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ABSTRACT:Using the classical pulse decay test to measure the permeability of tight rock such as serpentinized harzburgite can be time-consuming, often requiring hours or even days. This prolonged duration not only complicates experimental control but also introduces difficulties in maintaining stable environmental conditions. To address such challenges, a fast permeability measurement method has been developed based on an analytical solution that approximates the pressure distribution in the test specimen using parabolic arcs. This solution yields a simple linear regression formula, enabling rapid interpretation of rock permeability using data from only the initial stage of the pulse decay test. In this study, the proposed method is validated by numerical simulations using synthesized pulse decay test data. In addition, an experimental validation of this method using a serpentinized harzburgite is also presented. It is shown that the method is not only faster but also more accurate than the classical method, which ignores the storage of the rock specimen. 

ABSTRACT:Creation of a fracture network in a hydraulic fracturing process is essential for subsurface energy extraction and CO2 sequestration. It is facilitated by reactivation of pre-existing intersecting weak layers and cemented cracks in the rock. In this study, a poromechanical model is developed for the hydraulic fracturing process in rocks containing such pre-existing weak layers. Based on the mixture theory, the crack band model is used to simulate the growth of a crack system. The governing equations with the parameters for hydromechanical coupling are derived, to describe the evolution of the opening and branching of cracks caused by water injection. Microplane model M7 is adopted to characterize the deformation and fracturing of the solid skeleton of the rock, and the Poiseuille law is used to characterize fluid flow through the hydraulic fractures. Numerical simulations are performed to reproduce and interpret recently published laboratory-scale hydraulic fracturing experiments conducted at Los Alamos National Laboratory (LANL). In these experiments, the rock was represented by confined plaster slabs containing orthogonal intersecting weak layers of higher porosity. Numerical simulations reveal how poromechanical characteristics such as the Biot coefficient and the fluid injection rate lead to various typical fracture modes observed in the experiments. These modes include formation of one dominant planar crack or various orthogonal fracture networks. 

ABSTRACT:Long-term deep sequestration of CO2-rich brine in deep formations of ultramafic rock (e.g. Oman serpentinized harzburgite) will be feasible only if a network of hydraulic cracks could be produced and made to grow for years and decades. Fraccing of gas- or oil-bearing shales has a similar objective. The following points are planned to be made in the presentation in Golden. 1) A branching of fracture can be analyzed only if the fracture is modeled by a band with triaxial tensorial damage, for which the new smooth Lagrangian crack band model is effective. 2) To achieve a progressive growth of the fracture network one will need to manipulate the osmotic pressure gradients by changing alkali metal ion concentration in pore fluid. 3) A standardized experimental framework to measure rock permeability at various ion concentrations and various osmotic pressure gradients is needed, and will be presented. 1 INTRODUCTIONCarbon dioxide (CO2) emissions by human activities is the largest contributor to global warming; therefore, effective carbon sequestration technologies attract great amount of interest. One emerging and promising technology for storing CO2 in the subsurface permanently is through carbon mineralization in mafic and ultramafic rock (Kelemen and Matter, 2008). Despite the abundance of these types of rock in the Earth's upper crust (Matter et al., 2016), the rate of this process in nature is too slow to reduce CO2 emissions effectively (Seifritz, 1990). One of the key challenges to achieve a sustainable and large-scale storage of CO2 by mineralization is to engineer a progressive growth of a fracture network conveying water with dissolved CO2 to reach a gradually increasing volume of the mafic rock formation. The CO2 rich water often cannot penetrate the tight matrix of silica-rich serpentinized harzburgites because under high concentrations of CO2, the wetting angle of CO2 -bearing water-rock-rock interface exceeds the critical value of 60 degrees. Therefore, the presence of a family of cracks is the only means by which CO2 -bearing fluids can interact with matrix of ultramafic rock (Bruce Watson and Brenan, 1987). Lateral fracture branching from a major fracture provides a sustainable fluid pathway and therefore is essential for continued rock-water geochemical reactions that lead to mineralization of carbonate minerals. Realistic computational modeling of hydraulic fractures in peridotite or basalt must involve lateral fracture branching and account for stress distribution changes between solid and fluid phases under constant tectonic stress, triggered by pore exposure to fluid pressure in hydraulic cracks. 

Abstract Generation of a large network of hydraulic cracks is of key importance not only for the success of fracking of shale but also for the recent scheme of sequestration of CO2 in deep formations of basalt and peridotite, which are mafic and ultramafic rocks that combine chemically with CO2. In numerical simulation of the creation of a fracture network in porous rock, an important goal is to enhance the rock permeability. The objective of this article is to calculate the effect of osmotic pressure gradients caused by gradients of concentration of the ions of Ca, Mg, Na, etc. on the effective permeability of the rock. The basic differential equations are formulated, and their explicit solutions for appropriate initial and boundary conditions are obtained under certain plausible simplifications. The main result is explicit approximate formulas for the critical time before which no water permeation through a test specimen can be observed. Depending on various parameters, this time can be unacceptably long, which is manifested as a zero water outflow. The solution may also explain the unreasonably small permeability values reported for some shales. 

Abstract A preceding 2023 study argued that the resistance of a heterogeneous material to the curvature of the displacement field is the most physically realistic localization limiter for softening damage. The curvature was characterized by the second gradient of the displacement vector field, which includes the material rotation gradient, and was named the “sprain” tensor, while the term “spress” is here proposed as the force variable work-conjugate to “sprain.” The partial derivatives of the associated sprain energy density yielded in the preceeding study, sets of curvature resisting self-equilibrated nodal sprain forces. However, the fact that the sprain forces had to be applied on the adjacent nodes of a finite element greatly complicated the programming and extended the simulation time in a commercial code such as abaqus by almost two orders of magnitude. In the present model, Smooth Lagrangian Crack Band Model (slCBM), these computational obstacles are here overcome by using finite elements with linear shape functions for both the displacement vector and for an approximate displacement gradient tensor. A crucial feature is that the nodal values of the approximate gradient tensor are shared by adjacent finite elements. The actual displacement gradient tensor calculated from the nodal displacement vectors is constrained to the approximate displacement gradient tensor by means of a Lagrange multiplier tensor, either one for each element or one for each node. The gradient tensor of the approximate gradient tensor then represents the approximate third-order displacement curvature tensor, or Hessian of the displacement field. Importantly, the Lagrange multiplier behaves as an externally applied generalized moment density that, similar to gravity, does not affect the total strain-plus-sprain energy density of material. The Helmholtz free energy of the finite element and its associated stiffness matrix are formulated and implemented in a user’s element of abaqus. The conditions of stationary values of the total free energy of the structure with respect to the nodal degrees-of-freedom yield the set of equilibrium equations of the structure for each loading step. One- and two-dimensional examples of crack growth in fracture specimens are given. It is demonstrated that the simulation results of the three-point bend test are independent of the orientation of a regular square mesh, capture the width variation of the crack band, the damage strain profile across the band, and converge as the finite element mesh is refined. 

ABSTRACT:The mechanism of formation of natural cracks in sedimentary rocks in the geologic past is an important problem in hydraulic fracturing. Why are the natural cracks roughly parallel and equidistant, and why is the spacing in the order of 10 cm rather than 1 cm or 100 cm? Fracture mechanics alone cannot answer these questions. Here it is proposed that fracture mechanics must be coupled with the diffusion of solute ions (Na+ and Cl− are considered here), driven by an osmotic pressure gradient. Parallel equidistant cracks are considered to be subcritical and governed by the Charles-Evans law. The evolution in solute concentration also affects the solvent pressure in the pores and cracks, altering the resistance to frictional sliding. Only steady-state propagation and periodic cracks are studied. An analytical solution of the crack spacing as a function of the properties of the rock as well as the solvent and solute, and the imposed far-field deformation is obtained. Finally, the stability of the growth of parallel cracks is proven by examining the second variation of free energy. Stability of the periodic growth state is also considered. 1 INTRODUCTIONThe deep layers of sedimentary rocks such as shale and sandstone are usually intersected by systems of nearly parallel natural cracks either filled by mineral deposits or closed by creep over a million year life span. Their spacing is roughly uniform and is on the order of 0.1 m (rather than 1 m or 0.01m). These cracks likely play an important role in hydraulic fracturing for gas or oil recovery (aka fracking, fraccing or frac) (Rahimi-Aghdam et al., 2019, e.g.). Therefore, understanding the mechanism of their formation in the distant geologic past is of interest.What controls the spacing of the nearly parallel cracks in shale? According to the fracture mechanics alone, the crack spacing is arbitrary. If propagating parallel equidistant cracks are in a critical state, stability analysis shows that many cracks would have to stop growing, causing a great increase of their average spacing, which was obviously not the case (Bažant et al., 2014). 

Abstract The crack band model, which was shown to provide a superior computational representation of fracture of quasibrittle materials (in this journal, May 2022), still suffers from three limitations: (1) The material damage is forced to be uniform across a one-element wide band because of unrestricted strain localization instability; (2) the width of the fracture process zone is fixed as the width of a single element; and (3) cracks inclined to rectangular mesh lines are represented by a rough zig-zag damage band. Presented is a generalization that overcomes all three, by enforcing a variable multi-element width of the crack band front controlled by a material characteristic length l0. This is achieved by introducing a homogenized localization energy density that increases, after a certain threshold, as a function of an invariant of the third-order tensor of second gradient of the displacement vector, called the sprain tensorη, representing (in isotropic materials) the magnitude of its Laplacian (not expressible as a strain-gradient tensor). The continuum free energy density must be augmented by additional sprain energy Φ(l0η), which affects only the postpeak softening damage. In finite element discretization, the localization resistance is effected by applying triplets of self-equilibrated in-plane nodal forces, which follow as partial derivatives of Φ(l0η). The force triplets enforce a variable multi-element crack band width. The damage distribution across the fracture process zone is non-uniform but smoothed. The standard boundary conditions of the finite element method apply. Numerical simulations document that the crack band propagates through regular rectangular meshes with virtually no directional bias. 

Abstract Whereas various simplistic microplane models of limited applicability, defined by stress–strain curves on the microplane, can function as either explicit or implicit, the explicit‐to‐implicit conversion of realistic versatile microplane models for plain or fiber‐reinforced concrete, shale and composites has remained a challenge for quarter century. The reason is that these realistic models use microplane stress–strain boundaries defined by inequalities. Here, we show how the conversion can be easily achieved on the microplane level and then transferred to a tangent stiffness tensor or an inelastic stiffness tensor to be used in Newton–Raphson iterations within a loading step. To ensure convergence, a minor adjustment in the M7 algorithm is introduced to achieve continuity. Power‐law convergence, almost quadratic in most cases, is also demonstrated. Seven examples of crack‐band finite element simulations of challenging laboratory tests document nearly identical implicit and explicit results, as well as good match of test data. Three of them, including the vertex effect in compression‐torsion tests, pure Mode II shear fracture, and the “gap test” of the crack‐parallel compression effect on Mode I load‐deflection curve, have not been reproduced by other models before. The coding of implicit M7 subroutine, usable in, for example, UMAT of ABAQUS, is posted for a free download. 

The 2023 smooth Lagrangian Crack-Band Model (slCBM), inspired by the 2020 invention of the gap test, prevented spurious damage localization during fracture growth by introducing the second gradient of the displacement field vector, named the “sprain,” as the localization limiter. The key idea was that, in the finite element implementation, the displacement vector and its gradient should be treated as independent fields with the lowest ( $C_0$) continuity, constrained by a second-order Lagrange multiplier tensor. Coupled with a realistic constitutive law for triaxial softening damage, such as microplane model M7, the known limitations of the classical Crack Band Model were eliminated. Here, we show that the slCBM closely reproduces the size effect revealed by the gap test at various crack-parallel stresses. To describe it, we present an approximate corrective formula, although a strong loading-path dependence limits its applicability. Except for the rare case of zero crack-parallel stresses, the fracture predictions of the line crack models (linear elastic fracture mechanics, phase-field, extended finite element method (XFEM), cohesive crack models) can be as much as 100% in error. We argue that the localization limiter concept must be extended by including the resistance to material rotation gradients. We also show that, without this resistance, the existing strain-gradient damage theories may predict a wrong fracture pattern and have, for Mode II and III fractures, a load capacity error as much as 55%. Finally, we argue that the crack-parallel stress effect must occur in all materials, ranging from concrete to atomistically sharp cracks in crystals.