Parameter

All realizations

Geometric mean hydraulic conductivity, K _g (m/day)

10

ln K variance

2.0

Correlation scale, λ (m)

2

Porosity, n

0.3

Local pore-scale dispersivity

0

Head difference (m)

1

Size of model (m)

200 x 50

Covariance function

Exponential

Approximate initial plume size (m)

10

Grid

401 x 407

Cell size, Δx (m)

0.5 x 0.5

Time step, Δt(days)

1

The model parameters and inputs are explained below:

• The ln K field is a normally distributed random field. Therefore, there is an equal probability of the plume encountering a ‘low' K zone or a ‘high' K zone.
• The heterogeneous conductivity field is uniquely characterized by the following set of statistical parameters: geometric mean ( K _g ) of K, correlation scale, variance and covariance.
• The geometric mean, ln K variance, correlation scale ( λ), and covariance function are constant for all cases.
• The cell size, Δx, is selected such that the correlation scale, λ, is resolved; typically, Δx is at least 3 to 4 times smaller than the correlation scale.
• The time step, Δt, is selected such that a particle of the plume travels a distance that is less than λ in one time step.

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 Problem Statement

This video demonstrates 4 different realizations of hydraulic conductivity heterogeneity. The spreading of a conservative solute plume through these realizations is shown. The modeling domain consists of constant head boundaries on the left and right extremes, and no-flow boundaries at the top and bottom. Details are provided in Table 4.3.

 Key Observations

The following observations can be made from the video:

• Despite the plume being larger than the scale of heterogeneity, different realizations cause significantly different plume spreading.
• The different plumes vary in all aspects, such as size, shape, mean displacement, and overall extent of spreading.

 Additional Observations

Even though all the realizations have the same set of statistical parameters, the eventual patterns of spreading exhibited by the plumes are significantly different. The plumes differ not only in their eventual extents of spreading, but also in their shape, size, and mean displacement. There is no way to compare the patterns of spreading of the different realizations using a generalization. However, if a large number of such realizations are considered, the mean of all those realizations will be a Gaussian distribution.

Uniquely characterizing the plume's behavior from a single realization is difficult. This uncertainty in plume behavior is a result of uncertainty in aquifer properties. Therefore, stochastic methods such as Monte-Carlo simulations need to be used in order to perform a systematic probabilistic analysis that can be used for risk-estimation.