Pumping: Borehole Modelling

If we consider the borehole as an impervious tube, when pumping the water level will drop as the flowrate Q [m3/h] divided by the hole section area Aw [m²].

On the other hand, the re-filling of the well from the surrounding porous medium is a diffusive process. One can admit as a reasonable hypothesis that the refilling flowrate is proportional to the stress, i.e. the drawdown dynamic head.

Under these hypotheses the real level in the well (or HD evolution) will obey the following equation:

\(dHD / dt = - 1/τ * HD + Q (t) / Aw\)

One can easily see that for steady-state conditions (dHD / dt = 0), this equation leads to a drawdown height HD linear with the flowrate. Indeed, compared to a reference case, we have for any flowrate:

\(HD = Q * HDref / Qref\)

Under this hypothesis, the ratio HDref/Qref is a characteristics of the well, which we will call the "specific drawdown" (expressed in [meter / m³/h]).

This parameter is mainly related to the geologic properties of the surrounding ground (permeability, storage capacity), and the construction technique of the borehole. It may be measured rather easily, using a portable engine-pump and measuring the water depth and flowrate in stabilized conditions.

Borehole parameter in PVsyst

As a matter of fact, a pumping test is often performed for measuring the borehole performance, which yields essentially 3 parameters: the static level (HS), a reference flowrate available from the well Qref, and the corresponding dynamic level (HDref). Navarte (2000)¹ reports several results of such tests in Africa, of which we give some examples.

We can observe from these examples that the Dynamic contribution is not to be neglected !

The recovery time τ (corresponding to a 1/e re-filling) is easily calculated from the steady state conditions:

\(τ = Aw * HDref / Qref\)

For example, in the case of a borehole of diameter 0.15 m in Rotunda, this is about 4 minutes. Therefore this dynamic model describes the short term behavior of the well.

Medium-term (annual) variations are likely due to modifications of the groundwater (phreatic water) level along the seasons. They may be introduced in PVsyst by specifying a monthly profile of the static head HS.

Long-term exhausting effects caused by an excessive water drain involve complex (and not sustainable) phenomena which are not modelled here in PVsyst.

Finally, the simulation (as well as the real system regulation) should take the maximum head Hmax, i.e. the inlet level of the pump, into account for stopping the pump, avoiding dry-running.