A daily rise and fall in a well is a clue to a water balance, not a direct plant-water meter
A shallow well falls during daylight and recovers overnight. Plant water use may help explain the pattern, but the hydrograph is still a record of water level. Turning its movement into groundwater evapotranspiration requires assumptions about storage, background groundwater supply and other influences. The daily cycle is a starting observation, not the final flux.
This guide builds a simplified White-style daily account from conservation, using a sign convention that makes rises and falls unambiguous. It then changes one assumption at a time: specific yield, night-time uptake, pumping recovery, stream stage, sampling and spatial representation. All numerical cases are original fictional examples. No value is a recommended coefficient for a real aquifer.
The method concerns an inferred groundwater contribution to evapotranspiration near an appropriate unconfined water table. It does not automatically measure total ecosystem evapotranspiration, recharge or a catchment-wide water loss. The existing recharge guide addresses a different inference from water-level rises; distinguishing the two avoids counting an ordinary overnight recovery as new rainfall recharge.
Hydrology subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1: Understand the daily account · Chapters 1–3
Route 2: Test storage and recovery · Chapters 4–6
Route 3: Challenge the signal · Chapters 7–9
Route 4: Carry uncertainty and scale · Chapters 10–11
Route 5: Practise independent transfer · Chapter 12
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Understand the daily account · Chapters 1–3
Identify the variable the well actually measures
Start from water storage conservation
Derive the night-recovery estimate explicitly
Test storage and recovery · Chapters 4–6
Choose specific yield for the response being studied
Estimate a recovery slope rather than picking two convenient points
Test the assumption of negligible night-time uptake
Challenge the signal · Chapters 7–9
Separate rain, recharge and other short events
Check pumping, stream stage and pressure effects
Avoid daily aliasing and inconsistent day boundaries
Carry uncertainty and scale · Chapters 10–11
Propagate uncertainty without disguising assumptions
Do not multiply one well into a landscape total
Practise independent transfer · Chapter 12
CHAPTER 1 OF 12
1. Identify the variable the well actually measures
A water-level record commonly describes hydraulic head or depth to water. First check which one is plotted. Increasing head means the water surface rises; increasing depth below a fixed reference means it falls. A daily pattern can appear inverted when those axes are confused. Keep the original variable, units, sensor elevation and reference datum beside the record before calculating any slope.
The water-table fluctuation approach needs an appropriate unconfined setting in which level changes can be related to drainable storage. A pressure response in a confined aquifer is not interchangeable with movement of an unconfined water table. Well construction, screened interval, geology and hydraulic connection matter. A clean sinusoidal-looking signal does not by itself establish that the required storage model applies.
Use “diel” to mean a cycle over approximately a day. Daily variation can be consistent with vegetation-driven uptake, but other daily drivers exist. Pump schedules, stream operations, atmospheric effects and instrument temperature sensitivity can also repeat. A visually persuasive rhythm needs a mechanism check. Compare independent weather, pressure and operating records rather than assuming that every daytime decline is transpiration.
For transfer, keep the hydrograph shape but replace a shallow water-table well with a deep confined monitoring well. The graph may look similar while the storage relation is different. The same formula cannot be justified by appearance alone. State what construction and hydrogeological evidence would be needed before using the water-table method. Recognising this non-transfer is part of understanding the method.
CHAPTER 2 OF 12
2. Start from water storage conservation
For a simplified local daily account, let Δh be end-of-day head minus start-of-day head, positive for a rise. Let Sy be an appropriate dimensionless effective specific yield. Approximate change in drainable storage per unit area as Sy × Δh, in metres of water. This is already a model: it assumes the chosen yield represents the level range and response time relevant to the day.
Let G be the net background groundwater supply per unit area over the day, after the background flows included in its definition. Let E_g be groundwater evapotranspiration depth over the same day. With no other unaccounted influences, Sy × Δh = G − E_g. Rearranging gives E_g = G − Sy × Δh. A falling water table has negative Δh, so loss from storage adds to the supply available for evapotranspiration.
The equation does not yet identify G. A well hydrograph alone records the net result of several processes. The White-style step is to estimate background supply using a period when plant-related groundwater withdrawal is assumed negligible or sufficiently small. That assumption allows a recovery slope to stand in for a supply rate. It is not a direct observation of lateral groundwater inflow.
Check the signs without numbers. If storage is unchanged over a day, Δh is zero and estimated evapotranspiration equals inferred supply. If storage falls, estimated evapotranspiration exceeds that supply. If storage rises, some supply remains in storage. These relationships follow from the simplified account. They are useful sanity checks, but they do not prove that the account contains every real process.
CHAPTER 3 OF 12
3. Derive the night-recovery estimate explicitly
Let r be the positive hourly head recovery rate during a chosen night-time window, in metres per hour. If groundwater evapotranspiration is negligible then, and net background supply per hour is represented by Sy × r, extending that supply across a 24-hour day gives G ≈ Sy × 24r. Substituting into the daily balance gives E_g ≈ Sy × (24r − Δh), in metres per day.
This guide uses Δh as a signed net rise. Some presentations instead use a positive daily fall or a plus-or-minus storage term. Those expressions can be equivalent when their conventions are followed. Do not copy a plus sign from one source while inserting a signed rise defined by another. Write the conservation equation first; then the meaning of the sign remains visible.
For a small fictional example, r = 0.001 m/h, Δh = −0.012 m and Sy = 0.10. The head-equivalent balance is 24 × 0.001 − (−0.012) = 0.036 m. Multiplying by 0.10 gives 0.0036 m/day, or 3.6 mm/day. The 36 mm head-equivalent change is not itself 36 mm of evapotranspired water.
The estimate is conditional on both the night-window interpretation and extending its background rate through the day. A smooth recovery segment does not establish constant lateral supply. River stage, neighbouring wells or changing gradients may alter it. Treat the formula as a transparent approximation to test, not as a way of converting any daily wiggle into a validated flux.
CHAPTER 4 OF 12
4. Choose specific yield for the response being studied
Specific yield describes water released from drainable storage per unit bulk volume under relevant conditions; it is not simply total porosity. The effective amount released over a few hours can differ from a long-duration drainage value. Sediment texture, water-table depth and delayed drainage can therefore affect how a small daily head change translates into water depth. A generic textbook value should not be treated as a site measurement.
Loheide, Butler and Gorelick’s saturated–unsaturated flow assessment highlights the sensitivity of White-method estimates to yield and drainage conditions. The practical lesson is to justify the effective yield at the relevant response time, rather than multiplying by an apparently precise coefficient without checking its meaning. Their work does not supply a universal replacement value for every material or water-table depth.
In the earlier example, the head-equivalent term is 0.036 m. With Sy = 0.05, estimated groundwater evapotranspiration is 1.8 mm/day; with Sy = 0.15 it is 5.4 mm/day. The hydrograph is unchanged. This threefold range comes entirely from the storage conversion. Reporting only a central 3.6 mm/day would conceal the assumption most strongly controlling that illustrative result.
A changed-context check lowers the water table into a different layer while leaving the observed daily amplitude similar. The original yield may no longer apply, and the root–groundwater connection may also change. Reassess both the conversion and the mechanism. Similar head movement does not guarantee similar water use, because head is a state variable whose relation to stored water depends on the material and time scale.
CHAPTER 5 OF 12
5. Estimate a recovery slope rather than picking two convenient points
Plot the night segment with its actual timestamps and quality flags. A slope based on two points can be highly sensitive to noise or a brief disturbance. Fitting a suitable straight segment to multiple readings can expose curvature and outliers, but the fitted line remains an approximation. Examine residuals and the physical setting instead of assuming that a regression automatically identifies background groundwater supply.
Use a window chosen for defensible process reasons and test nearby plausible windows. Suppose illustrative recoveries are 0.0008, 0.0011 and 0.0015 m/h under three choices. With Sy = 0.10 and Δh = −0.010 m, estimates are 2.92, 3.64 and 4.60 mm/day. The variation is informative: window choice matters materially. Selecting only the window that agrees with an expected answer would hide that sensitivity.
A night segment may curve because gradients relax, drainage is delayed or uptake changes. Extending one local slope for 24 hours may then be inappropriate. A more detailed model or an alternative method may be needed. It is legitimate to reject a day from this simplified calculation when its assumptions fail; forcing a value for every date can reduce rather than improve the credibility of the resulting series.
For transfer, shift the window across a pump shutdown. The steep early recovery is not necessarily a baseline groundwater supply that continues all day. Compare operating records and unaffected periods before fitting. Keep a documented rejection reason, such as “recovery overlaps pumping transition”, instead of merely deleting a numerically inconvenient day. The reason is necessary for later interpretation of missing estimates.
CHAPTER 6 OF 12
6. Test the assumption of negligible night-time uptake
The recovery step assumes night-time groundwater evapotranspiration is negligible for its intended approximation. This is a condition to assess, not a universal biological fact. If uptake continues at night, the observed recovery represents background supply minus that uptake. Using it as the full supply rate can bias the daily estimate. The direction follows from the balance rather than from a general claim that all night recovery is reliable.
Use a fictional constant-background case. Actual background supply is 0.20 mm/h and night-time groundwater uptake is 0.05 mm/h. Net storage recovery is 0.15 mm/h. A method treating that net recovery as background supply extrapolates 3.6 mm/day instead of the actual 4.8 mm/day supply. Holding the daily storage change fixed, its groundwater evapotranspiration estimate is low by 1.2 mm/day under these exact assumptions.
The correction in that example is not a universal night-uptake adjustment. Real background supply and uptake can both vary. If independent measurements or a justified model are unavailable, report sensitivity scenarios rather than pretending the missing night term is known. A transparent conditional estimate can still be useful, especially when the question is whether plausible assumptions change the interpretation.
A changed-context exercise compares a cool, quiet night with a warm, windy night while holding the daily head change fixed. You cannot assert equal night uptake from the hydrograph alone. Ask whether the recovery-derived background estimate remains comparable. The transferable habit is to challenge the supposedly inactive reference period whenever the environmental conditions or vegetation behaviour differ.
CHAPTER 7 OF 12
7. Separate rain, recharge and other short events
Rainfall can disturb a daily cycle by changing near-surface storage, recharge timing and evaporative demand. An overnight rise after rain may include recharge rather than merely recovery under reduced uptake. Applying the simple recovery formula to that segment could treat new input as an all-day background flux. Review rainfall and hydrograph timing before deciding which days satisfy the approximation.
Recharge need not arrive at the water table at the same time as rain at the surface. Delay depends on pathways and antecedent conditions. A rain-free day can still contain a delayed recharge response, while a rainy day may produce little immediate water-table rise. Avoid a screening rule that accepts every dry calendar day without inspecting neighbouring events and the actual groundwater response.
The existing water-table recharge guide estimates a different quantity: water added to saturated storage through a recharge-attributed rise above an appropriate baseline. A diel evapotranspiration calculation uses a daily storage change and a background-supply assumption. The two can share observations and yield uncertainty. They should not be treated as independent flux measurements merely because the calculations have different names.
For transfer, insert a short rise caused by delayed recharge into the night window. Explain how the inferred r and therefore G would increase if the disturbance were ignored. Then ask whether removing the affected readings restores a physically credible baseline or leaves too little evidence. Excluding an event is justified only when the remaining record still supports the method, not simply when it produces a smoother result.
CHAPTER 8 OF 12
8. Check pumping, stream stage and pressure effects
A pump operating during daylight can mimic vegetation-driven decline, followed by recovery after shutdown. A daily reservoir or river-stage cycle can also propagate into nearby groundwater. These alternatives are especially important near managed systems. Obtain operating or stage records at a resolution capable of matching the groundwater cycle. Annual pumping totals cannot diagnose a daily schedule.
Pressure transducers require the appropriate atmospheric compensation and quality control. Atmospheric pressure changes, sensor drift, thermal response and datum changes can affect apparent level. A consistent daily pattern in a measurement error can resemble an environmental rhythm. Compare manual checks, instrument specifications and independent pressure records where relevant. Do not assume that a regular signal is free of instrumental structure.
Consider a fictional record with a 20 mm daytime head decline near a scheduled pump. Calling it plant uptake without examining the schedule would skip an obvious competing cause. If a nearby unaffected well lacks the pattern, that is useful comparative evidence but not final proof by itself. Differences in geology, screen depth and location also need consideration before attributing the contrast to one mechanism.
For a changed-context check, stop the pump for several comparable days while recording weather and river stage. Such operational evidence, if legitimately available, can help distinguish mechanisms; the educational point does not authorise changing real operations. Explain what each alternative predicts. A convincing attribution tests different predictions rather than merely noting that two curves look similar.
CHAPTER 9 OF 12
9. Avoid daily aliasing and inconsistent day boundaries
The storage term requires beginning and ending levels separated by the intended day length. Samples taken at different times on successive dates can mix the diel phase into the apparent daily change. Comparing one day’s morning high with the next day’s afternoon low can create an exaggerated loss. Use consistent timestamps and keep any interpolation or gap filling visible.
Sampling once per day can miss the cycle entirely. Sampling several times per day may reveal a pattern but still estimate a short recovery slope poorly. Match the interval to the process and instrument precision. A one-millimetre rise over four hours is difficult to interpret if the relevant combined measurement uncertainty is of comparable size. More decimal places in stored data do not establish that the slope is resolved.
Time zones and clock changes can matter when combining groundwater, meteorological and operating records. For the simplified equation, 24r assumes a 24-hour interval. If the actual elapsed interval differs, use elapsed hours consistently rather than relying only on calendar labels. Singapore’s local time conventions do not remove problems in datasets collected elsewhere or assembled from instruments using different time references.
For transfer, move from hourly measurements to six-hourly measurements. The daily net change may remain estimable, while the night recovery segment becomes poorly constrained. You should not keep the same apparent precision for E_g. Distinguish which component lost information. This helps a reader see why adequate resolution for daily storage change is not automatically adequate resolution for the method’s background-supply estimate.
CHAPTER 10 OF 12
10. Propagate uncertainty without disguising assumptions
Write the estimate as E_g = Sy × B, where B = 24r − Δh. This separates the yield conversion from the head-equivalent balance. Test plausible ranges for both. The result may be uncertain because of measurement noise, a poorly identified night window, yield uncertainty or a wrong process model. Numerical propagation addresses only the uncertainties actually included; it cannot validate a missing mechanism.
For a fictional range, take Sy = 0.06–0.12, r = 0.0009–0.0013 m/h and Δh = −0.010 to −0.006 m. Under simple rectangular bounds, minimum B is 24 × 0.0009 + 0.006 = 0.0276 m and maximum B is 24 × 0.0013 + 0.010 = 0.0412 m. Estimated E_g then spans 1.656–4.944 mm/day. This is a sensitivity envelope, not a statistical confidence interval.
The bounds may not be independent. The same readings can influence both r and Δh, and material properties can affect yield and recovery dynamics together. A more formal uncertainty analysis should represent relevant dependence. Do not assume that averaging many daily estimates removes a shared yield bias. Repeated use of the same wrong coefficient can produce a stable mean that remains systematically wrong.
A negative estimate should prompt diagnosis. If the daily rise exceeds the extrapolated head-equivalent background supply, the simplified equation gives E_g below zero. That may indicate recharge, a changing background flux, sign errors or other assumption failures. Do not silently set it to zero and continue as though the method worked. Preserve the original calculation, investigate and flag an excluded or unresolved day explicitly.
CHAPTER 11 OF 12
11. Do not multiply one well into a landscape total
A well samples a local hydraulic response influenced by surrounding properties and boundaries. It does not directly provide a known vegetation footprint over which its inferred flux can be multiplied. Spatial variation in water-table depth, sediments, roots and lateral gradients can change the relationship across short distances. Map the conceptual system before claiming an area-average groundwater evapotranspiration rate.
Suppose two teaching sites yield 2 and 6 mm/day under locally justified assumptions. Their simple mean is 4 mm/day, but that is not necessarily the landscape mean. If the first represents 90% of the relevant area and the second 10%, an illustrative area-weighted mean is 2.4 mm/day. Those weights themselves require evidence. Two points do not establish their own representative areas.
Total evapotranspiration may also include water drawn from unsaturated soil storage or recent precipitation. A groundwater estimate should not be equated to the entire atmospheric water flux. Independent energy-balance or other observations can offer comparisons, but different footprints and definitions must be reconciled. Apparent disagreement can reflect different quantities, not necessarily a faulty instrument.
For transfer, keep vegetation cover constant but deepen the water table or change the capillary and rooting connection. The groundwater share of water use can change without an equal change in total evapotranspiration. This is why a diel well estimate is useful as a component hypothesis, not a universal ecosystem water-use meter. Name the component and support scale every time the number is reused.
CHAPTER 12 OF 12
12. Use the estimate as a testable component of evidence
A responsible report includes well construction, water-table setting, raw levels, pressure treatment, timestamps, selected windows, specific-yield justification, exclusions and uncertainty. Show the daily cycle and the equation with its sign convention. A reader should be able to reproduce the estimate and understand why some days were not estimated. Completeness of reporting is more important than completeness of the output column.
Compare against independent evidence with compatible definitions and scales. A broader evapotranspiration measurement can constrain plausibility, while vegetation, soil moisture and stream-stage records can test mechanisms. Agreement is not fully independent if both methods share key assumptions or calibration data. Explain which part of the inference each comparison actually tests rather than presenting several similar curves as automatic confirmation.
The practical transfer check changes one condition at a time. Halve Sy and the calculated flux halves if B is fixed. Reverse the sign of a net daily change by changing the actual storage behaviour, and the storage contribution changes accordingly. Add a pump-recovery disturbance and the background-supply interpretation may fail altogether. These are different responses: sensitivity, changed water balance and invalidated method.
End with a bounded conclusion such as: “Under the selected yield and stable-background assumptions, the cycle is consistent with a groundwater evapotranspiration range of X–Y for the accepted days.” Do not call that range a directly measured total ecosystem flux. State what evidence would reopen it, such as a new layer, changed pumping, a deeper water table or a revised instrument correction. That makes the estimate reusable without overstating it.
Independent worked case and evidence checks
A fictional shallow unconfined well is screened across the water table. Head at midnight is 10.000 m and at the next midnight is 9.988 m, measured against the same datum. Thus Δh = 9.988 − 10.000 = −0.012 m. A selected undisturbed night window has an estimated recovery slope r = 0.0012 m/h. An independently justified teaching assumption sets effective Sy at 0.08 for the relevant daily response.
Compute the head-equivalent background supply: 24r = 0.0288 m/day. Then B = 0.0288 − (−0.012) = 0.0408 m/day. Estimated groundwater evapotranspiration is 0.08 × 0.0408 = 0.003264 m/day, or 3.264 mm/day. Round according to evidence quality; the exact arithmetic does not make all four significant figures physically meaningful.
Check conservation in water-depth units. Inferred background supply is 0.08 × 0.0288 = 0.002304 m/day, or 2.304 mm/day. Storage change is 0.08 × (−0.012) = −0.00096 m/day, or −0.96 mm/day. Supply minus evapotranspiration is 2.304 − 3.264 = −0.96 mm/day. The account closes because the equation was constructed from that balance, not because an independent evapotranspiration measurement confirmed it.
Now retain the same recovery slope but let the water table rise by 0.006 m over the day. B becomes 0.0288 − 0.006 = 0.0228 m/day and estimated E_g becomes 1.824 mm/day. Some inferred supply remains in storage. Using a plus sign with this positive rise would yield 2.784 mm/day and violate the chosen balance. This is a sign-convention error, not a different legitimate interpretation of the same defined variables.
Test yield uncertainty next. With the original B of 0.0408 m/day, Sy between 0.05 and 0.11 produces 2.04–4.488 mm/day. The range does not include uncertainty in r, Δh or the process model. If those matter, the uncertainty assessment must expand. A yield-only range should be labelled as such so it is not mistaken for a complete interval around the true flux.
A second analyst selects a different night window and obtains r = 0.0007 m/h. Keeping Sy = 0.08 and original Δh = −0.012 m gives B = 0.0168 + 0.012 = 0.0288 m/day and E_g = 2.304 mm/day. The window change reduces the estimate by 0.96 mm/day. Inspect why the slopes differ: curvature, noise, uptake or operations may explain the disagreement. Do not average away the mechanism question.
An operating log then reveals that a nearby pump stopped shortly before the steeper recovery window. This does not automatically prove that all recovery was pumping-related, but it undermines the assumption that the chosen slope represents steady background supply. Reassess the day using appropriate evidence. If no uncontaminated window remains, report no defensible White-style estimate rather than inventing a replacement value.
Consider night uptake under a separate controlled teaching scenario. True hourly background supply is 0.14 mm and night groundwater uptake is 0.02 mm, so observed storage recovery is 0.12 mm/h. Extrapolating net recovery gives 2.88 mm/day background supply instead of 3.36. Holding daily storage change fixed, E_g is underestimated by 0.48 mm/day. This is conditional algebra, not a proposed universal night correction.
Check the opposite physical failure. A delayed recharge pulse raises the head by 0.040 m during a day with r = 0.0010 m/h and Sy = 0.08. The formula gives 0.08 × (0.024 − 0.040) = −0.00128 m/day, or −1.28 mm/day. The negative output flags that the simplified assumptions do not adequately describe the accepted inputs. It should not be published as negative plant water consumption or silently clipped.
Now change the spatial question. The first well’s 3.264 mm/day estimate is multiplied by a claimed 2 km² footprint, giving 6,528 m³/day. The multiplication is arithmetically correct because 0.003264 × 2,000,000 = 6,528. But the result is not defensible unless the footprint and representativeness are justified. Correct unit conversion cannot supply the missing spatial evidence.
Suppose independent mapping instead supports a two-zone illustrative model: 80% of an area at 2 mm/day and 20% at 5 mm/day. The area-average groundwater component is 0.8 × 2 + 0.2 × 5 = 2.6 mm/day. That calculation belongs to the two-zone model, not automatically to the original well. Retain the different evidence and assumptions rather than combining all numbers into an unexplained average.
Finish by writing what survives all these tests. The original accepted assumptions yield 3.264 mm/day, with explicitly demonstrated sensitivity to yield and recovery window. Pumping, delayed recharge and unjustified area scaling can invalidate particular interpretations rather than merely adding small error bars. Understanding that distinction is more useful than memorising the formula: some changes alter a parameter, while others remove the reason the formula was appropriate.
Worked comparison
| Change | Effect under the simplified equation | Interpretation check |
|---|---|---|
| Daily head fall, Δh negative | Adds storage release to inferred supply | Use a signed change consistently |
| Larger effective Sy | Larger inferred water-depth flux | Yield must match drainage time and material |
| Night uptake not negligible | Recovery is not full background supply | Conditional bias needs independent evidence |
| Pump shutdown in night window | Recovery slope may be contaminated | Method may fail rather than need a small adjustment |
| Rain-driven rise exceeds 24r | Negative calculated E_g | Investigate assumptions; do not silently clip |
Questions and practical checks
Does a daily groundwater cycle prove transpiration?
No. It can be consistent with vegetation water use, but pumping, stage changes, pressure effects and instrumental behaviour can also create daily patterns. Attribution needs independent evidence.
Why multiply by specific yield?
Water-level movement is not water volume per area. An appropriate effective yield converts the head-equivalent balance into a drainable water-depth estimate under the storage model.
Is this the same as recharge estimation?
No. Recharge inference attributes a rise to new input above a baseline. The diel method estimates a groundwater evapotranspiration component using daily storage change and a background-supply assumption. They may share uncertain inputs.
What if the estimate is negative?
Retain the calculation and investigate signs, recharge, changing background flow or other failures. A negative value is not evidence of negative plant uptake, and automatic clipping conceals the diagnostic information.
Can one well estimate total ecosystem evapotranspiration?
Not automatically. The inferred groundwater component, spatial representativeness and observation footprint need justification. Unsaturated soil water and recent rainfall can contribute to total evapotranspiration separately.
Continue with the connected subjects
Separate recharge, ecosystem dependence and total flux
Groundwater recharge estimation and why a rise needs attribution
Reading water-table recharge hydrographs and specific yield
Groundwater-dependent ecosystems and ecological drawdown
Water-budget closure and the difference between identity and validation
Eddy-covariance evapotranspiration and why flux footprints must be reconciled
Sources and scope
The numerical cases in this guide are original fictional teaching examples, not measurements from a real site. This is an educational explanation, not a field safety instruction, engineering approval or an official certification. For a real project, use qualified practitioners, current product documentation and site-specific evidence. Sources were checked on 29 October 2026.