Application of Optimal Control to NMR Well Logging Martin Hürlimann Schlumberger – Doll Research, Cambridge USA Email: [email protected] Quantum Cybernetics & Control Workshop, Nottingham, January 19-23, 2015
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Application of Optimal Control to NMR Well
Logging
Martin Hürlimann Schlumberger – Doll Research, Cambridge USA
Email: [email protected]
Quantum Cybernetics & Control Workshop, Nottingham, January 19-23, 2015
Typical Borehole dimensions Typical Conditions Diameter: 20 – 30 cm Temperature: up to 175o C Depth/Length: up to 12 km Pressure: up to 1400 atm
NMR Well Logging: Characterization of fluids in geological formations by NMR
measurements from the borehole
Outline Well Logging / Challenges of NMR Well Logging NMR Measurements in inhomogeneous fields
• Spin dynamics of CPMG • Optimization of Pulse Sequences
Characterizing the Reservoir
Wireline logging
Well logging
Accessing the Reservoir
Logging while Drilling
• Resistivity / Dielectric Measurements • Nuclear measurements γ-ray absorption / spectroscopy neutron thermalization • Sonic measurements • Nuclear Magnetic Resonance • …
NMR well logging Seeing the fluids inside rocks
Questions to address:
• Porosity: How much fluid is inside the rock?
• Pore size: How big are the pores, how easily can the fluid flow through the rock?
• Fluid quantification: What and how many types of fluids are inside the rock?
• Fluid identification: What is the composition of the fluids?
100 mm
200 mm
200 mm
100 mm 200 mm
NMR well logging Seeing the fluids inside rocks
Questions to address:
• Porosity: How much fluid is inside the rock?
• Pore size: How big are the pores, how easily can the fluid flow through the rock?
• Fluid quantification: What and how many types of fluids are inside the rock?
• Fluid identification: What is the composition of the fluids?
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
0
1
2
3
4
5
6
7
8
9
Chain Length: Retention Time (minutes)
Aro
maticity:
Rete
ntion T
ime (
seconds)
10
18
00
16
00
14
00
12
00
10
00
20
00
22
00
24
00
26
00
28
00
30
00
32
00
34
00
36
00
38
00
steranes
hopanes
napthalenes
isoprenoid
alkanes
branched &
normal alkanes
Complex Fluid Composition
Crude Oil: 2D Gas Chromatography
O. C. Mullins et al., Energy & Fuels, 22, 496 (2008)
Pulse Programmer
Filter, A/D Converter Data and Frequency
Analyzer
Low noise preamplifier
N S
Spectrometer
Magnet Tuned Probe with Sample
rf amplifier
Pulsed NMR Measurements
Magnetic field, B0
Ener
gy
wLarmor = g B0
Pulse Programmer
Filter, A/D Converter Data and Frequency
Analyzer
Low noise preamplifier
N S
Spectrometer
Magnet Tuned Probe
rf amplifier
Pulsed NMR Measurements
Magnetic field, B0
Ener
gy
wLarmor = g B0
Sample
Hardware • Magnet Configuration • RF Coil • Spectrometer/Electronics • Mechanical Robustness
Measurement • Pulse Sequence • Spin Dynamics • Calibration
Interpretation • Physics of Porous Media • Physics of Complex Fluids • Flow Properties
NMR Well Logging
M. D. Hürlimann, “NMR Well Logging”, Encyclopedia of Magnetic Resonance, 2012, John Wiley & Sons, Ltd. DOI: 10.1002/9780470034590.emrstm0593.pub2
Borehole NMR: Issues to overcome
Hardware • Inside - out - apparatus: sample is outside apparatus • Downhole compact spectrometer • Operation at 175o C and 1400 atm Solution: Permanent magnets (SmCo), customized apparatus and electronics
Measurement • NMR in weak and grossly inhomogeneous fields • Spectroscopy or conventional imaging not possible. Solution: Relaxation and diffusion measurements including 2d relaxation – diffusion measurements
Interpretation Physics of relaxation and diffusion Brownian motion • in porous media • of complex fluids
Borehole NMR Hardware: Wireline logging
Static field
Bo ~ 45 mT
Larmor frequency
fL ~ 2 MHz
R. L. Kleinberg et al., J. Magn. Reson. 97, 466, (1992).
Measurement Zone
Stabilizer
Annular Magnets
Drill Bit
Borehole NMR Hardware: Logging while drilling
J. Horkowitz et al., SPWLA Paper EEE (2002) J. Morley et al., SPE Paper 77477-MS (2002)
NMR Methodology:
Relaxation and Diffusion Measurements
L. Venkataramanan et al., Petrophysics 55, 6, 572 (2014).
Development based on laboratory measurements on cores
Relaxation: Pore size Distribution Pulse Sequence: CPMG Typ. parameters: 200 ms echo spacing, 8000 echoes
Diffusion - Relaxation: Fluid identification Pulse Sequence: Diffusion encoding – CPMG Typ. parameters: static gradient 13 G/cm
Pore
Rock
Surface relaxivity r
222 /exp )( )( TtTfdTtM
2
1
S
T Vr
M. D. Hürlimann et al., Magn. Reson. Imag. 21, 305 (2003).
20
0 m
NMR Logs
Hardware • Magnet Configuration • RF Coil • Spectrometer/Electronics • Mechanical Robustness
Measurement • Pulse Sequence • Spin Dynamics • Calibration
Interpretation • Physics of Porous Media • Physics of Complex Fluids • Flow Properties
NMR Well Logging
yxzo ItvItuIH
Hdt
di
)( )(
,
w
rr
rf pulses: control
Simple Hamiltonian: Independent spins in an inhomogeneous magnetic field
Governing Equation:
ωrf /2π = 950 MHz
Bo/Bo ~ 10-8 ωrf /2π = 150 kHz – 2 MHz
Bo/Bo ~ 1 grossly inhomogeneous fields
Pulsed NMR Measurements
Zeeman
kHz 1 - Hz 0.1 2 1
kHz 20 2
MHz 2 MHz 2 2 )(
2,1
1
g
g
T
B
rBo
011
1
1
11
2
0
),(
),()( )(),(),(
MtrM
trMtBrBtrMtrM
dt
d
TzT
T
o
g0
1
1M
T1
1T
2
1T
(t)rdtrMtS ),()(
static magnetic field
rf magnetic field
relaxation rates
detected signal:
Optimization Problem:
Determine best B1(t), i.e. pulse sequence to extract Mo, T1, T2 from signal S(t)
Governing equations: Bloch Equations
NMR in (slightly) inhomogeneous fields
Hahn Echo Sequence (1950)
90o 180o B1(t) echo
time
Magnetization
Signal S(t)
CPMG Sequence (Carr – Purcell – Meiboom – Gill,1958)
90o 180o 180o 180o 180o 180o 180o
time
Ech
o a
mp
litu
de
Time
Mo exp{-t/T2}
Spin dynamics in grossly inhomogeneous fields
90o 180o 180o 180o 180o 180o 180o
time
Bo << B1
Conventional NMR
Bo = B1
Bo = 3 B1
NMR well logging 1st echo 2nd echo 3rd echo 4th echo 5th echo 10th echo Bo >> B1
non-trivial spin dynamics
200 ms rf pulses
detected magnetization
MDH J. Magn. Reson. 148, 367 (2001)
Experimental results:
time
Initial amplitude Ao → Porosity Relaxation time T2,eff → Pore size,
Viscosity
After 3rd echo: • constant echo shape • exponential decay A(t) = Ao exp{-t/T2,eff}
YES, but WHY?
NMR measurements in grossly inhomogeneous fields Is it possible to make quantitative NMR measurements?
N - th Echo (fast pulsing regime):
MN
,ˆR B BB Nn zR A
Excitation pulse N refocusing cycles
RB ˆ n B, B RA
Spin dynamics of CPMG in inhomogeneous fields
Asymptotic CPMG Signal (N >> 1):
ˆ n Bˆ z
RA
RB M(0+)
)0(Mn n M BBasy
{M(0+)} ,ˆR B BB Nn =
Component of
B
B
n)(0M
n || )(0M
coherent refocusing: eigenvalue = 1
rapid dephasing
MDH et al. J. Magn. Reson. 143, 120 (2000)
static field strength
rf f
ield
str
engt
h
Asymptotic CPMG Signal (N >> 1):
)0(Mn n M BBasy
Standard CPMG
CPMG in inhomogeneous fields
MDH et al. J. Magn. Reson. 143, 120 (2000)
Experimental Verification Standard CPMG in a gradient magnetic field
Experiment Theory
)0(Mn n M BBasy
Echo shape (in asymptotic regime, i.e. N > 3)
Echo amplitude:
Borehole NMR Measurements: NMR in grossly inhomogeneous fields
Measurement with inside-out set-up Challenge: SNR
Composite pulses (phase modulated pulses)
Standard 180 inversion pulse 180x
Composite 180 pulse: 90x – 180y – 90x
Self correcting pulses
(Malcolm Levitt, 1981)
N - th Echo (fast pulsing regime):
MN
,ˆR B BB Nn zR A
Excitation pulse N refocusing cycles
RB ˆ n B, B RA
Optimization of CPMG in inhomogeneous fields
Asymptotic CPMG Signal (N >> 1):
ˆ n Bˆ z
RA
RB M(0+)
)0(Mn n M BBasy
{M(0+)} ,ˆR B BB Nn =
10
10B
, x )0(M
, x n
ww
ww
Goal: Refocusing pulse
Excitation pulse
MDH , J. Magn. Reson. 152, 109 (2001); Mandal et al., J. Magn. Reson. 237, 1 (2013)
Standard pulses OCT pulses
GRAPE algorithm: Khaneja et al. JMR 172, 296 (2005) Search for OCT broadband π refocusing pulse, Nt = 10 t180 , Ai ≤ ω1 Cost function: Average fidelity over ±1.6 ω1
Borneman et al. , J. Magn. Reson. 207, 220-233 (2010)
Application of optimal control to CPMG refocusing pulse design
t180
tE Tacq
90ox 180o
y 180oy 180o
y
SPA SPA SPA
Koroleva et al. , JMR 230, 64-75 (2013)
Optimization of short refocusing pulses
SPA = Symmetric Phase Alternating Pulse 27o
-y – 126o+y – 27o
-y
tp=t180
SPA 180o
Echo shapes
(N>3)
theory experiment
Koroleva et al. JMR 230, 64-75 (2013)
Strong 90o
Strong 90o
Broadband CPMG sequence with short refocusing pulses
Asymptotic magnetization Signal-to-Noise Ratio • perfect AMEX pulse Masy ~ nx SNR ~ < nx
2 >
• perfect 90 pulse Masy ~ nx2 SNR ~ < nx
4 >
21( )
2
T
T
SNR M t dtT
Optimization of excitation pulse
10B , n )0(M ww
For a given refocusing pulse, the optimal excitation pulse fulfills:
Asymptotic CPMG Signal (N >> 1): )0(Mn n M BBasy
ˆ n Bˆ z
RA
RB M(0+)
“Axis-matching excitation (AMEX) pulse”
Generalized CPMG condition
),(n z 10B ww
Mandal et al. JMR 237, 1 (2013)
Axis Matching Excitation Pulses
nx nz nx nz
Masy Masy
Axis-matching
excitation pulse
Mx(0+) Mz(0
+) Mx(0+) Mz(0
+)
Asymptotic
CPMG echo
Refocusing
pulse 0
ω1
-ω1
t180
0
ω1
-ω1
t180
SPA 180x
27-x-126x-27-x
nB
)0(Mn n M BBasy
)0(M
Mandal et al. JMR 237, 1 (2013)
Standard CPMG
Experimental Verification
S. Mandal et al., JMR 237, 1 (2013)
Standard CPMG
echo amplitudes
S. Mandal et al., JMR 237, 1 (2013)
Standard CPMG
first echo asymptotic echo
Experimental Verification
Optimization of duration of refocusing pulse Optimize SNR per unit time
𝜔1 ≤ 𝜔1,𝑚𝑎𝑥 optimize 𝑆𝑁𝑅𝑡𝑖𝑚𝑒= 𝑆𝑁𝑅𝑒𝑐ℎ𝑜
𝑡𝐸
As pulse duration tp is allowed to get longer, SNRecho will generally increase.
Key Questions:
• Does SNRtime also increase?
• Is there a maximum in SNRtime for a certain value of tp?
• What determines SNRtime ?
tE tp
tE tp
Cost function: <nx4>
Constraints: Ai = ω1 pure phase modulation
i = -N-I antisymmetric: ny = 0
Cost function: <nx4>
Constraint: Ai ≤ ω1
Cost function: <nx4>
Constraints: Ai = ω1 pure phase modulation
i = -N-I antisymmetric: ny = 0 i = +N-i symmetric:
nx (ω0)= nx (-ω0)
Cost function: <> Constraint: Ai ≤ ω1
t = t180 /10
tp = Nt
Optimize {Ai,i} using GRAPE algorithm with
S. Mandal et al., JMR 247, 54 (2014)
perfect 90
perfect AMEX
perfect AMEX
perfect 90
Best refocusing pulses:
(Anti)-symmetric phase alternating pulses
i= 0 or π
S. Mandal et al., JMR 247, 54 (2014)
Experimental Verification
S. Mandal et al., JMR 247, 54 (2014)
Comparison to standard CPMG sequence:
4.5 times higher SNR per echo
3.9 times higher SNR per unit time
Programmable Pulse generator
Transmitter Ct
Probe
RC
L
1: nTX
Transformer
VTX
RTX
Transmitter
Signal Receiver Ct
Probe
RC
L
1: nTX
Transformer
VTX
RTX
Transmitter
NMR Spin Dynamics
Practical Implementation Optimization of Complete NMR System
Take into account:
• Detailed Response of Transmitter and Receiver
• Variations of Environmental Parameters (e.g. Salinity, Temperature,…)
Optimization of
• Electronics / hardware (phase resolution, gain vs bandwidth,
receiver Q, …)
• Pulse Sequence
• Digital Filter
Acknowledgments:
Van Koroleva
Soumyajit Mandal
Troy Borneman
David Cory
Conclusion
• Quantitative NMR measurements with inside-out sensors provide quantitative results.
• This has enabled NMR well logging to become a successful commercial service.
• Optimal control is an essential tool in the further development of this technique.
Thank you
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