Hi Djamil,
On 05/22/2013 04:16 PM, dj.a...@gmx.de wrote:
Dear friends, I'm trying to solve the following *equations *for the self-potential distribution along a water injection in a well.
- sigma * Laplace(V) + L * Laplace (h) = 0
- L/sigma * grad(h) - grad(V) = 0
The weak form of your equations below seems to be wrong - as grad(h) and grad(V) are vectors, the second equation has to be tested by a vector test variable. Then this vector test variable should have a corresponding unknown vector variable (the velocity) - there should be IMHO one more equation with that.
Also, each one of the above equations should be tested by a single test variable - you are mixing it in the equations.
*Here h is the hydraulic head (pressure) distribution, V is the Self_Potential signal*. I'm using the *3d_cube_big_tetra mesh*. The code is attached.
The *boundary conditions* are: Left and right side of the cube h=100m Left side of the cube (reference point) V=0mV. Center cells of the cube: h=1000m (injection of water)
In general it should be very easy to find a solution with sfepy FE but I donĀ“t get any relation between V and h, although both should be coupled via the L/sigma parameter. L and sigma are constant in this *homogeneous case*(L=-1.5e-7, sigma=1).
Hydraulic head (h) is calculated correctly, but the SP signal (V) shows no connection to h. Changing the parameters L or sigma has absolutely no effect on the solution. It depends only on the boundary conditions.
So I tried to implement it with the following *equations*: * 1 'SP' : """dw_laplace.i3.Omega( fluid.sigma, w, v ) + dw_laplace.i3.Omega( fluid.C_sigma_rel, q, h ) = 0""", 2 'Coupling' : """dw_stokes.i3.Omega( fluid.C_sigma_rel, q, h ) - dw_stokes.i3.Omega( w, v ) = 0""", *
A probe image through the middle of the cube is attached.
So my questions:
- Am I using the right equations? I also tried it with dw_div_grad instead of dw_laplace (should be the same, no?) but unfortunately the calculation then doesn't converge.
Use dw_div_grad for vector variables, and dw_laplace for scalar variables.
- Why is the coupling between both Laplacian eqn. not given? There should be an effect of L and sigma even without defining eqn 2 (Coupling) but it has no effect. This is not a SP problem but more a general issue because there should be a coupling between both unless one part is zero.
I guess this is caused by the one equation that is IMHO missing...
That's why I'm afraid that having problems already with a homegeneous case leads to much more problems with inhomogeneous sigma and L distributions.
Thanks very much for any hints and suggestions. I'm really out of ideas now.
I some more observations/suggestions related to the example file:
- in field_2, use 'shape' : 'vector' instead of 'shape' : (2,), - that will work for both 2D and 3D meshes
- it does not run with current sfepy (fmf_sumLevelsMulF(): ERR_BadMatch (4 == 12, 1 == 1, 1 == 1)) - this is caused by using the scalar test functions in the second equation
- when I correct the test field of the second equation to be "s" (vector), the system assembles, but the matrix is singular, as there is no "u" in the equations.
r.