**3. Mathematical modeling**

The equation that best describes thermal conductivity is given by the relationship:

$$\frac{\partial t}{\partial \mathbf{r}} = a \left( \frac{\partial^2 t}{\partial \mathbf{x}^2} + \frac{\partial^2 t}{\partial \mathbf{y}^2} + \frac{\partial^2 t}{\partial \mathbf{z}^2} \right) \tag{15}$$

where *x*, *y*, and *z* are the heat dissipation directions, and *t* is the heat flow temperature, measured after time *τ*.

Sometimes it is used as a way of calculation, writing the thermal conductivity in spherical coordinates:

$$\frac{\partial t}{\partial \tau} = a \left( \frac{\partial^2 t}{\partial r^2} + \frac{1}{r} \frac{\partial t}{\partial r} + \frac{1}{r^2} \frac{\partial^2 t}{\partial \theta^2} + \frac{\partial^2 t}{\partial x^2} \right) \tag{16}$$

And from Eq. (16) it is written:

$$
\nabla^2 t = -\frac{1}{a} \frac{\partial t}{\partial \tau} = -\frac{A}{k} \tag{17}
$$

Given that in the stabilized regime *<sup>∂</sup><sup>t</sup> <sup>∂</sup><sup>τ</sup>* ¼ 0, ecuația 17 se Eq. (17) turns into Poisson's equation, namely:

$$
\nabla^2 t = -\frac{A(\mathbf{x}, y, z)}{k} \tag{18}
$$

where *A*(*x*,*y*,*z*) is the amount of heat that dissipates in the analyzed rock volume, and *k* represents the thermal conductivity of the rocks.


**Table 1.**

*Analysis of the productive states of the studied deposits (oil and gas) (Moesic platform) [9, 13, 14].*


**2.** *Equations for simulating thermal conductivity as a function of porosity and density and density as a function of porosity.*
