Simulation and interpretation of compaction patterns around boreholes excavated in high-porosity rocks

Abstract

Boreholes excavated in porous rocks exhibit a variety of inelastic deformation patterns. This paper investigates how nucleation, propagation, and coalescence of compaction zones, including those resulting from compaction localization, affect the mechanics of excavated boreholes in high-porosity rock formations. For this purpose, a recently developed controllability framework able to differentiate among multiple modes of compaction banding is used to guide the interpretation of numerical simulations for boreholes excavated in porous rock. The goal is to explain the link between heterogeneous deformation patterns forming around the excavated zone and in situ stress conditions. To carry out the analyses, the theory is combined with a strain-hardening constitutive law calibrated against Berea sandstone laboratory evidence, thus capturing the stress-dependence of both homogeneous and heterogeneous compaction. The resulting simulations of idealized boreholes show that the inelastic compaction around the excavated zone depends dramatically on the far-field stress state, leading to transitions from isotropic-distributed plastic deformation modes under in-plane isotropic stress states, to dog-ear-, slot-, and/or butterfly-shaped inelasticity in the presence of strongly anisotropic in situ stress. It was found that each of these scenarios can be explained in terms of controllability indices computed at locations exhibiting inelastic response, thus establishing a link between the computed modes of global borehole deformation and multiple types of compaction localization processes (e.g., compactive shear bands, shear-enhanced compaction bands, and pure compaction bands).

Appendices

A Mesh sensitivity analysis

Several numerical treatments have been proposed to overcome mesh-dependent issues in boundary value problems, including multiscale approach [25], enhanced assumed strain FEM [10], micro-polar theory [3], nonlocal models [8, 36], and strain gradient models [1, 37]. However, the above methods are usually complicated and require sophisticated numerical implementations. In this paper, a simple and computationally inexpensive viscoplastic regularization of the Perzyna-type [49] is used. The enhancement of the constitutive model has been implemented by modifying the plastic flow rule in the following manner:

$$\begin{aligned} \dot{\varepsilon }^{vp}_{ij}=\Phi (F)\frac{\partial Q}{\partial \sigma _{ij}}=\frac{1}{\omega }\left[ \frac{\left\langle F\right\rangle }{P_{co}}\right] \frac{\partial Q}{\partial \sigma _{ij}} \end{aligned}$$

(A.1)

where, $P_{co}$ is the initial hydrostatic yielding stress, the symbol $\left\langle \bullet \right\rangle$ indicates the Macaulay brackets, and $\omega$ is the viscosity factor. The value of the viscosity factor $\omega$ has been set to $\omega$=100MPas, with the aim to obtain a rate-dependent response as close as possible to that of the rate-independent version of the model, while preserving in the meantime the mesh independence of the numerical solutions.

The model has been implemented in the ABAQUS Finite Element code through a UMAT interface following a strategy similar to that detailed by Das and Buscarnera [16]. A plane strain compression sample is used to verify the viscoplastic regularization, as shown in Fig. 17. The formation of inelastic deformation bands is captured through the plastic volumetric strain map. Mesh objectivity is clearly evident in the simulations through the identical overall mechanical response (Fig. 17a) and consistent thickness of localization strain (Fig. 17b and c). It is worth noting that although the simulations of Berea sandstone at the material point level do not show strain-softening behavior, the occurrence of compactive deformation bands under full-field condition can induce softening, which can in turn be linked to a local loss of strength and negative values of the strain localization indices $A\left( \theta \right)$ and $H-H_\chi$.

Fig. 17

Analyses of the mesh dependency finite element numerical simulations for a plane strain compression sample: a global responses; b and c comparable simulations with coarse and finer meshes, showing the objectivity of band thickness

B Validation of the numerical model

Typically, when a borehole is just excavated, it is subjected to major and minor far-field principal stresses, i.e., $\sigma _H$ and $\sigma _h$. The corresponding stress distribution around the excavation area can be derived using the Kirsch equations that assume isotropic elasticity [27, 69] as follows:

$$\begin{aligned} \sigma _{\theta \theta }= & {} \frac{1}{2}\left( \sigma _h+\sigma _H\right) \left( 1+\frac{R^2}{r^2}\right)-\frac{1}{2}\left( \sigma _h-\sigma _H\right) \left( 1+\frac{3R^4}{r^4}\right) \cos 2\theta -P_w\frac{R^2}{r^2} \end{aligned}$$

(B.1)

$$\begin{aligned} \sigma _{rr}= & {} \frac{1}{2}\left( \sigma _h+\sigma _H\right) \left( 1-\frac{R^2}{r^2}\right) +\frac{1}{2}\left( \sigma _h-\sigma _H\right)\left( 1-\frac{4R^2}{r^2} +\frac{3R^4}{r^4}\right) \cos 2\theta +P_w\frac{R^2}{r^2} \end{aligned}$$

(B.2)

where, $\sigma _{\theta \theta }$ and $\sigma _{rr}$ are the tangential and radial stress, respectively, R is the radius of the borehole, r is the distance from the center of the borehole, $\theta$ is the angular direction measured counter clockwise from the $\sigma _h$ direction, and $P_w$ is the difference between the fluid pressure in the borehole and the pore pressure in the surrounding rocks ($P_w$ is ignored in this paper). Figure 18a shows the schematic diagram for stress state around a borehole after excavation.

Figure 18b shows the comparison between the analytical solutions and the stresses obtained from the established numerical model for four different stress conditions ($\theta =0$, i.e., the $\sigma _h$ direction, is chosen for comparison). The measured stresses from the numerical simulations match very well with the analytical solutions, indicating the applicability of the proposed numerical model.

Fig. 18

Schematic diagram for stress state around a borehole after excavation is shown in a. Comparison between the stresses obtained from the numerical simulations and the analytical solutions ($\theta =0$) are presented with b $\sigma _H$ = 40 MPa, $\sigma _h$= 20 MPa, and $\sigma _V$= 10 MPa; c $\sigma _H$= 40 MPa, $\sigma _h$ = 20 MPa, and $\sigma _V$= 20 MPa; d  $\sigma _H$ = 40 MPa, $\sigma _h$= 30 MPa, and $\sigma _V$= 10 MPa; and e $\sigma _H$= 40 MPa, $\sigma _h$= 30 MPa, and $\sigma _V$ = 20 MPa