welltest deconvolution

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12/09/05 19:25 Symposium for Alain Gringarten 1

Deconvolution in Well Test Analysis

Thomas von Schroeter


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



2



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

B

t

t

W

S

’ n

G p

’ n

p G ’ t

’ x ’ x ,x ,’ t t G ’ x ,’ t ’ t c

x ,t p

∫ ∫

∫ ∫

∂

∂∆−

∂

∆∂−

−φ

=∆

B

W



3


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

W

d)()( ∫ =

• Define a source function


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



4


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

t

d)()()(

0

−=∆ ∫ W ’ x ’ x ,x ,t G

c x ,t g

)(

1)( ∈φ

=

• Superposition principle (Duhamel 1833)


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.

t

p x ,t g ’ t x ,t’ g t p U

G

t

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



5


Derivative analysis (1983)

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

Derivative

Pressure drop

Estimate


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!



6


t 0

p U

b

Q

t

p

• 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

Q

0 b

Derivative analysis, taken (too) literally…


… and with a vengeance!

t

p

Q

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…



7


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:


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)



8


{ } ’ 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:


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)



9


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!


• 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)



10


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


Rate simulation

50000 100000 150000 200000 t

1

2

3

4

5

q(t) unperturbed

+ 1 % error (RMS)

+ 10 % error (RMS)



11


Pressure simulation

50000 100000 150000 200000 t

10

20

30

40

50

60

p(t)unperturbed

0.5% in ∆p

5% in ∆p


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



12


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

2

• ⇒ analytic expressions for

– Bias if λ > 0 (“stiffness”)

– Covariance matrix ⇒ confidence intervals

– λ controls trade-off: bias vs variance


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%



13


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%


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%



14


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


Well

HWellV

Seismic faults 0 500 1000

meters

Baffle

Proposed water injection wells

N

Field example [AG, SPE 93988]



15


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


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



16


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



17


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?

?


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



18


Where do we go from here?


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



19


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


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!




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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