unit5_curve fitting.pdf

Ho Chi Minh City University of Technology Faculty of Geology & Petroleum Engineering

Modeling & Simulation Division

Presenter: Dr. Do Quang Khanh Email: [email protected] Website: www.hcmut.edu.vn

CURVE FITTING (REGRESSION)

Presenter

Presentation Notes

Drillstring hay drill string, drillcollar hay drill collar? Ở các tiêu đề dùng plural hay singular. Hình nền slide đầu tiên. Hình substitutes

HCMUT

Least Squares

Dr. Do Quang Khanh 2

HCMUT

Least Squares

Sum of Squares as a measure of “how good the fit is”. (Other possible measures: Sum of Abs Deviations)

Other requirements: Smoothness, Least number of parameters, Extrapolating power, etc.

Two basic cases Model is based on “first principles” Model is just a convenient vehicle


HCMUT

Least Squares and its Num. Aspects

Two parameters or more to find? If two parameters Is it in straight line form?

• “Linear regression” Or can we transform it into straight line form?

• “Pseudo-linear regression:” Transformation to straight-line form

More than two parameters (linear):

Generalized least squares Nonlinear least squares: Gauss-Newton-

Marquardt Dr. Do Quang Khanh 4

HCMUT

)]([)]([)]([ 233

222

211 bmxybmxybmxyQMinimize +−++−++−=

)]([2)]([2)]([20 332211 bmxybmxybmxybQ

+−−+−−+−−=∂∂

=

Fitting the straight line (y = mx + b) to three points; Degrees of freedom: 1 );(

);();(

33

22

11

yxyxyx

)]([2)]([2)]([20 333222111 bmxyxbmxyxbmxyxmQ

+−−+−−+−−=∂∂

=

Select m and b to minimize the Objective Function

Two equations, two unknowns: m and b

Objective function:


HCMUT

3)()( 321321 xxxmyyyb ++−++

=

2321

23

22

21

321321332211

)()(3))(()(3

xxxxxxyyyxxxyxyxyxm

++−++++++−++

=

33221132123

22

21 )()( yxyxyxxxxbxxxm ++=+++++

321321 3)( yyybxxxm ++=+++

Multiply by 3 and by (x1+x2+x3), subtract, get m, then get b


HCMUT

Formulas for m & b, & programming

HW: Programming using arrays

2

11

2

111

−

−

=

∑∑

∑∑∑

==

===

n

ii

n

ii

n

ii

n

ii

n

iii

xxn

yxyxnm

n

xmyb

n

ii

n

ii

−

=∑∑== 11

How to improve the program efficiency for very large values of n? Think about calculating the same something several times!


HCMUT

Transformations to Straight Line


HCMUT

Means to Achieve SL Form

Only two unknown parameters, m and b

Number of points should be at least 3 (Degrees of freedom at least 1) Needs ingenuity


HCMUT

Examples

Flow-After-Flow Test of a Gas Well

Material Balance of Volumetric Dry-Gas Reservoir

n

wf

pp

qq

−=

2

max 1

−+=

2

max 1lnlnlnp

pnqq wf

pi

i

i

i GGzp

zp

zp

−=

−=

GG

zzpp p

i

i 1


HCMUT

Flow-After-Flow Test of a Gas Well: the Cast

Real World Straight Line World

−+=

2

max 1logloglogp

pnqq wf

nq

pp

q

wf

max

2

log

1log

log

−

mbxy

xmby ×+=

n

wf

pp

qq

−=

2

max 1

intercept

slope

Independent variable

dependent variable


HCMUT

Mat. Balance of Vol. Dry-Gas Res.: The Cast

Real World Straight Line World

( )ii

ii

p

Gzpzp

Gzp

//

/

mbxy

pi

i

i

i GGz

pzp

zp

−= xmby ×+=

Measured

From measured p; z is a known function of p

Measured

There are other forms


HCMUT

Programming services

• Add Trendline • Options: show equation • Select model • (Does not help you unless you

understand…) • For nonlinear least squares: Solver


HCMUT

Nonlinear least-squares

Minimize sum of squared deviation (residual)

Use Excel’s “Solver” Example: Hubbard curve (Egypt)

( )∑ − 2)(

:

ii xfy

functionObjective


HCMUT

Hubbert Model

Hubbert curve: Derivative of the logistics curve Production rate (q) vs. time

[ ]2)(

)(

)(

2

2

2

1

:

,

1

:Pr

:var

,

0/;

o

o

o

oo

ttao

ttao

o

oo

ttao

t

t

Q

Q

eNeaNQ

dtdQq

timetorespectwithQtingDifferntiaQ

QQNwhere

eNQQ

oductioneCummulativ

dta

QQQ

dQ

iablesSeparating

QQ

aaQdtdQ

QabThen

dtdQQQWhen

bQaQdtdQ

−−

−−

∞

∞

−−∞

∞

∞

∞

∞

+==

−=

+=

=

−

−+=

−=

==

+=

∫∫

• Logistics curve -Cumulative production (Q) Vs. time


HCMUT

Minimize objective function

Change variables

HCMUT

Weighting Factors

Account for the importance of each data point by using a weighting factor, wi

( )∑ − 2)(

:

iii xfyw

functionObjective


HCMUT

Straight-line: formation volume factor model 1 Given: pb = 2012 psi, bubble point pressure Data (observed): P, psi Bo, resBBL/STB 1500 1.262 1600 1.279 1800 1.298 Determine the parameters of the nonlinear model describing the Bo: What is the best estimate of the Bo at the bubble point?


ASSIGNMENTS, TEST PROBLEMS

HCMUT

Straight-line: formation volume factor model 2 Consider the following model of Formation Volume Factor, Bo as a function of pressure, p: Bo = aeb(p−pb) where Bo is in resBBL/STB, p in psi, and pb is the known bubble point press.(pb = 3007 psi). The model paras. a & b are to be find. The following lab. data are available: P, psi Bo, resBBL/STB 500 1.070 1500 1.175 2500 1.301

Determine the Formation Volume Factor at the bubble point (pb) using the above model.



HCMUT

Straight-line: Gas in place Production and static (field) pressure data for a gas field is given below. (Craft and Hawkins)



HCMUT

Straight-line: Flow-After-Flow Test (IPR) A frequently used IPR equation: Find the Absolute Open Flow Potential. Hint: fill out the following table first.



HCMUT

Nonlinear least squares: oil viscosity as a function of pressure and temperature

Consider the following model of oil viscosity (μo) for a certain field as a function of pore pressure, p and layer temperature T: The model parameters a, b & c are to be determined by the method of nonlinear least squares using a general purpose minimization program (e.g, Solver). The available data are:

Program to calculate the obj. function to be minimized. Dr. Do Quang Khanh 24


unit5_curve fitting.pdf

Documents