Fig 6 - uploaded by Waiching Sun

Content may be subject to copyright.

# Schematic of geometry and boundary conditions for the fracture problem. (a) The domain with explicitly modeled cavities; and (b) its homogenized counterpart.

Source publication

This paper presents an immersed phase field model designed to predict the fracture-induced flow due to brittle fracture in vuggy porous media. Due to the multiscale nature of pores in vuggy porous material, crack growth may connect previously isolated pores which lead to flow conduits. This mechanism has important implications for many applications...

## Contexts in source publication

**Context 2**

... contrast, our second domain in Fig. 6(b) is a homogenized representation of Fig. 6(a), where all the Table 1: The major and minor radii of the explicitly modeled elliptical cavities in Fig. ...

**Context 3**

... contrast, our second domain in Fig. 6(b) is a homogenized representation of Fig. 6(a), where all the Table 1: The major and minor radii of the explicitly modeled elliptical cavities in Fig. ...

**Context 4**

... contrast, our second domain in Fig. 6(b) is a homogenized representation of Fig. 6(a), where all the Table 1: The major and minor radii of the explicitly modeled elliptical cavities in Fig. ...

**Context 5**

... [1993] and by assuming that the matrix shares the same material properties of those chosen 321 for Fig. 6(a), the effective bulk modulus (K hom ) and shear modulus (µ hom ) for the homogenized represen- ...

**Context 6**

... addition, since all the cavities in Fig. 6(a) are completely isolated, we adopt the following effective to those measured from the shear tests. After the first peaks shown in Fig. 8(a) and Fig. 8(b), cracks start to initiate from the tips of the pre-existing flaw since they experience higher stress concentration ...

## Similar publications

## Citations

... In order to model the fracture flow in a fluid-infiltrating porous media, we adopt the permeability enhancement approach that approximates the water flow inside the fracture as the flow between two parallel plates [138][139][140][141]: ...

... (2.46)] and relative permeability [Eq. (2.39)] yield relatively low pore water pressure inside the notch similar to the results shown in[141]. The phase transition process of pore water begins once the temperature at the top surface reaches the freezing temperature θ m in both the damaged and undamaged regions, however, since the proposed driving force for the Allen-Cahn equation in Eq. (2.27) leads to an intense growth of ice inside the fracture (i.e., ice lens) such that the phase field c tends to evolve faster inside the damaged region [Fig. ...

Global challenges associated with extreme climate events and increasing energy demand require significant advances in our understanding and predictive capability of coupled multi- physical processes across spatial and temporal scales. While classical approaches based on the mixture theory may shed light on the macroscopic poromechanics simulations, accurate forward predictions of the complex behavior of phase-changing geomaterials cannot be made without understanding the underlying coupling mechanisms among constituents at the microstructural scale. To precisely predict the multi-physical behaviors originated by smaller scales, fundamental understandings of the micromechanical interactions among phase constituents are crucial. Hence, this dissertation discusses mathematical and computational frameworks designed to capture coupled thermo-hydro-mechanical-fracture processes in phase-changing porous media that incorporate necessary microscopic details. To achieve this goal, this dissertation aims to introduce a practical way to investigate how phase transition and evolving microstructural attributes at small scales affect the applicability of meso- or macroscopic finite element simulations, by leveraging the phase field method to represent the regularized interfaces of phase constituents.
Firstly, a multi-phase-field microporomechanics model is presented to model the growth and thaw of ice lenses. In specific, we extend the field theory for ice lens that is not restricted to one-dimensional space. The key idea is to represent the state of the pore fluid and the evolution of freezing-induced fracture via two distinct phase field variables coupled with balance laws, which leads to an immersed approach where both the homogeneous freezing and ice lensing are distinctively captured. Secondly, a thermo-hydro-mechanical theory for geological media with thermally non-equilibrated constituents is presented, where we develop an operator-split framework that updates the temperature of each constituent in an asynchronous manner. Here, the existence of an effective medium is hypothesized, in which the constituents exhibit different temperatures while heat exchange among the phases is captured via Newton’s law of cooling. Thirdly, an immersed phase field model is introduced to predict fluid flow in fracturing vuggy porous media, where crack growth may connect previously isolated voids and form flow conduits. In this approach, we present a framework where the phase field is not only used as a damage parameter for the solid skeleton but also as an indicator of the pore space, which enables us to analyze how crack growth in vuggy porous matrix affects the flow mechanism differently compared to the homogenized effective medium while bypassing the needs of partitioning the domain and tracking the moving interface. Finally, we present a new phase field fracture theory for higher-order continuum that can capture physically justified size effects for both the path-independent elastic responses and the path-dependent fracture. Specifically, we adopt quasi-quadratic degradation function and linear local dissipation function such that the physical size dependence are insensitive to the fictitious length scale for the regularized interface, which addresses the numerical needs to employ sufficiently large phase field length scale parameter without comprising the correct physical size effect.

... While leak-off terms may have great impact on the propagating fracture and it surroundings, ,most approaches to modeling leak-off are heuristic. For example, some studies include a leak-off term as an input quantity to the model based on field observations [75][76][77]. ...

Recent experiments and physical evidence show that fractured porous media feature cracks and fluid capillary networks at various scales. We present a multi-physics macro-scale model that can distinguish between the mechanics and transport interactions. The porous media is represented by a poroelastic domain incorporating non-local damage and non-local transport. The evolution of each of these processes is governed by a unique length scale and driving force, which allows for better flexibility in modeling hydraulic-deformation network systems. For consistency the governing equations of the non-local multi-physics problem are derived from thermodynamics principles. Hence, a four-field () mixed finite element formulation is developed. The non-linear system of equations is linearized and solved using Newton’s method and a backward Euler scheme is used to evolve the system in time, for which a consistent Jacobian matrix and residual vector are derived analytically. Two benchmark examples are investigated: hydraulic fracturing of rocks and soil consolidation. The numerical examples show the viability of this model, and how the variation of the two length scales and damage parameters can be used to describe different physical phenomena.

... In order to model the fracture flow in a fluid-infiltrating porous media, we adopt the permeability enhancement approach that approximates the water flow inside the fracture as the flow between two parallel plates [Miehe and Mauthe, 2016, Mauthe and Miehe, 2017, Wang and Sun, 2017, Suh and Sun, 2021b: ...

This article presents a multi-phase-field poromechanics model that simulates the growth and thaw of ice lenses and the resultant frozen heave and thaw settlement in multi-constituent frozen soils. In this model, the growth of segregated ice inside the freezing-induced fracture is implicitly represented by the evolution of two phase fields that indicate the locations of segregated ice and the damaged zone, respectively. The evolution of two phase fields are driven by the driving forces that capture the physical mechanisms of ice and crack growths respectively, while the phase field governing equations are coupled with the balance laws such that the coupling among heat transfer, solid deformation, fluid diffusion, crack growth, and phase transition can be observed numerically. Unlike phenomenological approaches that indirectly captures the freezing influence on the shear strength, the multi-phase-field model introduces an immersed approach where both the homogeneous freezing and the ice lens growth are distinctively captured by the freezing characteristic function and the driving force accordingly. Verification and validation examples are provided to demonstrate the capacities of the proposed models.

... It has the value of COI I = 0 for intact porous solid with shrinking pores or fractured porous solid with closed cracks, and the value COI I = 1 for intact solid with enlarging pores or fractured solid with open cracks. The utilization of the Stokes-Darcy approach to describe the flow during crack propagation in saturated vuggy porous media was introduced by Suh and Sun [126]. With pre-existing larger-scale saturated pores, their approach was capable of capturing the coalescence of saturated brittle cracks with the pre-existing pores, which also lead to triggering both redistribution of flow and macroscopic softening. ...

Motivated by the successful implementation of the phase-field method (PFM) to simulate complicated fracture patterns at moderate computational costs in solid materials, many research groups have started since 2012 applying the PFM to model hydraulic fracturing, especially that occurs in porous geomaterials. These research works have contributed to the development of the PFM from different perspectives, especially in connection with the mathematical formulations of the hydro-mechanical processes and the numerical algorithms to solve the emerging coupled problems. In this regard, the underlying paper aims to review the significant scientific works that utilized the PFM to model fracturing caused mainly by fluid injection in a certain porous domain and, less common, by fluid extraction (e.g., drying) from a certain porous domain. This includes reviewing different approaches for deriving the phase-field evolution formulation (e.g. Ginzburg–Landau approach, thermodynamically consistent approaches, and microforce-based approach) and reviewing several formulations for the stiffness degradation function and that of the crack driving force. Besides, the paper will go through several methods to estimate the crack aperture width, in addition to reviewing different numerical approaches and implementations. The paper will be concluded by presenting a number of open topics and challenges to be addressed in future works.

This article presents a multi-phase-field poromechanics model that simulates the growth and thaw of ice lenses and the resultant frozen heave and thaw settlement in multi-constituent frozen soils. In this model, the growth of segregated ice inside the freezing-induced fracture is implicitly represented by the evolution of two phase fields that indicate the locations of segregated ice and the damaged zone, respectively. The evolution of two phase fields is induced by their own driving forces that capture the physical mechanisms of ice and crack growths respectively, while the phase field governing equations are coupled with the balance laws such that the coupling among heat transfer, solid deformation, fluid diffusion, crack growth, and phase transition can be observed numerically. Unlike phenomenological approaches that indirectly capture the freezing influence on the shear strength, the multi-phase-field model introduces an immersed approach where both the homogeneous freezing and the ice lens growth are distinctively captured by the freezing characteristic function and the driving force accordingly. Verification and validation examples are provided to demonstrate the capacities of the proposed models.

We present a hybrid model/model-free data-driven approach to solve poroelasticity problems. Extending the data-driven modeling framework originated from \citet{kirchdoerfer2016data}, we introduce one model-free and two hybrid model-based/data-driven formulations capable of simulating the coupled diffusion-deformation of fluid-infiltrating porous media with different amounts of available data. To improve the efficiency of the model-free data search, we introduce a distance-minimized algorithm accelerated by a k-dimensional tree search. To handle the different fidelities of the solid elasticity and fluid hydraulic constitutive responses, we introduce a hybridized model in which either the solid and the fluid solver can switch from a model-based to a model-free approach depending on the availability and the properties of the data. Numerical experiments are designed to verify the implementation and compare the performance of the proposed model to other alternatives.

We present a SE(3)-equivariant graph neural network (GNN) approach that directly predicting the formation factor and effective permeability from micro-CT images. FFT solvers are established to compute both the formation factor and effective permeability, while the topology and geometry of the pore space are represented by a persistence-based Morse graph. Together, they constitute the database for training, validating, and testing the neural networks. While the graph and Euclidean convolutional approaches both employ neural networks to generate low-dimensional latent space to represent the features of the micro-structures for forward predictions, the SE(3) equivariant neural network is found to generate more accurate predictions, especially when the training data is limited. Numerical experiments have also shown that the new SE(3) approach leads to predictions that fulfill the material frame indifference whereas the predictions from classical convolutional neural networks (CNN) may suffer from spurious dependence on the coordinate system of the training data. Comparisons among predictions inferred from training the CNN and those from graph convolutional neural networks (GNN) with and without the equivariant constraint indicate that the equivariant graph neural network seems to perform better than the CNN and GNN without enforcing equivariant constraints.