Department of Mathematics Education

EMAT 6690, J. Wilson

The Pipeline Problem

## By Matt Tumlin

A common problem encountered by the oil industry is determining the most cost effective pipeline route in connecting various wells in an oil fertile area.  The map below is a section of the U. S. Geographical Survey Contour Map of northeast Ohio with wetlands (swamp) area outlined for clarity.  An existing oil well is located approximately at the point labeled B.  If a new well is to be dug at point A, a pipeline instillation company must be directed as to where to lay connecting pipe.  In consultation with the instillation company, the following information has been obtained:

·        Straight, two-inch coated pie must be used at a cost of \$1.50/foot.

·        A maximum of two elbow joints may be used.  Assume that the elbow joints may be fabricated with any angle measure.

·        In crossing normal terrain, instillation cost is \$1.20/foot.

·        Installation in the wetland area requires an additional Track Hoe at a cost of \$60/hour.

·        In a 10 hour day, a Track Hoe can dig approximately 300 foot trench.

Determine the pipeline route connecting the new well at A to the well at B, which incurs the least cost.

Suggestions:  First, solve the problem as if the wetland separating A and B were a rectangle, and then improve on this solution by modeling the wetland area more accurately.  Also, reduce the number of paths to consider before you begin modeling.  For example, one need not consider a path around the swamp to the north since it is further than the path around the swamp to the south and both traverse only normal terrain.

Problem posted by John Ramsay, College of Wooster.

Rectangular Model

We can first begin to solve this problem by using a rectangle as a model for our wetland.  The rectangle we used has five separate measurements.

A.   The distance from A to the southern most point of the wetland.

B.    The vertical distance from B to the point perpendicular to A (also the vertical distance form A to the point perpendicular to B).

C.   The width of the wetland.

D.   The distance from A to northern most part of the wetland.

E.    The distance from B to the southern most part of the wetland.

By using these measurements, we can find a cost equation for a route the pipeline can take through our model.  Once we get this equation, we can find the route which incurs the minimum cost.

Our cost function is:

The variable, x, is the total distance on normal terrain.  This formula is deducted through many steps.  To follow the steps I used, CLICK HERE.

By minimizing this equation, we can find the cheapest route through our rectangle model, which is \$3644.83.  To follow the steps I used, CLICK HERE.

Other Routes for the Rectangular Model

The southern route around the wetland is the total distance around the wetland.

This is 970 ft + 190 ft + 400 ft, which equals 1560 ft.  We multiply that by 2.7, which is the total cost per foot on normal terrain.

970 + 190 + 400 = 1560

1560(2.7) = \$4212.00

The total cost of this route is \$4212.00, and since this is more expensive than our minimum cost for the rectangular function, this is not the route we want.  The northern route is more expensive because it was given that the northern route was longer than the southern route, thus is would cost more.

A direct route through the wetland area can be found by using our rectangle cost function just by using zero for x, since there is no distance traveled on normal terrain.

This is also more expensive than our minimum cost.

The cheapest way through the wetland alone is a straight line from A to B because the shortest distance between two points is a straight line.  Thus, any routes through the wetland alone with one or two elbow joints will cover more wetland and therefore be more expensive.

Other possible routes might be as follows:

See if you can find a cheaper price.

Trapezoidal Model

Since we now have the cheapest route thru our rectangular model, we can begin to remodel the region to more closely resemble the actual wetland area.

By using a trapezoid as the model for the wetland, we get closer the shape of the wetland because the eastern side tends to indent toward the north-west.

My model has the following dimensions (which were found by measuring the map):

A.   The distance from A to the southern most point of the wetland.

B.    The distance from B north to the point created by the angled portion of the trapezoid on the eastern side.

C.   The width of the wetland.

D.   The distance from A north to the point created by the angled portion of the trapezoid on the western side.

E.    The distance from B to the southern most part of the wetland.

F.    The distance from the northern most point on the western side to the northern most point on the eastern side of the model.

Q. The measure of the northern angle in the trapezoid.

For the trapezoid model, the cost function is:

The variable, x, is the distance from the northern most point in the western side to the point created where the pipeline intersects the angled side.  This formula is deducted through many steps.  To follow the steps I used, CLICK HERE.

By minimizing this function we can find the minimum cost, which is \$3245.55.  To follow the steps I used, CLICK HERE.

This equation does basically every logical path.  However, any path in the trapezoid that does not emerge on the angled portion, in other words, x>1030, can be used in the rectangle model.

For example:

In conclusion, the cheapest route is the one found with the use of the trapezoid model.

The cost of this route is \$3245.55.