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Analytical solutions of 2D cresting models using the hodograph method

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Abstract

Three new applications of the hodograph method to the problem of gas and water cresting towards horizontal wells are presented:

  1. (1)

    A numerical technique, based on the hodograph method, is used to find the shape of the oil/water interface and values of critical heights for the problem of water cresting towards a horizontal well beneath an impermeable plane.

  2. (2)

    By correct implementation of the boundary conditions in the lateral edge drive model, a hodograph solution is found for which the water crest tends to a horizontal asymptote far from the well, rather than tending towards a parabolic curve as in previous approaches. The solution yields integral representations for the lengths of boundary segments and enables an explicit expression for the critical rate to be derived in terms of the distance to the constant potential boundary.

  3. (3)

    The problem of simultaneous gas and water cresting towards a horizontal well in a thin oil column reservoir is solved using the hodograph method, providing shapes of the free interfaces and an expression for the optimal placement of the well with respect to the interfaces.

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Abbreviations

H :

height of oil leg

S :

sink point

G :

point on the interface immediately above the sink

W :

point on the interface immediately below the sink

\(\overline {GG'} \) :

oil-gas interface

\(\overline {WW'} \) :

oil-water interface

\(\overline {W'G'} \) :

constant potential boundary

V :

volumetric flow rate

Μ :

viscosity

γ o , γ w :

specific gravities of flowing/stationary fluid phases

k :

permeability of the porous medium

y :

elevation above some arbitrary level

p :

pressure

q :

specific discharge vector

Φ :

potential function

u, v :

horizontal/vertical components of the specific discharge vector

ψ :

stream function

f :

complex potential function

Ω :

hodograph function

K :

parameter

z :

complex coordinate function

\(\bar \omega \) :

inverse hodograph function

ζ :

auxiliary function

x e :

distance from the sink to the constant potential boundary

L :

parameter which determinesx e

y e :

height of the cusp above the asymptote of the oil-water interface

b :

critical height from the top of the cusp to the sink

References

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Authors and Affiliations

  1. BHP Research - Melbourne Laboratories, Applied Mechanics Group, PO Box 264, 3168, Clayton, Vic., Australia

    J. F. McCarthy

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  1. J. F. McCarthy
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McCarthy, J.F. Analytical solutions of 2D cresting models using the hodograph method. Transp Porous Med 15, 251–269 (1994). https://doi.org/10.1007/BF00613281

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  • DOI: https://doi.org/10.1007/BF00613281

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