1. An example moving boundary problem Dry porous media Saturated porous media x = 0 x = s(t) h(0) = L Fixed Head If water head remains at fixed value L at x = 0 front between saturated and dry will move toward the right Governing Equations In saturated region Darcy equation applies Volume Flux m 3 -m -2 -s -1 Hydraulic Conductivity m-s -1 And If K constant, mass (volume) continuity gives the governing equation With conditions Moving boundary condition

2. Dimensionless form

3. Analytical Solution (1) General solution of (1) is Solution satisfying first two boundary conditions (2) Sub into (2) to get following initial value ODE in s Solution

4. An example moving boundary problem Dry porous media Saturated porous media x = 0 x = s(t) h(0) = 1 Fixed Head If water head remains at fixed value L at x = 0 front between saturated and dry will move toward the right

5. A Numerical Solution With a Fixed Uniform Grid with spacing 1 i=0 1 2 3 4 5 6 7 ------------------- 1 Key assumption: While Node number So while front s is in the volume around node j and

6. So the time taken for the jth volume to fill (to become saturated) is And Fixed values while Volume around node j becomes saturated A Numerical Solution With a Fixed Uniform Grid with spacing 1 i=0 1 2 3 4 5 6 7 ------------------- 1 Key assumption: While Node number So while front s is in the volume around node j

7. i=0 1 2 3 4 j j+1 ------------------- 1 Half Step Numerical time to reach position s = j+0.5 Analytical time to reach this position So when s = j+0.5 Analytical solution for time is (A) (B) Constant time error of 0.125 between analytical and numerical