Environmental Engineering Reference
In-Depth Information
The mathematical model solution allows knowing the pressure, temperature,
steam quality, and flow patterns of the water-steam mixture, as well as the ther-
mal energy that is dissipated to the surroundings. Although a steady-state drift-flux
was used, such solution is on semi-unsteady-state due to the accumulation term in the
2D heat diffusion equation allows us to evaluate temperature changes in the porous
medium as time elapses.
On the other hand, to get a closed system, some auxiliary equations were used to
evaluate: (1) the gas volume fraction, (2) the drift velocity, (3) flowpatterns transitions
and (4) fluids properties. Peng-Robinson and Valderrama-Patel-Teja state equations
were also implemented to evaluate steam and air density.
3 Numerical Implementation
Finite differencemethodwas used to resolve the steam injectionmathematical model.
A staggered mesh (see Fig. 1 ) was implemented for the hydrodynamic model, so
most of variables were evaluated at the center of each cell, but superficial and mixture
velocities were evaluated at the boundary of the cells. The transient 2D heat diffusion
equation was discretized with an implicit scheme, with an irregular grid in r direction
and regular spacing in z direction (see Fig. 1 ). Spatial derivatives were discretized
applying the standard Godunov first-order upwinding scheme, while for the time
derivative in the heat diffusion equation we used a forward first-order scheme. A
computational algorithmwas built in the FORTRAN language to solve the discretized
system of equations. We refer to the reader to the work of Bahonar et al. ( 2009 )for
detail about the flowchart that shows the steps of the numerical algorithm that has
been used in this work. The following steps proposed by Bahonar et al. ( 2009 ) yield
a more detailed description of the numerical solution procedure:
1. At first grid block, P sat and
w are known, so the properties x ,
˙
ˁ l ,
ˁ g , v sl , v sg , h l
and h g can be calculated using correlations or an state equation.
2. For the first iteration, the properties of the previous grid block are used at the
current grid block. However, from second iteration the values of the previous
iteration are used at the current grid block.
3. Knowing P sat ; T sat , h l , h g ,
ˁ g are calculated.
4. The overall heat transfer coefficient is calculated as a function of depth using an
iterative scheme. For the first iteration, U to is calculated based on both, the initial
temperature distribution along the well and the boundary formation. For other
iterations, U to takes the value from the previous iteration. Knowing U to , we can
calculate T to and T ci , and then know a new value for U to . If there is difference
between them (old and new value for U to )
ˁ l and
, the iterative process has to continue
until convergence is achieved.
5. For the first iteration a superficial gas velocity, v sg k , is assumed and it is approx-
imated to v sg k + 1 / 2 . Then the superficial liquid velocity v sl k + 1 / 2 is calculated using
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