Analytical solution for a radial advection-dispersion equation including both mechanical dispersion and molecular diffusion for a steady-state flow field in a horizontal aquifer caused by a constant rate injection from a well

Summary Productivity-index (PI) optimization by means of optimal fracture design for a vertical well in a circular reservoir is a canonical problem in performance optimization for hydraulically fractured wells. Recent availability of the exact analytical solution for the pseudosteady-state (PSS) flow of a vertically fractured well with finite fracture conductivity in an elliptical drainage area provides an opportunity to re-examine this fundamental problem in a more-rigorous manner. This paper first quantitatively estimates the shape-approximation-induced error in the PI when the exact solution for an elliptical drainage area is applied to a circular drainage area. It is shown that the shape-approximation-induced error in the PSS-flow PI is less than 1% for fracture penetration ratios up to 53%, and this error decreases significantly as the fracture conductivity is increased. PI optimization is then performed with the highly accurate analytical solution for this range of the penetration ratios. The results show that the optimal fracture conductivity increases linearly from 1.39 to 1.71 when the proppant number is increased from 0.0001 to 0.6. PI for the steady-state flow and a popular ad hoc PSS-flow PI are compared with the analytical PSS-flow PI. It is found that both the steady-state and the ad hoc PIs deviate significantly from the analytical PSS-flow PI. In particular, the optimal fracture conductivity for the steady-state flow and the ad hoc PIs decreases with the proppant number, opposite to the trend observed for the optimal fracture conductivity for the PSS flow. It is suggested that the ad hoc PI should be abandoned in favor of the more-rigorous analytical PSS-flow solution.

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Steady state flow/field model of solar wind interaction with Venus: Global implications of local effects

Journal of Geophysical Research Atmospheres ◽

10.1029/ja092ia07p07289 ◽

1987 ◽

Vol 92 (A7) ◽

pp. 7289 ◽

Cited By ~ 31

Author(s):

P. A. Cloutier ◽

H. A. Taylor ◽

J. E. McGary

Keyword(s):

Solar Wind ◽

Steady State ◽

Flow Field ◽

Field Model ◽

Steady State Flow ◽

Solar Wind Interaction ◽

Local Effects ◽

State Flow

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Steady state flow for inside porous flat channel with using analytical solution to fourth-grade fluid

Journal of the Brazilian Society of Mechanical Sciences and Engineering ◽

10.1007/s40430-014-0204-5 ◽

2014 ◽

Vol 37 (2) ◽

pp. 525-531 ◽

Cited By ~ 2

Author(s):

M. Ebrahimpour ◽

D. D. Ganji ◽

H. Foroughnia

Keyword(s):

Steady State ◽

Analytical Solution ◽

Fourth Grade ◽

Flat Channel ◽

Steady State Flow ◽

State Flow ◽

Grade Fluid ◽

Fourth Grade Fluid

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Dusty Gas Flow in a Converging-Diverging Nozzle

Journal of Fluids Engineering ◽

10.1115/1.2823554 ◽

1999 ◽

Vol 121 (4) ◽

pp. 908-913 ◽

Cited By ~ 7

Author(s):

O. Igra ◽

I. Elperin ◽

G. Ben-Dor

Keyword(s):

Steady State ◽

Flow Field ◽

Gas Flow ◽

Jet Flow ◽

Mach Disk ◽

Solid Particles ◽

Steady State Flow ◽

Free Jet ◽

State Flow ◽

Free Jet Flow

The flow in a converging-diverging nozzle is studied numerically. The flowing medium is a suspension composed of gas seeded with small, spherical, solid particles. The solution covers the entire flow history, from its initiation and until a steady state flow is reached. The covered flow domain includes both the flow field inside the nozzle and part of the free jet flow outside of the nozzle exit plane. The solution is repeated for different solid particle diameters, ranging from 0.5 μm to 50 μm, and different dust loading ratios. It is shown that the presence of solid particles in the flow has a significant effect on the developed flow field, inside and outside the nozzle. In particular, by a proper choice of particles diameter lateral pressure waves and the secondary shock wave can be significantly attenuated. The solid particles size has also a marked effect on the position and size of the Mach disk appearing in the free jet flow. It is also shown that in a suspension case a steady state flow is reached faster than in a similar pure gas flow.

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Electrohydrodynamics of a liquid drop: the development of the flow field

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1973.0096 ◽

1973 ◽

Vol 334 (1598) ◽

pp. 343-356 ◽

Cited By ~ 11

Keyword(s):

Steady State ◽

Flow Field ◽

Electric Field ◽

Liquid Drop ◽

Drop Surface ◽

Steady State Flow ◽

Field Stress ◽

State Flow ◽

Developing Flow ◽

Set Up

In this paper we consider the development of the flow field in and about a liquid drop immersed in a conducting fluid, induced by an electric field stress. We place special emphasis to the case when the applied electric field is a d.c. field. We assume that the electric field stress is set up instantaneously and investigate the development of the flow field as the drop is being deformed. Thus, the present work is an extension of Sir Geoffrey Taylor’s work concerning the steady state flow field set up by a d.c. field and the author’s work concerning the quasi-steady state flow generated by an a.c. field. In the case of a d.c. field, the fluid circulation in the proximity of the drop surface initially forms closed loops which eventually propagate to infinity. Also, in the proximity of the drop surface, the developing flow field may be more intense and even directed in opposite sense in comparison with that of the steady state. In the limit, when the time t tends to infinity, the solution presented here converges to the solutions established in the papers referred to above.

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