Need help solving PDE, numerically which contains some transformations

[re444]

Hi, hope this is a right place to ask this question. I work in the soil physics field and this problem has taken lots of my energy for a while! let's state it:

Unsaturated horizontal water flow in 2 layer soil:

we have, M(for Moisture), K (for hydraulic conductivity), h (for hydraulic head), x , t (for time).

M(x,t,h) , K(x,t,h) , h(x,t).

PDE is of the form: ∂M/∂t = ∂/∂x (K ∂h/∂x ) .

on the domain (x=[0,xL] ) is divided into 2 layers (x1 & x2) each with its own M and K (2 soil type).
So I have M1, K1 for first layer and M2 & K2 for the second. Instead of solving these two PDE, by applying some linear transformations of the kind:

M*= k1.M ; K*= k2.K ; h*= k3.h ; x1*=k4.x1 ; x2*= k5.x2 ; t*= t .

I'm solving one unique PDE* in the domain (x1* $\cup$ x2*) with transformed boundary(at the upper of 1st layer and lower of 2nd layer) and initial conditions. Now here is my problem: when I get PDE* solutions and then transforms theme back, I get answers different from when not applying this whole transformation procedure.

I guess the question is: when I do a series of [linear] transformations to a PDE is it true to transform the solution back using the inverse of that transformations?

tried my best to express the problem,
thanks

 

[Chestermiller]

See Freeze and Cherry, "Groundwater."

 

[re444]

See Freeze and Cherry, "Groundwater."

I want to look at it from Differential Equations point of view. Just consider the last question at the bottom of 1st post.

 

[Chestermiller]

Let me see if I understand correctly. When you say that the flow is unsaturated, you mean that, within your two soil layers, there is an interface between the water below and the air above, and that this interface is moving as a function of time. Correct?

Chet