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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?

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  • $\begingroup$ I wonder if the broader question would get more response in Math SE, but seems a good one to stay here for the moment. $\endgroup$ Nov 20 at 8:41
  • $\begingroup$ 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. $\endgroup$ Nov 20 at 8:43
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    $\begingroup$ @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 :) $\endgroup$ Nov 20 at 11:04
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    $\begingroup$ 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? $\endgroup$
    – Erik
    Nov 20 at 12:30
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    $\begingroup$ 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. $\endgroup$
    – Erik
    Nov 20 at 12:33
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The proposition of this question says nothing about the size of the domain, or how the domain has been discretized. And yes, the numerical model will converge if the necessary boundary conditions between model elements have been properly described. In order to do this, all the dimensions of all elements must be properly characterized. Presumably, a description of hydraulic head is desired in this exercise, and a visualization of how this change in head will evolve over time given various time- and pumping/recharge-driven scenarios.

Appropriate description of the aquifer is the single most important aspect in developing the model. Boundary conditions occur everywhere within the domain of the model and are maintained by appropriately accounting for the inflow and outflow across discretized model elements. Presumably, this model was competently written. Nevertheless, the elements may have been incorrectly characterized regarding their dimensions, or the mathematics may be in error. Because the characterization given in the question provides no definition regarding scale, or a representation of the model output, the determination that the model did not converge may be in error. The model may, in fact, have converged! The dimensional scaling of discretized elements in the model may be wrong.

One of the easiest and best ways to determine if the characterization of the modeled domain is correct, is to set up an analogous analytic model for the same domain. The Dupuit-Forscheimer relationship is quite strongly determined, and the outcome of such a model should be representative of the domain both in dimensions spatially and in time. That outcome can be compared with the numerical model and differences noted. One caution, though, is that the analytic model will have to appropriately use the method of images to simulate the no-flow boundaries that define the edges of the modeled domain. This will entail an appropriate accounting of the images for pumping and recharge.

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