An eulerian-eulerian approach for oil&gas separator design conference Paper

Assumption on the fluid-dynamics
Regarding the fluid-dynamics aspects, our assumptions deal with two aspects: 
1. Phase physical properties: the mixture is composed of three phases (gas, oil and water), 
which are pure and with constant properties; 
2. Flow characteristic: the flow is incompressible and adiabatic with absence of physical 
reactions. Moreover, it can span only the disperse and separate flow regime; 
PRELIMINARY RESULTS 
Before testing the code in a separator, the code has been validated in order to ensure the correctness 
of the algorithm in addressing the previously explained numerical aspects. For this purpose, two 
analytical benchmarks have been used: water faucet problem and phase-redistribution problem. 
Water Faucet problem 
The Water faucet problem [8] represents a standard benchmark for two-phase flow solvers. Even 
though the present code has been implemented with the capabilities of handling three independent 
phases, if two of them represent the same phase with phase fractions equally divided, the overall 
code should degenerates to a two phase equivalent solver. This test consists of a vertical tube of 12 
m length and 1 m in diameter. The tube is initially filled with an air-water homogeneous mixture with 
a liquid phase fraction 
𝛼
𝑙
0
= 0.8
. At the inlet a mixture of the same composition enters the tube with 
a liquid velocity of 
𝑣
𝑙
= 10 𝑚/𝑠
, whereas the gas is at rest (i.e., 
𝑣
𝑔
= 0 𝑚/𝑠).
Due to gravity 
acceleration and mass conservation, the liquid vein diameter decreases and a phase fraction 
discontinuities propagates downward. Here the momentum transfer term is negligible (i.e., drag and 
virtual mass terms absent) and the dominant effect is the gravity force. The analytical solution for 
𝛼
𝑙
is given by: 

---

10 
𝛼
𝑙
(𝑥, 𝑡) =
𝛼
𝑙
0
𝑣
𝑙
0
√2𝑔𝑥 + (𝑣
𝑙
0
)
2
𝑖𝑓 𝑥 ≤ 𝑢
𝑙
0
𝑡 +
𝑔𝑡
2
2
In figure 6, we plotted the analytical solution against the numerical one, computed with two 
discretization schemes: Upwind (1
st
order), Van Leer (2
nd
order). Due to time-dependent nature of 
the solution, we considered the results at 
𝑡 = 0.5
s. As expected, 2
nd
order scheme produces better 
results in shock-capturing abilities reducing the numerical diffusion across the interface. On the other 
hand, 1
st
order schemes show the highest tendency to smear the discontinuities. Alongside with the 
good agreement with the analytical benchmark, this test proves that the first two numerical aspects 
previously explained are correctly implemented in the algorithm. 
Figure 6: Water Faucet problem