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2D hp Finite Element Simulation of Sonic Measurements in the Borehole Pawe l J. Matuszyk, Leszek F. Demkowicz, and Carlos Torres-Verd n The University of Texas at Austin JUBILEE SCIENTIFIC CONFERENCE Practical Applications of

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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´

w, May 15, 2015

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

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Some personal story

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

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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

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One of difficult applications that initiated this line of research was simulation of induction tools in presence of casing.

344 Computing with hp-ADAPTIVE FINITE ELEMENTS

r=0m r=0.1m 1.5m Borehole 0.25m CASED WELL Casing Thickness: 0.013m Casing Relative Permeability: 85 Casing Resistivity: 0.00000023 Ohm−m d Receiver 1 Receiver 2 Transmitter

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 moving along the vertical direction.

Matuszyk et al. () 2D hp-FEM for Sonic Logging PK 2015 4 / 1

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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.

1P. 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

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Methods Problem formulation

Multiphysics setting: coupled acoustic-elastic problem

Acoustic domain: borehole fluid (Vf , ρf )

iωp + V 2

f ρf ∇ · v = 0

iωρf v + ∇p = 0 Elastic solid: rock, tool, casing (ρs, Vp, Vs, . . . )      = ∇ · σ + ρsω2u σ = Cε, ε = 1

∇u + ∇Tu
C

= C (ρs, Vi, Qi, . . .) Coupling: (AE) nf · ∇p = ρf ω2nf · u (EA) ns · σ = −pns Acoustic source: nf · ∇p = 1 Anelastic damping (Aki-Richards model): V (ω, Q) = V0

1 πQ ln ω ω0 1 + i 2Q

z STEEL CASING FLUID TOOL FORM ATION 1 FORM ATION 2 FORM ATION ... RECEIVERS SOURCE PM L PM L PM L PM L CEM ENT BOND

Matuszyk et al. () 2D hp-FEM for Sonic Logging PK 2015 6 / 1

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Methods Problem formulation

Weak formulation: coupled acoustic-elastic problem Find (u, p) ∈ (uD, pD) + W × Q such as:

bAA(p, q) + bAE(u, q) = lA(q)

∀q ∈ Q, bEA(p, w) + bEE(u, w) = 0 ∀w ∈ W, where:

bAA(p, q) =

ΩA
∇p · ∇q − ω2

c2

pq

dΩA

bAE (u, q) = −

ΓAE

ρf ω2qnf · u dΓAE bEA(p, w) =

ΓEA

pns · w dΓEA bEE (u, w) =

ΩE
ε(w) : C : ε(u) − ρsω2u · w
dΩE

lA(q) =

Γex

qgex dΓex

ΩΕ ΩΑ Γex ΓΑΕ ΓΕΑ ΓAD ΓED ΓAD ΓED ΓED

PML PML PML FLUID SOLID

Matuszyk et al. () 2D hp-FEM for Sonic Logging PK 2015 7 / 1

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Methods Treatment of multipole sources

Treatment of multipole sources

Monopole n = 0

+ −

Dipole n = 1

+ − + −

Quadrupole n = 2

p0 cos(nθ) = p0 2 einθ

+ p0 2 e−inθ

g−

Multipole source of order n = collection of 2n monopoles placed periodically along a circle of radius r0, 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(nθ) 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:

p = p+ cos(nθ) u =   u+

r cos(nθ)

u+

θ i sin(nθ)

u+

z cos(nθ)

 

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Methods Treatment of multipole sources

Modification of hp-algorithm due to multiphysics settings Let u = su¯

u and p = sp¯ p ⇒ scaling for linear and bilinear forms:      sp su bAA(¯ p, q) + bAE(¯ u, q) = 1 su lA(q) ∀q ∈ Q, bEA(¯ p, w) + su sp bEE(¯ u, w) = 0 ∀w ∈ W, ⇒ (norm scalling) |p|A = sp|¯ p|A |u|E = su|¯ u|E. Now, if sp = |p |A and su = |u |E then the energy norms of the rescaled variables are of

rder 1 and comparison of relative errors is meaningful!

Modified automatic hp algorithm has the form:

set sp = su = 1,

solve the problem for (¯ u, ¯ p)

save new values s′

p = |¯

p|A and s′

u = |¯

u|E

FOR each hp step DO

set sp ← s′

p, su ← s′ u,

perform classical hp-step, calc new solution (¯ u, ¯ p),

save new values s′

p = sp|¯

p|A and s′

u = su|¯

u|E

Matuszyk et al. () 2D hp-FEM for Sonic Logging PK 2015 9 / 1

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Numerical results LWD Logging in Cased Boreholes

LWD Logging in Cased Boreholes: Geometry

0.3m 3.048m/ 10' 1.6764m/ 5.5' 8.51cm 30.0cm 14.0cm FORM ATION PM L PM L PM L

(a) No tool

2.44cm 0.3m 3.048m/ 10' 1.6764m/ 5.5' 8.51cm 30.0cm 14.0cm TOOL FORM ATION PM L PM L PM L

(b) LWD, no casing

2.44cm 0.3m 3.048m/ 10' 1.6764m/ 5.5' 8.51cm 30.0cm 14.0cm TOOL FORM ATION PM L PM L PM L

(c) Well-bonded cas- ing

2.44cm 0.3m 3.048m/ 10' 1.6764m/ 5.5' 8.51cm 30.0cm 14.0cm TOOL FORM ATION PM L PM L PM L

(d) Poor-bonded casing

Matuszyk et al. () 2D hp-FEM for Sonic Logging PK 2015 10 / 1

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Numerical results LWD Logging in Cased Boreholes

LWD Logging in Cased Boreholes: Data Monopole, dipole and quadrupole sonic source. Ricker wavelet, fc = 8 kHz. Tool size: φin = 1.92”, φout = 6.7”. Borehole size: rB = 5.5”.

P-wave S-wave Material ρ Vp Sp Qp Vs Ss Qs kg/m3 m/s µs/ft m/s µs/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

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Numerical results LWD Logging in Cased Boreholes

LWD Logging in Cased Boreholes: Monopole modes

5 10 15 20 25 100 120 140 160 180 200 220 240

Stoneley and pseudo-Rayleigh modes in fast formation

no tool, no casing LWD, no casing LWD, well-bonded casing LWD, poor-bonded casing

Slowness, us/ft Frequency, kHz 5 10 15 20 25 100 120 140 160 180 200 220 240 260 280 300 320 Stoneley modes in slow formation

no tool, no casing LWD, no casing LWD, well-bonded casing LWD, poor-bonded casing

Slowness, us/ft Frequency, kHz

Matuszyk et al. () 2D hp-FEM for Sonic Logging PK 2015 12 / 1

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Numerical results LWD Logging in Cased Boreholes

LWD Logging in Cased Boreholes: Dipole modes

5 10 15 20 25 80 100 120 140 160 180 200 220 240 260 280

1st, 2nd and 3rd flexural modes in fast formation

no tool, no casing LWD, no casing LWD, well-bonded casing LWD, poor-bonded casing

Slowness, us/ft Frequency, kHz 5 10 15 20 25 100 120 140 160 180 200 220 240 260 280 300 320 1st flexural modes in slow formation

no tool, no casing LWD, no casing LWD, well-bonded casing LWD, poor-bonded casing

Slowness, us/ft Frequency, kHz

Matuszyk et al. () 2D hp-FEM for Sonic Logging PK 2015 13 / 1

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Numerical results LWD Logging in Cased Boreholes

LWD Logging in Cased Boreholes: Quadrupole modes

5 10 15 20 25 80 100 120 140 160 180 200 220 240 260

1st and 2nd screw modes in fast formation

no tool, no casing LWD, no casing LWD, well-bonded casing LWD, poor-bonded casing

Slowness, us/ft Frequency, kHz 5 10 15 20 25 100 120 140 160 180 200 220 240 260 280 300 320 1st screw modes in slow formation

320

no tool, no casing LWD, no casing LWD, well-bonded casing LWD, poor-bonded casing

Slowness, us/ft Frequency, kHz

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Numerical results LWD Logging in Cased Boreholes

Poor-bonded casing, fast formation Monopole Dipole Quadrupole

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Numerical results LWD Logging in the Formation with Fracture

LWD Logging in the Formation with Fracture: Geometry

2.44cm 0.3m 1.829m/ 6' 1.067m/ 3.5' 8.51cm 25.0cm 10.795cm TOOL FAST FORM ATION PM L PM L PM L 0.4m 1.5m FAST FORM ATION 2.5cm FRACTURE

(a) Fracture below re- ceivers

2.44cm 0.3m 1.829m/ 6' 1.067m/ 3.5' 8.51cm 25.0cm 10.795cm TOOL FAST FORM ATION PM L PM L PM L 0.4m 2.4m 2.5cm FRACTURE FAST FORM ATION

(b) Fracture facing re- ceivers

2.44cm 0.3m 1.829m/ 6' 1.067m/ 3.5' 8.51cm 25.0cm 10.795cm TOOL PM L PM L PM L 0.4m 3.2m 2.5cm FRACTURE FAST FORM ATION

(c) Fracture above re- ceivers

Matuszyk et al. () 2D hp-FEM for Sonic Logging PK 2015 16 / 1

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Numerical results LWD Logging in the Formation with Fracture

LWD Logging in the Formation with Fracture: Data

TI homogenous formation (Cotton Valley shale): c33 = 58.84 GPa c66 = 29.99 GPa ε = 0.135 γ = 0.18 δ = 0.205

Monopole and quadrupole sonic source, fc = 8 kHz. Tool size: φin = 1.92”, φout = 6.7”. Borehole size: rB = 4.25”. The same parameters for mandrel and fluid as previous.

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Numerical results LWD Logging in the Formation with Fracture

LWD Logging in the Formation with Fracture

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Numerical results LWD Logging in the Formation with Fracture

LWD Logging in the Formation with Fracture Monopole ⇐ Monopole Quadrupole ⇒ Quadrupole

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