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= Some explanation = 
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work as Distance*Elevation Crossin et al. 2004 or something more ?

Van Ginneken et al. (2008) estimate a cost of transport (COT)in kj.kg-1.km-1 do the amount of energy will be proportional to the distance covered by silver eel. It makes sense to say that the energy expenditure is relative to the distance covered (E1)
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Après avoir regardé la publi indiquée par Laurent, il semble que l'hypothèse que l'énergie soit relative à la distance est juste

Wheis, 1978 [[BR]]
Work for constant speed energy [[BR]]
Thrust equal to hydrodynamic resistance R [[BR]]
R=a/2*Af*Cd*V^2  [[BR]]
a density of the water [[BR]]
With Af=frontal area [[BR]]
Cd=drag coefficient [[BR]]
V relative velocity between the water and the fish [[BR]]
Total work=R*D where D is the distance covered [[BR]]
The total energy requirement inclusdes in addition to the energy expenditure the maintenance of bodily functions
=standard resting metabolic rate [[BR]]
Esm=M*t [[BR]]
where M is the standard resting metabolic rate and t is time [[BR]]

total work=propulsive eneregy + standard metabolic rate energy=
Epe+ Esm= [[BR]]
R*D +Mt=R*D+M*D/V [[BR]]
=a/2*Af*Cd*V*V*D+M*D/V  [[BR]]
on retrouve bien pour une vitesse relative constante une grandeur relative à la distance parcourue
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 The additional distance linked to altitude (water current) is relative to
 Vitesse d'écoulement :

      vitesse instantanée: u = dx/dt
      vitesse moyenne : V
Dans un chenal, relation empirique de Chezy:
V= C (h S)1/2
                                    C : coefficient de friction de Chezy
                                    h : profondeur
                                    S : pente
The additional energy (E2)due to slope is related to the current
distance additional=v*t
~c(hS)1/2*t
~alpha t (E/D)1/2 with alpha a multiplying coeff depending on depth, C
t is proportional to the distance
so E2= beta(DE)1/2

Energy=D+sqrt(DE) which is correct for dimensions (m) Energy ~ distance ~ mgh (potential energy)