Fast Inversion of Logging-While-Drilling (LWD) Resistivity Measurements David Pardo 1 Carlos Torres-Verdín 2 1 University of the Basque Country (UPV/EHU) and Ikerbasque, Bilbao, Spain. 2 The University of Texas at Austin, USA 17 Oct. 2013 BCAM Workshop on Computational Mathematics, Bilbao, Spain 1
formation evaluation Surface measurements on the sea. Marine seismic measurements. 2
formation evaluation Surface measurements on the sea. Marine controlled source electromagnetic (CSEM) measurements. 3
formation evaluation Surface measurements on land. Seismic measurements. 4
formation evaluation Surface measurements on land. Magnetotelluric (MT) measurements. 5
formation evaluation Logging measurements Multiphysics Logging while drilling in a deviated well. 6
formation evaluation Logging measurements Multiphysics Dip Angle Logging while drilling in a deviated well. 7
formation evaluation Logging measurements Multiphysics Dip Angle Borehole eccentricity Logging while drilling in a deviated well. 8
formation evaluation Logging measurements Multiphysics Dip Angle Borehole eccentricity Invasion Logging while drilling in a deviated well. 9
formation evaluation Logging measurements Multiphysics Dip Angle Borehole eccentricity Invasion Anisotropy Logging while drilling in a deviated well. 10
formation evaluation Logging measurements Multiphysics Dip Angle Borehole eccentricity Invasion Anisotropy Fractures Logging while drilling in a deviated well. 11
formation evaluation Logging measurements Multiphysics Dip Angle Borehole eccentricity Invasion Anisotropy Fractures Logging while drilling in a deviated well. Different Logging Devices 12
main areas of expertise Resistivity Measurements: Marine CSEM measurements. Magnetotelluric (MT) measurements. Galvanic and induction devices. Cased wells. Cross-well and borehole-to-surface measurements. Deviated wells. Borehole eccentered tools. Hydrofracture characterization. Sonic Measurements: Wireline and logging-while-drilling. Borehole-eccentered tools. Inversion of Resistivity Measurements: One-dimensional model reduction. Rapid inversion of logging-while-drilling measurements. 13
main areas of expertise Resistivity Measurements: Marine CSEM measurements. Magnetotelluric (MT) measurements. Galvanic and induction devices. Cased wells. Cross-well and borehole-to-surface measurements. Deviated wells. Borehole eccentered tools. Hydrofracture characterization. Sonic Measurements: Wireline and logging-while-drilling. Borehole-eccentered tools. Inversion of Resistivity Measurements: One-dimensional model reduction. Rapid inversion of logging-while-drilling measurements. 14
inversion of LWD measurements Motivation and objectives. Assumptions. Forward Problem. Inverse Problem. Numerical Results. Conclusions. 15
motivation and objectives Goal: Inversion of LWD resistivity measurements. 16
motivation and objectives Goal: Inversion of LWD resistivity measurements. We want the inversion algorithm to be: Efficient. Inversion in real time using 1D model reduction. 17
motivation and objectives Goal: Inversion of LWD resistivity measurements. We want the inversion algorithm to be: Efficient. Inversion in real time using 1D model reduction. Flexible. It should enable the dynamic selection of a subset of measurement and/or unknowns during inversion. 18
motivation and objectives Goal: Inversion of LWD resistivity measurements. We want the inversion algorithm to be: Efficient. Inversion in real time using 1D model reduction. Flexible. It should enable the dynamic selection of a subset of measurement and/or unknowns during inversion. Robust. It should always converge to physically meaningful solutions. 19
motivation and objectives Goal: Inversion of LWD resistivity measurements. We want the inversion algorithm to be: Efficient. Inversion in real time using 1D model reduction. Flexible. It should enable the dynamic selection of a subset of measurement and/or unknowns during inversion. Robust. It should always converge to physically meaningful solutions. Reliable. It should provide error bars. 20
motivation and objectives Goal: Inversion of LWD resistivity measurements. We want the inversion algorithm to be: Efficient. Inversion in real time using 1D model reduction. Flexible. It should enable the dynamic selection of a subset of measurement and/or unknowns during inversion. Robust. It should always converge to physically meaningful solutions. Reliable. It should provide error bars. Useful. It should work for any commercial LWD instrument with actual field measurements. 21
assumptions We assume a planarly TI layered media with piecewise constant resistivities. We assume no borehole effects and no mandrel effects. True Vertical Depth (m) 0 1 2 3 4 Model Problem and Well Trajectory 1050 1100 Horizontal Depth (m) 1000 1050 1100 1150 We know the bed boundaries a priori. We know the dip and azimuthal angles of intersection a priori. 22
forward problem Magnetic field H produced by a magnetic dipole is obtained using a semi-analytical solution for a 1D planarly layered TI media (Kong, 1972). 23
forward problem Magnetic field H produced by a magnetic dipole is obtained using a semi-analytical solution for a 1D planarly layered TI media (Kong, 1972). A) Hankel transform in the horizontal plane. 24
forward problem Magnetic field H produced by a magnetic dipole is obtained using a semi-analytical solution for a 1D planarly layered TI media (Kong, 1972). A) Hankel transform in the horizontal plane. B) Analytical solution of the resulting ordinary differential equation in the vertical direction. 25
forward problem Magnetic field H produced by a magnetic dipole is obtained using a semi-analytical solution for a 1D planarly layered TI media (Kong, 1972). A) Hankel transform in the horizontal plane. B) Analytical solution of the resulting ordinary differential equation in the vertical direction. C) Numerical inverse Hankel transform (integration). 26
forward problem Magnetic field H produced by a magnetic dipole is obtained using a semi-analytical solution for a 1D planarly layered TI media (Kong, 1972). A) Hankel transform in the horizontal plane. B) Analytical solution of the resulting ordinary differential equation in the vertical direction. C) Numerical inverse Hankel transform (integration). Result: Magnetic field H. 27
forward problem CASE I: Triaxial Induction. H = H xx H xy H xz H yx H yy H yz H zx H zy H zz. 28
forward problem CASE II: Conventional LWD resistivity tool. H q := log HRX 1 zz H RX 2 zz }{{} ATTENUATION + i [ph(h RX 1 zz ) ph(h RX 2 zz )] }{{} PHASE DIFFERENCE Attenuation Log Scale 5 2.5 1.2 10 2 10 0 10 2 Resistivity (Ohm m) Log Scale Phase Diff. Log Scale 10 0 10 1 10 2 10 2 10 0 10 2 Resistivity (Ohm m) Log Scale 29
forward problem CASE II: Conventional LWD resistivity tool. H q := log log HRX 1 zz H RX 2 zz }{{} ATTENUATION + i log [ph(h RX 1 zz ) ph(h RX 2 zz )] }{{} PHASE DIFFERENCE Attenuation Log Scale 5 2.5 1.2 10 2 10 0 10 2 Resistivity (Ohm m) Log Scale Phase Diff. Log Scale 10 0 10 1 10 2 10 2 10 0 10 2 Resistivity (Ohm m) Log Scale 30
forward problem To accelerate computations, we employ a WINDOWING system: 0 Model Problem and Well Trajectory 1 True Vertical Depth (m) 2 3 4 1000 1050 1100 1150 Horizontal Depth (m) 31
forward problem To accelerate computations, we employ a WINDOWING system: 32
forward problem To accelerate computations, we employ a WINDOWING system: 33
forward problem To accelerate computations, we employ a WINDOWING system: 34
forward problem To accelerate computations, we employ a WINDOWING system: 35
forward problem To accelerate computations, we employ a WINDOWING system: 36
forward problem To accelerate computations, we employ a WINDOWING system: 37
forward problem To accelerate computations, we employ a WINDOWING system: 38
forward problem To accelerate computations, we employ a WINDOWING system: 39
forward problem To accelerate computations, we employ a WINDOWING system: 40
forward problem To accelerate computations, we employ a WINDOWING system: 41
forward problem To accelerate computations, we employ a WINDOWING system: 42
forward problem To accelerate computations, we employ a WINDOWING system: 43
forward problem To accelerate computations, we employ a WINDOWING system: 44
inverse problem (formulation) Cost Functional: C W (s) = H(s) M 2, lw 2 }{{ M } MISFIT where s is either the conductivity σ, the resistivity ρ, or log ρ, H(s) is the set of simulated measurement for s, M is the set of actual (or synthetic) field measurements, HD(m) 1000 1020 1040 1060 Goal: To find s := arg mín C W (s). s 1 10 Resistivity (Ohm-m) 45
inverse problem (formulation) Cost Functional: where C W (s) = H(s) M 2 l 2 W M }{{} MISFIT + λ s s 0 2, L 2 Ws }{{ 0 } REGULARIZATION s is either the conductivity σ, the resistivity ρ, or log ρ, H(s) is the set of simulated measurement for s, M is the set of actual (or synthetic) field measurements, λ is a regularization parameter, and s 0 is an a priori distribution of s. Goal: To find s := arg mín C W (s). s 46
inverse problem (sol. method) We select the following deterministic iterative scheme: s (n+1) = s (n) + δs (n). Using a Taylor s series expansion of first order of H: ( H(s H(s (n+1) ) H(s (n) (n) ) ) ) + δs (n). s }{{} J Solving C W (s (n+1) ) δs (n) = 0, we obtain Gauss-Newton s method: δs (n) := Re(J, H(s (n) ) M) l 2 WM + λ(i, s (n) s 0 ) L 2 Ws0 (J, J) l 2 WM + λ(i, I) L 2 Ws0. 47
inverse problem (jacobian) To compute the Jacobian, we employ: The chain rule: J = H(s) s j = H(s) ρ j. ρ j s j }{{} J ρ The definition of derivative: J ρ = H(s) ρ j H(ρ + hδρ j) H(ρ) h (h small). Only one Jacobian matrix is computed for any variable s. 48
inverse problem (jacobian) Misfit( %) Dip Angle = 82. Thinnest Bed: 0.37 m. ρ 11,35 % HD(m) 1 10 Resistivity (Ohm-m) 1000 1020 1040 1060 49
inverse problem (jacobian) Misfit( %) Dip Angle = 82. Thinnest Bed: 0.37 m. ρ 11,35 % HD(m) 1000 1020 1040 1060 1 10 Resistivity (Ohm-m) 50
inverse problem (jacobian) Misfit( %) Dip Angle = 82. Thinnest Bed: 0.37 m. ρ 11,35 % HD(m) 1000 1020 1040 1060 1 10 Resistivity (Ohm-m) σ 11,32 % 1 10 100 Resistivity (Ohm-m) 51
inverse problem (jacobian) Misfit( %) Dip Angle = 82. Thinnest Bed: 0.37 m. ρ 11,35 % HD(m) 1000 1020 1040 1060 1 10 Resistivity (Ohm-m) σ 11,32 % 1 10 100 Resistivity (Ohm-m) 52
inverse problem (jacobian) Misfit( %) Dip Angle = 82. Thinnest Bed: 0.37 m. ρ 11,35 % HD(m) 1000 1020 1040 1060 1 10 Resistivity (Ohm-m) σ 11,32 % 1 10 100 Resistivity (Ohm-m) log ρ 7,87 % 1 10 Resistivity (Ohm-m) 53
inverse problem (jacobian) Misfit( %) Dip Angle = 82. Thinnest Bed: 0.37 m. ρ 11,35 % HD(m) 1000 1020 1040 1060 1 10 Resistivity (Ohm-m) σ 11,32 % 1 10 100 Resistivity (Ohm-m) log ρ 7,87 % 1 10 Resistivity (Ohm-m) 54
inverse problem (jacobian) Misfit( %) Dip Angle = 82. Thinnest Bed: 0.37 m. ρ 11,35 % HD(m) 1000 1020 1040 1060 1 10 Resistivity (Ohm-m) σ 11,32 % 1 10 100 Resistivity (Ohm-m) log ρ 7,87 % 1 10 Resistivity (Ohm-m) Best 6,58 % 1000 1 10 Resistivity (Ohm-m) 1020 1040 1060 55
inverse problem (error bars) Once we achieve convergence, we have: δs (n) 0. Considering new noisy measurements of the type: M := M + N and using these new measurements in our Gauss-Newton method, we obtain the following new correction δs (n) : δs (n) := Re(J, N) l 2 WM (J, J) l 2 WM + λ(i, I) L 2 Ws0 Error bars: [s (n) δs (n), s (n) + δs (n) ]. 56
numerical results (synthetic 1) 0 Resistivity (Ohm-m) Model Problem and Well Trajectory 2 True Vertical Depth (m) 4 6 8 10 1 10 Dip Angle: 82. Thinnest bed: 0.37m 1 10 Resistivity (Ohm-m) 1000 1020 1040 1060 57
numerical results (synthetic 1) TOOL 1 TOOL 2 TOOL 3 TOOL 4 TOOL 5 0 Resistivity (Ohm-m) Resistivity (Ohm-m) 0 Resistivity (Ohm-m) 0 Resistivity (Ohm-m) 0 Resistivity (Ohm-m) 2 2 2 2 True Vertical Depth (m) 4 6 8 True Vertical Depth (m) 4 6 8 True Vertical Depth (m) 4 6 8 True Vertical Depth (m) 4 6 8 10 10 10 10 1 10 1 10 1 10 100 1 10 1 10 Dip Angle = 82. Thinnest bed: 0.37m. 58
numerical results (synthetic 1) TOOL 1 TOOL 2 TOOL 3 TOOL 4 TOOL 5 0 2 Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) 0 0 0 2 2 2 True Vertical Depth (m) 4 6 8 True Vertical Depth (m) 4 6 8 True Vertical Depth (m) 4 6 8 True Vertical Depth (m) 4 6 8 10 10 10 10 1 10 1 10 1 10 100 1 10 1 10 Dip Angle = 82. Thinnest bed: 0.37m. 59
numerical results (synthetic 1) Sensitivity with respect to the Dip Angle. Thinnest bed: 0.37m. 0 30 45 60 82 89 Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) 0 0 0 2 2 2 2 True Vertical Depth (m) 4 6 8 True Vertical Depth (m) 4 6 8 True Vertical Depth (m) 4 6 8 True Vertical Depth (m) 4 6 8 10 10 10 10 1 10 1 10 1 10 1 10 1 10 100 60
numerical results (synthetic 1) 0 Resistivity (Ohm-m) Model Problem and Well Trajectory 2 True Vertical Depth (m) 4 6 8 10 1 10 100 Dip Angle: 82. Thinnest bed: 0.37m Anisotropy. 1000 1 10 100 Resistivity (Ohm-m) 1020 1040 1060 61
numerical results (synthetic 1) 30 HD(m) 1000 1002 1004 Vertical well > R h. 1006 1 10 100 Resistivity (Ohm-m) 62
numerical results (synthetic 1) 30 HD(m) 1000 1002 1004 Vertical well > R h. 1006 1 10 100 Resistivity (Ohm-m) 82 Horizontal well > R v. 1 10 100 Resistivity (Ohm-m) 63
numerical results (synthetic 2) 0.0 Resistivity (Ohm-m) Model Problem and Well Trajectory 0.2 True Vertical Depth (m) 0.4 0.6 0.8 1.0 1.2 1.4 1 10 Dip Angle: 82. Thinnest bed: 0.05m. 1000 1002 1004 1 10 Resistivity (Ohm-m) 1006 1008 1010 64
numerical results (synthetic 2) TOOL 1 TOOL 2 TOOL 3 TOOL 4 TOOL 5 Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) 0.0 0.0 0.0 0.0 0.2 0.2 0.2 0.2 True Vertical Depth (m) 0.4 0.6 0.8 1.0 True Vertical Depth (m) 0.4 0.6 0.8 1.0 True Vertical Depth (m) 0.4 0.6 0.8 1.0 True Vertical Depth (m) 0.4 0.6 0.8 1.0 1.2 1.2 1.2 1.2 1.4 1.4 1.4 1.4 1 10 1 10 100 1 10 1 10 100 1 10 65
numerical results (synthetic 2) TOOL 1 TRIAXIAL TRIAXIAL TRIAXIAL TRIAXIAL NO NOISE NO NOISE 5 % NOISE 10 % NOISE NO NOISE Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) Resistivity (Ohm-m) 0.0 0.0 0.0 0.0 0.2 0.2 0.2 0.2 True Vertical Depth (m) 0.4 0.6 0.8 1.0 True Vertical Depth (m) 0.4 0.6 0.8 1.0 True Vertical Depth (m) 0.4 0.6 0.8 1.0 True Vertical Depth (m) 0.4 0.6 0.8 1.0 1.2 1.2 1.2 1.2 1.4 1.4 1.4 1.4 1 10 1 10 1 10 100 1 10 100 1 10 100 66
numerical results (field 1) 0 Resistivity (Ohm-m) Model Problem and Well Trajectory 1 True Vertical Depth (m) 2 3 4 1 10 Almost horizontal. Field data. 1 10 Resistivity (Ohm-m) 1000 1050 1100 1150 67
numerical results (field 1) 0 Resistivity (Ohm-m) Model Problem and Well Trajectory 1 True Vertical Depth (m) 2 3 4 1 10 Almost horizontal. Field data. 1 10 Resistivity (Ohm-m) 1000 1050 1100 1150 68
numerical results (field 1) 1.5 Resistivity (Ohm-m) Model Problem and Well Trajectory 2.0 True Vertical Depth (m) 2.5 3.0 3.5 4.0 1 10 Almost horizontal. Field data. Zoom. 1110 1120 1130 1 10 Resistivity (Ohm-m) 1140 1150 69
numerical results (field 2) 0 Resistivity (Ohm-m) Model Problem and Well Trajectory 2 True Vertical Depth (m) 4 6 8 Dip Angle: 79.3. Field data. 1 10 1000 1010 1020 1 10 Resistivity (Ohm-m) 1030 1040 70
numerical results (field 2) HD(m) 1000 1010 1020 1030 1040 1 10 Resistivity (Ohm-m) Inversion results (blue) are similar to those obtained by Dr. Olabode Ijasan (red). 71
conclusions We have developed a library for the fast inversion of LWD resistivity measurements. 72
conclusions We have developed a library for the fast inversion of LWD resistivity measurements. The library enables any well trajectory and any logging instrument. We assume a 1D planarly layered TI media. 73
conclusions We have developed a library for the fast inversion of LWD resistivity measurements. The library enables any well trajectory and any logging instrument. We assume a 1D planarly layered TI media. The library automatically selects the regularization parameter, stopping criteria, and inversion variable. 74
conclusions We have developed a library for the fast inversion of LWD resistivity measurements. The library enables any well trajectory and any logging instrument. We assume a 1D planarly layered TI media. The library automatically selects the regularization parameter, stopping criteria, and inversion variable. It enables to first invert a subset of measurements and/or a subset of resistivities. 75
conclusions We have developed a library for the fast inversion of LWD resistivity measurements. The library enables any well trajectory and any logging instrument. We assume a 1D planarly layered TI media. The library automatically selects the regularization parameter, stopping criteria, and inversion variable. It enables to first invert a subset of measurements and/or a subset of resistivities. Numerical results illustrate the stability of the proposed inversion algorithm. 76
future work Computational cost of one forward simulation: COST = C N POSITIONS N LAYERS N FREQ. N TX N RX 77
future work Computational cost of one forward simulation: COST = C N POSITIONS N LAYERS N FREQ. N TX N RX Computational cost of building the Jacobian: COST = C N POSITIONS N LAYERS 2 N FREQ. N TX N RX 78
future work Computational cost of one forward simulation: COST = C N POSITIONS N LAYERS N FREQ. N TX N RX Computational cost of building the Jacobian: COST = C N POSITIONS N LAYERS 2 N FREQ. N TX N RX Can we eliminate the factor N RX? I think so! Can we eliminate the factor N TX? To some extend! Can we eliminate the square on the factor N LAYERS? Perhaps! 79
change of coordinates To employ a Model Reduction algorithm based on Cartesian (C) coordinates and obtain results for Borehole (B) coordinates, we employ: H BB = J BC H CC J CB, where: H CC and H BB are the model reduction algorithms for the Cartesian and Borehole systems of coordinates, respectively, cos θ 0 sin θ cos φ sin φ 0 J CB = 0 1 0 sin φ cos φ 0 sin θ 0 cos θ 0 0 1 J BC = J 1 CB, θ is the dip angle, and φ is the azimuthal angle. 80
inverse problem (reg. param.) We have: C (n) W (s) = H(s) M 2 l 2 W M } {{ } MISFIT + λ (n) s s 0 2, L 2 Ws }{{ 0 } REGULARIZATION 81
inverse problem (reg. param.) We have: C (n) W (s) = H(s) M 2 + λ (n) s s lw 2 0 2, L 2 }{{ M Ws }}{{ 0 } MISFIT REGULARIZATION 90 % 10 % We want the regularization term to contribute with 10 % to the total cost functional. 82
inverse problem (reg. param.) We have: C (n) W (s) = H(s) M 2 + λ (n) s s lw 2 0 2, L 2 }{{ M Ws }}{{ 0 } MISFIT REGULARIZATION 90 % 10 % We want the regularization term to contribute with 10 % to the total cost functional. Then: λ (n) := 0,1 H(s (n) ) + Jδs (n) λ (n) M 2 l 2 W M s (n) + δs (n) λ (n) s 0 2 L 2 Ws 0 We perform a fixed-point iteration to obtain the value of λ (n). 83
inverse problem (reg. param.) We have: C W (s (n+1) ) = H(s (n+1) ) M 2 +λ (n+1) s (n+1) s λ (n) λ (n) lw 2 λ (n) 0 2 L 2 M Ws 0 H(s (n) )+Jδs (n) λ (n) M 2 l 2 W M +λ (n) s (n) +δs (n) λ (n) s 0 2 L 2 Ws 0. We want the regularization term to contribute with 10 % to the total cost functional. Then: λ (n) := 0,1 H(s (n) ) + Jδs (n) λ (n) M 2 l 2 W M s (n) + δs (n) λ (n) s 0 2 L 2 Ws 0 We perform a fixed-point iteration to obtain the value of λ (n). 84
inverse p. (stopping criteria) We stop the inversion process when both the relative data misfit and regularization term do not vary significantly. Mathematically, we require the following two conditions to be satisfied: H(s (n+1) ) M 2 H(s (n) ) M 2 lw 2 l 2 100 M W M M 2 0,5 % lw 2 M And: s (n+1) s 0 2 s (n) s 100λ (n) L 2 0 2 L Ws 2 0 Ws 0 s 0 2 5 %. L 2 Ws 0 85
inverse problem (formulation) Cost Functional: C(s) = H(s) M 2 l 2 +λ s s 0 2 L 2. We want to weight all measurements and resistivities so equal relative errors will contribute equally to the cost functional. 86
inverse problem (formulation) Cost Functional: C(s) = H(s) M 2 l 2 +λ s s 0 2 L 2. We want to weight all measurements and resistivities so equal relative errors will contribute equally to the cost functional. Weighted cost functional: C W (s) = H(s) M 2 l 2 W M +λ s s 0 2 L 2 Ws 0, Goal: To find s := arg mín s C W (s). 87