Modelling electromagnetic responses from seismic dataDieter Werthmüller [Dieter.Werthmuller@ed.ac.uk], Anton Ziolkowski, and David Wright

IntroductionGood estimates of background resistivit-ies are often crucial in controlled-source elec-tromagnetic (CSEM) feasibility studies and in-versions. Seismic data and well logs are of-ten available prior to CSEM acquisition, butelastic waves and electromagnetic wavesshare no physical parameter.

Contributions• We present a methodology to estimate res-istivities from seismic velocities.

• We apply known methods, including rockphysics, depth trends, structural in-formation, and uncertainty analysis.

• We show an example of the methodologywith data from the North Sea Harding field.

Rock PhysicsWe use a Gassmann-based relation (fG) forthe transformation from P-wave velocity v toporosity φ, and the self-similar model (fs) forthe transformation from porosity φ to resistiv-ity ρ (e.g. Carcione et al., 2007):

ρ = fs(ρs, ρf,m, φ) , where

φ = fG(Ks,Kf, Gs, %s, %f, κ, v) ,

m is the cementation exponent, K and G arebulk and shear moduli, % is density, κ is theKrief exponent, and subscripts s and f standfor solid and fluid fraction (see Fig. 5).

EM-Line

b-7a-3

b-11

b-A01

b-86570000

6575000

412500

417500

4

0.0 0.1 0.2 0.3 0.4

Porosity (-)

1.5

2.5

3.5

4.5

Vel

oci

ty(k

m/s)

Gassmann

10−1

100

101

Res

isti

vit

y(Ω

m)

self-similar

5

The transform is done in three steps:1) Calibrate transform (incl. depth trend,box below) with a well log nearby (b-8).2) Apply to seismic velocity in area of in-terest (including uncertainty, box left).3) Check transform with well log in area ofinterest (if available).

1 10

1

2

3

Dep

th(k

m)

b-8

1 10

b-7

1 10

Resistivity (Ωm)

b-11

1 10

b-A01

1 10

a-3

ρs

mode

±2σ

6

Start – EM-Line – End

1

2

3

Dep

th(k

m)

Grid Sandstone

Seabed

Balder Formation

Base Cretaceous

Background resistivity model [Mode (Ωm)]

0.4

0.7

1.3

2.3

4.17

1D Modelling

1200 1300 1400 1500CMP Position

0

0.2

0.4

0.6Dep

th(k

m)

Start model ρm (Ωm)

1

2

3

4

59a

1200 1300 1400 1500CMP Position

0

0.2

0.4

0.6 Dep

th(k

m)

Final model ρm (Ωm)9b

This resistivity model (box left) has two weak-nesses:1) anisotropy (λ =

√ρv/ρh, ρm =

√ρvρh),

2) resistivities outside well control.CSEM impulse (IR) and step (SR) responseshave different sensitivities to anisotropy(Fig. 10). Only if the anisotropy factor is cor-

rect, inversion of IR and SR yield the same res-ult (Fig. 11). Short offset 1D inversions ofmeasured CSEM data, with correct aniso-tropy factor, improve the background resistivitymodel in the shallow section, were we have nowell control (Fig. 9); the resulting resistivitiesare in this case lower.

0.0 0.5 1.0 1.5 2.0 2.5

Time (s)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Am

pli

tud

e(Ω/(m

2s)

)

×10−10

ρm = 1.0

ρm = 2.0

ρm = 3.0

λ = 1.0

λ = 1.5

λ = 2.0

10a

0.0 0.5 1.0 1.5 2.0 2.5

Time (s)

0

1

2

3

4

5

6

7

Am

plitu

de

(Ω/m

2)

×10−11

10b

1.0 1.5 2.0 2.5

Anisotropy (−)

0

2

4

6

8

10

NR

MSD

(%)

NRMSD IR

NRMSD SR

Model (SR - IR)

0

2

4

6

8

10

∆M

odel

IR-S

R(Ω

m)

11

Uncertainty Analysis

0.4 0.6 0.8 1.0

2

6

10

14

Pro

b.

den

sity

(-) Parameter

0.4 0.6 0.8 1.0

Resistivity (Ωm)

Model

0.4 0.6 0.8 1.0

2

6

10

14

Pro

b.

den

sity

(-)Parameter & Model

1a 1b 1c

-1 0 1∆v : v(z)−vs (z)

pdf

data

0

1

2

3

4

5

Pro

b.

den

sity

(-)

2 3 4Vel. (km/s)

0.6

1.0

1.4

1.8D

ep

th (

km

)

v

vs

2

Data, rock physics model, and rock phys-ics parameter have errors. Chen and Dick-ens (2009) describe a methodology to accountfor the uncertainties related to rock physicsparameters and the rock physics model it-self. They describe the rock physics model asgamma distribution in a Bayesian frame-work, with a defined error E, and the rockphysics parameters as distributions,

f(ρ|θ) =βαxα−1

Γ(α)exp (−βx) ,

where θ is a vector containing all model para-meter distributions, α = 1/E2, and β = (α −1)/ρrp. Here, ρrp is one realization of the rockphysics model with a random set of model para-meters.

We define the distribution of the velo-city from the data themselves (see Fig. 2).The distribution is defined as the differencebetween the measured log values and the val-ues of the smoothed log, v(z) − vs(z). Thisyields resistivity as a probability densityfunction, instead of a deterministic resistivityvalue (Fig. 3c).

1 10

Resistivity (Ωm)

0.6

1.0

1.4

1.8

Dep

th(k

m)

Grid Sandstone

ρs

mode

±σ±2σ3a

vf 2.0 3.0 4.0 vs

Velocity (km/s)

ρf

1

10

ρs

Res

isti

vit

y(Ω

m) deterministic

0.6 0.8 1.0 1.2

Resistivity (Ωm)

1

3

5

Pro

b.

den

sity

(-)

3b

3c

Depth TrendRock parameters are a function of, e.g., litho-logy and depth. We include the followingdependences in our model:• Depth: Ks, Gs, κ, ρs

• Temperature: ρf• Porosity: m• Lithology: Grid Sandstones(delineated with seismic horizons)

1 10

Resistivity (Ωm)

ρ

ρs

ρ(φ[v])

2 3 4

Vel. (km/s)

0.6

1.0

1.4

1.8

Dep

th(k

m)

v

vs

10 30

(GPa)

Ks

Gs

1 10

(GPa)

ρf

ρs

2 3

(-)

m

κ

Grid Sandstone

8

AcknowledgmentWe thank PGS for funding the research andthe Harding partners, BP and Maersk, for per-mission to use the data.

ReferencesCarcione, J. M., B. Ursin, and J. I. Nordskag, 2007, Cross-property

relations between electrical conductivity and the seismic velocityof rocks: Geophysics, 72, E193–E204, doi: 10.1190/1.2762224.

Chen, J., and T. A. Dickens, 2009, Effects of uncertainty inrock-physics models on reservoir parameter estimation usingseismic amplitude variation with angle and controlled-sourceelectromagnetics data: Geophysical Prospecting, 57, 61–74,doi: 10.1111/j.1365-2478.2008.00721.x.

Werthmüller, D., A. Ziolkowski, and D. Wright, 2012, Backgroundresistivity model from seismic velocities: SEG Technical ProgramExpanded Abstracts, 31, doi: 10.1190/segam2012-0696.1.

Conclusions• This method yields the range of back-ground resistivity models, consistentwith the known seismic velocities.• This model provides an additional dataset, which can be used for integrated ana-lysis, or as a starting point for a detailedCSEM feasibility study or inversion.• We will use this background resistivity model

for 3D CSEM modelling for comparison withmeasured data.