**2. Estimation of the equivalent thermal conductivity of petroleum fluid deposits**

Thermal conductivity is that property defined as the heat flux (amount of heat) *Q*, which flows in a time *τ*, through a body with a given cross section and length, whose opposite faces are at temperatures *t*<sup>1</sup> and *t*2:

$$
\lambda = \frac{\Delta Q}{\Delta \tau} \bullet \frac{L}{A \Delta T} \tag{6}
$$

The estimation of the thermal conductivity of the oil fluid deposits was made on the basis of theoretical models to describe this property and following the use of practical methods of determination in the laboratory.

The estimation of thermal conductivity can best be done based on virtual models of behavior of oil and gas fields.

The first model used to determine thermal conductivity was the seriesdeveloped geological layer model, with heat flow Q being directed perpendicular to the layers.

In this case, the thermal conductivity calculation relation can be written as:

$$\frac{1}{\lambda\_x} = \frac{\Phi}{\lambda\_f} + \frac{1-\Phi}{\lambda\_s} \tag{7}$$

Where:

• *λ<sup>f</sup>* is the coefficient defining the thermal conductivity of the reservoir fluid, W/(m K),

*Estimation of Equivalent Thermal Conductivity Value Using Correlation Relationships… DOI: http://dx.doi.org/10.5772/intechopen.106453*


This model considers imperfections to be nonexistent and therefore a model developed for geological layers parallel to the direction of flow of the thermal flux was chosen.

The thermal conductivity equation of the petroleum fluid reservoir can be written as follows:

$$
\lambda\_x = \Phi \lambda\_{\hat{f}} + (1 - \Phi)\lambda\_t \tag{8}
$$

Where:


This model takes into account the porosity of the rocks, but there are errors in the conductivity estimate due to the possibility of material inclusions in the reservoir's protective rocks.

A cumulative variant of the two models is expressed by the weighted geometric model, which considers that the oil fluid constituting the collecting rocks has the largest weight in the calculation of thermal conductivity:

$$
\lambda\_x = \lambda\_f^{\Phi} \lambda\_s^{1-\Phi} \tag{9}
$$

where:


Paper 14 presents a model for determining thermal conductivity, based on the Maxwell equation, which assumes the reduction of petroleum fluids to interacting spheres (Euken):

$$
\lambda\_x = \lambda\_f \frac{2\Phi\lambda\_f + (3 - 2\Phi)\lambda\_s}{(3 - \Phi)\lambda\_f + \Phi\lambda\_s} \tag{10}
$$

Where:


For rocks with porosity Φ <0.5 and the thermal conductivity ratio of rocks and fluids, λs/λf, in the range 1 ÷ 300, the thermal conductivity was expressed by Beck's relation:

$$
\lambda\_x = \lambda\_s \left[ \frac{\left(2\frac{\dot{\lambda\_s}}{\dot{\lambda\_f}} + 1\right) - 2\Phi\left(\frac{\dot{\lambda\_s}}{\dot{\lambda\_f}} - 1\right)}{\left(2\frac{\dot{\lambda\_s}}{\dot{\lambda\_f}} + 1\right) + \Phi\left(\frac{\dot{\lambda\_s}}{\dot{\lambda\_f}} - 1\right)}\right] \tag{11}
$$

Where:


The model developed by Vries is based on the idea that the constituent rocks of the oil reservoir consist of ellipsoidal particles dispersed in the oil fluid being analyzed.

$$
\lambda\_x = \frac{\Phi \lambda\_f + (1 - \Phi)G\lambda\_s}{\Phi + (1 - \Phi)\overline{\varphi}G} \tag{12}
$$

Where:

$$\mathbf{G} = \frac{1}{3} \sum\_{j=1}^{3} \left[ \mathbf{1} + \left( \frac{\lambda\_s}{\lambda\_f} - \mathbf{1} \right) \xi\_j \right]^{-1} \xi i \sum\_{j=1}^{3} \xi\_j = \mathbf{1} \tag{13}$$

and:


*Estimation of Equivalent Thermal Conductivity Value Using Correlation Relationships… DOI: http://dx.doi.org/10.5772/intechopen.106453*

**Figure 1.**

*Schematic of the model for calculating the thermal conductivity of petroleum fluid deposits taking into account the possibility of considering a resistor-type structure (Wyllie and Southwick) [14].*

In relation 13, ξ<sup>j</sup> is a parameter that defines the shape of the particles.

If ξ\_1 = ξ\_2 = ξ\_3. then Eq. 13 reduces to Eq. 10. Also de Vries considered that ξ\_1 = ξ\_2 = 1/8, ξ\_3 = 3/4, the oil deposit consists of ellipsoids of resolution with the long axis equal to six times the short axis.

The model developed by Woodside and Messmer (**Figure 1**) is a model resulting from the determination of conductivity by heating rocks and oil fluids with electromagnetic bridges and determining this property from the idea that the oil reservoir is made up of a series of particles and electolite over which geological structures are arranged in parallel.

$$
\lambda\_x = \frac{\zeta\_1 \lambda\_\prime \lambda\_f}{\lambda\_\prime (1 - \zeta\_4) + \zeta\_4 \lambda\_f} + \zeta\_2 \lambda\_\sigma + \zeta\_3 \lambda\_f \tag{14}
$$

Where:


In Eq. 14, the coefficients *ζ*1, *ζ*2, *ζ*3, *ζ*<sup>4</sup> have values given by certain forms of rock and electrolyte arrangement.

These values of the parameters *ζ*1, *ζ*2, *ζ*3, *ζ*<sup>4</sup> are required for the calculation of the equivalent electrical conductivity.

The use of the relation (14) for the calculation of the thermal conductivity of an unconsolidated oil fluid field leads to values very close to those obtained experimentally, when the following coefficients are used:

$$
\zeta\_2 = 0, \zeta\_3 = \Phi - 0, \text{03, } \zeta\_1 = 1 - \zeta\_3, \zeta\_4 = \frac{(1 - \Phi)}{\zeta\_1} \tag{15}
$$

And this model has errors in calculating the real value of thermal conductivity, based on the idea of forming the oil field from electrolytic systems (optimal heat transfer).

Analyzing 165 thermal conductivity values, where the optimal calculation porosity was between 0.215 and 0.476, *Krupiczka* was found that in 76% of the values calculated with relation 16, the difference between the values obtained experimentally and those calculated has errors of �30%,

The calculation formula for calculating the thermal conductivity proposed by this model is:

$$
\lambda\_x = \lambda\_{\hat{f}} \left(\frac{\lambda\_{\hat{\iota}}}{\lambda\_{\hat{f}}}\right)^{A' + B' \log \frac{\lambda\_{\hat{\iota}}}{\hat{f}}} \tag{16}
$$

Where:

$$A^\circ = 0, \text{\textquotedblleft}0 - 0, \text{\textquotedblleft}57 \text{log}\Phi \text{ și} B^\circ = -0, \text{\textquotedblleft}57 \text{\textquotedblright} \tag{17}$$

and:


## **3. Evaluation of thermal conductivity simulation models**

Following the experiments on the cores extracted from potential areas supplying oil fluids, we managed to determine the conductivity of the rocks, their porosity and density.

The data obtained were used to simulate the thermal conductivity of deposits saturated with gas, water (density is 1000 kg/m<sup>3</sup> ), pure crude oil (density is 790 kg/ m3 ), and crude oil mixed with 35% water ((density is 863 kg/m<sup>3</sup> ).

At the same time, we created our own model for simulating thermal conductivity, based on the statistical interpretation of the data obtained from the calculations performed with five analyzed models (**Tables 2**–**5**).

*Estimation of Equivalent Thermal Conductivity Value Using Correlation Relationships… DOI: http://dx.doi.org/10.5772/intechopen.106453*

**Figure 2.** *Thermal conductivity tester.*

The model developed in this paper starts from the idea that the analyzed rock is not pure (i.e., the density of the analyzed rock was introduced in the calculation).

The model was tested on the cores analyzed in the apparatus of **Figure 2** and shown in the tables below for all existing fluids in the field.

We also introduced porosity in the calculation as a basic factor in the calculation of total conductivity.

$$
\lambda\_x = \frac{1}{\left(\rho\_{r)}} \bullet \left(\boldsymbol{\lambda}\_f^{(1-\theta)} \bullet \boldsymbol{\lambda}\_f^{\theta}\right) \tag{18}
$$

Where:


**Figures 3**–**6** give the differences between the calculated values of thermal conductivity with the six models, with porosity and thermal conductivity (kr) determined using the apparatus in **Figure 2**.

**Figures 7** and **8** show the calculated thermal conductivity values with the six models according to literature data.


**Table 2.** *The values of thermal conductivity*

 *calculated with simulation*

 *models, the fluid in the rock is crude oil.*

*Drilling Engineering and Technology - Recent Advances, New Perspectives and Applications*

*Estimation of Equivalent Thermal Conductivity Value Using Correlation Relationships… DOI: http://dx.doi.org/10.5772/intechopen.106453*

#### **Figure 3.**

*The variation of the conductivity parameters of the analyzed deposits of the fluid contained in the rocks is crude oil.*


**Table 3.**

*The values of thermal conductivity calculated with simulation models, the fluid in the rock is water and crude oil.*

#### **Figure 4.**

*The variation of the conductivity parameters of the analyzed deposits of the fluid contained in the rocks is water and crude oil.*


#### **Table 4.**

*The values of thermal conductivity calculated with simulation models, the fluid in the rock is water.*

*Estimation of Equivalent Thermal Conductivity Value Using Correlation Relationships… DOI: http://dx.doi.org/10.5772/intechopen.106453*

**Figure 5.**

*The variation of the conductivity parameters of the analyzed deposits of the fluid contained in the rocks is water.*


**Table 5.**

*The values of thermal conductivity calculated with simulation models, the fluid in the rocks is natural gases.*

The relative error (**Tables 7**–**9**) of the computationally determined data with the six models compared with the literature data shows that the Chiș model is closest to the values determined in other laboratories.

The error is due to the fact that the analyzed samples are impure.

**Figure 6.**

*Variation of the conductivity parameters of the analyzed deposits of the fluid contained in the rocks is natural gases (Table 6).*


#### **Table 6.**

*Thermal conductivity values calculated with simulation models and taken from the literature (Table 1 and references [10]) for saturated tiles.*

**Figure 7.**

*Thermal conductivity values calculated with simulation models and taken from the literature for saturated tiles.*

*Estimation of Equivalent Thermal Conductivity Value Using Correlation Relationships… DOI: http://dx.doi.org/10.5772/intechopen.106453*


#### **Table 7.**

*Relative error of the chosen model calculation based on literature values for saturated tiles.*

#### **Figure 8.**

*Thermal conductivity values calculated with simulation models and taken from the literature for chalk.*


#### **Table 8.**

*Thermal conductivity values calculated with simulation models and taken from the literature (Table 1 and references [15]) for chalk.*


#### **Table 9.**

*Relative error of the chosen model calculation based on literature values for chalk.*
