Unit 10 PTRL1010 Notes
Shale
General stuff in slides:
- Shale Gas and Shale Oil resource plays are tight reservoirs that require stimulation(fraccing)
to enable production
at economic rates
- Other tight
reservoirs which
require stimulation
may include
sa
LECTURE 32
SYSTEMS OF DIFFERENTIAL EQUATIONS
Systems of dierential equations y = Ay may be easily solved by implementing the eigenvalues
and eigenvectors of A.
If A is a 3 3 matrix with linearly independent eigenvectors v1 , v2 and v3 , and associated
eig
MATH2019 PROBLEM CLASS
EXAMPLES 7
MATRICES
1991
&
1994
1.
a) Find the eigenvalues and the corresponding eigenvectors of matrix
322
A= 2 2 0 .
204
b) Find an orthogonal matrix P such that
D = P 1 AP
is a diagonal matrix and write down the matrix D.
c) Usin
LECTURE 30
SPECIAL MATRICES
A matrix A is said to be symmetric in A = AT .
The eigenvectors from dierent eigenvalues of a symmetric matrix are mutually perpendicular.
A matrix Q is said to be orthogonal if QT Q = I or equivalently Q1 = QT .
The columns of
LECTURE 26
FORCED OSCILLATIONS AND RESONANCE
When simple periodic forcing is added to the mechanical or electrical system studied earlier,
we have to solve an equation like
my + cy + ky = F0 sin wt
(1)
where, as before, m > 0, c > 0, k > 0.
This models a
MATH2019 PROBLEM CLASS
EXAMPLES 6
ORDINARY DIFFERENTIAL EQUATIONS
1998 1.
a) Find the general solution of the dierential equation
4y
dy
=
dx
x(y 3)
satisfying the initial condition y (1) = 1.
1
b) Use the substitution y = z 3 where y and z are both functi
LECTURE 24
HOMOGENEOUS SECOND ORDER DIFFERENTIAL EQUATIONS
To solve the homogeneous second order constant coecient dierential equation
ay + by + cy = 0
rst form the auxiliary (also called characteristic) equation
a2 + b + c = 0.
The auxiliary equation is
LECTURE 25
NON-HOMOGENEOUS SECOND ORDER DIFFERENTIAL EQUATIONS
To solve the non-homogeneous second order constant coecient dierential equation
ay + by + cy = r(x)
we rst solve the homogeneous problem ay + by + cy = 0 to obtain a homogenous solution yh .
W
LECTURE 20
FURTHER VOLUMES
Recall that for a region in the x y plane and a surface z = f (x, y ) in R3 the double integral
f (x, y )dy dx.
evaluates the volume of the solid above and below z = f (x, y ).
If the solid is pinned between two surfaces z = f1
MATH2019 PROBLEM CLASS
EXAMPLES 8
LAPLACE TRANSFORMS
2001 1.
a) The Laplace Transform, F (s), of f (t) is denoted and dened by
F (s) = L(f (t) =
f (t)est dt .
0
i) Prove directly from the above denition that
L(tf (t) = F (s) .
ii) Find the Laplace transfo
LECTURE 1
PARTIAL DIFFERENTIATION
Suppose z = f (x, y ). Dene
f
)
= lim f (x+x,yxf (x,y)
x0
x
f
)
= lim f (x,y+yyf (x,y)
y 0
y
Notation
f
= fx = zx ,
x
f
= fy = zy
y
Hello and welcome to Math2019.
We will be meeting for 5 lectures per week and you will al
Methods of petroleum exploration:
Wild cat well: first well we drill in a new area -slide 5
Prospects: they have hydrocarbons which may be economically viable to produce
- d/p= drilling and production
denser the material the faster it moves through- seism
Unit 7 Ptrl 1010 Notes
First Well: Drakes well at
Titusville: 20 m deep, and 20 bbls
per day
Simple diagram of a drilling rig
and its basic operation SLIDE 4
Types of drill rigs
Onshore
o Land Rigs
Offshore
o Drillship
o Semi-Submersible
o Jack up
Onshore
Unit 6 PTRL 1010 Notes
Methods of Petroleum Exploration
Initial aim of petroleum exploration is to establish a viable petroleum system- this
involves the essential elements (source rock, migration route, reservoir rock, seal rock
and trap) and processes (
Chapter 4 PTRL 1010 Notes
Traps
A trap is a structural or stratigraphic feature where impermeable seals occur above and around
reservoir rocks so as to stop upward migration of (continuous phase) oil and gas, resulting in
their accumulation.
Form when buo
Unit 5 PTRL 1010 Notes
Reservoir Description
Firstly, we want to estimate the amount of hydrocarbons in place in the field.
This is done by:
o Original Oil in Place
o Original Gas in Place
OOIP is calculated by:
o trap Volume x Net/Gross (N/G) x Porosity
Unit 2 PTRL 1010 Notes
Planet Earth
Planet Earth is an oblate spheroid (flattened at both polar ends) due to the centrifugal force
resulting from axis rotations
In order to understand the internal structure of the earth, three principles of geophysics can
Unit 1 PTRL 1010 Notes
Introduction
85% of world energy consumption satisfied by coal oil and natural gas, 6% by nuclear energy and
9% by renewable sources
Until 2030, fossil fuels will meet 95% of worlds energy needs
Fossil Fuels are coal gas and oil
The
Unit 3 PTRL 1010 Notes
The origin of Oil and Gas generation and migration
Theories of Oil and Gas generation and migration
Origin and accumulation of petroleum are two completely different things which are completely
mutually exclusive.
The location and f
PTRL1010-LEC 3
Slide 9: catagenis is where oil is more likely and in the lower end there will be more
gas(metagenisis)
Mainly oil in the catagenisis region
Slide 10- there is a lot of overburden rock creating a lot of pressure
Slide 11- spill point- not e
MATH2019 PROBLEM CLASS
EXAMPLES 5
DOUBLE INTEGRALS
APPLICATIONS AND POLAR COORDINATES
1997 1. Evaluate the following integral by changing to polar coordinates:
1
I=
1998 2.
x= 2 / 2
1 x2
dy dx .
y =0
a) Dene
1 x
1
(4 x 2y ) dy dx .
V=
0
0
i) Evalulate V .
LECTURE 10
VECTOR AND SCALAR FIELDS
= grad =
i+
j+
k.
x
y
z
F = div F =
F1 F2 F3
+
+
.
x
y
z
i
F = curl F =
j
y
z
div : vector to scalar
k
x
grad : scalar to vector
curl : vector to vector
F1 F2 F3
Where the vector dierential operator is given by
=
i+
LECTURE 38
DES VIA LAPLACE TRANSFORMS
LAPLACE TRANSFORMS
est f (t)dt = F (s)
Lcfw_f (t) =
0
f (t)
1
F (s)
1/s
t
1/s2
tm
m!/sm+1
t , ( > 1)
( + 1)/s +1
eat
1/(s + a)
sin bt
b/(s2 + b2 )
cos bt
s/(s2 + b2 )
sinh bt
b/(s2 b2 )
cosh bt
s/(s2 b2 )
sin bt bt co
LECTURE 37
PARTIAL FRACTIONS
LAPLACE TRANSFORMS
est f (t)dt = F (s)
Lcfw_f (t) =
0
f (t)
1
F (s)
1/s
t
1/s2
tm
m!/sm+1
t , ( > 1)
( + 1)/s +1
eat
1/(s + a)
sin bt
b/(s2 + b2 )
cos bt
s/(s2 + b2 )
sinh bt
b/(s2 b2 )
cosh bt
s/(s2 b2 )
sin bt bt cos bt
2b3
LECTURE 36
THE SHIFTING THEOREMS
LAPLACE TRANSFORMS
est f (t)dt = F (s)
Lcfw_f (t) =
0
f (t)
1
F ( s)
1/s
t
1/s2
tm
m!/sm+1
t , ( > 1)
( + 1)/s +1
eat
1/(s + a)
sin bt
b/(s2 + b2 )
cos bt
s/(s2 + b2 )
sinh bt
b/(s2 b2 )
cosh bt
s/(s2 b2 )
sin bt bt cos bt
LECTURE 28
REVISION OF MATRIX THEORY
If A is an m n matrix and B is a p q matrix then AB exists i n=p and the product is
m q.
The identity matrix I serves as the 1 of matrix theory.
The transpose of A (denoted by AT ) has the columns of A as its rows.
LECTURE 29
EIGENVALUES AND EIGENVECTORS
Given a square matrix A, a non-zero vector v is said to be an eigenvector of A if Av = v for
some R. The number is referred to as the associated eigenvalue of A.
We rst nd eigenvalues through the characteristic equa
LECTURE 19
DENSITY, MASS AND CENTRE OF MASS
Consider a lamina of varying composition (for example a thin sheet of metal) in
the x y plane with density (x, y ) at the point (x, y ). Then
Mass() = M =
(x, y )dA.
If the centre of mass of is (, y ) then
x
x=
LECTURE 13
LINE INTEGRALS
Line integrals are used to calculate the work done in moving a particle P from A
to B along a path C in a force eld F.
F dr =
(F1 dx + F2 dy + F3 dz )
C
C
In general, this integral depends not only on F but also on the path C we
LECTURE 12
VECTOR CALCULUS
Before having a look at how the theories of calculus may be further applied to vectors,
we have some nal applications of grad, div and curl.
Example 1 Find the tangent plane and the normal line to the surface
x4 + y 4 + 3z 4 = 2