Viet T. Chau

Growth model for large branched three-dimensional hydraulic crack system in gas or oil shale.

Recent analysis of gas outflow histories at wellheads shows that the hydraulic crack spacing must be of the order of 0.1 m (rather than 1 m or 10 m). Consequently, the existing models, limited to one or several cracks, are unrealistic. The reality is 105–106 almost vertical hydraulic cracks per fracking stage. Here, we study the growth of two intersecting near-orthogonal systems of parallel hydraulic cracks spaced at 0.1 m, preferably following pre-existing rock joints. One key idea is that, to model lateral cracks branching from a primary crack wall, crack pressurization, by viscous Poiseuille-type flow, of compressible (proppant-laden) frac water must be complemented with the pressurization of a sufficient volume of micropores and microcracks by Darcy-type water diffusion into the shale, to generate tension along existing crack walls, overcoming the strength limit of the cohesive-crack or crack-band model. A second key idea is that enforcing the equilibrium of stresses in cracks, pores and water, with the generation of tension in the solid phase, requires a new three-phase medium concept, which is transitional between Biot’s two-phase medium and Terzaghi’s effective stress and introduces the loading of the solid by pressure gradients of diffusing pore water. A computer program, combining finite elements for deformation and fracture with volume elements for water flow, is developed to validate the new model. This article is part of the themed issue ‘Energy and the subsurface’.

Wave Dispersion and Basic Concepts of Peridynamics Compared to Classical Nonlocal Damage Models

The spectral approach is used to examine the wave dispersion in linearized bond-based and state-based peridynamics in one and two dimensions, and comparisons with the classical nonlocal models for damage are made. Similar to the classical nonlocal models, the peridynamic dispersion of elastic waves occurs for high frequencies. It is shown to be stronger in the state-based than in the bond-based version, with multiple wavelengths giving a vanishing phase velocity, one of them longer than the horizon. In the bond-based and state-based, the nonlocality of elastic and inelastic behaviors is coupled, i.e., the dispersion of elastic and inelastic waves cannot be independently controlled. In consequence, the difference between: (1) the nonlocality due to material characteristic length for softening damage, which ensures stability of softening damage and serves as the localization limiter, and (2) the nonlocality due to material heterogeneity cannot be distinguished. This coupling of both kinds of dispersion is unrealistic and similar to the original 1984 nonlocal model for damage which was in 1987 abandoned and improved to be nondispersive or mildly dispersive for elasticity but strongly dispersive for damage. With the same regular grid of nodes, the convergence rates for both the bond-based and statebased versions are found to be slower than for the finite difference methods. It is shown that there exists a limit case of peridynamics, with a micromodulus in the form of a Delta function spiking at the horizon. This limit case is equivalent to the unstabilized imbricate continuum and exhibits zero-energy periodic modes of instability. Finally, it is emphasized that the node-skipping force interactions, a salient feature of peridynamics, are physically unjustified (except on the atomic scale) because in reality the forces get transmitted to the second and farther neighboring particles (or nodes) through the displacements and rotations of the intermediate particles, rather than by some potential permeating particles as on the atomic scale. [DOI: 10.1115/1.4034319]

Direct Testing of Gradual Postpeak Softening of Fracture Specimens of Fiber Composites Stabilized by Enhanced Grip Stiffness and Mass

Static and dynamic analysis of the fracture tests of fiber composites in hydraulically servo-controlled testing machines currently in use shows that their grips are much too soft and light for observing the postpeak softening. Based on static and dynamic analysis of the test setup, far stiffer and heavier grips are proposed. Tests of compact-tension fracture specimens of woven carbon-epoxy laminates prove this theoretical conclusion. Sufficiently stiff grips allow observation of a stable postpeak, even under load-point displacement control. Dynamic stability analysis further indicates that stable postpeak can be observed under CMOD control provided that a large mass is rigidly attached to the current soft grips. The fracture energy deduced from the area under the measured complete load-deflection curve with stable postpeak agrees closely with the fracture energy deduced from the size effect tests of the same composite. Previous suspicions of dynamic snapback in the testing of composites are dispelled. So is the previous view that fracture mechanics was inapplicable to the fiber-polymer composites.

The Enigma of Large-Scale Permeability of Gas Shale: Pre-Existing or Frac-Induced?

The existing commercial programs for simulation of hydraulic fracturing (aka fracking, or frac) of gas (or oil) shale predict parallel vertical cracks to spread in vertical parallel planes, with no lateral branching. These cracks emanate from the perforation clusters on the horizontal wellbore casing, typically spaced 10 m apart or more. For such a large spacing, the rate of gas production observed at the wellhead can be explained only upon making the hypothesis that the large-scale (or regional) permeability of shale is (even at 3 km depth) about 10,000 times higher than the gas permeability of shale measured in the lab on drilled (nondried) shale cores under confining pressures corresponding to shale at the depth of about 3 km. This hypothesis has recently been rendered doubtful by a new three-phase medium theory that takes into account the body forces due to pressure gradients of pore water diffusing into the pores. This theory predicts the fracking to produce a dense system of branched vertical hydraulic cracks with the spacing of about 0.1 m. This value matches the crack spacing deduced from the gas production rate at wellhead based on the actual lab-measured permeability. It is calculated that, to boost the permeability 10,000 times, the width of the pre-existing open (unfilled) natural cracks or joints (whose ages are distributed from one to several hundred million years) would have to be about 2.8lm (not counting possible calcite deposits in the cracks). But this width is improbably high because, over the geologic time span, the shale must exhibit significant primary and secondary creep or flow. It is shown that the creep must close all the cracks tightly (except for residual openings of the order of 10 nm) even if the cracks are propped open by surface asperities. The inevitability of secondary creep (or steady-state flow) is explained theoretically by activation of new creep sites at stress concentrations caused by prior creep deformation. The time of transition from primary to secondary creep is taken equal to the Maxwell time estimate from geology. The overall conclusion is that the 10,000-fold increase of large-scale permeability is most likely not pre-existing but fracinduced. Although this conclusion will make little difference for long-term forecasts, it would make a major difference for the understanding and control of the frac process. [DOI: 10.1115/1.4036455]

Recent Advances in Global Fracture Mechanics of Growth of Large Hydraulic Crack Systems in Gas or Oil Shale: A Review

This chapter reviews the recent progress toward computer simulation of the growth of vast systems of branched hydraulic cracks needed for the efficient extraction of gas or oil from shale strata. It is emphasized that, to achieve significant gas extraction, the spacing of parallel hydraulic cracks must be on the order of 0.1 m, which means that the fracturing of the entire fracking stage would require creating about a million vertical cracks. Another emphasized feature is that the viscous flow of fracking water along the hydraulic cracks must be combined with Darcy diffusion of a large amount of water into the pores and flaws in shale. The fracture mechanics on the global scale is handled by the crack band model with gradual postpeak softening and a localization limiter in the form of a material characteristic length. Small scale computer simulations demonstrate that the computational approach produces realistically looking results.