### Appendices

### Appendix 1

The notations are as following: *T* is the temperature [K], \(p_{\text{w}}\) the pore pressure [Pa], \(\phi \) the Eulerian porosity, \(\mathbf {\epsilon }\) the linearized strain tensor, \(\mathbf {F}^{\mathbf {m}}\) the massic body force [\(\text {N kg}^{-3}\)], *r* the homogenized specific mass [\(\text {kg m}^{-3}\)], \(m_{\text{w}}\) the brine mass content [\(\text {kg m}^{-3}\)], \(\mathbf {M}_{\text{w}}\) the water surface mass flow [\(\text {kg m}^{-2} \text {s}^{-1}\)], \(\theta _{\text{rad}}\) the heat source due to the radioactivity of rocks [\(\text {W m}^{-3}\)], \(h^{\text{m}}_{\text{w}}\) the fluid specific enthalpy [\(\text {J kg}^{-1}\)], *Q* the rate of internal energy neither resulting from convection nor conduction [\(\text {J m}^{-3}\)], \(\mathbf {q}\) the heat conductive flow [\(\text {J m}^{-2} \text {s}^{-1}\)], \(\sigma \) the Cauchy stress tensor [Pa], \(\sigma '\) its effective counterpart, \(\sigma _{\text{p}}\) the hydraulic stress, \(\mathbb {C}\) the drained elasticity tensor [Pa], \(\mathbf {1}\) the unit tensor, \(\alpha _0\) the linear thermal dilation of the dry material [\(\text {K}^{-1}\)], \(\lambda \) the thermal conductivity [\(\text {W m}^{-1} \text {K}^{-1}\)], \(K_{\text{int}}\) the intrinsic permeability [m\(^2\)], \(\mu _{\text{w}}\) the fluid dynamic viscosity [Pa s] and \(\rho _{\text{w}}\) the fluid density [\(\text {kg m}^{-3}\)]. The balance equations correspond to the mechanical equilibrium, brine mass and energy balance:

$$\begin{aligned} \nabla \cdot \sigma +r\,\mathbf {F}^{\mathbf {m}}= 0 \end{aligned}$$

(3)

$$\begin{aligned} \frac{\partial {m_{\text{w}}}}{\partial {t}} + \nabla \cdot \mathbf {M}_{{\text{w}}}= 0 \end{aligned}$$

(4)

$$\begin{aligned} \mathbf {M}_{{\text{w}}} \cdot \mathbf {F^m} + \theta _\text{rad}= \frac{\partial {Q}}{\partial {t}} + \nabla \cdot (h^{\text{m}}_{\text{w}}\mathbf {M}_{\text{w}}) + \nabla \cdot \mathbf {q}+ h^{\text{m}}_{\text{w}}\frac{\partial {m_{\text{w}}}}{\partial {t}} \end{aligned}$$

(5)

The poro-elastic behavior is governed by the following relations:

$$\begin{aligned} \sigma= \sigma ' + \sigma _{{\text{p}}}\mathbf {1},\end{aligned}$$

(6)

$$\begin{aligned} d\sigma _{\text{p}}= {} -b\,dp_{\text{w}}\end{aligned}$$

(7)

$$\begin{aligned} d\phi= {} (b - \phi )\left(d\epsilon _{\text{v}} - 3\alpha _0 dT + \frac{dp_{\text{w}}}{K_{\text{s}}}\right) \end{aligned}$$

(8)

$$\begin{aligned} m_{\text{w}}= {} \phi (1 + \epsilon _{\text{v}})\rho _{\text{w}} - \phi _0\rho _{\text{w}}^0 \end{aligned}$$

(9)

with \(\epsilon _{\text{v}}=Tr(\epsilon )\) the total volume strain, \(K_{\text{s}}\) the matrix bulk modulus [Pa], *b* the Biot coefficient, \(\rho _{\text{w}}^0\) the initial fluid density [\(\text {kg m}^{-3}\)] and \(\phi _0\) the initial porosity. The thermodynamic flows \(\sigma \), \(\mathbf {M}_\text{w}\), and \(\mathbf {q}\) are linearly linked to forces \(\epsilon \), \(\nabla p_{\text{w}}\), \(\nabla T\) according to:

$$\begin{aligned} d\sigma= \mathbb {C}:(d\epsilon - \alpha _{0}\,dT\, \mathbf {1}) \end{aligned}$$

(10)

$$\begin{aligned} \mathbf {M}_{{\text{w}}}= \frac{\rho _{\text{w}} K_{{\text{int}}}}{\mu _{\text{w}}}(-\nabla p_{\text{w}} + \rho _{\text{w}} \mathbf {F}^{\mathbf {m}}) \end{aligned}$$

(11)

$$\begin{aligned} \mathbf {q}= -\lambda \nabla T \end{aligned}$$

(12)

$$\begin{aligned} \frac{\partial {Q}}{\partial {t}}= 3 \alpha _0 K_0 T \frac{\partial {\epsilon }}{\partial {t}} -3\left[ (b-\phi ) \alpha _0+\alpha _{\text{w}} \phi \right] T \frac{\partial {p_{\text{w}}}}{\partial {t}}+\left( c_{\text{s}}-9 T K_0\alpha _0^2\right) \frac{\partial {T}}{\partial {t}} \end{aligned}$$

(13)

with \(K_0\) the bulk modulus of the skeleton and \(c_{\text{s}}\) the specific heat at constant stress. Note that \(\rho _{\text{w}}\) and \(\mu _{\text{w}}\) as well as \(\lambda \) and \(c_{\text{s}}\) are functions of temperature and/or pore pressure. For the Fourier’s law, the thermal conductivity of the dry rock is described by the classical mixing law:

$$\begin{aligned} \lambda _{\text{dry}}(T) = (1 - \phi _0)\lambda _{\text{s}}(T) + \phi _0\lambda _{\text{air}}(T) \end{aligned}$$

(14)

with \(\lambda _{\text{s}}\) (resp. \(\lambda _{\text{air}}\)) the thermal conductivity of solid grains (resp. air). We assume that the thermal conductivity of air is negligible. Consequently, the thermal conductivity of solid grains can be written as:

$$\begin{aligned} \lambda _{\text{s}}(T)=\frac{\lambda _{{\text{dry}}}(T)}{1-\phi _0}. \end{aligned}$$

(15)

Thermal conductivity of the dry material is assumed to depend linearly on temperature:

$$\begin{aligned} \lambda _{{\text{dry}}}(T)=a_{\lambda _{{\text{dry}}}}+b_{\lambda _{{\text{dry}}}} T \end{aligned}$$

(16)

with \(a_{\lambda _{\text{dry}}}\) and \(b_{\lambda _{\text{dry}}}\) empirical constant parameters obtained from experimental measurements. Finally, the homogenized thermal conductivity of the saturated porous media is expressed by using the same kind of mixing law as previously:

$$\begin{aligned} \lambda (T)=(1-\phi )\lambda _{\text{dry}}(T)+\phi \lambda _{\text{w}}(T). \end{aligned}$$

(17)

The specific heat for the dry medium is defined using a similar experimental correlation as Eq. (16):

$$\begin{aligned} c_{\text{dry}}(T) = a_{c_{\text{dry}}} + b_{c_\text{{dry}}} T. \end{aligned}$$

(18)

As proposed for the homogenized thermal conductivity, we can define the specific heat capacity and the initial specific mass as:

$$\begin{aligned} c_{\text{s}}(T)= \frac{c_{\text{dry}}(T) - \phi _0 c_{\text{air}}}{1 - \phi _0} \end{aligned}$$

(19)

$$\begin{aligned} r_0= \rho _{\text{dry}} + \phi _0\rho ^0_{\text{w}} \end{aligned}$$

(20)

with \(c_\text{air}\) the specific heat capacity of air.

### Appendix 2: Validation of gravity calculations using analytical cases

#### Validation by the V-sheet analytical case

We reproduce the gravity effects calculated from robust analytical expressions. The first analytical case concerns the gravity anomaly due to a buried vertical finite sheet body called “Thin V-sheet” (Hinze et al. 2013). Figure 10 illustrates the conceptual representation of the geometric parameters for modeling the gravity effect of the V-sheet. The analytical equation of the gravity effect noted \(\text {g}_{\text{vs}}\) from the V-sheet source is as following (Hinze et al. 2013):

$$\begin{aligned} g_{\text{vs}} = 2\mathcal {G}td\rho \ln \left( \frac{z^2_2 + x^2}{z^2_1 + x^2}\right), \end{aligned}$$

(21)

where \(\textit{g}_{\text{vs}}\) is the vertical gravity anomaly in Gal (\(\text {m s}^{-2} = 10^5\,\text {mGal}\)) calculated at each point along *x*-axis on the surface due to the density contrast \(\textit{d}\rho \) (\(\text {kg m}^{-3}\)) at distances *x* (m) and *z* (m). The other parameters *t*, \(\textit{z}^2_1\) and \(\textit{z}^2_2\) are illustrated in Fig. 10.

For the analytical cases considered here, the heat transfer is assumed to be only driven by diffusion. All THM couplings are canceled including the temperature and fluid pressure dependencies for the fluid and rock properties. The reservoir is considered as a 2D cross-section of 10 km in width and 5.35 km in height and the V-sheet is 200 m in width and 1000 m in height. The sheet is set to have a specific mass contrast of \(300\,\text {kg m}^{-3}\) with the rest of the reservoir.

Figure 11 shows the comparison of the gravity effect profiles along the surface between the analytical expression and the simulation for the V-sheet case. First the dependence to element types is investigated. The simulated gravity anomaly is obtained from simulations with two different kinds of finite elements: triangle and quad. No significant effect of the element type is shown. The issue concerning the reference specific mass is addressed for the V-sheet analytical case. For the modeling with quad elements, two cases are considered: (i) the reference total homogenized specific mass \(\rho _\text{ref}\) is taken as the initial value \(\rho _0\); (ii) \(\rho _\text{ref}\) corresponds to the mean value in the spatial domain \(\rho _\text{m}\). The two simulations reproduce the analytical solution, demonstrating that the assumption “\(\rho _\text{ref} \approx \rho _{0} \approx \rho _{\text{m}}\)” is validated for low gravity effect in large-scale reservoir. Finally, all simulations from our modeling reproduce the analytical solution from the V-sheet gravity effect validating our calculation for this first analytical case.

#### Validation by the Mogi’s sphere analytical case

The Mogi’s sphere has been first introduced in the works of Mogi (1958) to study the link between the ground deformations and volcanic eruption and can be applied in a geothermal context (Heimlich 2017; Portier et al. 2018). Figure 12 illustrates the conceptual model of the buried Mogi’s sphere. The analytical equation of the gravity effect noted \(\text {g}_\text{m}\) is (Hinze et al. 2013):

$$\begin{aligned} g_\text{m} = 4\pi \mathcal {G}\frac{d\rho R^3}{3z^2(1+(\frac{x}{z})^2)^{\frac{3}{2}}}, \end{aligned}$$

(22)

where the same notations and the geometric parameters are shown in Fig. 12.

The hypotheses and reservoir geometry for the Mogi’s sphere model are the same than in the V-sheet case. The radius of the sphere is 500 m and its center is at 2.65 km in depth. It uses a density contrast of \(300\,\text {kg m}^{-3}\). Figure 13 illustrates the comparison between the analytical solution and the simulated gravity effect. The simulated gravity anomaly profile is well reproducing again the analytical solution. Our modeling approach of the gravity effect is then validated for two robust and well-known analytical cases.