Study Cases

Application of Integrals~ Week 13

Logo

1 Case Study 1

Area Between Curves – Reservoir Cross-Section: A reservoir engineer wants to estimate the cross-sectional area of an oil reservoir bounded by two geological layers. The profiles of the upper and lower layers are modeled as:

\[ \begin{eqnarray*} y_1(x) &=& 0.02x^2 + 5 \\ y_2(x) &=& 0.4x + 2 \end{eqnarray*} \]

where \(x\) and \(y\) are measured in meters.

Tasks

  1. Identify the integration limits for the bounded region.
  2. Compute the area between the two curves using definite integrals.

2 Case Study 2

Oil Zone Area Above Oil–Water Contact (OWC): The geometry of an oil zone is modeled by:

\[ y = 20 - 0.01x^2 \]

The oil–water contact is located at:

\[ y = 5 \]

Tasks

  1. Compute the area of the oil zone.
  2. Adjust the area using a porosity of 22%.

3 Case Study 3

Drainage Area of a Production Well: The drainage area around a production well is approximated by:

\[ y = 12 - x^2 \]

Tasks

  1. Determine the drainage area using integration.
  2. Estimate the initial oil volume if each \(1,m^2\) yields 0.8 barrels.

4 Case Study 4

Reservoir Volume – Disk Method: The vertical cross-section of a reservoir is given by:

\[ y = 10 - 0.005x^2 \]

The region is rotated about the \(x\)-axis.

Tasks

  1. Identify the appropriate volume method.
  2. Compute the reservoir volume.

5 Case Study 5

Volume of an Oil Storage Tank: The tank profile is described by:

\[ y = 0.02x^2 \]

The curve is rotated about the \(y\)-axis from \(y = 0\) to \(y = 8\) meters.

Tasks

  1. Compute the tank volume.
  2. Determine the effective oil volume if filled to 85%.

6 Case Study 6

Reservoir Layer Volume – Washer Method: A reservoir layer is bounded by:

\[ \begin{eqnarray*} y_1(x) &=& 18 - 0.02x^2 \\ y_2(x) &=& 5 \end{eqnarray*} \]

The region is rotated about the \(x\)-axis.

Tasks

  1. Identify the inner and outer radii.
  2. Compute the layer volume.

7 Case Study 7

Reservoir Volume – Shell Method: The reservoir geometry is defined by:

\[ x = 0.5y^2 \]

The region is rotated about the \(y\)-axis from \(y = 0\) to \(y = 10\) meters.

Tasks

  1. Identify the shell radius and height.
  2. Compute the reservoir volume using the shell method.
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