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Log Analysis by Computer


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Manual vs. Computer Methods

Manual Methods are mainly graphical limited to 2 dimensions

e.g., vs. Rt

b

vs. NLS

3 dimensions in case of MID Plot

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Graphical vs. Algebraic Methods crossplots have restricted dimensions

restricted number of logs to input

restricted number of lithology components

anything on a graph can be representedin equation form

equations are not restricted in number

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Crossplotting & Algebraic Methods unknown mixture of 3 known minerals

solve using D N S & MID Plot or, algebraically,

L1 = L1f + Vm1L1m1 + Vm2L1m2 + Vm3 L1m3



1 = + Vm1 + Vm2 + Vm3

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Simultaneous Equations

for D N S,

b = f + Vm1 m1 + Vm2 m2 + Vm3 m3N

= N

f + Vm1 N m1 + Vm2 Nm2 + Vm3 N m3 t = t f + Vm1 t m1 + Vm2 t m2 + Vm3 t m31 = + Vm1 + Vm2 + Vm3

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Matrix Algebraic Methods

In matrix notation, L = C x V

1

1N

n

n

Vm =

= +

N

in=1

= fLimL n+imV nL

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Matrix Algebra Solution

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The Perfect Martini Problem

Simultaneous EquationsAlcohol: 47G + 18D + 16S = 35

Sugar: 0G + 3D + 14S = 3.4Unity: G + D + S = 1

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The Perfect Martini Problem

Matrix Algebra Format[ 47 18 16 ] [ G ] [ 35 ]

[ 0 3 14] [ D ] = [ 3.4 ][ 1 1 1] [ S ] [ 1 ]

C V = L

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Matrix Solution

V = C-1 L

[ G ] [ 0.60 ]V = [ D ] = [ 0.20 ]

[ S ] [ 0.20 ]

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Martini Solution

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MineralMix D-N

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MineralMix S-N

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MineralMix M - N

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Algebraic Unique Solution

assuming limestone and dolostone solving 3 simultaneous equations:

point porosity vol. LS vol. Dol.

A 0.18 -0.13 0.95

B -0.02 0.04 0.98

impossible solution

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Problems with Unique Solutions

(deterministic answers)

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Problems with Unique Solutions Answers from any one data set may differ

from those from another because: incorrect choice of minerals

unknown clay types

unknown clay distribution modes uncertain log response equation

residual hydrocarbon effects

log errorscalibration

---borehole effects

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Some Errors Must be Assumed

in the log readings

in the tool response equations

in interpretation transforms

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Log Errors inadequate calibrations

sloppy calibration techniques poor operating techniques

tool malfunctions

operator error environmental effects

borehole size and condition

bed thicknessinvasion effects

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Real Problems of PhysicsConditions Solution Approaches

1. under-determined guesstimistic2. uniquely-determined deterministic

3. over-determined stochastic

Only stochastic approach really makes use of the

capabilities of computers,

other than as drudgery reduction devices.

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Guesstimistic Solution

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Deterministic Solution

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Deterministic Approaches

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Over-Determined Condition

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Equation Solution Approaches

uniquely-determinedDeterministic

over-determined

Stochastic

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Stochastic Approaches

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Stochastic Logics

make allowance for errors

least squares error solution

require over-determination

forward modeling concept

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Stochastic Logics

add an error term in toolresponse equations

N

in=1

= fLimL n+imV nL +i

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Li

= Li f +VclLcl++ iVmnLimn

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Forward Modeling Concept

1. Measured Log Dataunknown volume fractions plus errors

2. Theoretical Log Response Equation

guesstimated volume fractions

( ) ), , , ,ith i n w xo iL f Vm Vcl S S f X= =

Li=

Li

f +Vcl

Lcl

++i

Vmn

Li

mn

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Error Statement

l

l

)i i iL f X =

( )1 , , ,b D n xof Vm Vcl S =

( )2 , , ,N N n xof Vm Vcl S =

, , ,i t R n w

R f Vm Vcl S =

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Forward

Modeling

Concept

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Error Minimization

Instruct computer . . .

Minimize !

least error solution

i

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Error Squared Minimization

least squares error minimization

( )2

2

1 1

!I I

i i i

i i

Minimize E L f X= =

= =

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Least Squares Error Minimization

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Constraints

Certain constraints might be added

such as

Vmn 0

0

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Previous Example

Error Minimization Solution

Point Vol.LS Vol.Dol. Vol. Anhy. Res. Error

A 0.201 0 0.203 0.596 3.28

B 0.030 0.042 0 0.928 1.47

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Weighted Least Squares

weighting of each log reading accordingto its reliability / uncertainty

Minimize !

weighted least squares error solution

i

i

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Total Weighted Error

or,

2 2 2

1 2

1 2

n

n

E

= + + +

iii

2

1

n

i

i i

E

=

=

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Error ( uncertainty) Bands

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Weighted Least Squares ErrorMinimization

Minimize ! ( )

2

21

I

i i

w

i i

L f XE=

=

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Weighted Least Squares Error

Minimization

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Real Problems of PhysicsConditions Solution Approaches

1. under-determined guesstimistic

2. uniquely-determined deterministic

3. over-determined stochastic

Only stochastic approach really makes use of

the capabilities of computers,

other than as drudgery reduction devices.

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Stochastic Software

ELAN

ULTRA

MULTIMIN

Etc.

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ULTRAoutput

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Allowing for

Feldspar &

Mica

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Effect of

Pyrite

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Residual

Error

Check

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Date post:	04-Jun-2018
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