Posted Wed, 01 Dec 2021 23:43:21 GMT by Alan Jang

I am conducting simple simulations using multi-layer pumping wells. For your review, the model domain and other settings are shown in the attached.
Two cases are considered:
- Case 1. The model is solved by assuming the aquifer is confined.
- Case 2. The model is solved as an unconfined aquifer system: Slice #1 is set as "Phreatic" and Sy=0.1
Cases 1 and 2 use the same settings except the aquifer type.

[Case 1]
The simulation of Case 1 run well (e.g., dt increased from 1.0e-5 (a few steps at the beginning) up to 31day), and the modeled hydraulic heads look ok (see the attached).

[Case 2]
For Case 2, the model has difficulties in solving the unconfined well problem. The length of modeled time-steps stays at 1.0e-5 day, which is too small to run for the whole modeling period. My understanding of key difference between Cases 1 and 2 is the magnitude of relative conductivity (k): k= 1 constant for Case 1 while k= variable for Case 2, depending on the elevation of water table in model elements (saturation degree). In Case 2, the residual water depth is set as 0.25 (about 10% of the layer thickness).

My question is what causes the convergence difficulty in Case 2?
Could you share any ideas, suggestions, or experience to solve this issue?

Posted Thu, 02 Dec 2021 06:49:47 GMT by Peter Schätzl Grundwassermodellierer
Indeed the main difference is the variability of the transmissivity, which requires the so-called outer iteration in each time step, iterating over the step until the solution converges. In a confined simulation, this outer iteration is not necessary, the equation system can be directly solved (or with an iterative solver). So the time step length you end up with depends on the (dimensionless) error criterion you have specified plus the actual variation of the solution compared to the criterion. So if your criterion is appropriate for the application, the solution varies a lot - which could point towards an oscillation. Oscillations can occur, for example, when wells with a high abstraction rate are running in relatively low-permeable environments, high recharge rates are applied to phreatic elements with low relative conductivity, etc.
Posted Wed, 08 Dec 2021 21:09:18 GMT by Alan Jang
Thanks, Peter.

I tested the Unconfined-aq case with different solution convergences. With large convergence criteria, the solutions of the Unconfined aq. case were not correct. Considering the analytical solution and the Confined-aq case (see below), the modeled pumping rate might not be too big.

This modeling is for a simple pumping-well simulation:
- A single well in the homogeneous isotropic medium using a constant pumping rate.
- A steady-state analytical solution is available (Thiem-Dupuit formula, see (1)).

For your review, I have attached three files:
(1) Model setting and results - comparison between the analytical solution and the results of Confined-aquifer case. (pdf file)
(2) Model file:  Confined-aquifer case
(3) Model file:  Unconfined-aquifer case  --> the dt plot is shown in (1)

Could you review the above three files and help me understand the following?
(a) Are there some missing items/settings in the model file (2)?
(b) How to solve the convergence issue in (2)?


You must be signed in to post in this forum.