# Can we know if a numerical model will converge or not according to the boundary conditions?

Consider a square 2D domain and a steady-state simulation. According to the geology of the domain, the North, South, West and East limits must be conditions of zero flow (Neumann). The aquifer receives a uniform recharge R throughout the area. In the center of the aquifer, there is a well which pumps at the rate Q. The aquifer is considered to be homogeneous and isotropic. The equation in the (unconfined) aquifer is therefore:

K * hxx + K * hyy = R + Q


With hxx : second derivative of the hydraulic head following x direction & K: hydraulic conductivity.

I tried to build a numerical simulation model with the above boundary conditions but it does not converge. Is this normal?

Geology forces me to impose a zero flux limit on all boundaries of the real domain. Should I depart from these real limits and impose arbitrary conditions that make the model converge?

• I wonder if the broader question would get more response in Math SE, but seems a good one to stay here for the moment. Nov 20, 2021 at 8:41
• Knowing nothing about the topic... I'd think your Q should be a negative since it's leaving the system? If you got more recharge than you are pumping out, I'd think in the long run you'd be gaining water constantly? But I have never looked at the equations of hydrology to this level, so I'm not a good input if all that is fine and it's a deeper nuance. Nov 20, 2021 at 8:43
• @JeopardyTempest yes Q is an output, so it is negative. It's also true, if I have more recharge than pumping, the system will store more water. In hydrogeology, when a very poorly permeable soil is in contact with a permeable soil (aquifer), the contact between these two soils can be considered as zero flow. (soil here = geological formation). Thank you for your comments :) Nov 20, 2021 at 11:04
• I'm not sure if the solution you're describing (a field for 'h' that is flat [no gradient] at the boundaries, with constant amount of water added into the system everywhere, and a constant draw of water from the system at one point) can be described with a steady-state equation. If you're removing more water from the system than that enters it, a steady-state situation will never be achieved?
– Erik
Nov 20, 2021 at 12:30
• Furthermore, if you would find a solution u, then adding any constant to it, e.g., (u+c), would also be a valid solution, as the constraints on your solution are only given with the first and second derivatives. That may also prevent you from achieving convergence of your solution.
– Erik
Nov 20, 2021 at 12:33