[2330.2.1] The basic idea of the analysis below
is to approximate the travelling wave profile
for long times

(35) |

can also be regarded as a function of the similarity
variable

[2331.1.1] In the crudest approximation
one can split the total profile
for sufficiently large

(36) |

of an imbibition front

(36) |

located at

(36) |

both moving with the same speed

[2331.2.1] The two equations (28) become coupled,
if eq. (20) holds true, because
then there is only a single wave speed

(37) |

and the second equality (with colon)
defines the function

(37) |

defines the drainage front velocity as a function

(38) |

obtained from equating eqs. (37b) and (37a)
(See Fig. 1a).
[2331.2.5] The wave velocity

[page 2332, §1]

[2332.1.1] Equation (38) provides a necessary condition for
the existence of a travelling wave solution of the form of
eq. (36) with velocity

(39) |

while for

(40) |

[2332.1.5] In this case, for plateau saturations

Parameter | Symbol | Value | Units |
---|---|---|---|

system size | 1.0 | m | |

porosity | 0.38 | – | |

permeability | m | ||

density |
1000 | kg/m | |

density |
800 | kg/m | |

viscosity |
0.001 | Pa | |

viscosity |
0.0003 | Pa | |

imbibition exp. | 0.85 | – | |

drainage exp. | 0.98 | – | |

end pnt. rel.p. | 0.35 | – | |

end pnt. rel.p. | 1 | – | |

end pnt. rel.p. | 0.35 | – | |

end pnt. rel.p. | 0.75 | – | |

imb. cap. press. | 55.55 | Pa | |

dr. cap. press. | 100 | Pa | |

end pnt. sat. | 0 | – | |

end pnt. sat. | 0.07 | – | |

end pnt. sat. | 0.045 | – | |

end pnt. sat. | 0.045 | – | |

boundary sat. | 0.01 | – | |

boundary sat. | 0.60 | – | |

total flux | 1.196 10 |
m/s |