# Analysis of Pore Pressure and Stress Distribution around a Wellbore Drilled in Chemically Active Elastoplastic Formations

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## Abstract

Drilling in low-permeable reactive shale formations with water-based drilling mud presents significant challenges, particularly in high-pressure and high-temperature environments. In previous studies, several models were proposed to describe the thermodynamic behaviour of shale. Most shale formations under high pressure are expected to undergo plastic deformation. An innovative algorithm including work hardening is proposed in the framework of thermo-chemo-poroelasticity to investigate the effect of plasticity on stresses around the wellbore. For this purpose a finite-element model of coupled thermo-chemo-poro-elastoplasticity is developed. The governing equations are based on the concept of thermodynamics of irreversible processes in discontinuous systems. In order to solve the plastic problem, a single-step backward Euler algorithm containing a yield surface-correction scheme is used to integrate the plastic stress–strain relation. An *initial stress* method is employed to solve the non-linearity of the plastic equation. In addition, super convergent patch recovery is used to accurately evaluate the time-dependent stress tensor from nodal displacement. The results of this study reveal that thermal and chemical osmosis can significantly affect the fluid flow in low-permeable shale formations. When the salinity of drilling mud is higher than that of pore fluid, fluid is pulled out of the formation by chemical osmotic back flow. Similar results are observed when the temperature of drilling mud is lower than that of the formation fluid. It is found that linear elastic approaches to wellbore stability analysis appear to overestimate the tangential stress around the wellbore and produce more conservative stresses compared to the results of field observation. Therefore, the drilling mud properties obtained from the elastoplastic wellbore stability in shales provide a safer mud weight window and reduce drilling cost.

## Keywords

Thermo-chemo-poro-elastoplasticity Water-active rocks Osmotic flow Wellbore stability analysis## List of Symbols

*c*Cohesion

*c*^{T}Thermal diffusivity

*C*^{D}Average diluent mass fraction in formation

*C*_{M}Average solute mass fraction in drilling mud

*C*_{S}Average solute mass fraction in formation

*D*_{e}Elastic modulus tensor

*D*Solute diffusion coefficient

*D*^{T}Coefficient of thermal diffusion

*dλ*Plastic multiplier

*f*Yield function

*G*Shear modulus of rock

*H*_{i0}Constant isotropic hardening module

*J*_{1}First stress invariant

*J*_{2}^{′}Second invariant of the deviatoric stress

*k*Permeability

*κ*Isotropic hardening parameter

*K*_{f}Fluid bulk module

*K*^{T}Thermal osmosis coefficient

*K*_{S}Solid bulk module

*M*^{S}Molar mass of the solute

*n*Number of nodes

*N*_{P}Pressure shape functions

*N*_{u}Displacement shape functions

*N*_{C}^{S}Mass fraction shape functions

*N*_{T}Temperature shape functions

*p*Pressure

*P*_{i}Initial reservoir pressure

- \( \vec{P} \)
Pore pressure vector

*Q*_{L}Plastic potential

*R*Universal gas constant

*s*_{0}Specific fluid entropy

*T*_{s}Shale temperature

*T*_{m}Drilling mud temperature

*T*Temperature

*T*_{a}Absolute temperature

*t*Time

*u*Displacement

- \( \vec{U} \)
Displacement vector

*α*Biot’s coefficient

*α*_{f}Thermal expansion coefficient of fluid

*α*_{m}Thermal expansion coefficient of solid

*ɛ*Strain tensor

*ζ*Variation of the fluid content

*μ*Viscosity

*ν*Poisson’s ratio

- \( \mathop {\rho_{f} }\limits^{ - } \)
Fluid density

*σ*Stress tensor

- \( \sigma^{\prime}_{p} \)
Plastic effective stress

*σ*_{y}^{0}Initial uniaxial yield stress

*σ*_{h}Minimum in situ horizontal stress

*σ*_{H}Maximum in situ horizontal stress

*ϕ*Porosity

- φ
Internal friction

*ω*^{S}Chemical swelling parameter for solute

*ω*^{D}Chemical swelling parameters for diluent

- ℜ
Standard solute reflection coefficient

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