Front Controllability in Two-Phase Porous Media Flow
The propagation of the front (i.e. the interface) between two immiscible fluids flowing through a porous medium is governed by convection, i.e. by the fluid velocities at the front, which in turn are governed by the pressure gradient over the domain. We investigated a special case of immiscible two-phase flow that can be described as potential flow, in which case the front is sharp and can be traced with a simple Lagrangian formulation. We analyzed the controllability of the pressure field, the velocity field and the front position, for an input in the form of slowly time-varying boundary conditions. In the example considered in this paper of order one equivalent aspect ratio, controllability of the pressures and velocities at the front to any significant level of detail is only possible to a very limited extent.Moreover, the controllability reduces with increasing distance of the front from the wells. The same conclusion holds for the local controllability of the front position, i.e. of changes in the front position, because they are completely governed by the velocities. Aspect ratios much lower than one (for instance resulting from strongly anisotropic permeabilities) or geological heterogeneities (for instance in the form of high-permeable streaks) are an essential pre-requisite to be able to significantly influence subsurface fluid flow through manipulation of well rates.
KeywordsPorous Medium Singular Value Decomposition Line Source Singular Vector Front Position
Unable to display preview. Download preview PDF.
- Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publishers (1979)Google Scholar
- Brouwer, D.R.: Dynamic Water Flood Optimization With Smart Wells Using Optimal Control Theory. Ph. D. thesis, Delft University of Technology (2004)Google Scholar
- Brouwer, D.R., Jansen, J.D.: Dynamic optimization of waterflooding with smart wells using optimal control theory, SPE Journal (SPE 78278) 9(4), 391−402 (2004)Google Scholar
- Fyrozjaee, M.H.: Control of Displacement Fronts in Potential Flow using Flow-Rate Partitioning. Ph. D. thesis, University of Southern California (2008)Google Scholar
- Fyrozjaee, M.H., Y. C. Yortsos, Y.C.: Control of a displacement front in potential flow using flow rate partition. In: SPE Intelligent Energy Conference and Exhibition (SPE 99524), Amsterdam (2006)Google Scholar
- Stengel, R.F.: Stochastic Optimal Control: Theory and Application. John Wiley & Sons (1986)Google Scholar
- Van Doren, J.F.M.: Model Structure Analysis for Model-Based Operation of Petroleum Reservoirs. Ph. D. thesis, Delft University of Technology (in preparation)Google Scholar
- Yang, Z., Yortsos, Y.C., and Salin, D.: Asymptotic regimes of unstable miscible displacements in random porous media. Adv. Water Res., Special anniversary issue 25, 885−898 (2002)Google Scholar
- Zandvliet, M.J., Bosgra, O.H., Jansen, J.D., Van den Hof, P.M.J., Kraaijevanger, J.F.B.M.: Bang-bang control and singular arcs in reservoir flooding. Journal of Petroleum Science and Engineering 58(1&2), 186−200 (2007)Google Scholar