The Role of Temperature in Shear Instability and Bifurcation of Internally Pressurized Deep Boreholes

Abstract

This paper investigates localized shear deformation around a borehole due to internal pressure in the well such as by fluid injection. Using an elasto-visco-plastic formulation combined with damage mechanics for the effect of shear cracking, we first benchmark the model against analytical solutions and then provide bifurcation criteria for the onset of localized cracking at different temperature conditions. We report that at increased temperatures of the rock formation, hot fluid injection promotes shear stimulation, while cold fluid suppresses it. This counter-intuitive result can offer new pathways of effective stimulation in high-temperature environments, like those encountered in enhanced geothermal systems.

Appendices

Appendix A: Case Study for Damage Mechanics During Loading and Unloading

A simple numerical case study on a square body is presented here to illustrate the damage mechanisms described in Sect. 2.3 under loading and unloading tests. The boundary conditions are summarized in Fig. 14. The left boundary of the body is fixed, while the top and the bottom are free boundaries. On the right boundary, as shown in Fig. 14, uniformly distributed stress is applied and only horizontal displacement is allowed. The material is assumed pressure insensitive, and the evolution of internal damage variable (representing local concentration of micro-cracks within the material) follows Eq. 11.

Figure 15a demonstrates the stress–strain curve obtained from loading–unloading procedures. In the loading phase, the modified Young’s modulus of the body, i.e. the slope of the stress–strain curve, decreases due to internal damage evolution with local $\dot{D}$ being positive (see Eq. 11, Eq. 12b). No damage evolution occurs during the unloading phase, and hence, the corresponding stress–strain curve is linear, meaning a constant elastic modulus. The residual strain (small but still visible from Fig. 15a) after the loading–unloading cycle results largely from (irreversible) micro-cracking processes in the material. Damage evolution with respect to normalized time and normalized strain is shown in Fig. 15b, c, respectively. The middle point of the square body is chosen as a representative. The damage increases during loading as $\dot{D}$ stays positive, while it remains a constant during unloading since no micro-cracking occurs.

Fig. 14

A square body with the left boundary fixed and the right boundary subject to loading/unloading. The top boundary and bottom boundary are set free

Fig. 15

Behaviour of the material subject to a loading–unloading cycle: a stress–strain curve and evolution of damage at the central point of the body, with respect to b normalized time and c normalized strain

Appendix B: Derivation of the Bifurcation Characteristic Equation

The incremental constitutive equation in tensorial form is expressed as

$$\begin{aligned} \dot{\tau }_{ik}=C_{ijkl}\dot{\epsilon }_{kl} \end{aligned}$$

(24)

The stiffness tensor $C_{ijkl}$ in 2D configuration has the following properties:

$$\begin{aligned} C_{ij12}&=C_{ij21}, \end{aligned}$$

(25a)

$$\begin{aligned} C_{12kl}&=C_{21kl} \end{aligned}$$

(25b)

For elasto-plastic material, constitutive Eq. 24 is more often rewritten as

$$\begin{aligned} \dot{\sigma }_{11}&=L_{11}\dot{\epsilon }_{11}+L_{12}\dot{\epsilon }_{22}, \end{aligned}$$

(26a)

$$\begin{aligned} \dot{\sigma }_{22}&=L_{21}\dot{\epsilon }_{11}+L_{22}\dot{\epsilon }_{22}, \end{aligned}$$

(26b)

$$\begin{aligned} \dot{\sigma }_{12}&=2G\dot{\epsilon }_{12} \end{aligned}$$

(26c)

where $L_{ij}$ are coefficients dependent on material properties and hardening parameters, including friction coefficient, dilatancy coefficient, hardening modulus, etc. Eq. 24 reads

$$\begin{aligned} \dot{\sigma }_{11}&=C_{1111}\dot{\epsilon }_{11}+(C_{1112}+C_{1121})\dot{\epsilon }_{12}+C_{1122}\dot{\epsilon }_{22}, \end{aligned}$$

(27a)

$$\begin{aligned} \dot{\sigma }_{22}&=C_{2211}\dot{\epsilon }_{11}+(C_{2212}+C_{2221})\dot{\epsilon }_{12}+C_{2222}\dot{\epsilon }_{22}, \end{aligned}$$

(27b)

$$\begin{aligned} \dot{\sigma }_{12}&=C_{1211}\dot{\epsilon }_{11}+(C_{1212}+C_{1221})\dot{\epsilon }_{12}+C_{1222}\dot{\epsilon }_{22} \end{aligned}$$

(27c)

Combining Eq. 26 and Eq. 27, with Eq. 25, we have

$$\begin{aligned} C_{1111}&=L_{11},\, C_{1122}=L_{12},\, C_{2211}=L_{21},\, C_{2222}=L_{22}, \end{aligned}$$

(28a)

$$\begin{aligned} C_{1112}&=C_{1121}=C_{2212}=C_{2221}=0, \end{aligned}$$

(28b)

$$\begin{aligned} C_{1212}&=C_{1221}=G \end{aligned}$$

(28c)

Hence, the condition for localization Eq. 19 can be written as

$$\begin{aligned} \left| \begin{bmatrix} L_{11}n_{1}^2+G n_{2}^2&(L_{12}+G) n_{1} n_{2} \\ (L_{21}+G) n_{1} n_{2}&L_{22}n_{2}^2+G n_{1}^2 \end{bmatrix} \right| =0, \end{aligned}$$

(29)

that is,

$$\begin{aligned} G L_{11}n_{1}^4+(L_{11}L_{22}-L_{12}L_{21}-G L_{12}-G L_{21})n_{1}^2n_{2}^2+G L_{22} n_{2}^4 = 0 \end{aligned}$$

(30)

Substituting ${\mathrm{tan}} \theta = -(n_1/n_2)$ ($\theta$ being the shear band inclination angle) into Eq. 30, we have the characteristic Eq. 20.