~ ws / , under conditions of large efficiency and small fluid lag. After demonstrating the intimate connection between the tip aperture and the fracture propagation regime, we report the results of hydraulic fracturing laboratory experiments in PMMA and glass blocks that employ a novel optical technique to measure the fracture opening. These experiments provide incontrovertible evidence that the power law index, characterizing the fracture aperture near the tip, depends on the propagation regime in accordance with theoretical findings. Finally, we demonstrate that a coarsely-meshed planar hydraulic fracture simulator can produce accurate results relative to benchmark solutions provided that the appropriate tip behavior is embedded in the algorithm. Through theoretical, experimental, and computational considerations, these results make it clear that advances in the accuracy and efficiency of fracture simulators critically depend on a sophisticated treatment of the near-tip aperture that goes beyond basic linear elastic fracture considerations.

On the Production Analysis of a Multi-Fractured Horizontal Well

This paper investigates the post fracture transient analysis of multi-fractured horizontal wells under the assumption of infinitely large fracture conductivity. Most of the existing studies of multi-fractured wells have considered finite fracture conductivity, when the dynamic fluid pressure drop in the flow within fractures is a part of the solution. This led to computationally intensive solution methods, particularly when a reasonably large number of fractures representative of current field applications is considered. In this work, we limit our consideration to low-permeability, tight (e.g., shale) reservoirs, when pressure losses in propped fractures can be neglected. This assumption allows to develop a rigorous, accurate, and computationally efficient solution method based on the fundamental problem of a unit step pressure decline in an array of identically sized and equally spaced fractures. The study of this fundamental problem is analogous to the well testing analysis of a fractured well produced at constant bottom-hole pressure conditions. The solution for a unit step pressure decline is used within the Green’s function framework to formulate and solve for the transient pressure response of a multi-fracture array produced at a constant volumetric flow rate. We also explore two simplified approaches to the production analysis of multi-fractured wells based on (1) the infinite fracture array approximation for finite arrays, and (2) an extension of the ad hoc method of Gringarten et al. (Soc Pet En J 14(04):347–360), respectively. We show that both methods lead to very good approximations of the rigorous solution for a finite fracture array problem, thus allowing to further simplify the transient analysis of multi-fracture wells.

Diffuse Vs. Localized Instability in Compacting Geomaterials Under Undrained Conditions

The coupling between the pore fluid diffusion and deformation of geomaterials can alter the mechanical response and facilitate or delay material failure. Dilatant or contractive material behavior under conditions of limited drainage and/or of loading rate exceeding the pore fluid diffusion rate causes a reduction or increase in pore pressure, respectively. The latter, via the effective stress principle, results in dilatant strengthening or contractive weakening of frictional materials, namely, an increase or decrease in shear stress that can be sustained by the material from the corresponding value in the underlying drained response. In this work, we use perturbation theory to examine how the contractive weakening affects the stability of undrained deformation and what are the possible instability modes (localized vs. diffuse) on the example of a one-dimensional simple shear of a layer under partially strain-controlled boundary conditions. The effect of material rate-sensitivity on undrained stability is examined. INTRODUCTION Deformation of fluid-infiltrated solids is generally coupled with pore fluid flow. The presence of pore fluids can alter deformation processes and facilitate or delay material failure. Some of geomechanics applications where the coupling of inelastic deformation and pore fluid flow is important include the problems of slope stability and liquefaction in saturated soil deposits (Ishihara 1993), stability of slip in fault gouge zones in the Earth’s crust (Rice 1992), efficient underground storage of natural gas, and terrestrial sequestration of greenhouse gases (carbon dioxide) to mitigate adverse effects on the atmosphere (Rudnicki 1996). It has been recognized for a long time that dilation or contraction of frictional geomaterials in the course of inelastic shear is the key mechanism to cause the coupling. The micro-mechanism of macroscopic inelastic volume changes varies from shear-induced opening of micro-fissures in tight rocks (Brace et al. 1966) to the rearrangement of grains into more loose or dense configuration depending on the initial packing density in sands. 1In Proceedings of the First Japan-U.S. Workshop on Testing, Modeling and Simulation in Geomechanics, Boston, Massachusetts, USA, June 27-29, 2003. 2Department of Civil and Environmental Engineering, Clarkson University, 8 Clarkson Ave., Potsdam NY 13699-5710, garagash@clarkson.edu