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# 4 Approximate Analytical Solution

## 4.1 Step Function Approximation

[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.

## 4.2 Travelling wave solutions

[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]

## 4.3 General overshoot solutions with two wave speeds

[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
Table 1: Parameter values, their symbols and units