6 Numerical Applications
6.3 The Ninth SPE Benchmark Problem
The Ninth SPE Benchmark Problem 87
Model Grid Time Steps NEWTON
-Iterations
Wall Time/[h]
Immiscible 3D 133 1114 69.54
Immiscible 2D 115 929 0.24
Immiscible, non-isothermal 2D 115 932 0.49
PVS 2D 1066 12278 7.21
PVS, non-isothermal 2D 1744 20569 20.34
NCP 2D 1032 11817 12.22
NCP, non-isothermal 2D 1012 11503 17.43
Table 6.3: Performance metrics for various approaches on a single core of an3.5 GHzIntelR CoreTMi7-3770K CPU.
functions directly, although this is only barely enough to compensate the additional costs of the additional equations required by the NCP model.
Summarizing this example, we can conclude that the most effective approach to model radially symmetric problems is to use a two-dimensional grid in conjunction with radial domain extrusion. Further, Figure 6.6d shows that the results obtained using models that include miscibility are easily distinguishable from the ones neglecting it, even if miscibility is quite low as is the case for CO2injection scenarios. At the same time, it indicates that we do not need to consider energy as a conservation quantity for CO2injection scenarios if the temperature of the injected CO2is close to the temperature of the reservoir. Finally, we observe that the scalability of parallel runs of such simulations usually is not very good, and that this is primarily caused by the part which solves the linear systems of equations.
(a) The Grid, Initial Oil SaturationSoand Well Locations
(b) Intrinsic PermeabilitykKk/m2
(c) Oil Saturation After9000Days
(d) Oil Pressurepo/PaAfter9000Days
Figure 6.8: The simplified SPE-9 benchmark problem.
The Ninth SPE Benchmark Problem 89
t/d qW/(100 stb/d)
Black-oil NCP
0 2250 4500 6750
8.0 13.5 19.0 24.5 30.0
t/d qW/(100 stb/d)
Black-oil NCP
0 75 150 225
8.0 13.5 19.0 24.5 30.0
(a) Water Injection Rates
t/d
−qO/(100 stb/d)
Black-oil NCP
0 2250 4500 6750
7.5 9.6 11.8 13.9 16.0
t/d
−qO/(100 stb/d)
Black-oil NCP
0 75 150 225
7.5 9.6 11.8 13.9 16.0
(b) Oil Production Rates
t/d
−qG/(1000 m3/d)
Black-oil NCP
0 2250 4500 6750
35.0 46.2 57.5 68.8 80.0
t/d
−qG/(1000 m3/d)
Black-oil NCP
0 75 150 225
35.0 46.2 57.5 68.8 80.0
(c) Gas Production Rates
Figure 6.9: Predicted production and injection rates for the simplified SPE-9 benchmark:
The left side shows the production and injection curves for the whole simulated period of9000days, the graphs on the right show the same curves for the first300 days. The unit stb stands forstock tank barrel, and it is the amount of mass which is contained within one barrel at atmospheric pressure,i.e.,0.159 m3; the gas production rates are given in terms of the volume of the produced gas at atmospheric pressure.
Figure 6.10: Typical behavior of the NEWTON-RAPHSONmethod for the SPE-9 problem if using the PVS model: The phase presence pseudo variables oscillate. The value displayedχppis calculated using
χpp:= 20·p¯o+ 21·p¯w+ 22·p¯g
where p¯α is 1 if fluid phase α is present at a given spatial location, and zero otherwise. The phase presence initially displayed on the upper left is changed to the one on the upper right in the next NEWTON-RAPHSON iteration, then it assumes the value on the lower left, and finally goes back to the initial state as displayed on the lower right.
NCP Black-Oil
n[−] 84 60
nNEWTON[−] 536 327 tCPU[h] 0.77 0.19
Table 6.4: Comparison of the performance of the numerical models using the specialized black-oil model and the NCP model incorperating non-linear complementarity functions for the SPE-9 problem. Here,ncorresponds to the number of time steps required to simulate the setup for9000days,tCPU is the required computation time of a single core of an IntelR CoreTMi7-3770K CPU, andnNEWTONis the total number of iterations of the NEWTON-RAPHSONmethod required for the whole simulation.
model [19]. The parameters used are a skin factor of zero, a borehole radius of7.5 cm, and a bottom-hole pressure of275 barfor the injection well and206 barfor the production well.
This setup was simulated using the PVS and the NCP models as well as with the black-oil model. The obtained results for the water injection rate as well as for the black-oil and gas production rates agree very well as we can see in Figure 6.9. Inspecting this figure, we also note that the production and injection rates for the PVS model are not included. The reason for this is that the NEWTON-RAPHSON algorithm is very unstable for the SPE-9 problem when using the PVS model. This caused a non-recoverable breakdown at the beginning
The Fifth SPE Benchmark Problem 91
(a) (b)
(c) (d)
Figure 6.11: The SPE-5 benchmark problem: (a)The grid, initial oil phase saturation, and the locations of the wells.(b)Oil phase saturation after three years of production as seen from above. (c)The oil phase saturation after three years of production as seen from below. (d)The water phase saturation after three years of only oil production and one subsequent year of water injection plus oil production as seen from below.
of the simulation. The reason for this is illustrated in Figure 6.10. It displays the values of the phase-presence pseudo primary variables for successive iterations of the NEWTON -RAPHSON algorithm of a failing time step: The presence of the phases keeps oscillating between iterations instead of reaching a stable state. This behavior is likely caused by the similar values of the fugacity coefficients for components which are not preferred by the gas and the water phases. Since the NCP model directly embeds the presence of the fluid phases in the non-linear system of equations to be solved, it is less vulnerable to this issue and does not suffer a breakdown.
Moreover, we can infer from Figure 6.9 that the production and the injection rates calculated using the specialized black-oil model agree quite well with the ones computed using the generic NCP model. As outlined in Table 6.4, the computational effort for the generic NCP model is higher, though.
Componenth·iκ pκcrit [bar] Tcritκ [K] ωκ Water (H2O) 220.64 647.10 0.344 Methane (C1) 460.43 190.56 0.013 Propane (C3) 424.92 369.83 0.1524 Hexane (C6) 301.23 507.44 0.3007 Decane (C10) 209.60 617.67 0.489 Pentadecane (C15) 137.89 705.56 0.65 Icosane (C20) 111.69 766.67 0.85
Table 6.5: PENG-ROBINSONparameters used for the fifth SPE benchmark problem. All binary interaction coefficients¯kακλare zero, except for C1interacting with C15or C20, and C3interacting with C15or C20. In the former case, the interaction coefficient is assumed to be0.05while in the latter casek¯ακλ= 0.005is used.