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7/21/2019 welltest Deconvolution http://slidepdf.com/reader/full/welltest-deconvolution 1/20 1 12/09/05 19:25 Symposium for Alain Gringarten 1 Deconvolution in Well Test Analysis Thomas von Schroeter 12/09/05 19:25 Symposium for Alain Gringarten 2 Alain’s early laurels Type curve analysis 1960’s Derivative analysis (1983); WTA software Simultaneous downhole measurements; PC’s 1980’s Green’s functions (1971) Electronic pressure gauges 1970’s Straight line analysis Mechanical pressure gauges 1950’s Methods of Analysis Technology Time
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12/09/05 19:25 Symposium for Alain Gringarten 1

Deconvolution in Well Test Analysis

Thomas von Schroeter

12/09/05 19:25 Symposium for Alain Gringarten 2

Alain’s early laurels

Type curve analysis1960’s

Derivative analysis (1983);WTA software

Simultaneous downholemeasurements; PC’s

1980’s

Green’s functions (1971)Electronic pressure gauges1970’s

Straight line analysisMechanical pressure gauges1950’s

Methods of AnalysisTechnologyTime

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12/09/05 19:25 Symposium for Alain Gringarten 4

Green’s functions (Alain’s PhD, 1971)

• Fields: ∆p pressure drop, source strength, n unit normal

• Constants: Porosity φ , compressibility c , diffusivity η = k  /(φµ c )

• Green’s function G (t – t’ ,x ,x’ ) ≡ pressure drop at (t,x ) due to aninstantaneous point source of unit strength going off at (t’,x’ )

• G can (but need not) be adapted to the shape of the reservoir

• Origin in the theory of heat conduction: Minnigerode (PhD thesis 1862)

termsBoundary

0

partfieldFree

0

dd 

d)()(d 

1)(

’ x 

’ n 

G p 

’ n 

p G ’ t 

’ x ’ x ,x ,’ t t G ’ x ,’ t ’ t c 

x ,t p 

∫ ∫ 

∫ ∫ 

∂∆−

∆∂−

−φ

=∆

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Uniform sources

’ t x ,’ t t S ’ t c x ,t p  d)()( 

1)(

t

0

−φ=∆ ∫ 

• Then the free field part of the pressure drop is the convolution in time ofthe source strength with the source function:

• Assumption: The source strength (t’,x’ ) is independent of x’ ∈ W 

’ x ’ x ,x ,t G t,x S 

d)()( ∫ =

• Define a source function 

12/09/05 19:25 Symposium for Alain Gringarten 6

Product rule for source functions

• Product rule: The source function for a Cartesian product W 1×W 2 is theproduct of source functions for W 1 and W 2.

= xW W 1 W 2

   

  

    −−

π=

’ x x 

’ x ,x ,t G 4

exp)(4

1)(

2

2 / 3  ( ) 2

22

12

21 x x x ,x    +=

• Reason: The simple form of the free field Green’s function,

• Leads to a catalogue of analytic solutions for simple source geometries

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Superposition in time

• Clear conceptual distinction between:

 – Effects of the production schedule (Q ) and

 – Reservoir behaviour (characterized by its impulse response g )

 – No such thing as “buildup/drawdown behaviour”! (An artefact ofcertain signal processing techniques.)

• Assumptions:

 – Uniform sources: (t’,x’ ) is independent of x’ ∈ W 

 – No reservoir boundaries, or:

 – Impermeable boundaries, and Green’s function adapted to theboundary such that ∂G/ ∂n x’ = 0 for x’ ∈ B (no loss of generality)

’ t x ,’ t t g ’ t Q x ,t p 

d)()()(

0

−=∆ ∫  W ’ x ’ x ,x ,t G 

c x ,t g 

 )(

1)( ∈φ

=

• Superposition principle (Duhamel 1833)

12/09/05 19:25 Symposium for Alain Gringarten 8

Derivative type curves

• Advantage: Radial flow regime shows up as horizontal stabilization.

• Convention: Classify reservoirs by their pressure response to constant

production of Q ≡ 1 rate unit.

p x ,t g ’ t x ,t’ g t p  U 

G U d

d )( d)()(

0

=⇒=

∫ 

• Normalized pressure drop at the gauge (x = x G ):

t x ,t g t t 

p G 

U   vs )( lnd

d=

• Diagnostic plot: Log-log plot of p U and its logarithmic time derivative

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Derivative analysis (1983)

Source: Bourdet, Ayoub & Pirard, SPEFE June 1989, p. 296

Derivative

Pressure drop

Estimate

12/09/05 19:25 Symposium for Alain Gringarten 10

Well test analysis

• Procedure:

1. Estimate p U (t ) and its derivative dp U  /d ln t from the data

2. Diagnostic plot: p U and dp U  /d ln t vs time

3. Compare with a catalogue of type curves

4. Match model parameters to data by regression• Steps 2–4 are well understood:

 – A large library of analytic models exists

 – Regression on model parameters is now routinely performed by

WTA software

• Step 1 has long been underestimated in its complexity!

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t 0

p U 

• Analysis of a 1st drawdown: – Differentiate pressure signal

numerically wrto log of time

 – Divide by rate

• Analytically correct, butnumerically inaccurate! – Loss of information by cancellation of

leading digits

 – Result: Amplification of measurementerrors

 – Subsequent smoothing may hide thetrue scale of uncertainty and causefurther artefacts

0 b 

Derivative analysis, taken (too) literally…

12/09/05 19:25 Symposium for Alain Gringarten 12

… and with a vengeance!

b 0 c 

p U 

p U 

• Analysis of subsequent flowperiods:

 – Differentiate wrto Horner time /

superposition function

 – Divide by last rate change

 – Log-log plot against the elapsed time

• Not even analytically correct!

 – The data sample more than just theelapsed time interval!

 – Hence the true radius of investigation

is underestimated

 – Model bias: Horner time and

superposition function are based on

the assumption of radial flow

 – Plus: all the disadvantages of

numerical differentiation…

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Estimating the derivative without taking it

• Integrate to find the rate-normalized pressure drop p U 

• Hence, essentially a deconvolution problem!

g Q    ∆p 

{ } ’ t ’ t t g ’ t Q t g Q t p t 

)d()()()(0

−≡∗=∆ ∫ • Means:

•   ∆p and Q known (up to measurement errors), g unknown

)( lnd

dt g t t 

p U  =

• The desired derivative:

12/09/05 19:25 Symposium for Alain Gringarten 14

Deconvolution

• Deconvolution problems occur in many areas of science:

 – Tomography

 – Seismics

• Yet each deconvolution problem is different:

 – Statistical signals with zero average (e.g. seismics)

 – Signals characterized by trend plus noise (e.g. tomography) – Physical constraints on the solution space

• In well test analysis:

 – Problem first formulated by Hutchinson & Sikora (1959)

 – Two main categories:• Time domain approaches

• Spectral approaches

 – About 20 publications to date (see survey in SPEJ 9, 375)

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{ } ’ t ’ t t g ’ t Q t g Q t p t 

)d()()()(0

−≡∗=∆ ∫ • Start from superposition principle (SP):

Ingredients (1): Physical constraints

• Discretizing the integral and solving for the impulse response g cannot guarantee g > 0 (Hutchinson & Sikora 1959, others 80’s)

• Optimization with explicit constraints can only ensure g  ≥ 0 [Coats& al. 1964, Kuchuk & al., 1990’s]

• Our approach (2001/4): Use the encoding from the diagnostic plotln {t g (t )} = Z (τ) where   τ = ln t 

τττ−≡∆ ∫ ∞−d))(exp())exp(( )(

ln

Z t Q t p t 

• However, this sacrifices the linearity of SP:

12/09/05 19:25 Symposium for Alain Gringarten 16

Ingredients (2): Error models

Least Squares• The error model behind the

standard optimization approach:

ming  || ∆p  – g ∗ Q ||

• Implicit assumption: Onlypressure affected bymeasurement errors

• In reality, much more uncertaintyin the rate data!

g Q    ∆p  g Q    ∆p 

δ 

Total Least Squares

• Common in signal processing:

min δ, g || ∆p – g ∗(Q +δ) ||2 + ν||δ||2

• Better adapted to relative size oferrors

• Enables joint estimate of ratecorrection and response (datapermitting)

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Ingredients (3): Regularization

• With field data, an estimate based on nonlinear encoding + TLSerror measure alone is usually uninterpretable

• Regularization: Constrain

 – the sign of p U and its derivatives (Coats & al. 1964, Kuchuk & al. 1990)

 – the mean squared slope between nodes and the autocorrelationfunction (Baygün & al. 1997)

 – or add a penalty based on the mean squared curvature of the solutiongraph (vS & al. 2002/4).

• Advantage of using curvature:

 – Slopes carry information and should be preserved!

• Advantage of penalties over constraints:

 – Constrained optimization is much harder numerically!

12/09/05 19:25 Symposium for Alain Gringarten 18

• Data: p , q (as vectors)

• Estimate: y : linear parameters (initial pressure & rates)

z : nonlinear parameters (coeffs of deriv. interpolation)• G ( ): a matrix-valued function reflecting sampling and interpolation

• Regularization: constant matrix D & vector k such that ||Dz –k ||2 is ameasure of the total curvature of the response graph

• Weights: ν, λ (default choices & user intervention)

• A “separable nonlinear Least Squares problem” (Björck 1996)

• Efficient implementation: Variable Projection algorithm (Golub &Pereyra 1973)

curvaturematchratematchpressure

)()( min22

22

22 k z D q y y z G p z ,y E 

z ,y −λ+− ν+−=

The NTLS approach (vS & al. 2002,4)

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Simulated example

1 2 3 4 5 60.01

0.1

1

10

log10 t 

p U (t ), t g (t )

radial flow

CD=100

Skin S = 5

Sealing fault

d = 300 r w

bestinterpolation

longest period test duration

invisible to

conv. analysis

12/09/05 19:25 Symposium for Alain Gringarten 20

Rate simulation

50000 100000 150000 200000 t

1

2

3

4

5

q(t) unperturbed

+ 1 % error (RMS)

+ 10 % error (RMS)

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Pressure simulation

50000 100000 150000 200000 t

10

20

30

40

50

60

p(t)unperturbed

0.5% in ∆p 

5% in ∆p 

12/09/05 19:25 Symposium for Alain Gringarten 22

Typical Results

1 2 3 4 5 6 log10t0.01

0.05

0.1

0.51

5

10

t g(t) homogeneousstart

fault

unperturbed

0.5% in ∆p 

+ 1% in rates

+ 10% in rates

+ 5% in ∆p 

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Error analysis

• Linearize TLS residue about true reservoir model

• Assumptions:

 – Errors in p and q normally distributed

 – Zero mean, variances σ p 2 and σ q 

•   ⇒ analytic expressions for

 – Bias if λ > 0 (“stiffness”)

 – Covariance matrix ⇒ confidence intervals

 –   λ controls trade-off: bias vs variance

12/09/05 19:25 Symposium for Alain Gringarten 24

Confidence intervals, λ = 10-2 λdef

10 100 1000 10000 100000 1060.01

0.05

0.1

0.51

5

10

t g(t)

t

Error levels p,q 

0.5%   —

0.5% 1%

0.5% 10%

5% 10%

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Confidence intervals, λ = λdef

10 100 1000 10000 100000 1060.01

0.050.1

0.5

1

5

10

t g(t)

t

Error levels p,q 

0.5%   —

0.5% 1%

0.5% 10%

5% 10%

12/09/05 19:25 Symposium for Alain Gringarten 26

Confidence intervals, λ = 102 λdef

10 100 1000 10000 100000 1060.01

0.05

0.1

0.51

5

10

t g(t)

t

Error levels p,q 

0.5% —

0.5% 1%

0.5% 10%

5% 10%

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Work flow

Data Initial guess (y 0, z 0)

Compute default weights: νdef , λdef

Minimize error measure

Optimum rate / response (y , z )

Data honoured & response interpretable ?

Done

Adapt λ

12/09/05 19:25 Symposium for Alain Gringarten 28

Well

HWellV

Seismic faults 0 500 1000

meters

Baffle

Proposed water injection wells

N

Field example [AG, SPE 93988]

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DST Extended Well test

Pressure

Rate

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

-1 0 1 2 3 4

   P  r  e  s  s  u  r  e   (  p  s   i  a   )

Elapsed time (yrs)

0

10000

20000

30000

40000

   O   i   l   R  a   t  e   (   S   T   B   /   D   )

Well shut-in

Well test data

12/09/05 19:25 Symposium for Alain Gringarten 30

Pressure

Rate

3000

4000

5000

6000

7000

8000

9000

2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5

   P  r  e  s  s  u  r  e   (  p  s   i  a   )

Elapsed time (yrs)

0

10000

20000

30000

40000

   O   i   l   R

  a   t  e   (   S   T   B   /   D   )

FP 208

FP187

FP178

FP

112

FP124

FP118

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DST

DERIVATIVE 

PRESSURE 

10-2 10-1 1 10 102 103 104

Elapsed time (hrs)   R  a   t  e   N  o  r  m  a   l   i  s

  e   d   P  r  e  s  s  u  r  e   D  r  o  p  a  n   d   D  e  r   i  v  a   t   i  v  e   (  p

  s   i   )

104

103

102

10

1

FP 208FP 144

FP 178

FP 124

A

C

B

D FP 118

Main features

Unit slope

12/09/05 19:25 Symposium for Alain Gringarten 32Time from start of pressure measurements, hours

   F   l  o  w

  p  e  r   i  o   d   d  u  r  a   t   i  o  n ,

   h  o  u  r  s Minimum duration

for interpretation

FP 112

23.3 days

FP 187

3.2

months

FP 208

8.5 monthsFP 178

2.6 months

1 10 102 103 104

104

103

102

10

1

10-1

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   D  e  c  o

  n  v  o   l  v  e   d  n  o  r  m  a   l   i  z  e   d   d  e  r   i  v  a   t   i  v  e

FP 112 (3 weeks)

FP 118 (5 weeks)

FP 124 (7 weeks)FP 144 (8 weeks)

FP 178 (11 weeks)

FP 208 (37 weeks)

[112,144,178,187,208]

All production data

Elapsed time, hrs

FP 112 (3 weeks)

FP 208

FP 187FP 178

FP 144

FP 124

FP 118

FP 112

10-3 10-2 10-1 1 10 102 103 104 105 106

10

1

10-1

10-2

10-3

10-4

10-5

   D   E  C

  O   N   V  O

   L   V   E   D

    D   E   R

   I   V  A   T   I   V   E  S , 

    F   P   1  1   8   -

   2   0   8

Which FPs contain the unit slope?

?

12/09/05 19:25 Symposium for Alain Gringarten 34

   D  e  c  o  n  v  o   l  v

  e   d  n  o  r  m  a   l   i  z  e   d   d  e  r   i  v  a   t   i  v  e ALL PRODUCTION DATA

Elapsed time, hrs

FP 208FP 187FP 178FP 144FP 124FP 118FP 112

10-3 10-2 10-1 1 10 102 103 104 105 106

FINAL BUILD-UP

10

1

10-1

10-2

10-3

10-4

10-5

Comparison of estimates

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Where do we go from here?

12/09/05 19:25 Symposium for Alain Gringarten 36

Extension to multiple wells

δ 1

δ 2   ε 2

ε 1

g 11Q 1   ∆p 1

Q 2

g 12

g 21

g 22  ∆p 2

W 2

W 1

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Interference ?

W 2

W 1

?

ε 2

g 11Q 1   – p 1

δ 1

Q 2

g 12

g 21

g 22

δ 2

ε 1

 – p 2

– p 02

– p 01

12/09/05 19:25 Symposium for Alain Gringarten 38

Conclusions

• The need for deconvolution in WTA has long been recognized.

• But stable, efficient, and flexible algorithms have only recentlybeen developed.

• Similar ideas can be applied to a variety of long-standingchallenges in well test analysis, including the problem ofinterfering wells.

• The unifying aspect behind these ideas is the method of Green’sfunctions, which has proved immensely fruitful for well testanalysis.

• Alain and his PhD supervisor Henry Ramey had the vision tointroduce these methods into well test analysis!

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References

• Baygün, Kuchuk & Ar kan (1997), SPEJ Sept. 1997, 246.

• Bourdet & al. (1983), World Oil 196, 97.

• Bourdet & al. (1989), SPEFE June 1989, 293.

• Björck (1996), Numerical Methods for Least Squares Problems . SIAM.

• Coats & al. (1964), Trans. AIME 231, 1417.

• Golub & Pereyra (1973), SIAM J. Num. Anal . 10, 413.

• Gringarten & Ramey (1973), SPEJ Oct. 1973, 285. Paper SPE 3818.

• Gringarten (2005), EAGE Madrid, Paper SPE 93988.

• Hutchinson & Sikora (1959), Trans. AIME 216, 169.

• Kuchuk & al. (1990), SPEFE December 1990, 375.

• von Schroeter, Hollaender & Gringarten:

 – (2001) SPE 71574.

 – (2002) SPE 77688.

 – (2004) SPEJ 9 (December 2004), 375.


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