Petroleum Engineering -Q40

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This is an individual assignment due Thursday at the start of class. Solve the pressures for the 1-D steady-state diffusivity equation where permeability varies in the x-direction. The initial pressure is 3000 psia, and the well pressure is held at 5000 psia at x=0 ft. The production well pressure at the outlet (x=300 ft) is fixed at the 500 psia.

You will need to use the Thomas algorithm to solve this problem that will be uploaded to Canvas.
a) Make a schematic of the grid block system and the imaginary grid blocks carefully labeling the grid block numbers. The imaginary grid blocks have the same permeability as their neighboring grid block. Show your matrix system and
equations for blocks 1, 2, 3, and N, where N is the number of grid blocks.
b) Solve the pressure diffusivity equation using a central finite difference for the case when there are 1000 grid blocks. The permeability varies according to k =10exp(−x /100) as determined from core analysis. Plot the solved pressures as a function of distance (a profile plot). Use the harmonic mean for the permeability. That is, ( ) 1/2 1 1 2 / i i i i i k kk k k ± ± ± = + . We will show later why this average should be used for permeability at grid-block boundaries.

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c) Comment on the pressures that you calculate above. What is the slope in the pressure at each location equal to (this is the pressure gradient)? Can you explain this result using Darcy’s law where the rate is constant with time (steady-state after all)?
d) Change your calculations for 10 and 100 grid blocks and plot the result on the same graph. Comment on the differences.
e) Post your code used in the above solutions online at the Canvas website.
Product Code- Petroleum Engineering -Q40
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