Area Under a Curve, 3D, with known end points and curve radius

In summary, the conversation is about finding a method for determining the area below a curve given its endpoints and curvature in cartesian space. The conversation also discusses using a wellbore trajectory and directional survey to find the net area above and below a certain depth plane. The suggested method involves projecting the curve onto a linear equation and integrating it.
  • #1
geetar_king
26
0
I am trying to find out a method of determining the area below a curve.

The end points of the curve are known in cartesian space, and the curvature of the curve is known.

A diagram of the curve is here, shown in the images belowthis webpage

Minimum-Curvature-Method.jpg


Minimum-Curvature-Method-2.jpg


ß must be in radians

Where;
MD = Measured Depth between surveys in ft
I1 = Inclination (angle) of upper survey in degrees
I2 = Inclination (angle) of lower in degrees
Az1= Azimuth direction of upper survey
Az2 = Azimuth direction of lower survey
RF = Ratio Factor
ß is the dog leg angle.


I'm trying to find the area between the curve and a line projected downwards onto the bottom plane.

I have a wellbore trajectory which gives x,y,z, coordinates (northing,easting,vertical depth)and the angle DL, and I am trying to find the net area above and below a certain depth plane by using the wells directional survey.

Any suggestions would be appreciated, thanks!
 
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  • #2
The plane along which you project is given by a linear equation. It can be written in a form ##z=\ldots ## or another variable. This substituted into the equation of the curve gives a new equation of the curve with only two coordinates. Easiest would be to write the curve as ##y=f(x)## and integrate it.
 

1. What is the formula for calculating the area under a curve?

The formula for calculating the area under a curve is the definite integral of the curve's function from the starting point to the ending point. This can be represented as ∫f(x)dx, where f(x) is the function of the curve and dx represents the change in x.

2. How do you calculate the area under a 3D curve?

To calculate the area under a 3D curve, you need to first find the surface area of the curve. This can be done by dividing the curve into smaller sections and using the formula for the surface area of a shape, such as a rectangle or triangle, for each section. Then, add up all the surface areas to find the total area under the curve.

3. What are the known end points and curve radius?

The known end points refer to the starting and ending points of the curve, which are often denoted as a and b in the formula for calculating the area under a curve. The curve radius refers to the radius of the curve at any given point, which can be found using the formula for the derivative of a curve at that point.

4. How does the curve radius affect the area under the curve?

The curve radius can affect the area under the curve in a few different ways. If the curve radius is small, the curve will be steep and the area under the curve will be small. On the other hand, if the curve radius is large, the curve will be more gradual and the area under the curve will be larger. Additionally, if the curve radius changes throughout the curve, this can also affect the overall area under the curve.

5. Can the area under a curve be negative?

No, the area under a curve cannot be negative. The area under a curve represents the total amount of space between the curve and the x-axis, and this space cannot be negative. However, if the curve dips below the x-axis, the area under that section will be considered negative and will offset any positive area above the x-axis.

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