By Raymond L. Straub Jr., PG
Solving the three-point problem.
I came across a fellow groundwater professional working in the same area as me recently while I was on a field study.
We stopped to talk. I had never met him but recognized his name from other projects. After a few moments of conversation, he said, “Aren’t you the guy who writes the Field Notes articles?”
I was a little surprised by the question but equally flattered. It was the first time I had been asked about the article series from someone I had never met. He went on to tell me he enjoys the articles, but he thinks they are too complicated sometimes for anything he may need.
Following the conversation, I considered the context and focus of these articles. In the course of contemplation, I asked a friend who is a groundwater scientist if he could explain how to work a “three-point problem.” After a few minutes of attempting to do so, he finally said, “I know how to work the problem; I just can’t explain it.”
That reinvigorated me. You see, the goal of my articles is to provide a useful tool to help everyday working groundwater professionals understand the subsurface. This is regardless of your skill set and whether you work on the drilling, service, or geoscience side of the equation.
The Three-Point Problem
So let’s work this month on understanding the “three-point problem.” Estimating anything below the surface can be challenging. Being able to predict the location of a specific target is near impossible—but not completely impossible.
Whether you are working with an outcrop, geology map, boring logs, or groundwater—through the use of simple mathematics, planar targets can be recognized and potentially predicted with a degree of accuracy.
The three-point problem, also known as the three-point method, can be used to solve for specific strike and dip of a given formation or even for the direction of movement and hydraulic gradient of groundwater.
To graphically solve the problem requires the use of three known points on a common plane with corresponding elevations referenced to a common datum and not aligned in a straight line. In other words, you just need to know three known locations not in a straight line with known common elevations associated with the top or bottom of a specific formation contact or planar surface. You got that?
Solving the Problem
To solve the three-point problem graphically, you must first create a scaled map in plan view of the planar surface that includes the three points of measurement.
You can use a topographic map, a road map, computerbased surface map, or even a piece of grid paper. Once you have acquired your map, locate and mark the three points of known contact measurement.
If measuring a formational contact, whether measuring an outcrop or from a boring log or geophysical log, pick the top or bottom of the common formation to be measured that is best represented at all three locations. Keep in mind, it is best to use a common datum for the X, Y, and Z axis of measurement throughout the process (feet, meters).
Once you have set up the graphical problem, it is time to measure the distances. I should point out a graphical solution accuracy is directly dependent on skill of measurement and attention to detail when plotting.
Determining the strike and dip of a formation or planar surface is accomplished by finding the strike line or intermediate point along the line between the highest and lowest points corresponding to the elevation of the mid-point. To find the strike line, first draw a line connecting the highest contact point, C, to the lowest contact point, A. You can then use the following equation to find the strike line mid-point, D, that corresponds to mid-point, B.
or as shown in Figure 1:
Once you have located the strike line mid-point D along Line AC, draw a connecting line from contact point B to point D along Line AC. The strike line or hinge point is located along the line from contact point B to point D. To find the direction of the dip, draw a line at a right angle to the strike line or 90 degrees to the strike line through the lowest point at contact point A. The Line AE is the dip line and represents the direction of the dip of the formation or planar surface.
To find the dip angle, you can use the following equations:
You can use tangent tables to assist in solving the first equation. If you have a calculator with the Cotangent function, the second equation can be useful.
Going back to Figure 1 and using tangent tables, the dip angle is 3 degrees, 30 minutes.
Using a compass or protractor to measure the direction of Line AE will give the direction of dip, or measuring the direction Line BD will give the direction of the strike line. Following the right hand rule for strike and dip notation, which puts the dip direction to the right of the strike line, in this exercise would show the formation contact planar surface striking at 228° with a dip of 3.51° dipping to the northwest quadrant. A more simplistic notation of this is 228°/3.5° NW.
Dealing with groundwater, hydraulic gradient is often an important consideration. As described by Bates and Jackson in their Dictionary of Geological Terms, hydraulic gradient in an aquifer is referred to as: “The rate of change of total head per unit of distance of flow at a given point and in a given direction.” (Bates and Jackson 1984)
To determine the direction and hydraulic gradient of groundwater, the three-point problem is set much the same as for a geologic surface. However, instead of using a geologic contact for measurement, the static water level in three wells are measured. It is best to use wells installed and screened similarly within the same aquifer and not in a straight line. Groundwater gradient can shift on a local scale; therefore, measurement wells should be located close enough to accurately realize local effects.
Determining the static water level elevation is accomplished by subtracting the depth to water from the measuring point at the well surface from the elevation of the measuring point. For instance, if the static water level is 100 feet below the measuring point at the top of the casing, and the top of casing elevation is 2000 feet mean sea level (MSL), then the static water level elevation is 1900 feet MSL.
Determining hydraulic gradient can be accomplished graphically by using the three-point problem with the addition of the following equation:
When notating groundwater movement, the direction of the groundwater gradient is utilized instead of the contour or strike line. Therefore, as shown in the example in Figure 2, the hydraulic gradient would be expressed as:
Hydraulic gradient lines are often considered the direction of flow lines while the perpendicular lines to the gradient are considered equipotential lines or lines of equal head. The determination of the hydraulic gradient is a key component in the Darcian equation for groundwater flow and determining groundwater velocity, but we will save that exercise for another article.
Oftentimes in field work we come across problems we may not have encountered recently, and having access to the right tools can make the difference between success or failure. It is my hope through this article and others like it I’ve helped in some small way to provide you with useful tools—and these Field Notes help you in your daily work.
Bates, Robert L., and Julia A. Jackson. 1984. Dictionary of Geological Terms–Third Edition. New York: Anchor Books.
Compton, Robert R. 1962. Manual of Field Geology. New York: John Wiley & Sons Inc.
Heath, Ralph C. 1983. Basic Ground-Water Hydrology, U.S. Geological Survey Water-Supply Paper 2220. Water- Supply Paper, Reston, Virgina: U.S. Geological Survey.
Raymond L. Straub Jr., PG, is the president of Straub Corp. in Stanton, Texas, a Texas-registered geoscience firm and specialized groundwater services firm. He is a Texas-licensed professional geoscientist and holds master driller licenses in Texas and New Mexico and a master pump installer license in Texas. He can be reached at firstname.lastname@example.org.