# Fluid Gradients (Dual)

Fluid gradients (dual) is an external profile, where the unknown is the external pressure profile of the tubing. The pressure profile consists of the hydrostatic pressure from a fluid above and a fluid below the top of cement (TOC).

NOTE!
In this documentation we denote any tubular as casing or tubing. All calculations however encompass any tubular, such as tubings, casings, liners, tie-backs etc.

## Inputs#

The following inputs define the fluid gradients (dual) external profile:

1. The true vertical depth (TVD) along the wellbore as a function of measured depth. Alternatively, the wellbore described by a set of survey stations, with complete information about measured depth, inclination, and azimuth.

2. The true vertical depth (TVD) of:

a. The rig RKB, $\text{TVD}_{\text{RKB}}$

b. The hanger of the casing / tubing, $\text{TVD}_{\text{hanger}}$

c. The shoe of the tubing, $\text{TVD}_{\text{shoe}}$

3. The fluid density above top of cement, $\rho_m$

4. The fluid density below top of cement, $\rho_{c}$

## Calculation#

The external pressure profile above and including TOC, parametrized by TVD, of the casing / tubing is then given by:

$p_{\text{TOC}} = \rho_{m}\, g\, \text{TVD}_{\text{TOC}}$

where $g$ is the gravitational constant, and $p_{\text{TOC}}$ is the hydrostatic fluid pressure at top of cement. The pressure below TOC and including the casing / tubing shoe is thus:

$p_{\text{shoe}} = \rho_{c}\, g\, \left(\text{TVD}_{\text{shoe}} - \text{TVD}_{TOC} \right)+ p_{TOC}$