Extrapolation Traps with Water Wells

It’s important to not confuse actual measurements from projections.

By Marvin F. Glotfelty, RG

During our work as groundwater professionals, we are frequently confronted with limits in the information we rely upon to conduct our tasks.

Figure 1. Examples of extrapolated and interpolated values.

Both water well drillers and hydrogeologists undertake a wide variety of tasks that require data to form the basis of our interpretations or next actions—and in many cases the necessary information is inadequate or incomplete.

This means groundwater professionals are making an educated guess as to the missing information, which involves applying an extrapolation or an interpolation.

An extrapolation is the process of projecting, extending, or expanding known (measured) data or experiences into an area that is unknown or has not been experienced. An interpolation is the process of estimating a value or condition in an unknown area that is situated in between two or more known values or conditions that were actually measured or experienced.

Figure 1 shows a graphical representation of how we might fill in missing information through extrapolated or interpolated data projections.

Both extrapolations and interpolations are common during groundwater projects, and they’re both useful and appropriate, since they enable us to arrive at a reasonable interpretation of values or conditions that we may be unable to measure or experience directly.

It’s appropriate to apply extrapolations or interpolations to fill in data gaps, but it is inappropriate to disregard the stark difference between a fact that we measured or experienced, versus a projection that is based on extensions of things we know for sure but can’t directly measure.

There are many extrapolations that routinely occur around well drilling sites. One of the most common of these extrapolations is the weight measurement of drilling mud or cement slurry with a mud scale.

Figure 2. Mud scales are used to measure the weight of a small volume (130 milliliters, or a little more than ½ cup) of drilling fluid or cement, so the fluid’s “pounds per gallon” value is an extrapolation.

Mud scales provide a reliable and accurate measurement of fluid density, which is achieved with a bar scale equipped with a sliding counterweight and bubble level (Figure 2). The standardized lid cover on mud scale cups is outfitted with a hole in it to allow excess fluid to pass outward, so that a uniform volume of fluid is measured. That uniform volume is approximately 130 milliliters (about ½ cup, or 0.034 gallons).

Therefore, when a mud scale is used to measure drilling fluid or cement slurry in units of pounds per gallon (lb/gal), the weight measurement is being extrapolated by a ratio of about 29:1. In other words, it would take about 29 mud scale cups to fill a gallon jug, even though we’re measuring the fluid density in units of lb/gal.

Although we recognize that mud scales provide a reliable and accurate measurement of fluids, that attribute is predicated on proper use and maintenance of the scale.

For example, if a mud scale is not kept clean and has splatters of clay or cement on the scale’s bar or cup, the extrapolated fluid measurement will be flawed. Similarly, if the hole in the scale cup lid is clogged, the excess air or fluid in the cup may not be properly evacuated, so a non-uniform volume of fluid
may be measured, which will provide an inaccurate weight value.

To confirm the calibration of a mud scale, the user can simply measure the weight of clear water which weighs 8.33 lb/gal (with slight variations depending on water temperature).

Victim of Human Nature

Figure 3. Multiple wells in an area (top) may be assumed to have consistent subsurface properties, but the geologic complexity and variability of well designs could make that extrapolation erroneous (bottom).

Groundwater professionals must be careful to avoid becoming victims of our own human nature, which gives us the tendency to extrapolate things that we can see and experience on the land surface, as being representative of subsurface conditions.

For example, we may stand at a high viewpoint and look over a wellfield where we can observe individual wellheads and the distances in between them, as shown on the top image of Figure 3. Our human nature is to consider this view of the wellfield as an array of data points on a sheet of paper.

Each of the wellheads does indeed represent a data point, but rather than being repetitive points of information on a two-dimensional layer, the wellheads are actually surface expressions of a complex array of unique well structures with differing depths, diameters, screened intervals, annular seals, etc.

In addition to the variability of the well designs, geologic inconsistencies between the well locations add a level of complexity to the data, as shown on the bottom image of Figure 3. Although some of the wells in the figure are closely spaced, their differing structures and hydrogeologic conditions will potentially introduce irregularities that must be resolved by the groundwater professional.

Figure 4. It’s tempting to extrapolate the properties of Well A to represent conditions at nearby Well B (top). However, that extrapolation can be erroneous in stream deposits (bottom).

A groundwater professional may decide to bypass sampling some closely spaced wells to avoid duplication of effort that will extend the sampling schedule and increase the project budget. Although selective sampling might be appropriate in some cases, the scenario shown in the bottom image of Figure 3 represents conditions where nearby wells can represent different conditions, so caution of extrapolating observed conditions is warranted.

Even when groundwater professionals consider the variability of well designs and dimensions, there can be inadvertent extrapolations when evaluating, siting, or installing water wells.

For example, a hydrogeologic environment that is notorious for abrupt subsurface changes is fluvial (stream-deposited) depositional environments. The cut-and-fill stratigraphy of fluvial deposits is established as streams meander to different locations over geologic time and lay down coarse-grained strata of sand and gravel in the high-energy channel deposits.

Directly adjacent to those channel deposits, the fluvial system also lays down layers of fine-grained silt and clay in the low-energy floodplain deposits. So, as the channel migrates from place to place over millennia, the channel and floodplain deposits cut into one another and are deposited upon one another in a complex and unpredictable manner.

Therefore, we should be careful to avoid extrapolating the characteristics of one fluvial aquifer well onto other nearby wells within that same aquifer.

Figure 4 shows an example of how we may assume Well A, with good groundwater production, to produce water similar to nearby Well B. However, the subsurface details that we can’t observe may cause Well B to have poorer groundwater production even though it has the same screened interval and is located close to Well A.

An Important Difference

Figure 5. Groundwater elevations in feet above mean sea level represent measured data (top). However, groundwater elevation contours and flow directions (bottom) are interpretations and not actual data, so they should be considered accordingly.

We commonly prepare groundwater elevation contour maps to show water table conditions and flow directions of an area, and it’s good to remember that those contours are interpolations of the water-table data, but not actual data themselves. This difference may seem subtle but represents the difference between measured values that we know for sure versus interpretations of probable conditions that we’ve developed on the basis of an imperfect understanding of the aquifer system.

That difference between measured values we know for sure and probabilities that represent our best interpretation of conditions can be important. For example, consider the area shown in Figure 5. The top map shows the locations of water wells with the elevations of their static water level (in feet above mean sea level).

Those values are measured data, so for the date they were measured those groundwater elevations can be considered reliable facts. From that water-level data we can create a groundwater elevation contour map (the bottom map in Figure 5) by interpolation of the water-level values. The white groundwater elevation contours and blue groundwater flow direction arrows are reasonable, and they honor all the water-level data points.

If we were to drill a new well near the center of the map along the 1200-foot elevation contour line, we could use that water-table elevation as a reasonable estimate of the depth at which we might encounter the water table.

However, since the hypothetical well has not yet been drilled and its water-table location is only an interpolation of surrounding data, we should remain mindful that the actual water-table location could be higher or lower than predicted due to a previously undocumented mound or drawdown cone in the water table at that location.

A hydrograph is a x-y plot of a single well showing variability in the static water level versus time. Hydrographs provide another example of data presentations that sometimes require extrapolation or interpolation to fill in missing data in areas of interest, so in this case we sometimes need to fill in both the date and the water-level measurement.

We sometime find ourselves needing to project the changes of depth-to-water trends into the future (an extrapolation), but we also commonly wish to clarify trends within the existing data set (an interpolation).

An example hydrograph that would encourage us to interpolate water-level change trends is shown in Figure 6.

Figure 6. Hydrograph showing depth to groundwater versus time. Abundant data are available after 2005 but earlier data are sparse, so interpolation may be required.

This hydrograph is from a well with abundant water-level measurements after 2005. However, earlier water-level measurements are limited to the years 1987, 1989, 1991, 1997, and 2002. The periods with missing information are reasonably represented with straight lines, but since the steep rise of water levels in the earlier years (1987-1991) differs from the more gradual rise of water levels in later years (after 2005), a clarification of the missing details would require an interpolation.

Both extrapolations and interpolations are commonplace and useful tools, and both of these projections provide a method for groundwater professionals to fill in data gaps to meet our information needs.

While both these data-estimation techniques are appropriate, it’s important to keep them separated in our minds from actual facts, so we avoid confusing quantified data or experiences that we actually measure or observe with probable conditions that we’ve projected from values obtained at nearby places or in previous times.

Learn The Art of Water Wells

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Marvin F. Glotfelty, RG, is the principal hydrogeologist for Clear Creek Associates, a Geo-Logic Associates Co. He is a licensed well driller and registered professional geologist in Arizona, where he has practiced water resources consulting for more than 35 years. He is author of The Art of Water Wells (NGWA Press, 2019) and was the Groundwater Foundation’s 2012 McEllhiney Lecturer. Glotfelty can be reached at mglotfelty@geo-logic.com.