HolmesCo: Groundwater Assessment

Regression and extrapolation — Week 8 handout with instructor notes

HolmesCo Groundwater Assessment Report

This is a fictional document for teaching purposes.

HolmesCo

Geological Consultants Since 2019

“Building Confidence in the Ground Beneath You”™

GROUNDWATER RISK ASSESSMENT

Prepared for: Teesdale Aggregates Ltd

Project: Bollihope Quarry Expansion, County Durham

Report No.: HC-2026-0512

Date: March 2026

Classification: CONFIDENTIAL

1. Introduction

HolmesCo was commissioned by Teesdale Aggregates Ltd to assess groundwater conditions at Bollihope Quarry, near Frosterley, County Durham. The quarry operator is seeking planning permission to deepen extraction by 8 metres. The Environment Agency requires evidence that (a) the controls on groundwater levels are understood, and (b) the deepened quarry will not drain the local aquifer.

2. Data

HolmesCo installed a monitoring borehole at the quarry in January 2022. Groundwater levels and monthly rainfall have been recorded continuously since then — a total of 48 monthly observations (January 2022 to December 2025).

3. Analysis Part A: What controls groundwater level?

HolmesCo fitted a linear regression model to determine whether rainfall controls groundwater levels:

groundwater_level = β₀ + β₁ × monthly_rainfall + ε

Results

Parameter	Estimate	Std. Error	p-value
Intercept (β₀)	142.3 m AOD	1.8	< 0.001
Rainfall (β₁)	+0.038 m per mm	0.008	< 0.001

R² = 0.45 F(1,46) = 37.6, p < 0.001

Diagnostic plots

(Instructor: display the four diagnostic plots on screen. Key features described below for reference.)

Residuals vs Fitted:

     +2 ┤ ·  ·        ·  ·         ·  ·        ·  ·
        │  ·  ·        ·  ·         ·  ·        ·
      0 ┤       · ·         · ·          · ·
        │          · ·         ·  ·         · ·
     -2 ┤            ·  ·        ·  ·          ·  ·
        └──────────────────────────────────────────────
          Low fitted              →            High fitted

There is a clear wave pattern in the residuals, repeating approximately every 12 data points.

HolmesCo’s conclusion

“The regression is highly significant (p < 0.001), confirming that rainfall is the dominant control on groundwater levels at this site. R² = 0.45 demonstrates a strong relationship. Higher rainfall leads to higher groundwater levels, as expected.”

4. Analysis Part B: Future groundwater trend

During a site visit in December 2025, the quarry manager noted that groundwater levels had been declining. HolmesCo plotted the monthly groundwater level over time and fitted a linear trend:

groundwater_level = β₀ + β₁ × months_since_start + ε

Results

Parameter	Estimate	Std. Error	p-value
Intercept	148.2 m AOD	0.4	< 0.001
Trend	−0.21 m/month	0.05	< 0.001

R² = 0.52

Extrapolation

HolmesCo projected the linear trend forward:

Date	Predicted level (m AOD)	Status
December 2025 (observed)	138.5	Quarry floor at 140 m — water 1.5 m below
June 2026	137.2	Below quarry floor
December 2027	134.7	Well below quarry floor
March 2028	133.0	Aquifer effectively empty at quarry depth

HolmesCo’s conclusion

“The groundwater trend is unambiguous: levels are declining at a rate of approximately 0.21 metres per month and will reach critically low levels by early 2028. At current rates, the aquifer at quarry depth will be effectively exhausted within two years.

This is actually favourable for the proposed quarry expansion — the declining water table means deepened extraction is unlikely to encounter significant groundwater problems. HolmesCo recommends proceeding with the expansion.”

This report has been prepared by HolmesCo for the exclusive use of Teesdale Aggregates Ltd. HolmesCo accepts no liability for losses arising from use of this report by others.

Instructor notes

This handout contains two separate problems for students to identify, corresponding to the two halves of the Week 8 content session.

Part A: The regression — “Rainfall is the dominant control”

Use with: Exercise 2 (concept block 2: assumptions and diagnostics).

What students should spot:

The residuals have a seasonal wave pattern. This means the model is structurally wrong — it’s missing a seasonal component. Groundwater responds to rainfall with a time lag (winter rain recharges the aquifer over weeks to months), and there’s a seasonal cycle (high in spring after winter recharge, low in late summer after evapotranspiration). A simple rainfall → level model doesn’t capture this.
R² = 0.45 means 55% of variation is unexplained. HolmesCo calls this a “strong relationship” and “dominant control.” It isn’t. The majority of the variation is unaccounted for. What else might matter? Evapotranspiration, upstream abstractions, quarry dewatering pumps, antecedent soil moisture.
“Highly significant” conflates statistical significance with explanatory power. The p-value is tiny because 48 observations is enough to detect a real-but-partial relationship. The p-value says rainfall matters somewhat; it does not say rainfall is dominant. This echoes the Week 7 lesson: significance ≠ importance.
The diagnostic plot is the smoking gun. A well-specified model has residuals that look like random noise. A wave pattern means the model is systematically wrong — not just noisy.

Discussion prompt: “What would a better model look like?” (Include a lagged rainfall term, a seasonal indicator or month variable, or fit a time-series model like ARIMA. Even adding month as a predictor would capture much of the remaining variation.)

Part B: The extrapolation — “The aquifer will be empty by 2028”

Use with: Concept block 3, failure mode 2 (extrapolation).

What students should spot:

Linear extrapolation of a 4-year trend is reckless. The monitoring period (2022–2025) may capture a temporary phenomenon rather than a long-term trend. HolmesCo is predicting 2+ years into the future from 4 years of data, with no consideration of whether the trend will continue.
Seasonal recharge is ignored. Groundwater levels in the Northern Pennines follow a seasonal cycle: they decline in summer (evapotranspiration > recharge) and recover in winter (recharge > evapotranspiration). A linear fit to monthly data averages over these cycles. If the monitoring period happened to end during a dry spell, the linear trend will be steeper than the long-term reality.
The decline may be caused by temporary pumping. Quarries routinely pump groundwater during active extraction phases. If Bollihope was dewatering during the monitoring period, the observed decline reflects pumping, not a natural trend. When pumping stops (e.g., for the expansion permitting period), levels may recover.
Extrapolation to “aquifer effectively empty” is physically absurd. Aquifer levels don’t decline linearly to zero. As the head drops, the hydraulic gradient changes, recharge becomes relatively more important, and the system approaches a new equilibrium. The linear model has no mechanism for this.
HolmesCo’s spin is remarkable. They reframe a declining water table — which might concern a regulator — as good news for their client. The Environment Agency asked for a risk assessment; HolmesCo delivered an argument for proceeding. Students should notice whose interests the conclusion serves.

Key message: A model’s quality in-sample tells you nothing about its quality out-of-sample. “What is my model not capturing?” is the question that separates careful analysis from HolmesCo.

Discussion prompt: “What data would you need to make a credible prediction of future groundwater levels?” (Longer monitoring record, seasonal decomposition, pumping records, regional borehole data for comparison, a process-based groundwater model rather than a simple regression.)