Tutorial 16: Monte Carlo Flow Simulation

1Monte Carlo Stochastic Flow

Step 1 — Enter Synthetic Mode

Click 'Go to Synthetic Case Area' to create a blank domain.

Step 2 — Add River Boundaries

Click to add river zones on left (prescribed head = 0 m) and right (prescribed head = -10 m) edges.

Step 3 — Assign K as a Random Field

Click the 'Zone=DM' button to create a domain-wide zone. Assign hydraulic conductivity as a random field with the desired probability distribution parameters (mean, variance, correlation scales) — same as Tutorial 15.

Step 4 — Enable Monte Carlo Simulation

Click to access Simulation Settings in the Domain Attributes menu, then click 'Solver Options'. Check the box next to 'Monte Carlo Simulation'. Set the number of realizations — the default is 10. Each realization generates a new random K field, solves the flow equations, updates the running statistics, and moves on. More realizations = smoother, more reliable statistics.

Step 5 — Enable MCS Display

Click to access Display Settings. Check 'MCS Display' to enable Monte Carlo visualization. Change the Main Display to 'Vectors Only' for a clear view of how mean flow vectors evolve as realizations accumulate.

Step 6 — Run the Monte Carlo Simulation

Click to submit. Watch the simulation proceed realization by realization:

Realizations 1-2: Only the current realization's flow vectors are displayed
Realization 3+: Both the current realization AND the statistical mean flow vectors appear
Each realization: The mean head and mean flow vectors update in real time — you watch the statistics converge

Step 7 — View Mean Head and Flow Vectors

After all realizations complete, the map displays the mean head field and mean flow vectors — the expected (average) outcome across all realizations. The mean field is smoother than any single realization — the random fluctuations average out, revealing the underlying regional flow pattern.

Step 8 — View Head Variance

Access Display Settings again. Under 'MCS Display', check 'Head Variance' and uncheck 'Mean Head'. Click save. The display now shows the head variance at every cell — computed from all 10 realizations. High variance = high uncertainty (the head value changes a lot between realizations). Low variance = well-constrained (the head is similar regardless of which K field is used). Variance is typically highest in the interior and lowest near fixed boundaries.

Step 9 — Add a Monitoring Well with Probability Tracking

Add a well near the right edge. In the well properties, check the box next to 'Monitoring Probability'. This tells IGW-NET to record the head value at this location for every realization — building a probability distribution of possible heads.

Step 10 — View Statistics at the Monitoring Well

Click 'Display Charts' under Analysis to view results at the well location. The chart shows the probability distribution of head — a histogram of head values across all realizations. This tells you: "At this location, head is most likely between X and Y, with a standard deviation of Z." This is the foundation of risk-based decision making.

Monte Carlo setup showing synthetic domain with river boundaries, domain-wide random K zone, and solver options dialog with Monte Carlo Simulation checked and 10 realizations configured — Figure 2-3: Monte Carlo setup — river boundaries, random K zone, and Solver Options with Monte Carlo enabled for 10 realizations. Each realization generates a new K field, solves flow, updates statistics.

Display Settings showing MCS Display options checked, with Mean Head and Vectors Only selected for Monte Carlo visualization — Figure 4: MCS Display settings — enable Monte Carlo visualization to see mean head, mean vectors, head variance, and realization-by-realization updates.

Monte Carlo simulation in progress showing current realization flow vectors alongside the accumulating mean flow vectors, demonstrating how the mean converges as more realizations complete — Figure 5: Monte Carlo in progress — current realization vectors (one pattern) alongside the accumulating mean vectors (smoother, converging). With each realization, the mean becomes more stable and the random fluctuations average out.

Head variance map showing spatial distribution of uncertainty across the model domain — higher variance in the interior where K variability has the most impact, lower variance near the fixed-head boundaries where the solution is constrained — Figure 6: Head variance map — the spatial distribution of uncertainty. High variance (warm colors) in the interior where the random K field has maximum influence. Low variance (cool colors) near boundaries where prescribed heads constrain the solution. This map tells you WHERE your model is most uncertain.

Monte Carlo results showing mean head field with flow vectors overlaid, plus the monitoring well location with probability tracking enabled — Figure 5 (extended): Completed Monte Carlo mean head field — the expected flow pattern after averaging 10 realizations. Smoother than any single realization, this represents the "most likely" outcome.

Probability distribution chart at the monitoring well showing histogram of head values across all realizations, with mean, standard deviation, and confidence intervals — the quantitative basis for uncertainty-informed decision making — Figure 7: Probability distribution at the monitoring well — a histogram of head values across all realizations. The mean is the expected head; the spread quantifies uncertainty. This is how you go from "the head is about -5 m" to "the head is -5.2 ± 0.8 m with 95% confidence."

Key Concepts

Constant-memory streaming: IGW-NET computes running mean and variance using Welford's algorithm — each realization is generated, simulated, incorporated into the running statistics, and discarded. Running 10,000 realizations uses the same memory as running 10. This is unique among groundwater modeling tools and makes large-scale Monte Carlo feasible in a browser environment.

Convergence: With few realizations, the mean is noisy and the variance estimate is unreliable. As you add more realizations, the statistics converge — the mean stabilizes and the variance estimate becomes more precise. In practice, 100-500 realizations are often sufficient for mean head; variance and tail probabilities may need more. Watch the mean display — when it stops changing visibly, you're approaching convergence.

Variance = uncertainty map: The head variance map is a spatial uncertainty map. It answers: "Where in my model is the prediction most uncertain?" Areas near fixed boundaries are well-constrained (low variance). Interior areas far from constraints have high variance — these are where you need more data, more characterization, or more conservative design assumptions.

From variance to risk: At a monitoring well with probability tracking, you get a full distribution — not just a single number. This enables risk-based decisions: "There's a 90% probability that head at this well exceeds 5 m" or "The 95th percentile drawdown is 3.2 m." Regulators and decision-makers need these probability statements, not single deterministic predictions.

2What's Next

With Monte Carlo flow mastered, continue the learning path:

Tutorial 17: MC Transport Simulation — Monte Carlo for contaminant plumes, not just heads
Tutorial 18: Probabilistic Capture Zone — capture zone uncertainty from stochastic realizations
Tutorial 19: Automatic Parameter Estimation — optimize parameters across realizations

This tutorial covers