The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

Plasticity : Modeling & Computation

Since the beginning of the US shale gas revolution in 2005, the development of unconventional oil and gas resources has gathered tremendous pace around the world. This book provides a comprehensive overview of the key geologic, geophysical, and engineering principles that govern the development of unconventional reservoirs. The book begins with a detailed characterization of unconventional reservoir rocks: their composition and microstructure, mechanical properties, and the processes controlling fault slip and fluid flow. A discussion of geomechanical principles follows, including the state of stress, pore pressure, and the importance of fractures and faults. After reviewing the fundamentals of horizontal drilling, multi-stage hydraulic fracturing, and stimulation of slip on pre-existing faults, the key factors impacting hydrocarbon production are explored. The final chapters cover environmental impacts and how to mitigate hazards associated with induced seismicity. This text provides an essential overview for students, researchers, and industry professionals interested in unconventional reservoirs.

Unconventional Reservoir Geomechanics: Shale Gas, Tight Oil, and Induced Seismicity

In this, the twentieth Campbell Memorial lecture, I propose to discuss that aspect of metallurgy which has come to be known as mechanical metallurgy. This kind of metallurgy comprises our efforts to understand the relationships among the measured mechanical properties of metals and their mechanical behavior in service, and to understand the way in which these mechanical properties and service characteristics are controlled by chemical composition and structure. This field was not one which received a large amount of consideration from Professor Campbell, but I am sure that if he were alive today it would command his attention. In his day, the factors that influence the structure of alloys engaged the talents of some of the best men, and Professor Campbell and his contemporaries nobly inaugurated the work which has given us such a complete understanding of how to control the structure of metals, that many of us now feel free to turn to other problems. Of course this does not mean that no problems in that field remain to be solved. Rather, progress there has been so great that it now seems profitable to turn to other fields. I have chosen to concentrate on the problem of trying to understand quantitatively how alloying elements in solid solution, and undissolved compounds distributed in these solid solutions, affect mechanical properties. Broadly there are two aspects to what we mean by mechanical properties. These are succinctly called strength and ductility. By strength we mean the resistance of a substance to distortion or fracture, and by ductility we mean how much we may distort it before it fractures. There are many ways to measure these. I have chosen to concentrate on the simple tension test, for I believe it yields as much or more information than any other test. Indentation hardness, which has been widely used, yields much less information, and those who have used it only have missed much of interest. Even the tension test alone is inadequate to complete understanding, but by and large other tests are useful to supplement what the tension test can tell us. This applies particularly to measurements of strength, for which the tension test is almost wholly adequate; but it does not apply so well to ductility, where the conditions of loading are much more important and much less well understood in their action. There are two aspects to what we mean by strength. These are, first, resistance to flow, and second, resistance to fracture [1]. Both are functions of numerous variables, and can only be plotted on two-dimensional space by ignoring or holding constant all the variables but one. Both must be thought of as surfaces in multi-dimensional space. Among the variables which may influence both the flow and fracture stress, aside from composition and structure which are our principal concern today, we may list strain, time rate of straining, and temperature as the ones that most commonly are taken into consideration. In Fig. 1, the resistance to flow and the resistance to fracture are plotted against strain. I like to refer to these two curves simply as the flow curve and the fracture curve, for the effect of deformation on Reprinted from Transactions of the American Society for Metals, 36, 30–60 (1946). Copyright 1946 by the American Society for Metals, Cleveland, Ohio.

Strength and Ductility

Enriched Galerkin finite elements for coupled poromechanics with local mass conservation

We formulate the problem of fully coupled transient fluid flow and quasi-static poroelasticity in arbitrarily fractured, deformable porous media saturated with a single-phase compressible fluid. The fractures we consider are hydraulically highly conductive, allowing discontinuous fluid flux across them; mechanically they act as finite-thickness shear deformation zones prior to failure (i.e., non-slipping and non-propagating), leading to ‘apparent discontinuity’ in strain and stress across them. Local nonlinearity arising from pressure-dependent permeability of fractures is also included. Taking advantage of typically high aspect ratio of a fracture, we do not resolve transversal variations and instead assume uniform flow velocity and simple shear strain within each fracture, rendering the coupled problem numerically more tractable. Fractures are discretized as lower-dimensional zero-thickness elements tangentially conforming to unstructured matrix elements. A hybrid-dimensional, equal-low-order, two-field mixed finite element method is developed, which is free from stability issues for a drained coupled system. The fully implicit backward Euler scheme is employed for advancing the fully coupled solution in time, and the Newton-Raphson scheme is implemented for linearization. We show that the fully discretized system retains a canonical form of a fracture-free poromechanical problem; the effect of fractures is translated to the modification of some existing terms as well as the addition of several terms to the capacity, conductivity and stiffness matrices, therefore, allowing the development of independent subroutines for treating fractures within a standard computational framework. Our computational model provides more realistic inputs for some fracture-dominated poromechanical problems like fluid-induced seismicity.

Fully Coupled Nonlinear Fluid Flow and Poroelasticity in Arbitrarily Fractured Porous Media: A Hybrid‐Dimensional Computational Model

Fluid perturbations play a pivotal role in triggering earthquakes. However, the role of fluid in the coseismic rupture process remains largely unknown. To this end, we develop a 2-D fully dynamic spontaneous rupture model for fluid-induced earthquakes. The effect of fluid in the preseismic quasi-static regime is modeled as either pore pressure diffusion or fully coupled poroelasticity, using our Jin and Zoback (2017, https://doi.org/10.1002/2017JB014892) computational model. The two approaches lead to radically different predictions on the time of earthquake nucleation. Correspondingly, the evolved fluid pressure or poroelastic stress on the fault, together with the spatially altered density of the fluid-saturated hosting rock, is passed to the dynamic regime. Under the assumption of an undrained coseismic fluid-solid system, we discretize the fully dynamic Cauchy equation of motion subjected to an exact fault contact constraint using a split-node finite element method in space and an implicit Newmark family finite difference method in time. Within each time step, a fully implicit Newton-Raphson scheme is implemented iteratively for linearizing the fully discrete equations. Within each Newton iteration, a physics-based and nonstationary preconditioner is designed to accelerate the convergence of the selected generalized minimal residual method iterative linear solver. The effect of fluid is highlighted throughout the discretization and computational procedures. Finally, by conducting a numerical experiment, we illustrate that a fully coupled poroelastic model can lead to significantly different predictions on coseismic rupture behaviors and wavefields compared to a decoupled model. Our computational model can also serve as one of the earliest full-physics modeling tool for fluid-induced earthquakes.

Fully Dynamic Spontaneous Rupture Due to Quasi-Static Pore Pressure and Poroelastic Effects: An Implicit Nonlinear Computational Model of Fluid-Induced Seismic Events

Modeling Induced Seismicity: Co-Seismic Fully Dynamic Spontaneous Rupture Considering Fault Poroelastic Stress

Fault-controlled damage zones have important implications for fluid flow in fractured reservoirs. We present here a methodology for identification of fault-controlled damage zones in microseismic data using a dataset from the Haynesville shale. We first develop a discrete fracture network (DFN) model of the pre-existing faults shear activated during hydraulic fracturing stimulation. We utilize the DFN to reveal fracture concentrations, diagnostic of fault damage zones. The DFN also reveals a planar zone of diminished microseismic events which we hypothesize are correlative with the fault itself. In support of this interpretation, we show that fault density in the damage zones drops off with distance from the fault according to a power law F=F0r which has been observed in faultcontrolled damage zones at other locations.

Identification of Fault-Controlled Damage Zones in Microseismic Data – An Example from the Haynesville Shale

Including a Stochastic Discrete Fracture Network into One-Way Coupled Poromechanical Modeling of Injection-Induced Shear Re-Activation

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