Impact of local thermal non-equilibrium on temporal thermo-hydro-mechanical processes in low permeable porous media

. The thermo-hydraulic-mechanical (THM) response of low permeable media is of crucial significance in thermal fracturing for production of unconventional shale oil, enhanced geothermal systems, and waste disposal. During such processes, pore pressures and stresses change in a spatiotemporal manner due to hydraulic and thermal loadings. From the viewpoint of the energy balance equation, the available theoretical studies can be classified as local thermal equilibrium (LTE), and local thermal non-equilibrium (LTNE) models. LTE models consider identical temperature for different phase of the porous system. LTNE models allow different temperature variations in solid and fluid phases of a porous medium. Current LTNE studies are weakly-coupled – not incorporating thermo-osmosis. This paper presents novel coupled LTNE thermo-poroelastic solutions in a transversely isotropic saturated porous medium, incorporating thermo- osmosis effect. Solutions are obtained for permeable and impermeable boundaries. Thermo-osmosis is found to have a very different effect in case of LTNE versus LTE, resulting in a fundamentally different THM response. LTNE effect analysis reveals different THM responses under different heat transfer properties at the solid-fluid interface in low permeable strata.


Introduction
The thermo-hydraulic-mechanical (THM) behavior of low permeable media is of crucial significant in numerous projects including unconventional shale oil and tight sand [1], enhanced geothermal, and waste disposal [2,3]. Numerous researchers have studied the THM response of the porous media under different conditions (loading, material anisotropy, wellbore inclination, and heat transfer mechanisms) using the thermo-poroelastic theory [4][5][6][7][8][9]. The majority of the relevant literature was developed assuming local thermal equilibrium (LTE), where identical temperature is assumed for both the solid and fluid phases. However, this assumption is contingent upon large interstitial heat transfer coefficients between different phases, and the porous formation to have higher ratios of pore surface area to pore volume.
Local Thermal Non-Equilibrium (LTNE) models are intended to elucidate the heat transfer between the solid phase and the fluid phase [10][11][12][13][14]. Due to the additional intricacy in the coupled response of the porous medium because of local thermal non-equilibrium, there are very few LTNE models developed, and are obtained based on a number of simplifying assumptions. No current LTNE model incorporates thermo-osmosis effects. Thermoosmosis is a coupled process that is of significance in certain porous media, and is analogous to the Sorêt effect in solutions, which describes the influence of temperature gradient on fluid flow [15][16][17][18][19][20][21].
The current paper presents a new thermo-poroelastic theory to describe the THM response of transversely isotropic saturated porous media based on the LTNE theory, incorporating thermal osmosis. Analytical solutions are obtained using Laplace transform. The solutions are verified versus the results of Gao, et al. [1]. Finally, a comprehensive analysis is conducted to assess THM processes under LTE versus LTNE.

Problem description
The problem of interest involves non-isothermal fluid injected into a saturated porous medium via a fully penetrating vertical wellbore. The porous formation is homogeneous, transversely isotropic, and linear elastic. Plane strain assumption is adopted. The permeability of formation is assumed < 1×10 -18 m 2 , making conduction the main heat transfer mechanism. Fluid and heat flow are assumed axisymmetric, and LTNE theory is adopted to consider the temperature difference between the solid phase and the pore fluid. Thermo-osmosis is considered in the formulations. The input parameters are those of Gao et al. [1,21] and presented in Tab 1.

Constitutive equation
The constitutive relation describing the correlation between induced total stresses ij σ , strains ij ε , induced pore pressures P , and the solid phase induced temperature s T for a transversely isotropic medium in a cylindrical coordinate system can be expressed as [22,23] 11 12 13 13 33  Note that tension is taken as positive sign convention.
Following the approach of Atefi Monfared and Rothenburg [24], we utilize the momentum equilibrium theorem, and the kinematic equations to obtain the following general differential equation Integrating Eq. (5) over radius results in the following differential equation for the radial displacement

Fluid diffusion equation
In the LTNE theory, pore pressures are related to the fluid content, rock deformation, rock temperature, and the pore fluid temperature as demonstrated in Eq. (8) [1] ( )

1.0
Geometry, Initial state, Source parameters Wellbore radius, a 0 . 1 m Initial pore pressure, P0 2 4 M P a Initial temperature of formation, T0 8 5 o C Wellbore pressure (Source), Pw 2 9 M P a Wellbore temperature (Source), Tw M is Biot's modulus; f T is pore fluid temperature; f K is the bulk modulus of the pore fluid; φ α is the thermic coefficient of the pore space; and ζ is the variation of the pore fluid content. The rate of change of fluid content is accompanied by divergence of fluid flux where ∇ ⋅ is the divergence vector operator, and r ∇⋅ = ∂ ∂ for axisymmetric flow; and f q is the total pore fluid flux vector. If we incorporate thermo-osmosis, f q becomes a function of pore pressure gradients and pore fluid temperature gradients as follows [15] PT f where ∇ is the gradient vector operator. PT K is referred to as the thermo-osmosis coefficient, and is related to enthalpy and the permeability coefficient. PT K may be positive or negative [18], with an absolute reported value ranging from 14 1 10 − × to 10 1 10 − × m 2 /(K‧s) [25]. Combining Eqs. (8), (12) and (13), results in the following equation relating the induced pore pressures, strains, and temperatures of the solid and fluid phases ( ) Combining Eqs. (6) and (14), we obtain the fluid diffusion equation

Heat diffusion equation
The total heat flux of the solid phase h s q can be described based on the thermal conductivity of solid phase s λ as Specific entropy of rock d s e is related to rock temperature, rock deformation, and pore pressure through [26,27]: where d s e is the specific entropy of rock; 0 s ρ is the initial rock density; s C is the specific heat of rock, and 0 s T is the initial rock temperature. Here we neglect the effect of rock deformation on d s e (heat sink effect). The heat energy conversation of solid phase based on the LTNE theory is described as The parameter int h is the heat transfer coefficient related to the solid-fluid interface, and expressed as [1] where a is the wellbore radius; * h is the nondimensional solid-fluid transfer coefficient and assumed to be 1 in this paper following Gao, et al. [1]. Later in this paper, we assess the sensitivity of the THM response based on the * h parameter. h c is the thermal diffusivity of the rock mass and computed using the proposed equation by Carson, et al. [28]. Gao et al. [1] obtained Eq. (19) by performing the non-dimensional operation on Eq. (20) and Eq. (24), where it was revealed that the hint parameter is impacted by the wellbore radius.
Combining Eqs. (16), (17) and (18) The total heat flux of the fluid phase can be expressed as where h f q is the total heat flux vector of pore fluid. The specific entropy of the fluid phase is related to the fluid temperature and pore pressure as In case of LTNE, the heat energy conversation of the fluid phase can be described as Combining Eqs. (21), (22) and (23) and after some manipulations, we are able to obtain the heat diffusion equation for the pore fluid as

General solution
Thus far, we derived general equations for fluid diffusion Eq. (15) where ( ) Applying Laplace transform to Eq.
It should be noted that we assume the initial state of the reservoir to be homogeneous. Pore pressures, and temperatures in Eq. (25) are induced values, thus the initial term due to Laplace transform will be zero. We can find from Eq. (32) that matrix Λ is independent of time and space. We assume Ψ can be expressed in terms of the modified Bessel function of the second kind in the following manner where Χ is a column vector with three rows, and . Hence, we need to obtain the eigenvalues and eigenvectors of matrix Λ . Assuming Λ is a non-singular matrix, it has three eigenvalues i q with corresponding eigenvectors i Χ (i=1, 2, 3). We define matrix Q to be composed of the eigenvectors i Χ . Therefore, the solution of Eq. (30) can thus be expressed as where the parameter vector ( )

Permeable boundary condition (PBC)
The PBC implies constant pore pressure and temperature at the wellbore, as expressed in Eq.
where 1 L − is the inverse Laplace transform.

Impermeable boundary condition (IMPBC)
Under this boundary condition, the pore pressure, rock temperature, and fluid temperature at the wellbore satisfy the following criteria ( 4 0 ) We adopt the approximate formulae proposed by Valsa and Brančik [29] to compute numerically the inverse Laplace transform. It should be noted that the h* value, which governs heat transfer between the solid and fluid phases, controls the status of LTNE. A very high h* value replicates LTE [9]. The LTE status for THM analysis in this paper is obtained as such.

Model verification
There is currently no LTNE model that incorporates thermo-osmosis. For model verification, we have adopted the results presented by Gao, et al. [1]. We choose the thermo-osmosis coefficient to be zero in our model, and adopt the special case solutions of Eq. (38) for this verification. Input parameters are presented in Tab 1.

LTNE results versus LTE results
To demonstrate the impact of LTNE on formation's response, temperatures, pore pressures, and stresses are computed under LTNE (h* = 1) and compared against those of LTE (h* = 1000). Thermo-osmosis has been incorporated for this assessment.  At early times subsequent to implementation of heat source -independent of the fluid boundary -the temperature of the pore fluid is notably lower compared to that of the solid phase under LTNE. This is due to higher thermal conductivity of the solid phase compared to the fluid, leading to a more rapid heat transmission. LTE results in in-situ temperatures that are slightly lower compared to those of the solid phase from LTNE, but significantly higher than those of the fluid phase. The temperature of the solid phase is much closer to the results from LTE compared to the fluid temperatures.
Next, we assess the significance of the thermoosmosis under LTNE. To achieve this, the corresponding values from LTE have been subtracted from the presented results. Fig. 3 illustrates induced pore pressures and effective stresses under both PBC and IMPBC boundary conditions for two case scenarios: conventional thermoporoelasticity (trivial thermal osmosis); and incorporating thermal osmosis. This analysis is conducted for 0.01 day and 0.1 day. Fig. 3a reveals that even under conventional thermo-poroelasticity (trivial thermo-osmosis), LTNE influences pore pressure and stress variations (maximum 3 MPa at 0.01 day). This observation is consistent with previous studies [30]. Lower pore pressures are generated under LTNE compared to LTE when ignoring the nontraditional thermal process. This is due to the notably smaller temperature gradients in the former as a result of the gradual heat exchange between the two phases. Fig. 3b reveals a much higher impact (maximum 20 MPa difference between induced stresses at 0.01 day) from thermo-osmosis. This thermal mechanism results in notably higher pore pressures under LTNE versus LTE. For later times (0.1 day), the results (Fig. 3c)

LTNE effect sensitivity
Next, we assess the impact of * h on the THM response of porous medium. Fig. 4 shows notably higher pore pressures induced under a smaller * h value compared to LTE results. This is due to higher solid temperatures and thermo-osmosis effect. LTNE results yield to those of LTE with * h value increasing. A higher * h value is required under IMPBC for the formation to reach LTE compared to PBC.

Conclusions
This paper presents new fully-coupled thermo-poroelastic solutions under LTNE, incorporating thermo-osmosis for the first time. Formulations are obtained for a transversely isotropic saturated porous medium, subjected to a vertical heat and fluid line source. Specific solutions are obtained for permeable and impermeable hydraulic boundary conditions. The following are key observations: • Under LTNE, a saturated porous formation subjected to heat source undergoes notably lower pore fluid temperatures compared to LTE, while the solid temperatures are slightly higher from the former. • Thermo-osmosis results in generation of notably higher pore pressures under LTNE versus LTE, whereas ignoring thermo-osmosis causes lower pore pressures induced under LTNE compared to LTE. • A higher solid-fluid transfer coefficient * h increases fluid temperature substantially, while slightly decreasing solid temperature. Notably higher pore pressures and stresses are induced under a smaller * h value ( * 1 h = ).