Vertical Lift Performance
=========================


**Definition:**
The Vertical Lift Performance (VLP) describes the relationship between the bottom-hole flowing pressure (:math:`p_{wf}`) and the production rate (:math:`q`). It represents the pressure required to lift fluids from the bottom-hole to the surface against gravity, friction, and acceleration.

Oil Well VLP (Single Phase)
~~~~~~~~~~~~~~~~~~~~~~~~~~~
For a single-phase liquid, the pressure gradient (:math:`dp/dz`) is the sum of hydrostatic (elevation) and frictional components.

**Differential Equation:**

.. math::

   \frac{dp}{dz} = \rho g \sin(\theta) + \frac{2f \rho v^2}{D}

Where:
- :math:`\rho` = fluid density (:math:`lb/ft^3`)
- :math:`g` = gravitational constant
- :math:`f` = Fanning friction factor
- :math:`v` = fluid velocity (:math:`ft/s`)
- :math:`D` = tubing internal diameter (:math:`ft`)

Numerical Example (ODEs with SepalSolver): We solve for :math:`p_{wf}` by integrating from surface pressure (:math:`p_{surf}`) to the
total depth (:math:`H`).


.. code-block:: csharp

   // Inputs
   double p_surf = 200; // psi (Wellhead Pressure)
   double depth = 2000; // ft
   double q_o = 500; // STB/day
                    // ODE Definition: dp/dz = gradient
   double pressureGradient(double z, double p, double q)
   {
       double density = 55.0; // lb/ft3 (Oil)
       double friction_grad = 0.00002 * Pow(q, 1.8); // Simplified friction term
       double hydro_grad = density / 144.0; // psi/ft
       return hydro_grad + friction_grad;
   }
   // Solve using SepalSolver Ode45
   // Integrate from z=0 (surface) to z=8000 (bottom-hole)
   var (Z, P) = Ode45((z, p)=>pressureGradient(z, p, q_o), p_surf, [0, depth]);
   double p_wf = P[^1]; // extract the pressure at the bottom
   Console.WriteLine($"Bottom-hole Flowing Pressure (Pwf) = {p_wf:F2} psi");



   //FullRange
   double pfun(double q_g)
   {
       var (Z, P) = Ode45((z, p) => pressureGradient(z, p, q_g), p_surf, [0, depth]);
       double p_wf = P[^1]; // extract the pressure at the bottom
       return p_wf;
   }
   ColVec Qrange = Linspace(0, 800);
   ColVec Prange = Arrayfun(pfun, Qrange);
   Plot(Qrange, Prange, "b", 2);
   Xlabel("Flowrate Q (STB/day)");
   Ylabel("Pressure P (psia)");
   Title("OilVLP");
   SaveAs("OilVLP.png");


Ouput

.. terminal::

   Bottom-hole Flowing Pressure (Pwf) = 3849.29 psi

.. figure:: images/OilVLP.png
   :align: center
   :alt: OilVLP.png


Gas Well VLP
~~~~~~~~~~~~
Gas VLP is more complex because gas density is highly dependent on pressure. As gas rises, it expands, increasing velocity and frictional losses.

Differential Equation:

.. math::

   \frac{dp}{dz} = \frac{p M}{z R T} g \sin(\theta) + \frac{2f \rho v^2}{D}


Numerical Example (ODEs with SepalSolver):
In this example, the gradient function must recalculate gas density (:math:`\rho_g = \frac{p M}{Z R T}`) at every step of the integration.


.. code-block:: csharp

   double p_surf = 500; // psia
   double depth = 10000; // ft
   double q_g = 5000; // Mscf/day
   // ODE Definition for Gas
   double pressureGradient(double z, double p, double q)
   {
       double MW = 20.0; // Gas molecular weight
       double T = 540 + (0.015 * z); // Temp profile in Rankine
       double Z = 0.85; // Average Z-factor
       double R = 10.73;

       // Density as a function of current Pressure (p)
       double rho_g = (p * MW) / (Z * R * T);

       double hydro_grad = rho_g / 144.0;
       double friction_grad = 1.5e-9 * (Pow(q, 2) / p); // Simplified gas friction
       return hydro_grad + friction_grad;
   }
   // Solve using SepalSolver Ode45
   // Integrate from z=0 (surface) to z=8000 (bottom-hole)
   var (Z, P) = Ode45((z, p)=>pressureGradient(z, p, q_g), p_surf, [0, depth]);
   double p_wf = P[^1]; // extract the pressure at the bottom
   Console.WriteLine($"Bottom-hole Flowing Pressure (Pwf) = {p_wf:F2} psi");


   //FullRange
   double pfun(double q_g)
   {
       var (Z, P) = Ode45((z, p) => pressureGradient(z, p, q_g), p_surf, [0, depth]);
       double p_wf = P[^1]; // extract the pressure at the bottom
       return p_wf;
   }
   ColVec Qrange = Linspace(0, 8000);
   ColVec Prange = Arrayfun(pfun, Qrange);
   Plot(Qrange, Prange, "b", 2);
   Xlabel("Flowrate Q (Mscf/day)");
   Ylabel("Pressure P (psia)");
   Title("GasVLP");
   SaveAs("GasVLP.png");


Ouput

.. terminal::

   Bottom-hole Flowing Pressure (Pwf) = 642.03 psi

.. figure:: images/GasVLP.png
   :align: center
   :alt: GasVLP.png