2D hp Finite Element Simulation of Sonic Measurements in the - PowerPoint PPT Presentation

2D hp Finite Element Simulation of Sonic Measurements in the Borehole Pawe� l J. Matuszyk, Leszek F. Demkowicz, and Carlos Torres-Verd´ ın The University of Texas at Austin JUBILEE SCIENTIFIC CONFERENCE “Practical Applications of Innovative Solutions Resulting From Scientific Research” Politechnika Krakowska, Krak´ ow, May 15, 2015 Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 1 / 1

Some personal story Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 2 / 1

If you open the preface to the second hp book, you will find: “The research on simulation of logging devices of David Pardo and Maciek Paszy´ nski has been financially supported by Baker Atlas, and The University of Texas at Austin’s Joint Industry Research Consortium on Formation Evaluation sponsored by Aramco, Baker Atlas, BP, British Gas, Chevron, ConocoPhillips, ENI E&P, ExxonMobil, Halliburton, Marathon, Mexican Institute for Petroleum, Norsk-Hydro, Occidental Petroleum, Petrobras, Schlumberger, Shell E&P, Statoil, TOTAL, and Weatherford International Ltd.” Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 3 / 1

One of difficult applications that initiated this line of research was simulation of induction tools in presence of casing. Computing with hp -ADAPTIVE FINITE ELEMENTS 344 r=0m r=0.1m CASED WELL Receiver 2 Casing Thickness: 0.013m 0.25m Casing Resistivity: 0.00000023 Ohm−m d Casing Relative Permeability: 85 Receiver 1 1.5m Transmitter Borehole Figure 13.4 2D cross -section of the geometry of our model axisymmetric through-casing instrument. A 1 . 3 cm thick uniform steel casing surrounds the borehole, where one transmitter and two receiver contact electrodes are Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 4 / 1 moving along the vertical direction.

Motivation Objectives Solution of the coupled elastic-acoustic problem with anelastic attenuation 1 — application to borehole sonic logging. Development and essential modification of the new multiphysics hp -adaptive Finite Element code to the presented coupled problem. Calculation of dispersion curves. 1 P. Matuszyk, L.D. and and C. Torres-Verdin, C., Solution of coupled acoustic-elastic wave propagation problems with anelastic attenuation using automatic hp -adaptivity Comput. Methods Appl. Mech. Engrg. , 213/216: 299-323, 2012. Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 5 / 1

Methods Problem formulation Multiphysics setting: coupled acoustic-elastic problem Acoustic domain: borehole fluid ( V f , ρ f ) STEEL CASING z � i ω p + V 2 f ρ f ∇ · v = 0 PM L i ωρ f v + ∇ p = 0 FORM ATION 1 Elastic solid: rock, tool, casing ( ρ s , V p , V s , . . . ) RECEIVERS FORM ATION 2  = ∇ · σ + ρ s ω 2 u 0   ε = 1 ∇ u + ∇ T u � � σ = C ε , PM L 2  C = C ( ρ s , V i , Q i , . . . )  TOOL FLUID Coupling: n f · ∇ p = ρ f ω 2 n f · u ( AE ) SOURCE FORM ATION ... ( EA ) n s · σ = − p n s Acoustic source: n f · ∇ p = 1 PM L PM L r Anelastic damping (Aki-Richards model): CEM ENT BOND � π Q ln ω 1 � � i � V ( ω, Q ) = V 0 1 + 1 + ω 0 2 Q Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 6 / 1

Methods Problem formulation Weak formulation: coupled acoustic-elastic problem Find ( u , p ) ∈ ( u D , p D ) + W × Q such as: � b AA ( p , q ) + b AE ( u , q ) = l A ( q ) ∀ q ∈ Q , b EA ( p , w ) + b EE ( u , w ) = 0 ∀ w ∈ W , where: Γ AD Γ ED � � ∇ p · ∇ q − ω 2 PML � b AA ( p , q ) = d Ω A pq c 2 Ω Ε Ω Α f Ω A � Γ ΑΕ ρ f ω 2 q n f · u d Γ AE b AE ( u , q ) = − Γ AE Γ ED Γ ex � PML b EA ( p , w ) = p n s · w d Γ EA Γ ΕΑ Γ EA � ε ( w ) : C : ε ( u ) − ρ s ω 2 u · w � � b EE ( u , w ) = d Ω E FLUID SOLID Ω E � PML l A ( q ) = qg ex d Γ ex Γ AD Γ ED Γ ex Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 7 / 1

Methods Treatment of multipole sources Treatment of multipole sources − p 0 cos( n θ ) = p 0 + p 0 + − + + + 2 e in θ 2 e − in θ − � �� � � �� � g + g − n n Monopole Dipole Quadrupole n = 0 n = 1 n = 2 Multipole source of order n = collection of 2 n monopoles placed periodically along a circle of radius r 0 , alternating in sign. The source can be approximated with a Fourier expansion in azimuthal direction θ ⇒ radiation pattern of the n -th order multipole source exhibits cos( n θ ) dependence. Due to symmetries and antisymmetries of linear and bilinear forms, it is sufficient to calculate only solution ( p + , u + ) for excitation g + n and set:   u + r cos( n θ ) p = p + cos( n θ ) u + u = θ i sin( n θ )   u + z cos( n θ ) Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 8 / 1

Methods Treatment of multipole sources Modification of hp -algorithm due to multiphysics settings Let u = s u ¯ u and p = s p ¯ p ⇒ scaling for linear and bilinear forms:  u , q ) = 1 s p  s u b AA (¯ p , q ) + b AE (¯ s u l A ( q ) ∀ q ∈ Q ,  p , w ) + s u b EA (¯ s p b EE (¯ u , w ) = 0 ∀ w ∈ W ,   ⇒ (norm scalling) �| p �| A = s p �| ¯ p �| A �| u �| E = s u �| ¯ u �| E . Now, if s p = � | p � | A and s u = � | u � | E then the energy norms of the rescaled variables are of order 1 and comparison of relative errors is meaningful! Modified automatic hp algorithm has the form: set s p = s u = 1, 1 solve the problem for (¯ u , ¯ p ) 2 save new values s ′ p = �| ¯ p �| A and s ′ u = �| ¯ u �| E 3 FOR each hp step DO 4 set s p ← s ′ p , s u ← s ′ u , 1 perform classical hp -step, calc new solution (¯ u , ¯ p ), 2 save new values s ′ p = s p �| ¯ p �| A and s ′ u = s u �| ¯ u �| E 3 Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 9 / 1

Numerical results LWD Logging in Cased Boreholes LWD Logging in Cased Boreholes: Geometry PM L PM L PM L PM L 1.6764m/ 5.5' 1.6764m/ 5.5' 1.6764m/ 5.5' 1.6764m/ 5.5' PM L PM L PM L PM L FORM ATION FORM ATION FORM ATION FORM ATION 3.048m/ 10' 3.048m/ 10' 3.048m/ 10' 3.048m/ 10' TOOL TOOL TOOL 0.3m 0.3m 0.3m 0.3m PM L PM L PM L PM L 8.51cm 14.0cm 30.0cm 2.44cm 8.51cm 14.0cm 30.0cm 2.44cm 8.51cm 14.0cm 30.0cm 2.44cm 8.51cm 14.0cm 30.0cm (a) No tool (b) LWD, no casing (c) Well-bonded cas- (d) Poor-bonded ing casing Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 10 / 1

Numerical results LWD Logging in Cased Boreholes LWD Logging in Cased Boreholes: Data Monopole, dipole and quadrupole sonic source. Ricker wavelet, f c = 8 kHz. Tool size: φ in = 1 . 92”, φ out = 6 . 7”. Borehole size: r B = 5 . 5”. P-wave S-wave Material ρ V p S p Q p V s S s Q s kg / m 3 m / s µ s / ft m / s µ s / ft Mandrel 5900 5862 52 100 2519 121 50 Casing 7500 6096 50 1000 3350 91 1000 Cement 1920 2822 108 40 1730 176 50 Fluid 1100 1524 200 100 – – – Fast formation 2300 4354 70 1000 2629 116 1000 Slow formation 2100 2540 120 1000 1270 240 1000 Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 11 / 1

Numerical results LWD Logging in Cased Boreholes LWD Logging in Cased Boreholes: Monopole modes Stoneley and pseudo-Rayleigh modes in fast formation Stoneley modes in slow formation 320 240 300 220 280 260 200 Slowness, us/ft Slowness, us/ft 240 no tool, no casing 180 220 LWD, no casing LWD, well-bonded casing 200 160 LWD, poor-bonded casing 180 no tool, no casing 140 160 LWD, no casing LWD, well-bonded casing 140 120 LWD, poor-bonded casing 120 100 100 0 5 10 15 20 25 0 5 10 15 20 25 Frequency, kHz Frequency, kHz Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 12 / 1

Numerical results LWD Logging in Cased Boreholes LWD Logging in Cased Boreholes: Dipole modes 1st, 2nd and 3rd flexural modes in fast formation 1st flexural modes in slow formation 280 320 no tool, no casing 260 LWD, no casing 300 LWD, well-bonded casing 240 280 LWD, poor-bonded casing 260 220 Slowness, us/ft Slowness, us/ft 240 200 220 180 200 160 180 140 160 no tool, no casing 120 140 LWD, no casing LWD, well-bonded casing 100 120 LWD, poor-bonded casing 80 100 0 5 10 15 20 25 0 5 10 15 20 25 Frequency, kHz Frequency, kHz Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 13 / 1

Numerical results LWD Logging in Cased Boreholes LWD Logging in Cased Boreholes: Quadrupole modes 1st and 2nd screw modes in fast formation 1st screw modes in slow formation 260 320 no tool, no casing 240 LWD, no casing 300 LWD, well-bonded casing 280 220 LWD, poor-bonded casing 260 200 Slowness, us/ft Slowness, us/ft 240 180 220 160 200 180 140 160 no tool, no casing 120 140 LWD, no casing LWD, well-bonded casing 100 120 LWD, poor-bonded casing 80 100 0 5 10 15 20 25 0 5 10 15 20 25 Frequency, kHz Frequency, kHz 320 Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 14 / 1

Numerical results LWD Logging in Cased Boreholes Poor-bonded casing, fast formation Monopole Dipole Quadrupole Matuszyk et al. () 2D hp -FEM for Sonic Logging PK 2015 15 / 1