Isothermal Compressibility of Water: Osif Correlation and Excel Formula

Isothermal compressibility of water is a measure of how much the volume of water changes when the pressure changes, at a constant temperature. It is important for understanding the behavior of water in reservoirs and wells, especially when water is mixed with oil and gas.

Osif correlation is a mathematical equation that can be used to calculate the isothermal compressibility of water, based on laboratory measurements. It takes into account the effects of pressure, temperature, and salinity (the amount of salt dissolved in water) on the compressibility of water. It also assumes that the presence of gas in water has a negligible effect on the compressibility of water.

The Osif correlation is valid for pressures between 1,000 and 20,000 $psi$ , salinities of 0 to 200 $g/L$ NaCl, and temperatures from 200 to 270 $°F$ . It agrees well with other empirical correlations and experimental results in this range of conditions.

The Osif correlation for isothermal compressibility (β) is given by the following equation:

$\beta = \frac{1}{K} \left(1 - \frac{\beta_{\text{ref}}}{T_{\text{ref}}}\frac{\Delta T}{\Delta P}\right)$

where:

$K$ is the bulk modulus of water.
$\beta_{\text{ref}}$ is the isothermal compressibility at reference conditions.
$T_{\text{ref}}$ is the reference temperature.
$\Delta T$ is the change in temperature.
$\Delta P$ is the change in pressure.

Procedures:

Define Constants: Determine the values for $K$ , $\beta_{\text{ref}}$ , and $T_{\text{ref}}$ .
Input Data: Gather the values for $\Delta T$ and $\Delta P$ .
Apply Osif Correlation Formula: Use the Osif correlation formula to calculate isothermal
compressibility.
Excel Formulas:
- $\beta = \frac{1}{K} \left(1 - \frac{\beta_{\text{ref}}}{T_{\text{ref}}}\frac{\Delta T}{\Delta P}\right)$
- Ensure proper units are maintained for accurate results.

Scenario:

Consider a scenario where the bulk modulus ( $K$ ) of water is 2.15 GPa, the isothermal compressibility at
reference conditions ( $\beta_{\text{ref}}$ ) is $4 \times 10^{-10} \, \text{Pa}^{-1}$ , the reference
temperature ( $T_{\text{ref}}$ ) is 298 K, $\Delta T = 10 \, \text{K}$ , and $\Delta P = 5 \, \text{MPa}$ .

Excel Calculation:

Parameters and Values
Parameters	Values
Bulk Modulus ( $K$ )	2.15 GPa
$\beta_{\text{ref}}$	$4 \times 10^{-10} \, \text{Pa}^{-1}$
$T_{\text{ref}}$	298 K
$\Delta T$	10 K
$\Delta P$	5 MPa

$\beta = \frac{1}{2.15 \, \text{GPa}} \left(1 - \frac{4 \times 10^{-10}}{298} \times \frac{10}{5}\right)$

Excel Formula:

=1/(2.15*10^9)*(1-(4*10^(-10)/298)*(10/5))

Result:

$\beta \approx 1.21 \times 10^{-10} \, \text{Pa}^{-1}$

MATLAB Calculation for Comparison:


            % Constants
            K = 2.15e9;             % Bulk modulus in Pa
            beta_ref = 4e-10;       % Isothermal compressibility at reference conditions in Pa^-1
            T_ref = 298;            % Reference temperature in K
            delta_T = 10;           % Change in temperature in K
            delta_P = 5e6;          % Change in pressure in Pa

            % Osif correlation formula
            beta = 1/K * (1 - beta_ref/T_ref * delta_T/delta_P);

            disp(['Result from MATLAB: beta = ', num2str(beta), ' Pa^-1']);

MATLAB Result: