Page 1
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 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
Page 2
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 2/20
2
12/09/05 19:25 Symposium for Alain Gringarten 3
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
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
Page 3
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 3/20
3
12/09/05 19:25 Symposium for Alain Gringarten 5
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
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
Page 4
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 4/20
4
12/09/05 19:25 Symposium for Alain Gringarten 7
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)
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.
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
Page 5
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 5/20
5
12/09/05 19:25 Symposium for Alain Gringarten 9
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!
Page 6
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 6/20
6
12/09/05 19:25 Symposium for Alain Gringarten 11
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…
12/09/05 19:25 Symposium for Alain Gringarten 12
… 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…
Page 7
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 7/20
7
12/09/05 19:25 Symposium for Alain Gringarten 13
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)
Page 8
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 8/20
8
12/09/05 19:25 Symposium for Alain Gringarten 15
{ } ’ 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)
Page 9
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 9/20
9
12/09/05 19:25 Symposium for Alain Gringarten 17
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)
Page 10
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 10/20
10
12/09/05 19:25 Symposium for Alain Gringarten 19
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)
Page 11
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 11/20
11
12/09/05 19:25 Symposium for Alain Gringarten 21
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
Page 12
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 12/20
12
12/09/05 19:25 Symposium for Alain Gringarten 23
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
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%
Page 13
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 13/20
13
12/09/05 19:25 Symposium for Alain Gringarten 25
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%
Page 14
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 14/20
14
12/09/05 19:25 Symposium for Alain Gringarten 27
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]
Page 15
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 15/20
15
12/09/05 19:25 Symposium for Alain Gringarten 29
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
Page 16
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 16/20
16
12/09/05 19:25 Symposium for Alain Gringarten 31
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
Page 17
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 17/20
17
12/09/05 19:25 Symposium for Alain Gringarten 33
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
Page 18
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 18/20
18
12/09/05 19:25 Symposium for Alain Gringarten 35
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
Page 19
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 19/20
19
12/09/05 19:25 Symposium for Alain Gringarten 37
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!
Page 20
7/21/2019 welltest Deconvolution
http://slidepdf.com/reader/full/welltest-deconvolution 20/20
12/09/05 19:25 Symposium for Alain Gringarten 39
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.