HR: 13:55h
AN: H23I-02    [Abstracts]
TI: Travel Time of Fluids in Porous Media: Bernoulli's Brachistochrone, Isoperimetric Estimates, and Optimal Design in Smart Hydroisolation of Horizontal Wells
AU: * Kacimov, A
EM: anvar@squ.edu.om
AF: Department of Soils, Water, and Agricultural Engineering, Sultan Qaboos University, Al- Khod-123, Muscat, POBox34, Oman
AU: Marketz, F
EM: Franz.Marketz@pdo.co.om
AF: Petroleum Development Oman, Muscat-113, Muscat, POBox81, Oman
AU: Pervez, T
EM: tasneem@squ.edu.om
AF: Department of Mechanical Engineering, Sultan Qaboos University, Al-Khod-123, Muscat, POBox34, Oman
AU: Yakimov, N
EM: nyakimov@inbox.ru
AF: Institute of Mathematics and Mechanics, Kazan University, University Str., 17, Kazan, 420028, Russian Federation
AB: Steady and transient 2,-3- D seepage in a rigid porous medium is studied by the methods of complex analysis and optimal shape design. Euler (1748) dichotomised the inverse-direct approaches. In the former, "surmised" functionals (in our case - travel time of a marked (neutral tracer) particle, seepage discharge, and acting seepage forces) are extremised under isoperimetric constraints, e.g., the "action formalis" (total head along streamlines of flow tubes) of Leibnitz (1860). Following Euler's ideas, the parameters of impervious hydraulic structures (dam, sheetpiling, well packer, seabed buried pipeline, etc.) are optimized. In the Verigin problem, the angle of inclination of a sheetpile providing minimal time along a bounding streamline is found. The maximum of the minimum of the travel time is searched between all streamlines originated in the upper pool. The calculated travel time distributions are also used in minimisation of the total volume of fluid, which arrives from the upper pool to the tailwater during a prescribed time span. In the class of arbitrary structures, time optimisation is carried out explicitly for the bounding streamline with a constraint of the wetted perimeter of a depressed structure. The found optimum (brachistochrone) is a semi-circle and the corresponding time coincides with that of a unidirectional flow between two constant head surfaces. A general theorem is proved that for an arbitrary 3-D flow tube the travel time along an arbitrary streamline is bounded from below by that in a 2-D Voshinin problem (line vortex). Generalisations to heterogeneous media (of varying hydraulic conductivity and porosity) and non-Darcian flows make possible isoperimetric estimates of breakthrough curves (i.e. a composition of marked particles as in Lagrange, 1760) in a purely advective transport of contaminants and advective dispersion with negligible transverse dispersivity. Seepage from one horizontal isobar to another with a tilted fracture intersecting the two isobars is studied for an impermeable seal of a constant length, which is placed as an expandable zonal isolation at the fracture-isobar intersection. The objective in optimization is either the total flow rate from the fracture or the travel time along the seal streamline. The optimal seal position is found for any given distance between isobaric horizons, tilt angle, pressure drop between isobars and hydraulic conductivity of the rock. For an impermeable depressed hydraulic structure three pore-pressure caused functionals (uplift and traction forces and the moment of the total force vector) are optimized in the class of semi-elliptical contours. In transient problems, a non-circular cylindrical no-flow boundary, partially buried into a porous bed and partially protruding into a sea layer where periodic surface waves cause 2-D pore pressure variations, is studied. The Polubarinova- Kochina model of transient impact on the structure delivers optimal shapes of cylinders for different frequencies of the imposed surface waves. References: Euler, L., 1748. Recherches sur les plus grands et les plus petits qui se trouvent dans les actions des forces. Mem. De l' Acad des Sciences de Berlin, t.4. Lagrange, J.L., 1760 Miscellanea Taurinensia, t.2 Leibniz G.W., 1860. Mathematische Schriften. Herasg. V. Gerchardt, t.II, III
DE: 1828 Groundwater hydraulics
DE: 1847 Modeling
SC: Hydrology [H]
MN: 2007 Fall Meeting