B.Tech Project Presentation on Wax Deposition

Wax Deposition

1. Indian Institute of Technology (Indian School of Mines) “ Solution of Diffusivity Equation for flow under 1D Linear and Bounded Hollow Cylindrical Case” Presented by : Aamir Iqbal (19JE0005) Department of Petroleum Engineering

2. Content Linear 1D Model Constant Pressure at x=0 and No Flow at Boundary Constant Production Rate at x=0 and No Flow at Boundary Solution using Numerical Method Comparison of Plots Hollow Cylinder Case

3. Constant Pressure at x=0 and No Flow at Boundary Solving this PDE involved splitting P( x,t ) in sum of two function one satisfying “Non Homogenous Dirichlet BC” due to constant pressure at x=0 and other having homogeneous BC. Then applied Separation of variable.

4. Constant Production Rate at x=0 and No Flow at Boundary Solving this PDE involved splitting P( x,t ) in sum of two function one satisfying “Non Homogenous Neumann BC” due to constant production rate at x=0 and other having homogeneous BC. Then applied Separation of variable, which leads to Non-Homogeneous PDE. Further solved using method of eigenfunction expansion

5. Solution using Numerical Method

6. Comparison of Plots

7. Hollow Cylinder Case Solution: According to Cauchy’s Theorem P D is equal to 2(pi) i times the sum of the residues at the poles of its integrand. First, value of at which poles will exist will de determined. In this case, =0 and = where are the roots of identified as poles. Next step will determining order of poles then calculation of corresponding residue function. = , are simple pole of order 1, while I figured out the nature of pole at =0. Calculation of Residue Function directly leads to the required function.   From Inversion Theorem

8. Thank You