It is widely recognized real world aquifers are inherently heterogeneous and small scale heterogeneities can large scale environmental management implications. However, modeling explicitly the effect of such small scale heterogeneities using traditional deterministic approaches is considered impractical because they cannot be realistically mapped using today's measurement technologies.
The alternative is to utilize a small number of scattered samples to estimate the variability of parameters in a statistical framework. That is, the spatial variation of a hydraulic property is characterized by its probability distribution estimated from samples. Research shows that the distribution of porosity data in an aquifer can be generally considered normal. The spatial distribution of storage coefficient might be lognormal. Hydraulic conductivity distributions are usually reported to be lognormal. Based on such a statistical approach, heterogeneous hydrogeological inputs or their logarithms can be represented as gaussian random variables and the uncertainty in groundwater flow and transport modelling can be systematically analyzed using Monte Carlo simulation.
However, recent analyses of hydraulic conductivity data showed that, although the hydraulic conductivity values vary significantly in space, the variation is not entirely random, but correlated in space. Such a correlated nature implies that the parameter values are not statistically independent in space and they must be treated as a stochastic process or a random field, instead of as a single random variable.
A. Watch the videos below and answer the following questions:
 What is a random field?

How can a Gaussian random field be characterized? how many parameters? physical meaning of these parameters?

How can the parameters characterizing a random field be estimated from limited field data?

What is a random field realization?

What is an ensemble?

What is the relationship (similarity and difference) between different random field realizations?
 Is hydraulic property (e.g., conductivity) at a given point in the subsurface environmental literally uncertain?
 What is the relationship between variability and uncertainty?
Video 1: Borden aquifer, Canada. (Top) Single K field realization based on observation data. (Bottom) Different simulated K field realizations for the same observed data. Model setup: model size: 12 x 1.75m; Geometric mean K: 9.75x10^{3} cm/sec; lnK varaiance: 0.38; correlation scales: 2.8m (xdirection); 0.12 (zdirection).
Video 2: Different realizations of an isotropic random field. Model setup: model size: 100 x 100m; Geometric mean K: 10 m/day; correlation scale: 5 m; lnK variance: 1.0.
Video 3: Different realizations, different degrees of heterogeneity (lnK). Model setup: model size: 100 x 100m; Geometric mean K: 10 m/day; correlation scale: 5 m
Video 4: Different realizations, different orientations of heterogeneity (correlation scales). Model setup: model size: 100 x 100m; Geometric mean K: 10 m/day; lnK variance: 10.
B. Develop a MAGNET random field model of a hydraulic conductivity field that reproduces the animations above, visualize the different realizations, and perform a sensitivity analysis of the random field model with respect to key parameters characterizing the random field. Discuss how the various sensitivity simulations can be used to mimic / represent randomly distributed heterogeneity in different types of depositional environment.
Write a 12 page memo that summarizes what you learn from your numerical experimentation on heterogeneity, data limitation, uncertainty, discussing the pros and cons of a stochastic approach to characterizing aquifer heterogeneity
MAGNET/Modeling Hints:
 Use 'Synthetic mode' to setup the model domain ( 'Other Tools' > 'Utilities' > 'Go to Synthetic Case Area' )
 You may follow the model setups described in the animation captions above.
 Use a zone feature of the entire aquifer area to assign the hydraulic conductivity as a random field