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Matlab & Mathematica

Solve the following initial/boundary value problem using your data set values. The reservoir length is 400 ft. The initial condition is p(x, 0)  1000 psi

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

Numerical & Computational Methods in Petroleum Engineering Coursework: Description

 

This coursework assessment is 40% of the total Numerical & Computational Methods in Petroleum Engineering (ENG513) assessment. Students should complete this coursework and submit one PDF Report. There are 100 marks available in this coursework.

The coursework involves four individual questions in which the student will require to write a Matlab Code in order to answer the questions. The report must include the Matlab Code as well. All the relevant material and tutorials to complete the coursework are given in the weekly lectures.

At the beginning of the coursework, each student will be given a unique set of parameter values. These parameters include permeability, porosity, fluid viscosity and total compressibility. These values will be required to answer questions 1, 2 and 3.

Please be noted that you have to program your code in Matlab using your unique values of permeability, porosity, viscosity and compressibility then answer the coursework questions based on the results obtained from Matlab software.

 Important Notes:

If the followings are not met in the submitted report, mark deductions will be applied in the relevant questions given in this coursework.

 

  • All plots should include axis title and legends if there is more than one curve in the

  • Axis title should clearly indicate the parameter name and its

  • If plot includes more than one curve, each curve should have a different It is also allowed to use different colour or marker options to edit each curve. All plots should have a white background.

  • The report should not be more than 1500 words. Only the pages equivalent to first 1500 words will be marked.

You need to provide the Matlab Code in your PDF report for all parts of the coursework.

Question 1: (35 Marks)

Given the diffusivity equation in one dimension as:

¶2 p =      fmc     ¶p

              

¶x2           0.00633k ¶t

 Solve the following initial/boundary value problem using your data set values. The reservoir length is 400 ft.

The   initial   condition   is p(x, 0) = 1000 psi.   

The   boundary   conditions   arep(0,t) = 0 psiandp(400,t) = 1000 psi . Model parameters are Dx = 100 ft and Dt = 5days .

  1. Run simulation for 80 days and show pressure profile for t=5, 10, 20, 30 and 80 days in a graph. Do you observe the end of unsteady-state flow regime in the reservoir? If yes, at what time approximately? [10 Marks]

  2. Run a sensitivity analysis on permeability and porosity and discuss the differences observed in pressure profile at t = 30days . Provide a graph for each parameter. [7 Marks]

  3. Show a comparison of the pressure profile at t = 30days

for different time step sizes of Dt5, 2.5,1.25 days and discuss the observed trend in your plot. [8 Marks]

  1. Assume the well is injecting water at node 4, Q= 1 BBL/DAY. Run simulation for 80 days and show your results using a graph for t=5, 10, 20, 30 and 80 What can you observe about pressures in the reservoir? [10 Marks]

 

¶2 p +¶x2Q =     fmc      ¶p0.00633k ¶t

 

Question 2: (30 Marks)

The equation for single-phase, slightly compressible flow through porous media after discretizing using a forward difference is given below.

 

n+1 =    næ a Bo Dt ö          æ a Bo Dt 

pi     pi+ ç  c    l           ÷qlsc  + ç  c    l           ÷ ´ ëT pi+1 - (T+ T ) pi+ T pi-1 ûæ        A k    önè Vbfcl øiè Vbfcl øi

 solve for pressure distribution in the reservoir

è        l    l           ø

The initial reservoir pressure is 6000 psia. The rock and fluid properties for this problem can be used from your

data set while

Dx = 1000  Dy = 1000 ft , Dz = 75 ft , Bl= 1RBSTBand Bo= 1RBSTB. The total length reservoir block is 5000 ft. consider production from this reservoir at fourth grid point of

qsc = -150STB / D . Vb = DxDyDz , Ax  = DyDz .

The boundary conditions have been given as

dp = 0 for both ends of the reservoir. Plot the pressure profile

dx

for each following time step and discuss the results. [20 Marks: Code and Results; 10 Marks: Discussion]

  1. Use time step size of Dt = 10days

  2. Use time step size of Dt = 15days

  3. Use time step size of

Dt = 30days

Question 3: (20 Marks)

Given similar information in question 2, consider the following changes and solve as appropriate.

The boundary conditions have been given as

P = 6000 psi for the beginning of the reservoir and a boundary

condition ofdp = 0dxat the end of the reservoir. The well is producing at grid block 7 and 8 withqsc = -75STB / Dfor both grids.

Dx = 500 ft

Use time step size of Dt = 10days . Provide the pressure changes plot versus time for each gridblock and explain the erratic behaviour of your results and suggest how to improve the results. [10 Marks: Code and Results; 10 Marks: Discussion]

 Question 4: (15 Marks)

One dimension unsteady PDE in [0,p ]

 

is given as:¶f =¶t¶2f ¶x2, subject to the following boundary and initial conditions:

Boundary conditions:f (t, 0) = e-t ,f (t,p ) = -e-t

Initial condition: f (0, x) = cos x

.  Consider the number of nodes to be 21 with  Dx = p20

. Given that themaximum time step based on stability criterion is Dtmax =

Dt = 0.8Dtmax(Dx)2 2. Plot the solution for 3

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