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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. 

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. 

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 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. 

Motivated by the extraordinary strength of nacre, which exceeds the strength of its fragile constituents by an order of magnitude, the fishnet statistics became in 2017 the only analytically solvable probabilistic model of structural strength other than the weakest-link and fiberbundle models. These two models lead, respectively, to the Weibull and Gaussian (or normal) distributions at the large-size limit, which are hardly distinguishable in the central range of failure probability. But they differ enormously at the failure probability level of 10−6 , considered as the maximum tolerable for engineering structures. Under the assumption that no more than three fishnet links fail prior to the peak load, the preceding studies led to exact solutions intermediate between Weibull and Gaussian distributions. Here massive Monte Carlo simulations are used to show that these exact solutions do not apply for fishnets with more than about 500 links. The simulations show that, as the number of links becomes larger, the likelihood of having more than three failed links up to the peak load is no longer negligible and becomes large for fishnets with many thousands of links. A differential equation is derived for the probability distribution of not-too-large fishnets, characterized by the size effect, the mean and the coefficient of variation. Although the large-size asymptotic distribution is beyond the reach of the Monte Carlo simulations, it can by illuminated by approximating the large-scale fishnet as a continuum with a crack or a circular hole. For the former, instability is proven via complex variables, and for the latter via a known elasticity solution for a hole in a continuum under antiplane shear. The fact that rows or enclaves of link failures acting as cracks or holes can form in the largescale continuum at many random locations necessarily leads to the Weibull distribution of the large fishnet, given that these cracks or holes become unstable as soon they reach a certain critical size. The Weibull modulus of this continuum is estimated to be more than triple that of the central range of small fishnets. The new model is expected to allow spin-offs for printed materials with octet architecture maximizing the strength–weight ratio.