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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...

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... or fractured porous medium), geophysics (e.g., magma-mantle interaction), petroleum engineering (e.g., production of hydrocarbons from an underground reservoir), and physiology (e.g., filtration of blood through arterial vessels), among many others. [1][2][3][4][5][6][7][8][9][10][11] Modeling this process requires the coupling of two different systems of partial differential equations-Navier-Stokes or Stokes equations that describe the motion of incompressible free fluid and Darcy's equation to capture the porous medium flow-through a set of transmissibility conditions. This often includes constraints that ensure the overall continuity of the fluid mass and the equilibrium of normal and shear forces exerted at the interface between the two regions. ...

... By focusing on the case where the Reynolds number is low (e.g., Re ≪ 1), this study assumes that the steady-state motion of an incompressible fluid in S is described by the momentum and mass balances in the Stokes equations, 11,45,46 for example, ...

... 66 Nevertheless, since either the modification of Equation (24) or its validation is out of the scope of this study, we assume that the Beavers-Joseph condition holds for the problems of interest, by following the previous works. 1,11,46,67,68 ...

The coupling between free and porous medium flows has received significant attention since it plays an important role in a wide range of problems from fluid-soil interactions to biofluid dynamics. However, modeling this coupled process remains a difficult task as it often involves a domain decomposition algorithm in conjunction with a special treatment at the interface. The problem can become more challenging under non-isothermal conditions because it requires the iterative procedure at every time step to simultaneously meet the transient mass continuity, force equilibrium, and energy balance for the entire system. This article presents a diffuse interface framework for modeling non-isothermal Stokes-Darcy flow and the corresponding finite element formulation that bypasses the need for explicitly splitting the domain into two, which enables the unified treatment for distinct regions with different hydrothermal flow regimes. To achieve this goal, we employ the Allen-Cahn type phase field model to generate the diffuse geometry, where the solution field can be seen as a regularized approximation of the Heaviside indicator function, allowing us to transfer the interface conditions into a set of immersed boundary conditions. Our formulation suggests that the isothermal operator splitting strategy can be adopted without compromising accuracy if the heat and mass transfer processes are decoupled by assuming that the density and viscosity of the phase constituents are independent to the temperature. Numerical examples are also introduced to verify the implementation and to demonstrate the model capacity. K E Y W O R D S coupled Stokes-Darcy flow, diffuse interface, finite element method, heat transport, immersed boundary condition, phase field model 1 INTRODUCTION Infiltration of free fluid through a permeable porous matrix plays an important role in several applications in various fields, such as energy geotechnics (e.g., geothermal energy exploitation), hydrogeology (e.g., fluid flow in vuggy This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

... The leakage may cause additional damage around the fracture [32] which is observed in the damage contours. This leakage is commonly referred to as leak-off [29], and it has been observed in many field cases and experiments [106,119,126]. Other formulations introduce an artificial flux to the hydraulic fracture boundary in order to account for the leak-off phenomena, e.g., phase-field [75], LEFM [117], cohesive element method [15,62], and XFEM [45]. ...

We present a unified non-local damage model for modeling hydraulic fracture processes in porous media, in which damage evolves as a function of fluid pressure. This setup allows for a non-local damage model that resembles gradient-type models without the need for additional degrees of freedom. In other words, we propose a non-local damage formulation at the same cost of a local damage approach. Nonlinear anisotropic permeability is employed to distinguish between the fluid flow velocity in the damage zone and the intact porous media. The permeability evolves as a function of an equivalent strain measure, where its anisotropic evolution behavior is controlled by the direction of principle strain. The length scale of the proposed model is analytically derived as a function of material point variables and is shown to be dependent on the pressure rate. A mixed finite element method is proposed to monolithically solve the coupled displacement–pressure system. The nonlinear system is linearized and solved using Newton’s method with analytically derived consistent Jacobian matrix and residual vector, and the evolution of the system in time is performed by a backward Euler scheme. Numerical examples of 1D and 2D hydraulic fracture problems are presented and discussed. The numerical results show that the proposed model is insensitive to the mesh size as well as the time step size and can well capture the features of hydraulic fracture in porous media.

... 17 The formation and propagation of macroscopic fractures during hydraulic fracturing have been studied in detail. [18][19][20][21][22] Cheng et al. 19 obtained the fracture growth rule of directional hydraulic fracturing technology through physical similarity simulation experiments and numerical analysis, and they revealed the fracture growth control mechanism of directional hydraulic fracturing technology by analyzing the change rule of the r H inside rock mass. Xu et al. 23 conducted triaxial hydraulic fracturing experiments using raw coal samples, and they found that the stress state plays an major role in the development of fractures; however, fractures do not always propagate along the direction of r H . Li et al. 14 conducted hydraulic fracturing experiments using molded coal samples, and they found that in situ stresses were the major factor affecting the development of fractures and that fractures always expanded along the direction of the r H . Renard et al. 24 found that hydraulic fracturing fractures tend to propagate in weaker or low-density areas under high in situ stress differentials. ...

Pores in coal is not only the main space for coalbed methane (CBM) occurrence, but also the space to be opened during CBM recovering. Therefore, the analysis of the impact of hydraulic fracturing on coal pores, especially the change of adsorption pores before and after hydraulic fracturing, is of great significance to the evaluation of hydraulic fracturing effect and CBM recovery. Hydraulic fracturing experiments and low-field nuclear magnetic resonance (NMR) technology were used to analyse changes in the T 2 curve, adsorption pore, and the effects of distance and in-situ stresses on pore modification in coal samples of the Sihe (SH) and the Chengzhuang (CZ) mine before and after Hydraulic fracturing. The results show that hydraulic fracturing can affect pores < 10 nm. The CZ coal samples exhibit stronger heterogeneity than the SH coal samples after hydraulic fracturing, the pore size distribution (PSD) anisotropy of the CZ samples is increased. For the SH coal samples with poor heterogeneity, the effect of hydraulic fracturing on pore transformation depends more on in-situ stresses. The smaller horizontal in-situ stresses difference facilitates the establishment of complex pore networks. After Hydraulic fracturing, when the pore diameter is 30 - 100 nm, the pore volume proportions (PVPs) of the SH samples in the directions of maximum horizontal principal stress (σ H ), minimum horizontal principal stress (σ h ), and vertical stress (σ V ), increase from 43.73% to 64.84%, 59.79%, and 60.16%, respectively. Hydraulic fracturing increases the anisotropy of the PSD of the CZ samples.

... 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.

Hydraulic fracturing is an efficient technology to extract hydrocarbon within natural caves. However, these caves can markedly affect the fracture propagation behavior. This paper proposes a novel hydraulic fracturing model to simulate the fracture propagation in poroelastic media containing the natural cave, utilizing the strengths of the phase-field method. By coupling the Reynolds flow with cubic law in fracture domain, free flow in cave domain, and low-permeability Darcy flow in reservoir domain, the fracture-cave-reservoir flow governing equations are established. The Biot poroelasticity theory and fracture width are the links of hydro-mechanical coupling. The smooth phase-field is introduced to diffuse not only the sharp fracture but also the sharp cave edge. The fully coupling model is solved by a staggered scheme, which independently solves the pressure field and displacement field in inner cycle, and then independently solves the phase field in outer cycle. The proposed model is verified by comparing with the Khristianovic–Geertsma–de Klerk (KGD) model and Cheng's hydraulic fracturing model. Then, the interaction between hydraulic fracture and natural cave is investigated through several two-dimensional and three-dimensional cases. The result shows that the cave effect can make the hydraulic fracture deflect and raise its propagation velocity. Increasing the fracture-cave distance, injection rate, and in situ stress difference can all decline the cave effect. The displayed cases also substantiate the capability and efficiency of the proposed model.

In various subsurface resource development or fluid piping problems, subsurface fluid-filled fractures often appear. Fracture location determination has always been critical in related fields. Acoustic wave reflection at the junction and boundary in the pipeline can carry information about the property of the system. By using the accompanying acoustic wave combined with the water hammer effect, the location of subsurface fractures can be estimated. A numerical fluid flow model for instantaneous shut-in is presented based on water hammer effect. Fluid penetration effects, wellbore storage effect and fluid inertial effect are considered. The fracture location estimation method called cepstral predominant peak (CPP) analysis is first proposed. By cepstral, we means the inverse Fourier transform of the logarithm of the estimated signal spectrum. Also the relationship between instantaneous shut-in pressure and cepstrum response is investigated in detail. To improve the robustness, CPP analysis based on Kaiser windowed cepstrum was used to identify the impulse period of fracture. Compared with original cepstrum, Kaiser windowed cepstrum has the better performance for CPP analysis. The proposed flow model is impactful as it can provide pressure data with known fracture locations. The data can be used to optimize and examine the performance of CPP analysis. A field experiment is conducted to validate the analysis about the acoustic wave in a pipeline system with fractures. The actual instantaneous shut-in pressure for an oil well is obtained. The experiment result shows the CPP analysis can get the fracture location efficiently and accurately, which can provide insights for engineering.

We present a SE(3)-equivariant graph neural network (GNN) approach that directly predicts the formation factor and effective permeability from micro-CT images. Fast Fourier Transform (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 microstructures for forward predictions, the SE(3) equivariant neural network is found to generate more accurate predictions, especially when the training data are 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 (CNNs) 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 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.