Thermal Impact on the Excavation Damage Zone Around a Supported Drift Using the 2nd Gradient Model

Abstract

The temperature increase induced by radioactive waste decay generates the thermal pressurisation around the excavation damage zone (EDZ), and the excess pore pressure could induce fracture re-opening and propagation. Shear strain localisation in band mode leading to the onset of micro-/macro-cracks can be always evidenced before the fracturing process from the lab experiments using advanced experimental devices. Hence, the thermal effects on the rock behaviour around the EDZ could be modelled with the consideration of development of shear bands. A coupled local 2nd gradient model with regularisation technique is implemented, considering the thermo-hydro-mechanical (THM) couplings in order to well reproduce the shear bands. Furthermore, the thermo-poro-elasticity framework is summarized to validate the implemented model. The discrepancy of thermal dilation coefficient between solid and fluid phases is proved to be the significant parameter leading to the excess pore pressure. Finally, an application of a heating test based on Eurad Hitec benchmark exercise with a drift supported by a liner is studied. The strain localisation induced by thermal effects is properly reproduced. The plasticity and shear bands evolutions are highlighted during the heating, and the shear bands are preferential to develop in the minor horizontal principal stress direction. Different shear band patterns are obtained with changing gap values between the drift wall and the liner. A smaller gap between the wall and the liner can limit the development of shear bands.

Highlights

• The formulation of a coupled local 2nd gradient model considering the thermo-hydro-mechanical (THM) couplings.

• Validation of the model with comparison with analytical solution of thermo-elastic problem.

• The prediction of strain localisation pattern induced by thermal effects around a large scale drift.

• The analysis of the gap distance (between the drift wall and the liner) on the strain localisation process under the thermal loading.

Appendix A Stiffness matrix

The stiffness matrix of the flow problem is:

$$\begin{aligned}&K^{\tau _1}_{{WW}{(3\times 3)}}= \begin{bmatrix} k_{11}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}} &{} k_{12}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}} &{} K_{WW_{1,3}}\\ k_{21}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}} &{} k_{22}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}} &{} K_{WW_{2,3}}\\ 0 &{} 0 &{} K_{WW_{3,3}}\\ \end{bmatrix} \end{aligned}$$

(A1)

where

$$\begin{aligned}&K^{\tau _1}_{WW_{1,3}}=\frac{\rho _w^{\tau _1}}{\chi _w}\bigg [\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}(k_{11}g_1+k_{12}g_2) \end{aligned}$$

(A2)

$$\begin{aligned}&+\frac{1}{\mu _w^{\tau _1}}(k_{11}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}}+k_{12}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}})\bigg ]\nonumber \\&K^{\tau _1}_{WW_{2,3}}=\frac{\rho _w^{\tau _1}}{\chi _w}\bigg [\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}(k_{21}g_1+k_{22}g_2)\nonumber \\&+\frac{1}{\mu _w^{\tau _1}}(k_{21}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}}+k_{22}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}})\bigg ]\nonumber \\&K^{\tau _1}_{WW_{3,3}}=\rho _w^{\tau _1}\bigg [(\frac{\dot{p}_w^{\tau _1}}{\chi _w}-\alpha _w^{\tau _1}\dot{T}^{\tau _1}\nonumber \\&-\frac{\dot{p}_w^{\tau _1}}{K_s})(\frac{b-\phi ^{\tau _1}}{K_s}+\frac{\phi ^{\tau _1}}{\chi _w}) \end{aligned}$$

(A3)

$$\begin{aligned}&+(\frac{\phi ^{\tau _1}}{\chi _w\cdot \Delta t}+\frac{b-\phi ^{\tau _1}}{K_s\cdot \Delta t})\nonumber \\&+\frac{b}{\chi _w}(\frac{\dot{p}_w^{\tau _1}}{K_s}+\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}})\bigg ] \end{aligned}$$

(A4)

The stiffness of the thermal problem is:

$$\begin{aligned}&K^{\tau _1}_{{TT}{(3\times 3)}}= \begin{bmatrix} \lambda _s(1-\phi ^{\tau _1})+\lambda _w\phi ^{\tau _1} &{} 0 &{} K_{TT_{1,3}}\\ 0 &{} \lambda _s(1-\phi ^{\tau _1})+\lambda _w\phi ^{\tau _1} &{} K_{TT_{2,3}}\\ 0 &{} 0 &{} K_{TT_{3,3}}\\ \end{bmatrix} \end{aligned}$$

(A5)

where

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{TT_{1,3}}=\alpha _d(\lambda _s-\lambda _w)(b-\phi ^{\tau _1})\frac{\partial T^{\tau _1}}{\partial x_1^{\tau _1}}\\&-c_{p,w}\rho _w^{\tau _1}f_{w,1}^{\tau _1}(1-\alpha _w^{\tau _1}(T^{\tau _1}-T_0))\\&-k_{11}c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)\bigg [\frac{1}{{\mu _w^2}^{\tau _1}}\frac{\partial \mu _w^{\tau _1}}{\partial T^{\tau _1}}(\frac{\partial {p_w}^{\tau _1}}{\partial {x_1}^{\tau _1}}+{\rho _w}^{\tau _1}g_1)\\&+(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_1+\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}})\alpha _w^{\tau _1}\bigg ]\rho _w^{\tau _1}-k_{12}c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)\\&\cdot \bigg [\frac{1}{{\mu _w^2}^{\tau _1}}\frac{\partial \mu _w^{\tau _1}}{\partial T^{\tau _1}}(\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}}+\rho _w^{\tau _1}g_2)\\&+(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_2+\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}})\alpha _w^{\tau _1}\bigg ]\rho _w^{\tau _1} \end{aligned} \end{aligned}$$

(A6)

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{TT_{2,3}}=\alpha _d(\lambda _s-\lambda _w)(b-\phi ^{\tau _1})\frac{\partial T^{\tau _1}}{\partial x_2^{\tau _1}}\\&-c_{p,w}\rho _w^{\tau _1}f_{w,2}^{\tau _1}(1-\alpha _w^{\tau _1}(T^{\tau _1}-T_0))\\&-k_{21}c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)\bigg [\frac{1}{{\mu _w^2}^{\tau _1}}\frac{\partial \mu _w^{\tau _1}}{\partial T^{\tau _1}}(\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}}+\rho _w^{\tau _1}g_1)\\&+(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_1+\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}})\alpha _w^{\tau _1}\bigg ]\rho _w^{\tau _1}-k_{22}c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)\\&\cdot \bigg [\frac{1}{{\mu _w^2}^{\tau _1}}\frac{\partial \mu _w^{\tau _1}}{\partial T^{\tau _1}}(\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}}+\rho _w^{\tau _1}g_2)\\&+(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_2+\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}})\alpha _w^{\tau _1}\bigg ]\rho _w^{\tau _1} \end{aligned} \end{aligned}$$

(A7)

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{TT_{3,3}}=c_{p,w}\bigg [-\rho _w^{\tau _1}(b-\phi ^{\tau _1})(T^{\tau _1}-T_0)\alpha _d\frac{1}{\Delta t}\\&+\rho _w^{\tau _1}(b-\phi ^{\tau _1})(\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}\\&+\frac{\dot{p}_w^{\tau _1}}{K_s^{\tau _1}}-\alpha _d\dot{T}^{\tau _1})-\rho _w^{\tau _1}\alpha _w^{\tau _1}(b-\phi ^{\tau _1})(T^{\tau _1}-T_0)(\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}\\&+\frac{\dot{p}_w^{\tau _1}}{K_s^{\tau _1}}\\&-\alpha _d\dot{T}^{\tau _1})+\rho _w^{\tau _1}\alpha _d (T^{\tau _1}-T_0)(\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}+\frac{\dot{p}_w^{\tau _1}}{K_s^{\tau _1}}-\alpha _d\dot{T}^{\tau _1})\\&-\rho _w^{\tau _1}\phi ^{\tau _1}\\&\cdot (T^{\tau _1}-T_0)(\frac{\partial \alpha _w^{\tau _1}}{\partial T^{\tau _1}}\dot{T}^{\tau _1}+\alpha _w^{\tau _1}\frac{1}{\Delta t})+\rho _w^{\tau _1}\phi ^{\tau _1}(\frac{\dot{P}_w^{\tau _1}}{\chi _w}-\alpha _w^{\tau _1}\dot{T}^{\tau _1})\\&-\rho _w^{\tau _1}(b-\phi ^{\tau _1})(T^{\tau _1}-T_0)\alpha _d(\frac{\dot{P}_w^{\tau _1}}{\chi _w}-\alpha _w^{\tau _1}\dot{T}^{\tau _1})\\&-\rho _w^{\tau _1}\alpha _w\phi ^{\tau _1}(T^{\tau _1}\\&-T_0)(\frac{\dot{P}_w^{\tau _1}}{\chi _w}-\alpha _w^{\tau _1}\dot{T}^{\tau _1})+\rho _w^{\tau _1}\phi ^{\tau _1}\frac{1}{\Delta t}-\rho _w^{\tau _1}\alpha _w\phi ^{\tau _1}\dot{T}^{\tau _1}\\&-\rho _w^{\tau _1}(b-\phi ^{\tau _1})\alpha _d\dot{T}^{\tau _1}-\rho _w^{\tau _1}\alpha _w\phi ^{\tau _1}(T^{\tau _1}-T_0)\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}\\&+\rho _w^{\tau _1}\phi ^{\tau _1}\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}\\&-\rho _w^{\tau _1}(b-\phi ^{\tau _1})\alpha _d(T^{\tau _1}-T_0)\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}\bigg ]\\&+c_{p,s}\bigg [-\rho _s^{\tau _1}\frac{\phi ^{\tau _1}}{1-\phi ^{\tau _1}}(b\\&-\phi ^{\tau _1})\alpha _d\dot{T}^{\tau _1}+\rho _s(b-\phi ^{\tau _1})\alpha _d(T^{\tau _1}-T_0)(\frac{b-1}{1-\phi ^{\tau _1}}\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}\\&+\frac{b-\phi ^{\tau _1}}{1-\phi ^{\tau _1}}\frac{\dot{p}_w^{\tau _1}}{K_s^{\tau _1}}-\frac{b-\phi ^{\tau _1}}{1-\phi ^{\tau _1}}\alpha _d\dot{T}^{\tau _1})+\rho _s^{\tau _1}(1-\phi ^{\tau _1})\frac{1}{\Delta t}\bigg ]\\ \end{aligned} \end{aligned}$$

(A8)

The stiffness matrices of the coupling between the flow and mechanical processes are:

$$\begin{aligned} K^{\tau _1}_{{MW}{(3\times 4)}}= \begin{bmatrix} K_{MW_{1,1}} &{} K_{MW_{1,2}} &{} K_{MW_{1,3}} &{} K_{MW_{1,4}}\\ K_{MW_{2,1}} &{} K_{MW_{2,2}} &{} K_{MW_{2,3}} &{} K_{MW_{2,4}}\\ A+\dot{M}_w^{\tau _1} &{} 0 &{} 0 &{} A+\dot{M}_w^{\tau _1} \end{bmatrix} \end{aligned}$$

(A9)

where

$$\begin{aligned}&K^{\tau _1}_{MW_{1,1}}=-k_{11}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}} \end{aligned}$$

(A10)

$$\begin{aligned}&K^{\tau _1}_{MW_{1,2}}=-k_{12}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}}+f_{w,2}^{\tau _1} \end{aligned}$$

(A11)

$$\begin{aligned}&K^{\tau _1}_{MW_{1,3}}=-k_{11}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}} \end{aligned}$$

(A12)

$$\begin{aligned}&K^{\tau _1}_{MW_{1,4}}=-k_{12}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}}-f_{w,1}^{\tau _1} \end{aligned}$$

(A13)

$$\begin{aligned}&K^{\tau _1}_{MW_{2,1}}=-k_{21}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _2}}-f_{w,2}^{\tau _1} \end{aligned}$$

(A14)

$$\begin{aligned}&K^{\tau _1}_{MW_{2,2}}=-k_{22}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}} \end{aligned}$$

(A15)

$$\begin{aligned}&K^{\tau _1}_{MW_{2,3}}=-k_{21}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}}+f_{w,1}^{\tau _1} \end{aligned}$$

(A16)

$$\begin{aligned}&K^{\tau _1}_{MW_{2,4}}=-k_{22}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}} \end{aligned}$$

(A17)

$$\begin{aligned}&A=\rho _w^{\tau _1}\bigg [(\frac{\dot{p}_w^{\tau _1}}{\chi _w}-\alpha _w^{\tau _1}\dot{T}^{\tau _1}-\frac{\dot{p}_w^{\tau _1}}{K_s})(b-\phi ^{\tau _1})+b(\frac{1}{\Delta t}-\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}})\bigg ] \end{aligned}$$

(A18)

$$\begin{aligned} K^{\tau _1}_{{WM}{(4\times 3)}}= \begin{bmatrix} 0 &{} 0 &{} -1\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} -1 \end{bmatrix} \end{aligned}$$

(A19)

The stiffness matrices of the coupling between the thermal and mechanical processes are:

$$\begin{aligned} K^{\tau _1}_{{MT}{(3\times 4)}}= \begin{bmatrix} K_{MT_{1,1}} &{} K_{MT_{1,2}} &{} K_{MT_{1,3}} &{} K_{MT_{1,4}}\\ K_{MT_{2,1}} &{} K_{MT_{2,2}} &{} K_{MT_{2,3}} &{} K_{MT_{2,4}}\\ D+\dot{S}_T^{\tau _1} &{} 0 &{} 0 &{} D+\dot{S}_T^{\tau _1} \end{bmatrix} \end{aligned}$$

(A20)

where

$$\begin{aligned}&K^{\tau _1}_{MT_{1,1}}=-(\lambda _s-\lambda _w)(b-\phi ^{\tau _1})\frac{\partial T^{\tau _1}}{\partial x_1^{\tau _1}}\nonumber \\&-c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{11}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}} \end{aligned}$$

(A21)

$$\begin{aligned}&K^{\tau _1}_{MT_{1,2}}=-c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{12}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}}+f_{T,2}^{\tau _1} \end{aligned}$$

(A22)

$$\begin{aligned}&K^{\tau _1}_{MT_{1,3}}=-c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{11}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}} \end{aligned}$$

(A23)

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{MT_{1,4}}=-(\lambda _s-\lambda _w)(b-\phi ^{\tau _1})\frac{\partial T^{\tau _1}}{\partial x_1^{\tau _1}}\\&-c_{p,w}\rho _w^{\tau _1}(T^{\tau 1}-T_0)k_{12}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}}\\&-f_{T,1}^{\tau _1} \end{aligned} \end{aligned}$$

(A24)

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{MT_{2,1}}=-(\lambda _s-\lambda _w)(b-\phi ^{\tau _1})\frac{\partial T^{\tau _1}}{\partial x_2^{\tau _1}}\\&-c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{21}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}}\\&-f_{T,2}^{\tau _1} \end{aligned} \end{aligned}$$

(A25)

$$\begin{aligned}&K^{\tau _1}_{MT_{2,2}}=-c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{22}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}} \end{aligned}$$

(A26)

$$\begin{aligned}&K^{\tau _1}_{MT_{2,3}}=-c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{21}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}}+f_{T,1}^{\tau _1} \end{aligned}$$

(A27)

$$\begin{aligned}&K^{\tau _1}_{MT_{2,4}}=-(\lambda _s-\lambda _w)(b-\phi ^{\tau _1})\frac{\partial T^{\tau _1}}{\partial x_2}\nonumber \\&-c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{22}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}} \end{aligned}$$

(A28)

$$\begin{aligned} \begin{aligned}&D=c_{p,w}\bigg [b\rho _w^{\tau _1}(T^{\tau _1}-T_0)(\frac{1}{\Delta t}-\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}})\\&-\rho _w^{\tau _1}(b-\phi ^{\tau _1})(T^{\tau _1}-T_0)(\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}\\&+\frac{\dot{p}_w^{\tau _1}}{K_s^{\tau _1}}-\alpha _d\dot{T}^{\tau _1})+\rho _w^{\tau _1}(b-\phi ^{\tau _1})(T^{\tau _1}-T_0)(\frac{\dot{p}_w^{\tau _1}}{\chi _w}-\alpha _w^{\tau _1}\dot{T}^{\tau _1})\\&+\rho _w^{\tau _1}(b-\phi ^{\tau _1})\dot{T}^{\tau _1}+\rho _w^{\tau _1}(b-\phi ^{\tau _1})(T^{\tau _1}-T_0)\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}\bigg ]\\&-c_{p,s}\rho _s^{\tau _1}\bigg [\frac{1-b}{1-\phi ^{\tau _1}}-(b-\phi ^{\tau _1})\bigg ]\dot{T}^{\tau _1}\\ \end{aligned} \end{aligned}$$

(A29)

$$\begin{aligned} K^{\tau _1}_{{TM}{(4\times 3)}}= \begin{bmatrix} 0 &{} 0 &{} -\alpha _d^{\tau _1}(C_{1111}+C_{1122})\\ 0 &{} 0 &{} -\alpha _d^{\tau _1}(C_{1211}+C_{1222})\\ 0 &{} 0 &{} -\alpha _d^{\tau _1}(C_{2111}+C_{2122})\\ 0 &{} 0 &{} -\alpha _d^{\tau _1}(C_{2211}+C_{2222})\\ \end{bmatrix} \end{aligned}$$

(A30)

The stiffness matrices of the coupling between the flow and thermal processes are:

$$\begin{aligned} K^{\tau _1}_{{WT}{(3\times 3)}}= \begin{bmatrix} K_{WT_{1,1}} &{} K_{WT_{1,2}} &{} K_{WT_{1,3}}\\ K_{WT_{2,1}} &{} K_{WT_{2,2}} &{} K_{WT_{2,3}}\\ 0 &{} 0 &{} K_{WT_{3,3}} \end{bmatrix} \end{aligned}$$

(A31)

$$\begin{aligned}&K^{\tau _1}_{WT_{1,1}}=c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{11}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}} \end{aligned}$$

(A32)

$$\begin{aligned}&K^{\tau _1}_{WT_{1,2}}=c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{12}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}} \end{aligned}$$

(A33)

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{WT_{1,3}}=-(\lambda _s-\lambda _w)\frac{b-\phi ^{\tau _1}}{K_s}\frac{\partial T^{\tau _1}}{\partial x_1^{\tau _1}}\\&-c_{p,w}\frac{\rho _w^{\tau _1}}{\chi _w}f_{w,1}^{\tau _1}(T^{\tau _1}-T_0)\\&+k_{11}c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_1\\&+\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}})\frac{\rho _w^{\tau _1}}{\chi _w}\\&+k_{12}c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_2\\&+\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}})\frac{\rho _w^{\tau _1}}{\chi _w}\\ \end{aligned} \end{aligned}$$

(A34)

$$\begin{aligned}&K^{\tau _1}_{WT_{2,1}}=c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{21}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}} \end{aligned}$$

(A35)

$$\begin{aligned}&K^{\tau _1}_{WT_{2,2}}=c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)k_{22}\frac{\rho _w^{\tau _1}}{\mu _w^{\tau _1}} \end{aligned}$$

(A36)

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{WT_{2,3}}=-(\lambda _s-\lambda _w)\frac{b-\phi ^{\tau _1}}{K_s}\frac{\partial T^{\tau _1}}{\partial x_2^{\tau _1}}\\&-c_{p,w}\frac{\rho _w^{\tau _1}}{\chi _w}f_{w,2}^{\tau _1}(T^{\tau _1}-T_0)\\&+k_{11}c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_1\\&+\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}})\frac{\rho _w^{\tau _1}}{\chi _w}\\&+k_{12}c_{p,w}\rho _w^{\tau _1}(T^{\tau _1}-T_0)(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_2\\&+\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}})\frac{\rho _w^{\tau _1}}{\chi _w}\\ \end{aligned} \end{aligned}$$

(A37)

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{WT_{3,3}}=c_{p,w}\bigg [\rho _w^{\tau _1}(b-\phi ^{\tau _1})(T^{\tau _1}-T_0)\frac{1}{K_s\cdot \Delta t}+\rho _w^{\tau _1}(b-\phi ^{\tau _1})(T^{\tau _1}\\&-T_0)(\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}+\frac{\dot{p}_w^{\tau _1}}{K_s^{\tau _1}}-\alpha _d\dot{T}^{\tau _1})(\frac{1}{\chi _w}-\frac{1}{K_s})+\frac{\rho _w^{\tau _1}}{\chi _w \cdot \Delta t}\phi ^{\tau _1}(T^{\tau _1}\\&-T_0)+\rho _w^{\tau _1}\frac{b-\phi ^{\tau _1}}{K_s}(\frac{\dot{p}_w^{\tau _1}}{\chi _w}-\alpha _w^{\tau _1}\dot{T}^{\tau _1})(T^{\tau _1}-T_0)+\frac{\rho _w^{\tau _1}}{\chi _w}\phi ^{\tau _1}(\frac{\dot{p}_w^{\tau _1}}{\chi _w}\\&-\alpha _w^{\tau _1}\dot{T}^{\tau _1})(T^{\tau _1}-T_0)+\frac{\rho _w^{\tau _1}}{\chi _w}\phi ^{\tau _1}\dot{T}^{\tau _1}+\rho _w^{\tau _1}(b-\phi ^{\tau _1})\frac{\dot{T}^{\tau _1}}{K_s}\\&+\frac{\rho _w^{\tau _1}}{\chi _w}\phi ^{\tau _1}(T^{\tau _1}-T_0)\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}+\rho _w^{\tau _1}\frac{b-\phi ^{\tau _1}}{K_s}(T^{\tau _1}-T_0)\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}\bigg ]\\&+c_{p,s}\bigg [\rho _s^{\tau _1}\frac{b-1}{K_s}(T^{\tau _1}-T_0)\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}}+\rho _s^{\tau _1}\frac{b-\phi ^{\tau _1}}{K_s}\frac{\phi ^{\tau _1}}{1-\phi ^{\tau _1}}\dot{T}^{\tau _1}\bigg ]\\ \end{aligned} \end{aligned}$$

(A38)

$$\begin{aligned} K^{\tau _1}_{{TW}{(3\times 3)}}= \begin{bmatrix} 0 &{} 0 &{} K_{TW_{1,3}}\\ 0 &{} 0 &{} K_{TW_{2,3}}\\ 0 &{} 0 &{} K_{TW_{3,3}} \end{bmatrix} \end{aligned}$$

(A39)

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{TW_{1,3}}=-k_{11}\bigg [\frac{1}{{\mu _w^2}^{\tau _1}}\frac{\partial \mu _w^{\tau _1}}{\partial T^{\tau _1}}(\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}}+\rho _w^{\tau _1}g_1)\\&+(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_1+\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}})\\&\cdot \alpha _w^{\tau _1}\bigg ]\rho _w^{\tau _1}-k_{12}\bigg [\frac{1}{{\mu _w^2}^{\tau _1}}\frac{\partial \mu _w^{\tau _1}}{\partial T^{\tau _1}}(\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}}+\rho _w^{\tau _1}g_2)\\&+(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_2 +\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}})\alpha _w^{\tau _1}\bigg ]\rho _w^{\tau _1} \end{aligned} \end{aligned}$$

(A40)

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{TW_{2,3}}=-k_{21}\bigg [\frac{1}{{\mu _w^2}^{\tau _1}}\frac{\partial \mu _w^{\tau _1}}{\partial T^{\tau _1}}(\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}}+\rho _w^{\tau _1}g_1)\\&+(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_1+\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_1^{\tau _1}})\\&\cdot \alpha _w^{\tau _1}\bigg ]\rho _w^{\tau _1}-k_{22}\bigg [\frac{1}{{\mu _w^2}^{\tau _1}}\frac{\partial \mu _w^{\tau _1}}{\partial T^{\tau _1}}(\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}}+\rho _w^{\tau _1}g_2)\\&+(\frac{2\rho _w^{\tau _1}}{\mu _w^{\tau _1}}g_2 +\frac{1}{\mu _w^{\tau _1}}\frac{\partial p_w^{\tau _1}}{\partial x_2^{\tau _1}})\alpha _w^{\tau _1}\bigg ]\rho _w^{\tau _1} \end{aligned} \end{aligned}$$

(A41)

$$\begin{aligned} \begin{aligned}&K^{\tau _1}_{TW_{3,3}}=-\rho _w^{\tau _1}\bigg [(\frac{\dot{p}_w^{\tau _1}}{\chi _w}-\alpha _w^{\tau _1}\dot{T}^{\tau _1}-\frac{\dot{p}_w^{\tau _1}}{K_s})[(b-\phi ^{\tau _1})\alpha _d+\alpha _w^{\tau _1}\phi ^{\tau _1}]\\&+(\frac{\alpha _w^{\tau _1}}{\Delta t}+\frac{\partial \alpha _w^{\tau _1}}{\partial T^{\tau _1}}\dot{T}^{\tau _1})\phi ^{\tau _1}+b\alpha _d(\frac{\dot{p}_w^{\tau _1}}{K_s}+\frac{{\dot{\Omega }}^{\tau _1}}{\Omega ^{\tau _1}})\bigg ] \end{aligned} \end{aligned}$$

(A42)

$G3_{(2\times 3)}^{\tau _1}$, same as $G1_{(2\times 4)}^{\tau _1}$ and $G2_{(2\times 3)}^{\tau _1}$, being the additional contribution of gravity volume force reads:

$$\begin{aligned} G1_{(2\times 4)}&= \begin{bmatrix} -b\rho _w^{\tau _1}g_1 &{} 0 &{} 0 &{} -b\rho _w^{\tau _1}g_1\\ -b\rho _w^{\tau _1}g_2 &{} 0 &{} 0 &{} -b\rho _w^{\tau _1}g_2 \end{bmatrix} \end{aligned}$$

(A43)

$$\begin{aligned} G2_{(2\times 3)}&= \begin{bmatrix} 0 &{} 0 &{} -\rho _w^{\tau _1}\big (\frac{b-\phi ^{\tau _1}}{K_s}+\frac{\phi ^{\tau _1}}{\chi _w}\big )g_1\\ 0 &{} 0 &{} -\rho _w^{\tau _1}\big (\frac{b-\phi ^{\tau _1}}{K_s}+\frac{\phi ^{\tau _1}}{\chi _w}\big )g_2 \end{bmatrix} \end{aligned}$$

(A44)

$$\begin{aligned} G3_{(2\times 3)}&= \begin{bmatrix} 0 &{} 0 &{} \rho _w^{\tau _1}\big [(b-\phi ^{\tau _1})\alpha _d+\alpha _w^{\tau _1}\phi ^{\tau _1}\big ]g_1\\ 0 &{} 0 &{} \rho _w^{\tau _1}\big [(b-\phi ^{\tau _1})\alpha _d+\alpha _w^{\tau _1}\phi ^{\tau _1}\big ]g_2\\ \end{bmatrix} \end{aligned}$$

(A45)