A variational minimization formulation for hydraulically induced fracturing in elastic-plastic solids

TL;DR

The paper develops a rigorous variational framework for hydraulically induced fracturing in elastic-plastic porous media by coupling phase-field fracture with a Drucker–Prager-type plasticity model and a Darcy–Biot-type fluid transport. A rate-type potential is formulated, decomposing energy and dissipation into solid, fluid, fracture, and plastic contributions, and a history-based fracture driving force ensures ductile fracture behavior through a modified energy balance. The model employs an $H(\mathrm{div})$-conforming finite-element discretization with enhanced-strain to overcome locking and uses a local return-mapping scheme for plastic updates, all integrated within a space-time, incremental variational framework. Numerical examples illustrate the method’s capabilities, showing how plasticity and fluid transport influence fracture initiation and propagation, as well as the significant effect of fracture-driving forces on crack length and internal pressures. Overall, the framework provides a robust tool for simulating hydraulically induced fractures in ductile porous media with potential applications in oil-and-gas operations and geomechanics.

Abstract

A variational modeling framework for hydraulically induced fracturing of elastic-plastic solids is developed in the present work. The developed variational structure provides a global minimization problem. While fracture propagation is modeled by means of a phase-field approach to fracture, plastic effects are taken into account by using a Drucker-Prager-type yield-criterion function. This yield-criterion function governs the plastic evolution of the fluid-solid mixture. Fluid storage and transport are described by a Darcy-Biot-type formulation. Thereby the fluid storage is decomposed into a contribution due to the elastic deformations and one due to the plastic deformations. A local return mapping scheme is used for the update of the plastic quantities. The global minimization structure demands a $H($div$)$-conforming finite-element formulation. Furthermore this is combined with an enhanced-assumed-strain formulation in order to overcome locking phenomena arising from the plastic deformations. The robustness and capabilities of the presented framework will be shown in a sequence of numerical examples.

Figures (17)

• Figure 1: The unknown fields for porous-elastic-plastic solids at fracture. The boundary $\partial {\cal B}$ is decomposed into Dirichlet and Neumann parts for the displacement$\partial {\cal B}_{\boldsymbol{\mathnormal u}} \cup \partial {\cal B}_{\boldsymbol{\mathnormal t}}$, the fluid flux$\partial {\cal B}_{\Inbb h} \cup \partial {\cal B}_\mu$ and the fracture phase-field$\partial {\cal B}_d \cup \partial {\cal B}_{\gothic k}$. For the fracture phase-field, zero Neuman boundary conditions are assumed. Here $c$ is a constant depending on the model formulation.
• Figure 2: Schematic representation of a) fluid flow in porous medium according to Dracy's law and b) within developing fractures according to Poiseuille--type law with fracture opening $w$.
• Figure 3: Hardening function $\beta(\alpha;d=0)$ with linear hardening $h\ne0$ (solid line) and without linear hardening $h=0$ (dashed line), where $\alpha^*=(\tfrac{h}{\sigma_y}+\omega)^{-1}$ in a). In b) the yield function in two-dimensional hydrostatic-deviatoric plane with and without regularization (dashed/solid lines) where $s^*_\text{max} = s_\text{max} - \sqrt{\tfrac{2}{3}} q_1$. In green with hardening and no hardening in purple.
• Figure 4: Visualization of the different energy contributions to the crack driving history field ${\mathcal{H}}$. For the representation of ${\mathcal{H}}$ in \ref{['eq:fracevol2']}$_2$ only the elastic energy and the energy arising from the hardening, as show in a) and b), will drive the crack. For ideal plasticity the plastic energy vanishes ($\psi^0_\text{plast}=0$) leading to pure brittle fracture, see a). By using \ref{['eq:crackdrive']} the crack is driven by the elastic energy and the full plastic work, see c).
• Figure 5: Rigid footing test: Geometry and boundary conditions. Bottom is mechanically fixed, left and right edge is mechanically fixed in horizontal direction. Bottom, left and right is impermeable.