# Introduction

• Deeper and consequently hoter holes
• the prediction and control of down hole mud properties depend in part upon our knowledge of temperature in the well bore
• the undrstood of the temperature distribution in a circulating drilling fluid improve the drilling operation

# Assumptions

• Axial conduction of heat in the fluid is negligible compared with axial convection.
•  No radial gradients in the fluid The fluid’s properties (heat capacity, density, and thermal conductivity) do not change significantly with temperature.
• Heat generation by viscous dissipation in the fluid is negligible

# Temperature behavior during circulation

There are 3 phases during the circulation of the mud drilling:

1. Fluid enters the drill pipe at the surface and passes down the drill pipe
2. Fluid exists the drill pipe through the bit and enters the annulus
3. Fluid passes up the annulus and exits the annulus at the surface

# Equations

In the pipe drill the temperature enters at Td0=25°C , this temperature is influenced by
• The heat exchanged between the drill pipe and the annulus
• The heat convected down the drill pipe
• The change with time
The equation which describes the temperature evolution into the pipe with the time and the depth is:

Boudary and initial conditions for this phase are

Td(Z=depth,t)=Ta(Z=depth,t)
Td(Z=0,t)=Td(t)

In the annulus the temperature depend on:
• Convection in the annulus
• The heat exchanged between the drill pipe and the annulus
• The heat exchanged between the formation and the annulus

The condition at the bottom well is
Td(Z=depth,t)=Ta(Z=depth,t)

We should take into account that the formation temperature has an influence on temperatures circulation.

The radial diffusivity equation that control the formation temperature is:
We suppose that the formation temperature at infinity is equal to geothermal temperature.

The last equation express the equality between the flux of heat out of the formation and the flux of heat into the annulus.
Results This figure shows the effect of time on temperature for a simulated well, we see that the outlet temperature rises rapidly to reach after a constant value. The bottom hole formation anfd fluid have the same evolution, indeed they fell rapidly from their initial value then they reach a constant level.

# Results

Here we have the profile of temperature as function of time and depth, we find that temperature increases withdepth, and after one or two circulation there are no change with time.

The formation temperature depend on radial position, time, and boundary condition on r=0, Here we see that there are a propagation of heat in the formation, but the temperature change only by a few degrees

The figure follow, we have the temperature as function of depth at 2 hours of circulation time, we notice that the mud system temperature depend mainly on the formation temperature.

we see here the effect of time on the bottom hole fluid and formation Temperatures, and the outlet temperature. We notice that at the begining the bottom hole temperature fail rapidly, and the outlet temperature increases rapidly. Furthermore, the temperaure change with time just at the begining of circulation.

- Bottom fluid temperature is much lower than the geothermal rock temperature
- The temperature do not change appreciably with time after one or two circulations
- The necessity to pursue the study to add the effect of formation temperature
- Outlet Temperature rises rapidely
- Bottom hole fluid and formation fell rapidly

# Combination of the three equation:

Now we ill take in consideration the change of formation temperature with time, so we found the results follow:

It's important to note that the maximum is not occurs at the botom but ruther in the annulus, after the fluid has made the turn and is exposed to the high formation temperature.

We see that at the begining, the temperature has a big values, and it is no stable, but after 3000s, the temperature get a normal behavior.