# Do the Math

### Know the calculations for proper well design and efficient drilling.

By Marvin F. Glotfelty, RG

There are several critical values that must be accurately determined and applied to the work at hand during the design of a water well and while the well’s borehole is being drilled.

Many groundwater professionals prefer using charts and tables to determine these values, and those tabulated references are available in the appendices of many textbooks and in handbooks from cement or drilling fluid suppliers.

This approach works well but relying on a printed reference is not without the risk since the wrong value can still be selected from the fine print of a reference table, or the reference document can be damaged or lost (e.g., dropped in the mud pit) altogether.

So, let’s address the alternative approach of using simple mathematical formulas to determine the same information. Although the reliance on a single sheet of paper to obtain the needed value is avoided with this approach, the potential for human error or miscalculation remains, meaning regardless of the approach, great care in determining such values is prudent.

As we consider the various calculations that enable us to determine the values of length, weight, pressure, volume, flow velocity, etc., we should remain mindful of the units of measure we’re dealing with. The groundwater industry uses units of measure that are somewhat intermingled with other units from associated disciplines such as engineering, surface water hydrology, and the oil and gas drilling industry.

Once we have obtained the number of units we need, we can convert that value into a familiar unit that we’re comfortable working with. For example, we may use a formula to determine the volume of a borehole to be 100 cubic feet (ft3), but we want to know the answer in gallons. Since we know that every cubic foot contains 7.48 gallons, we can easily convert that borehole volume to 748 gallons.

Some common unit conversions used in our industry are:

1 cubic foot = 7.48 gallons

1 acre-foot = 325,851 gallons

1 cubic foot per second = 448.8 gallons per minute (gpm)

1 acre-foot per year (AF/yr) = 0.62 gpm (continuous flow)

1 million gallons per day (MGD) = 694.44 gpm

1 psi of pressure = 2.31 feet of water head

water weight = 8.3 pounds/gallon

water viscosity = 26 seconds/quart

1 oilfield barrel = 42 gallons

1 cubic foot of sand or gravel ≈ 100 pounds

### Annular Volume Calculation

The word “annulus” refers to the area between two concentric circles, or in the case of a water well, the volume between two cylinders that are defined by the casing or tubing string in a boring, and the borehole wall.

We want to know the volume of material (filter pack sand, cement grout, etc.) that is to be placed in the annulus to assure the annular void has been properly and completely filled (Figure 1). The conceptual diagram showing the variables used for calculating an annular void is shown in Figure 2, and the formula for the annular volume calculation is:

ft3 = 0.005454 × (D2 – d2) × L

In this calculation, the “d” value is the diameter of the casing or pipe diameter, and the “D” value is the borehole diameter (Figure 2). The sump area below the base of the casing has only one diameter in the open borehole, so the “d” value is omitted, and the formula just becomes:

ft3 = 0.005454 × D2 × L

When the cubic feet of annular volume are known, that volume can be converted to other common units that are consistent with the material being installed (e.g., cement is commonly supplied in *cubic yards* and filter pack sand is commonly supplied in *tons*). Based on experience, we usually assume material volumes required to fill an annulus are 30% more than calculated volume, so we multiply the calculated volume by 1.3.

### Hydraulic Head Calculation

If excessive hydraulic pressures are exerted on a well casing, it will collapse. We generally know the collapse strength of the well casing from the casing supplier or from standard references such as the charts in American Water Works Association Standard A100. The hydraulic pressures applied to the outside of the well casing depend on the density of the liquid and the depth of submergence (Figure 1). Applying the fluid density (measured in the field) and depth (Figure 2), the formula for hydraulic pressure head calculation is:

psi = 0.052 × lb/gal × L

The hydraulic head formula is applicable to the hydraulic pressure head for any liquid, but we most commonly use this calculation during cement seal installation, since cement grout is generally the heaviest liquid being introduced to the annulus during well construction.

### Buoyancy Calculation

Some well installation projects involve telescoping casing diameters where a larger-diameter intermediate casing is cemented in place to isolate the upper borehole and prevent poor-quality groundwater from migrating down to the good-quality aquifer in the lower part of the well.

The intermediate casing can be sealed using the pressure grouting technique (Figure 3) to pump cement slurry down through the drill pipe and out to the annulus through a *float shoe* (a drillable check valve connected to the base of the casing). The inside of the intermediate casing is kept full of water during the cement placement to equilibrate hydraulic pressures inside and outside the casing. After the intermediate casing is sealed with the pressure grouted cement, the float shoe can be drilled out and the borehole advanced for installation of the screen and filter pack in the lower part of the well.

For situations like the one shown in Figure 3, there is a point in time when the steel casing filled with water may actually float upward due to the buoyancy forces that result from its submergence in the still-liquid cement slurry.

This is counterintuitive because it would not seem possible for a heavy steel casing filled with water (weighing tens of thousands of pounds) to be capable of floating. Nonetheless, because the water-filled casing is immersed in a much denser fluid (neat cement slurry generally weighs about 15.6 lb/gallon), it may float, just as with other heavy objects will float under the right conditions (e.g., icebergs and battleships).

Floating of a casing string introduces serious logistical and safety hazards and creates significant disruption to the integrity of the annular seal. The potential for floating of the intermediate casing can be easily mitigated by securing it at the land surface, but the driller needs to know that this is required before the cementing operations begin. Thus, a buoyancy calculation is a good idea prior to pressure grouting operations as illustrated in Figure 3.

The buoyancy calculation is more of a conceptual comparison than a pure mathematical formula. This analysis involves some visualization be made on the part of the groundwater professional.

The groundwater professional considers the imaginary scenario where the fluid in the borehole (neat cement in the example shown in Figure 4) is not displaced by water-filled steel casing, but rather a cylinder composed of that same fluid (neat cement).

For the example shown in Figure 4, we would imagine a 15.6 lb/gallon cylinder of neat cement that is within a borehole filled with 15.6 lb/gallon cement. Thus, the imaginary cylinder of cement is surrounded by the same material, so it will be in complete equilibrium and will neither float nor sink.

This imaginary situation enables us to characterize the two forces at play in this scenario: *gravity* (the downward force) and *buoyancy* (the upward force). The imaginary cylinder of 15.6 lb/gallon cement shown in Figure 4 is in equilibrium, so we know that the downward and upward forces are equal. We can calculate the downward force like this:

If we assume that we’ve got a 400-foot-long intermediate casing with a 16-inch OD and a 0.3125-inch wall thickness, the volume of the imaginary cement cylinder will be:

ft3 = 0.005454 × D2 × L

558 ft3 = 0.005454 × 16^{2} × 400

558 ft3 × 7.48 gal/ft3 = 4178 gallons

4178 gallons × 15.6 lb/gallon = 65,177 lb.

Therefore, 65,177 pounds is the buoyancy force since it is equal to the weight of the imaginary cement cylinder as calculated above. If the weight of the real-world steel water-filled casing is less than 65,177 pounds, then that casing will float.

If you apply the weight calculations for a 400-foot-long steel casing with a 16-inch diameter and a 5/16-inch wall thickness, which is filled with water, you’ll see that the downward force in this example is only 52,982 pounds. Thus, the casing in this example will float. The lesson from this counterintuitive scenario is that a casing can actually float. (I’ve seen it happen, and trust me, you don’t want to).

### String Weight Calculation

For heavier-walled casing materials or deeper wells, there are situations where the “string weight” of the casing and screen may exceed the safe hang weight of the casing string, or even exceed the mast capacity of the drilling rig. A good rule-of-thumb is to maintain a rig mast capacity that is no less than 1.5 times the string weight.

The pounds per linear foot of any casing or screen material is generally provided by the casing or screen supplier, but the variables in Figure 2 can be applied to the following formula to estimate the casing string weight:

lb/ft = (OD – WT) × (WT × K)

where:

K = 10.68 for mild steel

K = 10.93 for stainless steel

K = 2.04 for PVC

This string weight formula is applicable to blank casing only, and material suppliers should be consulted for detailed pounds per linear foot values and safe hang weights for well screens. This formula is broadly applicable, however, and handy for a quick double check on material weights.

### Drilling Fluid Circulation Time Calculations

There are several calculations that are commonly applied by drilling fluid engineers (mud engineers) to determine the time period required for the fluid to move from one location in the borehole to another. Some of the more common equations are described below.

The **uphole velocity** calculation provides a determination of the speed at which the drilling mud will flow as it moves up the borehole. For direct air rotary or reverse circulation drilling methods, the uphole velocity is high, so this calculation is generally applicable only for the direct mud-rotary drilling method. The formula for uphole velocity is:

ft/minute = (24.51 × gpm) ÷ (D2 – d2)

Notice the uphole velocity formula is similar to the annular volume formula in that both those calculations use the factor (D2 – d2) to address the cross-sectional area of the annulus. However, the constants in these two formulas are different (0.005454 versus 24.51), which can be confusing. Keep in mind, however, that the constants primarily just provide unit conversions.

If we plug in the values shown in Figure 5 and calculate the uphole velocity, we’ll come up with 3.83 ft/minute:

ft/minute = (24.51 × 10) ÷ (102 – 62)

ft/minute = 3.83

If we do the same thing by first calculating the annular volume and then applying the 10 gpm flow rate to it, we will get an identical result of 3.83 ft/minute. The uphole velocity formula provides a more direct method to determine uphole velocity, whereas the annular volume formula provides a more direct method to calculate the annular volume.

The** bottoms-up time** calculation enables us to determine the time period for the drilling fluid (and the cuttings it is carrying) to travel from the drill bit up to the land surface. This is illustrated in Figure 6(A).

We can calculate the bottoms-up time by using the uphole velocity formula with the borehole depth and drilling mud flow rate plugged in, but that flow rate is being generated by the mud pump, and positive displacement mud pumps (duplex or triplex) are almost never equipped with a flow meter. To determine the flow coming from the mud pump, we can use the formulas:

For duplex mud pumps:

gpm = 0.0068 × (2 (liner diameter)2 – (rod diameter)2) × (stroke length) × (strokes/minute) × (% efficient)

For triplex mud pumps:

gpm = 0.010206 × (liner diameter)2 × (stroke length) × (strokes/minute) × (% efficient)

Remember the strokes are counted in both the forward and backward directions on a duplex pump, but only in the forward direction on a triplex pump. Drillers often have reference charts that provide oilfield barrels per stroke (bbl/stroke), which can be converted to gpm by timing the strokes per minute and converting barrels to gallons (1 barrel = 42 gallons).

The **round-trip time** enables us to see the result of drilling fluid additives, as indicated by the return flow of fluids at the land surface, as is illustrated in Figure 6(B). The round-trip time calculation is the same as bottoms-up time, but with the travel time of fluid to displace the drill pipe added in.

A specified volume of drilling fluids (called a pill) can be circulated to a particular depth interval within the borehole (called spotting), so that the additives in the pill of drilling mud can address the borehole problem at a particular depth of the borehole. This is shown in Figure 6(C).

The calculation for **time required to spot a pill of drilling ****fluid** involves determining the pumping time (at the calculated flow rate) required to displace the fluid so that the drilling mud additives are located adjacent to the problematic interval. This approach is used by mud engineers to address problems such as lost circulation or stuck drill pipe.

### Calculations No Replacement for Knowledge and Experience

The formulas and calculations provided in this column and elsewhere provide important tools for us to quantify the variables we need for water well design and construction. However, it is important to remember that “doing the math” is not a replacement for applying professional knowledge and consideration to determine whether the mathematical result makes common sense.

Groundwater professionals should apply their years of experience to determine whether the mathematical results should be used as calculated, or whether adjustments to the results are appropriate to incorporate conservatism to the results in the face of data uncertainty.

*NGWA: Industry Connected*video. In the 10-minute video, Glotfelty shows that adding filter pack

sand increases the string weight by tens of thousands of pounds in some cases, and swabbing that filter pack to settle it further increases that string weight. Glotfelty also concludes that if there are multiple screened intervals, the uppermost filter pack intervals will have less impact than the lower one, and bentonite and cement annular seals both apply buoyancy effect that seem to relieve these stresses. Click here to watch the video.

*The Art of Water Wells*by Marvin F. Glotfelty, RG, a 2019 book from NGWA Press that is a comprehensive overview of well systems and delivers practical information applicable to real-world situations. The book is ideal for everyone working in the groundwater field. It provides practical information of water wells—covering everything from site selection to design, drilling methods, economics, and more. Click here to order it, call (800) 551-7379, fax (614) 898-7786, or email customerservice@ngwa.org.

**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.

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