[2330.2.1] The basic idea of the analysis below is to approximate the travelling wave profile for long times with piecewise constant functions (step functions). [2330.2.2] For large [page 2331, §0] (i.e. in the Buckley-Leverett limit) one may view this approximate profile as a superposition of two Buckley-Leverett shock fronts. [2331.0.1] This is possible by virtue of the fact, that the Heaviside step function

(35) |

can also be regarded as a function of the similarity variable of the Buckley-Leverett problem.

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

(36a) |

of an imbibition front

(36b) |

located at and a drainage front profile located at

(36c) |

both moving with the same speed , where resp. are the upper (inlet) resp. lower (outlet) saturations and is the maximum (overshoot) saturation. [2331.1.2] The quantity is the distance by which the imbibition front precedes the drainage front, i.e. the width of the tip (=overshoot) region.

[2331.2.1] The two equations (28) become coupled, if eq. (20) holds true, because then there is only a single wave speed for both fronts. [2331.2.2] At the imbibition discontinuity the Rankine-Hugoniot condition demands

(37a) |

and the second equality (with colon) defines the function . [2331.2.3] Similarly

(37b) |

defines the drainage front velocity as a function of the overshoot . Examples of the velocities used in the compuations are shown in Figure 1a. [2331.2.4] For a travelling wave both fronts move with the same velocity so that the mathematical problem is to find a solution of the equation

(38) |

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

[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 and overshoot . [2332.1.2] More generally, if the saturation plateau is larger or smaller than , one expects to find non-monotone profiles that are, however, not travelling waves. [2332.1.3] Instead the drainage and imbibition fronts are expected to have different velocities. [2332.1.4] The fractional flow functions with relative permeabilities from eqs. (31) and the parameters from Table 1 give for the result

(39) |

while for one has

(40) |

[2332.1.5] In this case, for plateau saturations , the leading (imbibition) front has a smaller velocity than the trailing (drainage) front. [2332.1.6] Thus the trailing front catches up and the profile approaches a single front at long times. [2332.1.7] For plateau saturations on the other hand the trailing drainage front moves slower than the leading imbibition front. [2332.1.8] In this case a non-monotone profile persists indefinitely, albeit with a plateau (tip) width that increases linearly with time.

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 | Pas | |

viscosity | 0.0003 | Pas | |

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 |