Geneus me!! at the description I didn't write at the boundaries "=0" to one of them. But I simplefied something.

In ordxer to make the boundary conditions understandable, I introduced fractional distance X that 0

s is a fraction of saturation that's why   takes only values from swi to (1-sor)

Then for  X=1 :  F(s)+k*sigma*F(s)*kro(s)*`∂`(J(s))/(U*L*`μo`*sqrt(k/phi)*`∂`(X)) = 0

and for  X=0 : s=swi

and the equation becomes:

`∂`(F(s))/`∂`(X) = -k*sigma*cos(theta)*`∂`(F(s)*kro(s)*`∂`(J(s))/`∂`(X))/(U*L*`μo`*sqrt(k/phi)*`∂`(X))

and the function remain the same

> krowi := 1; krwor := .1; aw := 1.16; ao := 3.69; alpha := 0.6e-1; U := 2.9374109836-10^(-7); L := 0.552566e-1; m := .5; sor := .3; swi := .25; k := 10^(-14); `μw` := 0.1e-2; `μo` := 0.5e-2; sigma := 0.72e-1; phi := .3; theta := 0;
> C := krowi*(1-sor-swi)^(-ao);
> A := krwor*(1-sor-swi)^(-aw);
> krw(s):=A*(s-swi)^(aw):
> kro(s):=C*(1-sor-s)^(ao):
> F(s):=((krw(s))/(muw))/((krw(s))/(muw)+(kro(s))/(muo)):
> J(s):=alpha*(((s-swi)/(1-sor-swi))^(-1/(m))-1)^((1-m)):

Thank you trying!

Thank you Preben. I fixed the issue with the muo-muw and the brackets.

I should have specified that this is about a waterfooding experiment.

x  is distance from the water inlet is a sample of total length L, and my unknown is "s" along the sample.

so I actually need a plot of   (s,x=0..L)