Risk of Exceeding Acute Water Quality Criteria

This example model demonstrates how to use a Nested Monte Carlo (2D Monte Carlo) simulation to estimate the risk of violating a specific water quality standard. It distinguishes between the uncertainty of the engineered system (the Design Space) and the natural variability of the environment (the Environmental Response).

Why use Nested Monte Carlo?

In water management, it is often necessary to separate epistemic uncertainty (what we do not know, such as the exact performance of a treatment plant) from aleatory variability (randomness inherent in the system, such as river flows).

This model uses guidance from the EPA Policy for Use of Probabilistic Analysis, specifically Condition 5, which states:

"For both the input and output distributions, variability and uncertainty are to be differentiated where possible."

By separating these into two distinct simulation loops, we can answer complex questions like: "Given the uncertainty in our effluent concentration, what is the probability that the downstream river will exceed regulatory limits?"

Model Logic

The model uses a SubModel element to run an "inner" simulation for every realization of the "outer" simulation.

1. The Outer Model: Design Space The main level of the model represents the Design Space. The parameters here are uncertain but treated as static for a given scenario.

• C_Eff: A Stochastic element representing the potential Effluent Concentration.

• Q_Eff: A Stochastic element representing the Effluent Flow rate. 

2. The SubModel: Environmental Response

The SubModel element (WQM) represents the natural environment. It runs multiple realizations (e.g., 100) for every step of the outer model.

• Q_Amb: Ambient flow of the receiving stream, sampled for each realization of the SubModel.

• C_Amb: Ambient concentration, correlated to the flow.

3. Mass Balance Calculation

Inside the SubModel, the downstream mixed concentration is calculated using a standard mass balance equation:

Note: This example uses a simple algebraic equation for demonstration purposes. In a real-world application, this SubModel would typically contain a complex dynamic simulation, such as a Contaminant Transport model or other time-dependent functions. 

4. Determining Compliance

• WQC: The downstream Water Quality Criterion is set to 4.8 ug/L.

• Is_Exceeded: An expression checks if C_Mixed > WQC.

Understanding the Results

The primary output is the Exceedance Risk.

The SubModel calculates the mean of the Is_Exceeded condition. Because Is_Exceeded is a Boolean (0 or 1), the mean represents the frequency (probability) of failure for that specific outer realization.

The final Multi-Variate Result plots Effluent Concentration vs. Exceedance Risk. This allows you to visualize how the risk of regulatory failure changes as the effluent concentration varies, accounting for all possible environmental conditions.

References

Simon, S. L., et al. (2015). The Two-Dimensional Monte Carlo: A New Methodologic Paradigm for Dose Reconstruction for Epidemiological Studies. Radiation Research, 183(1), 27-41.

U.S. Environmental Protection Agency (EPA). (1997). Guiding Principles for Monte Carlo Analysis. Risk Assessment Forum. Washington, DC. EPA/630/R-97/001. (See specifically Principle 5 regarding the differentiation of variability and uncertainty).

U.S. Environmental Protection Agency (EPA). (2001). Risk Assessment Guidance for Superfund: Volume III - Part A, Process for Conducting Probabilistic Risk Assessment. Office of Emergency and Remedial Response. Washington, DC. EPA 540-R-02-002. (See Appendix D for "Two-Dimensional Monte Carlo Analysis").

Vose, D. (2008). Risk Analysis: A Quantitative Guide (3rd ed.). Wiley. (See chapters regarding the separation of uncertainty and variability).

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