Sequential Methods for Coupled Geomechanics and Multiphase Flow
Sequential Methods for Coupled Geomechanics and Multiphase Flow
Sequential Methods for Coupled Geomechanics and Multiphase Flow
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210 CHAPTER 7. CONCLUSIONS<br />
7.2.3 Accuracy <strong>and</strong> Efficiency<br />
Given that the undrained <strong>and</strong> the fixed-stress splits have similar (very favorable) stability<br />
<strong>and</strong> convergence properties, which strategy should be used? Here, we have shown that <strong>for</strong><br />
problems of interest in reservoir engineering, the fixed-stress split converges significantly<br />
faster than the undrained split.<br />
The fixed-stress split requires few iterations to match the fully coupled solution, even<br />
when the coupling strength is quite high, while the undrained split requires many iterations<br />
per time step to achieve similar accuracy. When the fluid is incompressible, or nearly incom-<br />
pressible, the undrained split loses first-order accuracy. On the other h<strong>and</strong>, the fixed-stress<br />
split preserves first-order accuracy regardless of the fluid type. For the rate of convergence,<br />
the fixed-stress split needs only two iterations to converge <strong>for</strong> linear problems, regardless<br />
of coupling strength <strong>and</strong> pressure diffusivity. This assumes that we can estimate the local<br />
bulk modulus, Kdr, exactly. One cannot estimate the exact local Kdr in the flow problem<br />
in the presence of complex boundary conditions <strong>for</strong> the mechanics problem. However, our<br />
dimension based estimation of Kdr provides stability <strong>and</strong> first-order accuracy in time <strong>for</strong><br />
the fixed-stress split, which can be applied to an incompressible fluid.<br />
The undrained split also requires a robust linear solver, due to the fact that a stiffer<br />
mechanical problem must be solved, since the undrained moduli are used. In contrast, the<br />
fixed-stress split yields a less stiff mechanical problem (drained moduli are used) <strong>and</strong> a less<br />
stiff flow problem as well, due to the added pore-compressibility term.<br />
7.3 <strong>Coupled</strong> Mechanics <strong>and</strong> <strong>Multiphase</strong> <strong>Flow</strong><br />
In Chapter 6, we confirm that the fixed-stress split shows better numerical behaviors than<br />
the other sequential methods in the presence of multiple phases. In particular, we observed<br />
that the fixed-stress split with the staggered Newton scheme yields almost the same conver-<br />
gence rate as the fully coupled method. In fact, one or two iteration(s) with the fixed-stress<br />
split yield numerical solutions that match the fully coupled method. This implies that the
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