Page 163 - Fundamentals of Enhanced Oil and Gas Recovery

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Thermal Recovery Processes

This problem is solved explicitly in which the mean temperature of the last time
step is applied for the present time. The integral is converted to
f n 5 f n21 1 Δf PD (5.62)
PD PD
where n represents the time step and Δf PD is calculated as

n21
5:615 q o M o1 q w M w ÞðT avg 2 T R ÞΔt
ð
Δf PD 5 (5.63)
2Q max
The main steps for solving this problem are summarized as follows:
1. Initialize the model by inserting the fluid, reservoir, and operational data.
2. Calculate the initial values for the radius or thickness of the zone, fluid properties,
and saturations.
3. Using small time steps, calculate the water and oil production cumulatively or in
each time step. Determine the mean temperature for each cycle and finally deter-
mine the oil in place by checking the cumulative production in each cycle.
4. Check the requirements for the end of cycle by the number of time steps and
then continue.
5. Determine the amount of the remained heat and water in the reservoir.
6. It may require continuing calculations for another cycle, then go to step 2.
Otherwise this is end of calculations.

5.2.2.4 CSS 2 Boberg Lantz Model [36]
The most important assumptions of this model include
• Boberg Lantz model uses Marx Langenheim model to calculate radius of the
heated zone.
• The reservoir is assumed to heat to T s instantaneously.
• Initially, the entire heated zone is at T s while (remaining of) o/u and reservoir are at T R .
• Heat loss and production of hot fluid are considered.
• Although cold fluids enter the hot region, it has not been considered in energy
balance.
2
k h @ @T @ T @T
r 1 k h 5 ρC p (5.64)
r @r @r @z 2 @t
The initial conditions are reported as follows and represented in Fig. 5.3.

at t 5 t i ; T 5 T s for ðr w # r # r s and 0 # z # hÞ

at t 5 t i ; T 5 T R for ðr . r s and z , 0 and z . hÞ
The boundary conditions for r and z direction include
@T
at r 5 0; 5 0 (5.65)
@r