Pressure Volume Temperature PVT
===============================

**Definition:**
PVT modeling describes the changes in hydrocarbon fluid properties (volume, density, and phase) as a function of pressure and temperature. It is essential for converting surface volumes (STB) to reservoir volumes (RB).

Key Properties & Symbols 

- :math:`B_o`: Oil Formation Volume Factor (RB/STB) 
- :math:`R_s`: Solution Gas-Oil Ratio (scf/STB) 
- :math:`\mu_o`: Oil Viscosity (cP)
- :math:`\gamma_o, \gamma_g`: Specific gravities of oil and gas. 

The Bubble Point Pressure (:math:`P_b`) The pressure at which the first bubble of gas comes out of solution. Below this pressure, the fluid is "saturated." A common correlation used is Standing’s method.


.. math::

   P_b = 18.2 \left[ \left( \frac{R_s}{\gamma_g} \right)^{0.83} \times 10^{(0.00091 T - 0.0125 API)} - 1.4 \right]


Numerical Example:

- :math:`R_s = 500 \, \text{scf/STB}`
- :math:`\gamma_g = 0.65`
- :math:`T = 200 \, ^\circ\text{F}`:
- :math:`\text{API} = 35`


.. code-block:: csharp

   double Rs = 500, gamma_g = 0.65, T = 200,  API = 35, a = Pow(Rs / gamma_g, 0.83);
   double b = Pow(10, (0.00091 * T - 0.0125 * API));
   double Pb = 18.2 * (a * b - 1.4);
   Console.WriteLine($"Bubble Point Pressure = {Pb:F2} psia");


Ouput

.. terminal::

   Bubble Point Pressure = 2486.40 psia

2. Oil Formation Volume Factor(:math:B_o)Since oil shrinks as gas escapes, :math:B_o is almost always greater than 1.0.For pressures below the bubble point, we use the Standing correlation:

.. math::

   B_o = 0.9759 + 0.00012\left[R_s \left( \frac{\gamma_g}{\gamma_o} \right)^{0.5} + 1.25 T \right]^{1.2}


Numerical Example:

- :math:`\gamma_o = 0.85` (Typical for 35 API)
Using :math:`R_s`, 
- :math:`\gamma_g`, and
- :math:`T` from above:


.. code-block:: csharp

   double Rs = 500, gamma_g = 0.65, gamma_o = 0.85, T = 200;
   double F = Rs * Pow(gamma_g / gamma_o, 0.5) + 1.25 * T;
   double Bo = 0.9759 + 0.00012 * Pow(F, 1.2);
   Console.WriteLine($"Bo at Bubble Point = {Bo:F3} RB/STB");


Ouput

.. terminal::

   Bo at Bubble Point = 1.280 RB/STB

3.Gas Compressibility Factor(:math:`z`)
For gas modeling, the Ideal Gas Law fails at high pressure. We use the :math:`Z`-factor to correct it. The Hall-Yarborough or Dranchuk-Abu-Kassam methods are standard for coding this.

Linearization for Gas Density:

.. math::

   \rho_g = \frac{P \cdot MW_g}{ Z \cdot R \cdot T}


4. Gas Formation Volume Factor( :math:`B_g`) is the ratio of the volume of gas at reservoir conditions to the
volume of the same mass of gas at standard conditions. Because gas is highly
compressible, :math:`B_g` is always a very small number (typically :math:`< 0.01`).

Mathematical Expression:
Derived from the Real Gas Law (:math:`pV = nzRT`):

.. math::

   B_g = 0.02827 \frac{Z T}{p} \quad [\text{rcf/scf}]

Or in field units (res bbl/scf):

.. math::

   B_g = 0.005035 \frac{Z T}{p} \quad [\text{rb/scf}]

Where:

- :math:`p` = Reservoir pressure (psia)

- :math:`T` = Reservoir temperature (:math:`^\circ R`)

- :math:`Z` = Gas deviation factor at :math:`p` and :math:`T`

5.Isothermal Oil Compressibility(:math:`c_o`)
Above the bubble point (undersaturated), the oil volume changes only slightly due to pressure.

.. math::

   c_o = \frac{-1}{ V} \left( \frac{\partial V}{\partial P} \right)_T


Code Implementation for Undersaturated :math:`B_o`.
If :math:`P > P_b`, we adjust the :math:`B_{ ob}` (at bubble point) using compressibility:

.. code-block:: csharp

   double Bob = 1.32; // Bo at bubble point
   double co = 15e-6; // psi^-1
   double P = 5000;   // Reservoir pressure
   double Pb = 2500;  // Bubble point
   
   // Bo = Bob * exp(-co * (P - Pb))
   double Bo = Bob * Exp(-co * (P - Pb));
   Console.WriteLine($"Undersaturated Bo at {P} psi = {Bo:F3} RB/STB");


Ouput

.. terminal::

   Undersaturated Bo at 5000 psi = 1.271 RB/STB

Practical Application: Material BalanceWe combine these PVT parameters to calculate the Original Oil In Place (OOIP):

.. math::

   N = \frac{N_p B_o + (G_p - N_p R_s) B_g}{ B_o - B_{ oi} }