Sequentially Coupled Model

In the coupled model solved by sequential methods, the fluid-heat flow problem and the geomechanics problem are solved separately in two different linear systems. Pressure and temperature are first solved in one time step. After that, they are used along with the stress solution in the sequential coupling to update the coupled system.

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𝑑

𝑑𝑡∫ 𝑚Ω 𝑓𝑑Ω+ ∫ 𝐟Γ 𝑓∙ 𝐧 𝑑Ω= ∫ 𝑞Ω 𝑓𝑑Ω, (2.58)

where 𝑓 represents the fluid flow, 𝑚_𝑓 is the mass flow (accumulation), 𝐟_𝑓 is the mass flux term for the fluid flow, 𝑞_𝑓 is the sink/source for fluid flow, Ω is the domain, and Γ is the boundary. The mass flux for fluid flow 𝐟𝑓 can be further expressed by Darcy’s law as

𝐟_𝑓= −𝜌_𝑓 𝒌

𝜇𝑓(∇𝑝𝑓− 𝜌𝑓𝐠), (2.59)

where 𝜌_𝑓 is the fluid density, 𝜇_𝑓 is the fluid viscosity, and 𝐠 is the second order gravity tensor.

Energy conservation is used for the heat flow equation:

𝑑 𝑑𝑡∫ 𝑚 𝐻_𝑑Ω Ω + ∫ 𝐟 𝐻_{∙ 𝐧 𝑑Ω} Γ = ∫ 𝑞 𝐻_𝑑Ω Ω , (2.60)

where 𝐻 represents heat, 𝑚𝐻 _{is the heat flow (accumulation), 𝐟}𝐻 _{is the heat flux, and 𝑞}𝐻

is the sink/source term for heat. Heat accumulation 𝑚𝐻 can be written as 𝑚𝐻 = (1 − 𝜙) ∫ 𝜌𝑅𝐶𝑅𝑑𝑇

𝑇

𝑇0 + 𝜙𝜌𝑓𝑒𝑓, (2.61)

where 𝑇 is the temperature of rock, 𝑇0 is the reference temperature, 𝜌𝑅 is the density of

rock, 𝐶_𝑅 is the heat capacity of rock, and 𝑒_𝑓 is the specific internal energy of fluid. Heat flux 𝐟𝐻 can be further expressed as

𝐟𝐻 = −𝐊_𝐻∇𝑇 + ℎ_𝑓𝐰_𝑓, (2.62)

where 𝐊_𝐻 is the second order composite thermal conductivity tensor, ℎ_𝑓 is the specific enthalpy of the fluid, and 𝐰𝑓 is the mass flux.

Momentum balance (Hughes 1987) is used for the governing equation of geomechanics. The quasi-static assumption is used in the formulation.

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∇ ∙ 𝝈 + 𝜌𝑏𝐠 = 0, (2.63)

where 𝜌_𝑏 is the bulk density. Similar to the model presented in Section 2.1, the infinitesimal transformation assumption is used as

𝜀 = ∇𝑠𝒖 =1

2(∇

𝑇_{𝒖 + ∇𝒖), (2.64)}

Based on Biot (1941) and Coussy (1995), the constitutive relations for the coupling of fluid-heat flow and elastoplasticity are established as follow

𝛿𝝈 = 𝐂⏞ 𝑑𝑟𝛿𝛆 𝛿𝛔′ − 𝛼𝑓𝛿𝑝𝑓1 − 3𝛼T𝐾𝑑𝑟𝛿𝑇𝟏, (2.65) 𝛿𝑚𝑓 𝜌𝑓 = 𝛼𝑓𝛿𝜀𝑣+ 1 𝑀𝑓𝛿𝑝𝑓− 3𝛼𝑚,𝑓𝛿𝑇, (2.66) 𝛿𝑆̅ = 𝑠̅_𝑓𝛿𝑚_𝑓+ 3𝛼_𝑇𝐾_𝑑𝑟𝛿𝜀_𝑣 − 3𝛼_𝑚,𝑓𝛿𝑝_𝑓+𝐶𝑑 𝑇 𝛿𝑇, (2.67)

where 𝛿 represents the change caused by solid deformation, 𝛿𝛔′ is the effective stress in incremental form, 𝟏 is a second order identity tensor, 3𝛼_T is the volumetric skeleton thermal dilation coefficient. 3𝛼𝑚,𝑓 is further written as 3𝛼𝑚,𝑓 = 3𝛼𝜙+ 𝜙3𝛼𝑓, where 3𝛼𝜙

is the thermal dilation coefficient for porosity, and 𝜙3𝛼_𝑓 is the thermal dilation coefficient for fluid. 𝐶_𝑑 is the total volumetric heat capacity. 𝐶_𝑑 is further written as 𝐶_𝑑 = 𝐶_𝑅 + 𝑚𝑓𝐶𝑝,𝑓, where 𝐶𝑝,𝑓 is the fluid volumetric specific heat capacity at constant pressure. 𝑠̅𝑓

is the specific entropy of fluid.

In this context of thermos-poro-mechanics multiway coupling, the Biot’s coefficient is also written as 𝛼_𝑓 and the Biot modulus is written as 𝑀_𝑓. They are also expressed as

𝛼_𝑓= 1 −𝐾𝑑𝑟

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1

𝑀𝑓= 𝜙𝑐𝑓+ 𝛼𝑓−𝜙

𝐾𝑠 , (2.69)

where 𝐾_𝑠 is the solid grain bulk modulus.

Shear failure is taken into account using the Mohr-Coulomb model (Zoback 2007). This model is often used for the modeling of failure in cohesive frictional materials. The equations are as 𝑓 = 𝜏𝑚′ − 𝜎𝑚′ sin Ψ𝑓− 𝑐ℎcos Ψ𝑓 ≤ 0, (2.70) g = 𝜏_𝑚′ − 𝜎_𝑚′ sin Ψ_𝑑 − 𝑐_ℎcos Ψ_𝑑 ≤ 0, (2.71) 𝜎_𝑚′ =𝜎1′+𝜎3′ 2 , (2.72) 𝜏𝑚′ = 𝜎1′−𝜎3′ 2 , (2.73)

where 𝜎₁′ is the maximum principal effective stress, 𝜎₃′ is the minimum principal effective stress, 𝑐_ℎ is the cohesion, Ψ_𝑓 is the friction angle, Ψ_𝑑 is the dilation angle, 𝑓 is the yield function, and g is the plastic potential function.

After the mathematical formulations for the fluid-heat flow and elastoplasticity problems are presented, the numerical solution is then introduced. In the elastoplasticity problem, the space discretization is achieved using the nodal-based finite element method (Hughes 1987). In the fluid-heat flow problem, the space discretization is achieved using the finite volume method (Aziz and Settari 1979). The fluid-heat flow module is based on TOUGH2 (Pruess 1999) and the geomechanics module is based on TOUGH-ROCMECH (Kim et al. 2014). Backward Euler is used for the implicit method in time stepping.

The sequential coupling is achieved using the fixed-stress method. Kim et al. (2011) proved that the fixed-stress method can lead to good stability, efficiency, and accuracy in

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the numerical solution, and this method results in solutions comparable to the fully coupled method. In this sequential strategy, at each time step, the flow problem is first solved. When the flow problem is being solved, the stress in the geomechanics problem is fixed. Geomechanics is then solved based on the updated flow solutions. Lagrange porosity and its correction are used for the implementation of the fixed-stress method in the model as Φ𝑛+1− Φ𝑛 = (𝛼𝑓 2 𝐾𝑑𝑟+ 𝛼𝑓−Φ𝑛 𝐾𝑠 ) ⏟ Φ𝑛_𝑐 𝑝 (𝑝_𝑓𝑛+1− 𝑝_𝑓𝑛) + 3𝛼𝑇𝛼𝑓(𝑇𝑛+1− 𝑇𝑛) + 𝛼𝑓 𝐾𝑑𝑟(𝜎𝑣 𝑛_{− 𝜎} 𝑣𝑛−1) ⏟ ∆Φ , (2.74)

where 𝑛 is the time step, 𝑐𝑝 is the pore compressibility (Settari and Mourtis 1998), 𝜎𝑣 is

the total volumetric mean stress.

For the accuracy and stability of sequential coupling methods compared to fully coupled methods, Dean et al. (2006) pointed out that if a tight tolerance is used in the coupling process, explicit coupling, iterative coupling, and full coupling yield very similar simulation results for the nonlinear iterations in the coupled flow and geomechanics problem. This observation is honored in the fixed-stress coupling process in the coupled model in this subsection.