iMOD User Manual version 5.2 (html)

12.30MNW MultiNode Well Package

The existing Multi Node Well package of MODFLOW2005 ([Halford and Hanson(2002)], [Konikow et al.(2009)]) was implemented in iMOD. The Multi-Node Well package is used to simulate “long” wells that are connected to more than one model layer; the abstraction rate is vertically distributed proportional to the transmissivity adjacent to the well screen; e.g. when a hydraulic head gradually drops below the top of a well screen the yield of this shallow part of the well will also gradually drop.

MODFLOW computes the head at a block-centered node of a finite-difference grid on the basis of a fluid mass balance for fluxes into and out of the volume of the cell of interest, including flow in or out of a well located within the surface area (and volume) of that cell. However, because of differences between the volume of a cell and the volume of a well-bore, as well as differences between the average hydraulic properties of a cell and those immediately adjacent to a well, it is not expected that the computed head for the node of a finite-difference cell will accurately reproduce or predict the actual head or water level in a well at that location. Furthermore, if the length of the open interval or screen of a vertical well is greater than the thickness of the cell, then the head in the well would be related to the head in the ground-water system at multiple levels (and at multiple locations for a non-vertical well). Thus, if the user needs to estimate the head or water level in a well, rather than just the head at the nearest node, then additional calculations are needed to correct for the several factors contributing to the difference between the two.

As denoted by the name of the package, the advantage of the MNW package benefits mostly whenever wells are considered that discharge from a multi-aquifer system. Since the MNW package can deal with intra borehole flow and computes a realistic head loss at the well, this makes the package mostly applicable for multi-layered unconfined systems. In the case that the well falls dry, this is more realistic simulated with the MNW package, more-over, the total strength of a multi-layered extraction system remains intact as-much-as possible during a simulation. This example also demonstrates that the well discharge is not simply proportional to the transmissivities of the multiple aquifers screened by the well. For single penetrating, confined system the MNW is similar to the WEL package.

MNW computes a hydraulic head in the cell $h_n$ such that it equals the computed hydraulic head at the well minus a head loss term (e.g. the Thiem equation, we neglect in this tutorial head loss due to skin and local turbulence effects) for that particular cell, so:

$$\begin {array}{ll} h_{\rm WELL}-h_n &= \frac {Q_n}{2 \pi T}\rm {ln}\frac {r_0}{r_w} \\ \\ {\rm CWC}_n & = \frac {\rm {ln}\frac {r_0}{r_w}}{2 \pi T} \\ \\ Q_n &= \left (h_{\rm WELL}-h_n\right ){\rm CWC}_n \\ \\ h_{\rm WELL} &= \frac {{\rm CWC}_n h_n + Q_n }{{\rm CWC}_n} \end {array}$$

where ${\rm CWC}_n$ is the cell-to-well conductance m$^2$/d; $Q_n$ is the well rate (m$^3$/d), $T$ is transmissivity of the aquifer (m$^2$/d) at the well, $r_0$ is the effective radius of a finite-difference cell (m), this is assumed for isotropic conditions as $r_0=0.14\sqrt {\Delta x^2+\Delta y^2}$; $r_w$ is the actual radius of the well. Because $r_0$ is typically much greater than $r_w$, the head in a pumping well will be lower than the model-computed head.

The problem is solved by MODFLOW via estimations of $h_{\rm WELL}$ and $Q_n$ that lag an iteration behind estimated of $h_n$ because the above mentioned equation are solved explicitly. This causes slow convergence of the solver if the MNW cells are incorporated in MODFLOW as a general-head boundary (subtract CWC$_n$ from HCOF and subtract CWC$_n \times h_{\rm WELL}$ from RHS). Convergence is accelerated by alternately incorporating the MNW cells as specified rates in odd iterations (subtract $Q_n$ from RHS) and as general-head boundaries in even iterations.

Note: Multi-node wells with cell-to-well conductances that are “too great” tend to make MODFLOW numerically unstable. Cell-to-well conductances increase as cell size is decreased, which also decreases the effective external radius ($r_0$). Cell-to-well conductances become greater as $r_0$ approaches rw and are undefined if $r_0$ is less than or equal to $r_w$. For these small cells, a pumped well should be simulated as a high-conductivity zone as cell area approaches the cross-sectional area of a well.