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

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:

Where* ß̄*_{o }is the average thermal expansion between *T*_{1} and *T*_{2}; *v*_{1} and *v*_{2} are the oil volumes at *T*_{1} and *T*_{2}.

**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.

Where the values of regression fit are

a_{1} | = | -3.772241 |

a_{2} | = | -1.188914 x10^{-3} |

a_{3} | = | 7.666322 x10^{-4} |

a_{4} | = | 7.481999 |

a_{5} | = | 4.280123 x10^{-6} |

a_{6} | = | -3.864994 x10^{-7} |

a_{7} | = | -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.

$${\beta}_{o}={\beta}_{\mathit{ob}}+{a}_{6}{R}_{s}\left(p-{p}_{b}\right){+a}_{7}\frac{{R}_{s}{\gamma}_{g}}{{\gamma}_{o}}\mathrm{ln}\frac{p}{{p}_{b}}$$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 T

_{1}and T

_{2}without ignoring that

*p*

_{b}is different at different temperatures.

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