Thermal expansion is the unit change in oil volume with the temperature at constant pressure. In equation form, the thermal expansion,β_o, is defined as

$${\beta }_o=\frac{1}{v_o}{\left(\frac{\partial v_o}{\partial T}\right)}_P$$

The thermal expansion is a point function with the dimension of reciprocal temperature, 1/ºF. The following equation estimates the average thermal expansion coefficient between two temperatures at the same pressure:

$${\overline{\beta }}_o=\frac{1}{(v_1+v_2)/2}\left(\frac{v_2-v_1}{T_2-T_1}\right)$$

Where ß̄_o is the average thermal expansion between T₁ and T₂; v₁ and v₂ are the oil volumes at T₁ and T₂.

Al-Marhoun (2023)

The thermal expansion correlation was developed based on the following assumed general relationship:

$${\beta }_o=f({T,R}_s,{\gamma }_g,{\gamma }_o,p_b,p)$$

The following correlation describes the best relation to predicting thermal expansion.

$${\beta }_o={a_1+a}_2{R}_s{\gamma }_g+{a_{3}R}_s+a_4\frac{1}{{\gamma }_o}+a_5\frac{R_s{\gamma }_g}{{\gamma }_o}\left(T+459.67\right)+a_6{R}_s\left(p-p_b\right){+a}_7\frac{R_s{\gamma }_g}{{\gamma }_o}\ln \frac{p}{p_b}$$

Where the values of regression fit are

a1	=	-3.772241
a2	=	-1.188914 x10-3
a3	=	7.666322 x10-4
a4	=	7.481999
a5	=	4.280123 x10-6
a6	=	-3.864994 x10-7
a7	=	-8.469954 x10-4

For pressures equal to bubblepoint pressure, the thermal expansion at the bubblepoint is

$${\beta }_{\mathit{ob}}={a_1+a}_2{R}_s{\gamma }_g+{a_{3}R}_s+a_4\frac{1}{{\gamma }_o}+a_5\frac{R_s{\gamma }_g}{{\gamma }_o}\left(T+459.67\right)$$

If the bubblepoint thermal expansion is known, the thermal expansion at any pressure for the given bubblepoint condition is calculated using the following equation.

$${\beta }_o={\beta }_{\mathit{ob}}+a_6{R}_s\left(p-p_b\right){+a}_7\frac{R_s{\gamma }_g}{{\gamma }_o}\ln \frac{p}{p_b}$$

These equations predict the instantaneous or point thermal expansion. However, the average thermal expansion can be calculated by averaging βo estimated from these equations at T₁ and T₂ without ignoring that p_b is different at different temperatures.

$${\overline{\beta }}_o=\frac{1}{2}\left({\beta }_o\left(T_1\right)+{\beta }_o\left(T_2\right)\right)$$

Also, Eq. 18 can calculate the average thermal expansion by averaging βo estimated from the ANN model at T1 and T2.