HW1

Well Test Analysis  Assignment # 1  Due date: Friday, February 21st     Problem 1) Consider a core of length L, cross-sectional area A, and porosity φ is saturated with an incompressible fluid of viscosity μ at an initial pressure pi. While the outlet pressure is kept at pi, the inlet pressure is exposed to the following pressures.

Carefully draw schematic profiles of pressure in the core as a function of space and time.  

Problem 2) Write down the mathematical expression for hydraulic diffusivity. What is the physical significance of this term?   

Problem 3)  Consider the 1-D Radial Steady State Flow. The hydraulic diffusivity equation can be derived as: 

 

Mathematical model Steady State, Radial Flow •Choosing an appropriate element •Governing equation Mass balance

Input  Output  0

  A v     A v 

r  r

r

0

Darcy’s law

v

k p  r

Equation of state

   b expc ( p  pb ) 

 k p   k p     A      A  0  r  r   r  r r    k dp  k dp  d  k dp     A       A  r  ....     A dr dr dr dr  r      r r  r 

k 1   p  r  r r  r

   0   

k 1    p r  r  r  r

   p     r r  r 

  1  p  p    p r   r r   p r  r  r  r

  p   1   p  p   r     r  0 r  p r  r  r  r  

    0  

    0  

c

1  1 V   p V p

Negligible

  p  p  cr     r  r  r  r 2

   0 

Governing equation

  p r r  r

   0 

or

2 p r 2



1 p 0 r r  

 

a. Write down the initial and boundary conditions for this equation and solve it to obtain an expression for pressure distribution for Radial Steady State Flow. b. Obtain an expression for the volumetric flow rate, q. c. Obtain an expression for the volumetric flux (Darcy velocity), u.

Problem 4)  Given : k  100md

  0.2

ct  2  10 5 psi 1

  0.5cp

q  350 STB/D

 

(a) Find the time required to run a flow test on an exploratory well for sufficiently long period to ensure that the well will drain a cylinder of more than 1,000 ft radius. (b) What will be the time required if the flow rate is increased to 500 STB/D?

Problem 5)  Given : The well is producing only oil q  20 STB / D

  0.72 cp

rw  0.5 ft

k  0.10 md

Bo  1.475 RB / STB

ct  1.5  10 5 psi 1

h  150 ft

pi  3000 psi

  0.23

re  3000 ft

s0

Calculate : (a) p  ? at r  1 ft and t  3hrs (b) p  ? at r  10 ft and t  3hrs (c) p  ? at r  100 ft and t  3hrs