On the basics of groundwater modeling, there were two model types relevant to time: steady state and transient models. The first type assumes that flow is steady and boundary conditions and heads do not change over time, while the second takes into consideration changes in the boundary conditions, heads and the inflow/outflow from storage.

On the task of creating groundwater models with limited data, the steady state is the most common model type since it requires less input data (only K) and the calibration process is much simpler. However, when assessing an aquifer development trough numerical model the transient approach becomes unavoidable. For a transient model, a modeler needs to conceptualize a coherent and feasible time discretization approach, nevertheless searching for temporal discretization guidelines the Internet gives you digital noise: nothing relevant, nothing precise, nothing useful. These is the type of situation where you ask yourself: how many stress periods do I need, how many time steps do I require, which should be an appropriate size of the model output, or more important, will the size and number of stress periods and time steps have an effect on the model output?

It is not black box

Groundwater modeling has solvers, damping and relaxation coefficients, interblock transmissivities, and for many professionals and normal audience the task of groundwater simulation can be seen as a “black box” where twelves of parameters are declared and results comes without a great sense of the calculation method. It have to be declared that a groundwater flow simulation is not a black box, it is only a complex matrix calculation where we need to have control of how much water goes in and goes out and we have to be sure that the selected spatial and temporal discretization will allow a right representation of hydraulic heads and water balances.

Numerical simulation

MODFLOW uses the Darcy formula to solve the groundwater flow equation. As Darcy law says, flow is dependent on head gradient, so a good approach to analyze the impact of time discretization could be the analysis of head and water budget development.

For this tutorial we have used two temporal discretizations for 10 transient periods following a steady state period of 20 days:

The first approach has 10 transient stress periods of 5.5 days, this case will be called “Equal Interval”
The second approach has 10 increasing stress periods starting in 1 day and ending in 10 days on this order: [1,2,3,4…. 10], this case will be called “Increasing Interval”

Each stress period has 4 time steps and the impact was evaluated on heads on the wells cells and flow rates and accumulated volumes water balance.

A representation of the model discretization with the boundary conditions can be seen on the following figure:

Model output analysis

Head comparison

For the Observation Point 1 and as well as for the other observation points, there are differences on heads specially at the begging of the transient part of the model simulation. Since the increasing interval approach has more stress periods and time steps from day 20 to 30, the curve has more resolution and its more shaped, however at the end of the simulation there is no strong differences on heads among both time discretization approaches.

Rates comparison

The main inflows to the groundwater flow system come from storage and the boundary condition of River (RIV) and General Head Boundary (GHB). The inflow rates show different behavior according to the simulation time and to the time discretization.

Storage

Water depleted from storage at the beginning of the transient period is the main source of pumping wells. After hours or days of pumping, the inflow from storage decreases on a asymptotic behavior to the limit of reaching the steady state where inflow from storage will be zero. For our case, the phase when the model reaches the steady state was not considered.

While comparing the inflow from the two time discretization approaches we found some difference that we will try to explain carefully. This is the plot of inflows from storage for the two time discretization schemes:

On quick view, it seems that the “Increasing Interval” provides more water from storage, and that time discretization schema have impacts on the water balance, but we have to be careful and more analytical to take into consideration some facts that happened on the model.

It is important to remember that MODFLOW prints the flow rates at the end of the time series or stress period; therefore, if the time step / stress period is bigger, the points will be shifted to the right. A right comparison would be to have the flow rates at the middle of the time step/stress period.

The following figure is a zoom on the water inflow from storage after the beginning of pumping from wells.

On this zoom view, inflows from storage show a similar trend as the head drawdown. As long as the time step is smaller at the beginning of pumping, heads will decrease more and the storage will release more water; however, this happens only on the time step duration. On the “Equal Interval” approach the rates are and the gap is more significant on the beginning of pumping and reduces the progressively, after 12 days (Day 32) the gap is almost unnoticeable.

River (RIV)
On the next main inflow that is River inflow there are differences on both time discretization schemes. As the storage provides more water on the beginning of pumping, less water from river is required from the river.

General view

Zoom

General Head Boundary (GHB)

On the analysis of inflow from the General Head Boundary (GHB) there are differences on the inflow that don’t follow any trend. We assume that this difference could rely on selected values for the convergence criteria (RCLOSE and HCLOSE) and the analysis of this gap with different set of convergence criteria would be a wonderful topic for a post in our blog.

On the following figures the comparison of the inflow rates for the GHB boundary are being show as a general view and a zoom view 30 days after beginning of pumping.

Accumulated volumes comparison

Storage

Accumulated water volume that comes from storage for both schemes show a good correspondence on a general view as it can be seen on the following figure:

On a zoom view some differences are being notices for the accumulated volumes. The “Equal Interval” time discretization brings more water from storage. Difference on the accumulated volume for both schemes account for 0.19% of the total volume.

A zoom view of the accumulated volumes at the end of the simulation be seen on this plot.

It would be long and exhaustive to compare all the plots for every term on the water balance. For further analysis of the differences on the water balance for both time discretization schemes we have uploaded the entire set of scrips, spatial data and folder system to reproduce the related simulation.

Tutorial

Conclusions

This was not actually going to be a scientific publication, it was intended just to be a exploratory exercise of time discretization influence. From the numerical exercise there are some conclusions about the performance a MODFLOW model regarding time discretization.

Just two discretization schemes we tested on this exercise, we need more tests in order to provide a accurate diagnosis of the differences found on head drawdowns and water balances.
The shorter the time step at the begining of pumping, the greater the water released from storage.
There are many factors that provide discrepancy to groundwater models like time discretization, solver parameters and potentially spatial discretization (that we are going to study next).
On a general perspective, the model show low discrepancy to different approaches of time discretization. For this case was around 0.19%, however we need to assess how relevant is this discrepancy to represent the aquifer response at certain observation point, to calculate the water coming to a drain, or to determine the inflow/outflow to a boundary condition.