# Analytical modeling on the geo-stress and casing damage prevention with the thermo-hydro-mechanical (THM) coupling of multi-field physics

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## Abstract

Understanding the rule of squeezing force on casing changing with the working parameters of wells, such as steam injection pressure, steam injection temperature and bottom hole pressure, is very important to prevent casing damage in the complex thermal recovery process involving the interaction of temperature, stress and seepage field. According to the theory of heat transferring, seepage mechanics and elastic–plastic mechanics, the mathematical model considering seepage-stress-heat transferring coupling of thermal recovery was established. The numerical model of the thermal recovery for a block in GuTong oil field was built, and the stress sensitivity of the rock was tested. The research work indicates that the permeability is more sensible to the stress when the maximum stress is less than 35 MPa. The calculation results of the temperature, pressure and vertical displacement fit well with the monitoring data, and the error is less than 20%. The stress and displacement of well wall achieve peak value in the 7th–9th cycles. In the ninth cycle, the maximum horizontal displacement of the well wall occurs and reaches about 13 cm, and the casing damage is most likely to happen in the 7th–9th production cycles. The equivalent element method was used to calculate the extrusion force on the casing pipe. The pressure on casing under different steam injection rate, temperature of steam and bottom hole pressure was obtained and used for prevention of casing damage during the thermal recovery of heavy oil.

## Keywords

Thermal recovery of heavy oil Geo-stress Numerical simulation THM coupling## List of symbols

- \(\sigma_{\text{ij}}\)
The total stress tensor, MPa

- \(x_{\text{j}}\)
The coordinate at \(j\) direction, m

- \(f_{{{\text{x}}_{\text{i}} }}\)
The volume force of direction \(x_{\text{i}}\), N m

^{−3}- \(\sigma_{\text{ij}}^{'}\)
Effective stress tensor

- \(p\)
Pore pressure, MPa

- \(\varepsilon_{\text{ij}}\)
The strain tensor

- \(u\)
Displacement, cm

- \(G\,{\text{and}}\,\lambda\)
Lame constants

- \(\varepsilon_{\text{v}}\)
Volume strain

- \(u_{\text{i}}\)
The displacement at \(x_{\text{i}}\) direction, cm

- \(K_{\text{ij}}\)
Permeability tensor of the reservoir, μm

^{2}- \(K_{\text{ij}} {}^{0}\)
Initial permeability, μm

^{2}- \(Q_{\text{h}}\)
Energy input and output in unit time and unit volume of the reservoir, kJ m

^{−3}day^{−1}- \(\lambda_{\text{R}}\)
Thermal conductivity of the reservoir rock, kJ m

^{−1}day^{−1}°C^{−1}- \(C\)
Constant volume specific heat, kJ K

^{−1}m^{−3}- \((\rho C)_{\text{R}}\)
Rock heat capacity, kJ m

^{−3}°C^{−1}- \(\phi\)
Porosity of the reservoir

- \(\mu\)
Fluid viscosity, mPa s

- \(L_{\text{V}}\)
The latent heat of steam vaporization, kJ m

^{−3}- \(T_{0}\)
Initial reservoir temperature, °C

- \(\Delta T\)
The difference between saturated steam temperature and the reservoir temperature, °C

- \(e_{\text{in}}\)
The heat injected into the reservoir, kJ day

^{−1}- \(x_{\text{q}}\)
Steam quality at the well bottom

- \(\lambda_{\text{c}}\)
Rock thermal conductivity of the top and bottom, kJ day

^{−1}m^{−1}°C^{−1}- \(E\)
Elastic modulus, MPa

- \(v\)
Poison ratio

- \(\alpha\)
Thermal expansion coefficient, m

^{2}s^{−1}- \({\text{d}}\sigma_{\text{ij}}^{'}\)
The effective stress increment, Mpa

- \(D_{\text{ep}}\)
Elastic–plastic matrix, Mpa

- \({\text{d}}\varepsilon_{\text{ij}}\)
The strain increment

- \(I_{1}^{'}\)
The first non-variable for effective stress

- \(J_{2}^{'}\)
The second effective deviation stress invariant

- \(\sigma_{\text{x}}^{'} ,\,\sigma_{\text{y}}^{'} ,\,\sigma_{\text{z}}^{'}\)
Effective stress components at

*x*,*y*,*z*direction, MPa- \(\tau_{\text{xy}}^{'} ,\,\tau_{\text{yz}}^{'} ,\tau_{\text{zx}}^{'}\)
Effective shear stress components on the surfaces where the normal lines are

*x*,*y*,*z*- \(\beta \,{\text{and}}\,k_{\text{f}}\)
Shear strength parameters, Eq. (15)

- \(c\)
Cohesion, MPa

- \(\phi\)
Internal friction angle of rock

- \(L_{\text{j}}\)
The directional derivative of the boundary

- \(s_{\text{i}}\)
Distribution function for surface force, MPa

- \(g_{\text{i}}\)
Distribution function for the surface displacement, Eq. (17)

- \(\frac{\partial p}{\partial n}\)
Pressure derivative on the direction of outside the normal

- \(f_{1} (x,y,z,t)\)
A known function set artificially to represent the flow rate

- \(p(x,y,z,0)\)
Initial pore pressure of the reservoir, MPa

- \(S_{\text{w}}\)
Water saturation of the reservoir

- \(S_{\text{o}}\)
Oil saturation of the reservoir

- \([K]\)
The stiffness matrix

- \(\{ u\}\)
Node displacements matrix

- \(\{ F_{\text{v}} \}\)
Volume load matrix

- \(\{ F_{\text{s}} \}\)
Surface load matrix

- \(\{ F_{\text{p}} \}\)
Equivalent load matrix derived from pore pressure

- \(\{ F_{\text{T}} \}\)
The thermal load matrix caused by temperature changes

- \(\{ F_{{\upsigma_{0} }} \}\)
The initial stress matrix in situ

- \(\sigma_{\text{v}}\)
The vertical stress, MPa

- \(\sigma_{\text{H}}\)
The larger horizontal stress component, MPa

- \(\sigma_{\text{h}}\)
The smaller horizontal stress component, MPa

- \(p_{\text{wf}}\)
Fluid pressure in well, MPa

- \(p_{\text{co}}\)
The equivalent extrusion force on the casing pipe, Mpa

- \(r_{\text{ci}}^{{}}\)
The inner radius of the casing pipe, m

- \(r_{\text{co}}^{{}}\)
The equivalent outer radius of the thick wall cylinder, m

- \(\sigma_{\text{r}}\)
Radial extrusion force on the casing pipe, MPa

- Subscript \(1,2,3\)
The coordinate axis

*x, y*and*z,*respectively

## Notes

### Acknowledgements

This work was financially supported by National Natural Science Foundation of China (Grant Nos. 41702340 and 41602290), the youth scientific and technological innovation team of southwest petroleum university under (Grant No. 2018CXTD02), the major projects of national science and technology (Grant No. 2017ZX05013006-005). All the parameters of the rock and fluid and some monitoring data of casing pipe in this manuscript were provided by GuTong oil field, which is greatly appreciated.

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