three-dimensional joint petrophysical inversion of...

Three-dimensional joint petrophysical inversion of electromagnetic and seismic dataG. Gao∗, A. Abubakar, T. M. Habashy, and G. Pan, Schlumberger-Doll Research

SUMMARY

A method for obtaining porosity and fluid saturation distri-butions by inverting electromagnetic (EM) and seismic mea-surements is extended to three-dimensional (3D) geometries.For seismic modeling, we use the acoustic approximation. Asdemonstrated in the literature, the joint inversion of EM andseismic data reduces the non-uniqueness problem of deter-mining porosity and saturation distributions. As the inversionalgorithm we employ the preconditioned nonlinear conjugategradient method and an alternating minimization scheme. Theinversion algorithm is parallelized by using the message pass-ing interface (MPI) library. Application of the algorithm on a3D surface prospecting problem demonstrates that the methodis able to simultaneously estimate porosity and fluid saturationdistributions of the subsurface.

INTRODUCTION

It was shown that the use of petrophysical links for jointly in-verting EM and seismic data may reduce the non-uniquenessfor the determination of formation porosity and fluid satura-tion distributions (Hoversten et al., 2006; Gao et al., 2011,2012). Hence, the joint petrophysical inversion has a greatpotential to be an additional tool for reservoir characterizationapplications. The works in Gao et al. (2011, 2012) are lim-ited for two-dimensional (2D) geometries. In this paper, weextend the approach in Gao et al. (2012) to 3D for invertingcontrolled-source electromagnetic (CSEM) and seismic full-waveform data.

Modeling and inversion for 3D problems are extremely time-consuming and challenging. The Gauss-Newton method thatwe adopted in Gao et al. (2011, 2012) for 2D applications re-quires the construction and storage of the Jacobian matrix. Al-though a single-physics inversion can be carried out by com-bining the Gauss-Newton method with a compression schemefor reducing its memory requirement (Li et al., 2011), the useof the Gauss-Newton method for 3D multi-physics inversionscan still be very memory and computation intensive. Hence, inthis work, we employ the preconditioned nonlinear conjugategradient (P-NLCG) method investigated earlier by Mackie et al.(2007) and Hu et al. (2011). For seismic applications, Huet al. (2011) showed that the P-NLCG method produces in-version results comparable to those obtained by the Gauss-Newton method. To further increase the efficiency, we par-allelize the inversion algorithm using the MPI library.

To directly invert for porosity and fluid saturation distribu-tions from seismic and EM measurements, petrophysical trans-forms are required. In our 3D inversion algorithm, we employArchie’s equation (Archie, 1942) to link conductivity/resistivityto porosity and fluid saturations, and the fluid substitution equa-tions (Gassmann, 1951) coupled with the critical porosity model

of Nur (1992) to link P-wave velocity and mass-density toporosity and saturations. Gao et al. (2012) present an anal-ysis about how the uncertainty of parameters in these petro-physical transforms affects the inversion results. In this paper,we assume that those petrophysical relationships and their pa-rameters are known. In our implementation, for the seismicmodeling, we employ an acoustic approximation. We presenta numerical example to show that the proposed 3D joint in-version approach can help to efficiently and accurately obtainporosity and fluid saturation distributions from EM and seis-mic measurements.

THEORY

Petrophysical relationships

Archie’s equationArchie (1942) discovered that the formation conductivity (σ )can be expressed as a function of the porosity (φ ) and the watersaturation (Sw) according to the relationship:

σ =1a

σwφ mSnw, (1)

where a is a tortuosity factor, m is the porosity/cementationexponent, n is the saturation exponent, and σw is the conduc-tivity of the formation saline water. Archie’s equation is validfor clean sand formations. Since the development of Archie’sequation, other variants have been introduced to account forthe conduction of the clay content in rocks, such as Waxman-Smits equation (Waxman and Smits, 1968). Archie’s equationand its variants have served as important tools to calculate thehydrocarbon saturation from EM measurements.

Gassmann’s equationsOn the other hand, the fluid substitution model (Gassmann,1951) links seismic velocities to reservoir parameters, such asporosity (φ ), water saturation (Sw), and oil saturation (So) orgas saturation (Sg). In fluid-saturated rocks, the compressionalwave (P-wave) velocity VP is expressed as follows:

VP =

√Ksat

ρsat, (2)

where

Ksat = (1−β )Kma +β 2M, (3)

M =

(β −φKma

+φKf

)−1

, (4)

Kf =

(Cw

Sw

Kw+Co

So

Ko+Cg

Sg

Kg

)−1

, (5)

ρsat = (1−φ)ρma +φ(

Swρw +Soρo +Sgρg)

. (6)

In equations 3 and 4, β is the Biot coefficient, which in generalis a function of porosity. In this study, we chose the critical

© 2012 SEGSEG Las Vegas 2012 Annual Meeting

3D joint petrophysical inversion

porosity model of Nur (1992) for β ,

β =

φ/φc, 0≤ φ ≤ φc ,1 , φ > φc ,

(7)

where φc is the critical porosity above which solids becomesuspensions. In the above equations, Ksat and ρsat are the bulkmodulus and the bulk density of the fluid-saturated rock, Kf isthe bulk modulus of the pore fluid, Kma and ρma are the bulkmodulus and the density of the matrix (solid or grain), and Kw,Ko, and Kg are the bulk modulus of water, oil, and gas, re-spectively. ρw, ρo, and ρg are the density of water, oil, andgas, respectively. Cw, Co, and Cg are correction terms for wa-ter, oil, and gas, respectively. Note that the gas correction wasfound to be necessary by Hoversten et al. (2006) in the exis-tence of gas, while Cw and Co are usually set to unity. Due tothe assumption of acoustic approximation, the shear modulusis zero.

Inversion methodology

In our inversion algorithm, as forward solvers we employ 3Dfrequency-domain approaches for both EM ( Zaslavsky et al.,2011) and acoustic ( Pan et al., 2012) modeling. The unknownvectors representing porosity and water saturation distributionsare denoted by mφ and mSw . The total unknown vector is de-fined by m =

[mφ ;mSw

]T, where T denotes the matrix trans-

pose. The multiplicative cost function for the single-physicspetrophysical inversion is given by

Φ(m) = Φd (m)×Φm (m) . (8)

More details of the multiplicative cost function can be foundin Abubakar et al. (2008). The normalized data misfit Φd isgiven by

Φd (m) =

12

NF∑

k=1

η2k

∑NSi=1

∑NRj=1

∣∣wd; i, j,k[di, j,k− si, j,k(m)

]∣∣2

∑NSi=1

∑NRj=1

∣∣wd; i, j,kdi, j,k∣∣2

=12‖Wd [d− s(m)]‖2 , (9)

where NF, NS, and NR are the number of frequencies, sources,and receivers, respectively. The vector d is the measured EMor seismic data and s is the simulated EM or seismic responsevector for a given a model parameter m. The matrix wd; i, j,k isthe data weighting matrix whose diagonal elements are the in-verse of the estimates of the standard deviation of the measure-ment noise. For surface EM and seismic inversions, wd; i, j,k in-cludes the weighting based on the Jacobian of the initial modelto safeguard against the measurements with relative high am-plitudes that tend to dominate the inversion (Abubakar et al.,2009). The frequency weighting factor ηk is used for the seis-mic inversion as a way to prevent the high-frequency compo-nents from dominating the inversion process (Hu et al., 2009).The regularization cost function Φm is either the `2-norm orthe weighted `2-norm as described in Abubakar et al. (2008).

Preconditioned nonlinear conjugate gradient methodTo reduce the memory requirement we employ a NLCG methodusing the preconditioner as discussed in Mackie et al. (2007)

and Hu et al. (2011). In the P-NLCG method the CG directionv for each iteration n is computed as follows:

v0 = −C−10 g0 , (10)

vn = −C−1n gn +βnpn−1 , n = 1,2, · · · , (11)

where gn is the gradient of the cost function and Cn is thepreconditioning operator. Following Hu et al. (2011) the pre-conditioning operator is chosen as follows:

Cn = Φm (mn)diag(Hn)+Φd (mn)LTn Ln , (12)

where Ln is the regularization matrix operator, and Hn is theGauss-Newton Hessian matrix.

Because we employ the Polak-Ribiere scheme (Polak, 1971),βn is given by

βn =gT

n (C−1n gn−C−1

n−1gn−1)

gTn−1C−1

n−1gn−1. (13)

After the CG direction vn is obtained, the search vector is cal-culated by pn = γnvn, where γn is a scalar scaling factor ob-tained by minimizing the quadratic approximation of the costfunction as described in Habashy and Abubakar (2004). Then,the model parameter is updated as follows:

mn+1 = mn +αnpn , (14)

where αn is the line-search parameter for iteration n. Theline-search procedure is used to guarantee that the cost func-tion value is always reduced during the optimization process(Habashy and Abubakar, 2004). The action of the precondi-tioner on a vector is calculated by solving

Cnxn =−gn , (15)

using a conjugate-gradient least-square iterative method (Goluband van Loan, 1996).

Joint inversion algorithmIn the past, for joint petrophysical inversions we have devel-oped two different numerical schemes: an alternating approach,which runs EM and seismic inversion alternatingly in each it-eration (Gao et al., 2010); and a simultaneous approach, wherewe minimize the EM and seismic cost function simultaneously(Gao et al., 2012). The simultaneous inversion approach re-quires more memory storage and is computationally more ex-pensive. The inversion results reported in this paper are fromthe alternating inversion approach.

For the alternating approach to effectively work, a special strat-egy based on the data sensitivity must be included in the algo-rithm. According to the sensitivity analysis discussed in Gaoet al. (2012), seismic measurements are lacking sensitivity tothe water saturation. This means that if we use the EM in-version results for the seismic inversion and let the water sat-uration change freely, the water saturation may significantlychange after the seismic inversion step because seismic mea-surements are not very sensitive to the water saturation change.This will make the alternating minimization scheme non-convergent.On the other hand, EM data cannot uniquely determine poros-ity and water saturation. This means that both parameters may



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Figure 1: True petrophysical models.

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Figure 2: True geophysical models.

wildly change after the EM inversion step. To deal with thisproblem, we invert for the water saturation using tight boundsin the seismic inversion step, while the porosity is inverted forusing tight bounds during the EM inversion step. In addition,in each alternating inversion step, we allow both EM and seis-mic inversions to run more than one iteration to guarantee thatthe cost functions is always reduced compared to those in theprevious alternating inversion step. This significantly helps thealternating inversion approach.

INVERSION EXAMPLE

To demonstrate the proposed joint petrophysical inversion ap-proach using an alternating minimization scheme, we employ amarine configuration with two producing petroleum reservoirs.The objective is to obtain the porosity and fluid saturation dis-tributions of the reservoirs and their shapes and locations.

Model setupFigure 1 shows porosity and water saturation distributions ofthe test model, while Figure 2 shows the corresponding resis-tivity, P-wave velocity, and mass density distributions calcu-lated using the petrophysical relationships. For convenience,we only show three volume slices on the xz-plane, yz-plane,and xy-plane. The water depth is 1 km. The subsurface hassome specially structured porosity and saturation distributions

due to particular geological depositional process, of which themost interesting parts are the two oil-saturated reservoirs. Bothreservoirs are 2 km wide and located at a depth of 2 km. Theleft reservoir is 200 m thick, while the thickness of the rightreservoir is 100 m. The distance between the right edge of theleft reservoir and the left edge of the right reservoir is 2 km.Both reservoirs extend from -2 to +2 km in the y-direction. Theporosity of the reservoirs is 0.3, while that of the other portionbelow the seabed is 0.2. Both reservoirs are oil saturated witha water saturation of 0.2.

For the petrophysical transforms, we use the following Archie’sparameters: a = 1, m = 1.8, n = 2, and σw = 20.5 S/m. Theseabed resistivity is roughly 1 ohm-m, while the reservoir re-sistivities are roughly 11 ohm-m. For the rock-physics model,the following parameters are used: Kma = 37 GPa, µma = 44GPa, Kw = 2.56 GPa, Ko = 0.75 GPa, ρma = 2650 Kg/m3,ρw = 1050 Kg/m3, ρo = 750 Kg/m3, and φc = 0.4. All fluidcorrection terms are set to 1.

Survey setupFor the EM survey, sources are located at five y-lines (y=-3,-1.5, 0, 1.5, and 3 km) at a depth of 950 m and in each y-linethere are 19 inline electric sources spaced at 500 m apart. Wealso use five y-lines (y=-3, -1.5, 0, 1.5, and 3 km) for receiverslocated at the sea bottom, and in each y-line there are 19 inlineelectric receivers spaced 500 m apart. The source frequencyis 0.25 Hz. For the seismic survey, we use five y-lines (y=-3,-1.5, 0, 1.5, and 3 km) for sources, and in each y-line we have19 monopole sources located at the sea bottom and spaced 500m apart. We also use five y-lines (y=-3, -1.5, 0, 1.5, and 3 km)of receivers, and each y-line contains 19 hydrophones locatedat the sea-bottom and spaced 500 m apart.

We process the seismic data using a frequency-domain ap-proach. From the time-domain data, we choose frequencies:1 and 2 Hz for the inversion. We added 2% Gaussian randomnoise to both EM and seismic data.

For EM and seismic modeling, we use the same regular gridconfiguration: the grid number in x-, y-, and z-directions are100, 60, and 60, respectively. The grid size is 100 m, 100 m,and 50 m. As initial models, we use a homogeneous modelwith 0.1 for the porosity and 0.8 for the water saturation.

Single-physics inversion resultsFor comparison purpose, we first show the inversion resultsusing EM data only and seismic data only. Figure 3 showsinverted porosity and water saturation distributions using EMdata only, while Figure 4 shows inverted porosity and watersaturation distributions using seismic data only. The final datamisfit for EM inversion is 1.72% (after iteration 89), while thatof seismic inversion is 1.77% (after iteration 47). Because weare using the P-NLCG approach, the number of iterations is, ingeneral, larger than that of the Gauss-Newton approach. Bothinversions have been nicely converged. We observe that theseismic inversion recovers relatively good porosity distribu-tion; however, the water saturation image is not different thanthe initial model. The EM inversion reconstructs both porosityand water saturation distributions. The locations and shapes ofreservoirs are reasonably reconstructed. However, their poros-



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Figure 3: Inverted petrophysical models from EM data only.

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Figure 4: Inverted petrophysical models from seismic dataonly.

ity and water saturation values are not correct. This is under-standable because EM data is only sensitive to the product ofporosity and water saturation.

Joint inversion resultsFigure 5 shows the inverted porosity and water saturation dis-tributions using the alternating joint inversion algorithm. Fig-ure 6 shows the corresponding resistivity, P-wave velocity, andmass density distributions. After inversion, the seismic datamisfit is reduced to 1.77%, while the EM data misfit is reducedto 1.67%. The algorithm used 24 alternating iterations. In eachalternating inversion step, we set the maximum number of iter-ations to 10 and 6 for the EM inversion and seismic inversion,respectivly. Figures 5 and 6 clearly show that the joint inver-sion obtains both porosity and water saturation distributions.Note that the resolution of the water saturation image is lowerthan that of the porosity image. This is because of the low-resolution nature of the CSEM data.

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Figure 5: Inverted petrophysical model from joint inversion.

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Figure 6: Calculated geophysical model from the invertedpetrophysical models of the joint inversion.

CONCLUSIONS

We developed a three-dimensional joint inversion approach todirectly invert for porosity and saturation distributions fromelectromagnetic and seismic full-waveform data. The approachuse a preconditioned nonlinear conjugate gradient (P-NLCG)algorithm. The inversion algorithm is enhanced by using theMPI-based parallelization technique, which is carefully de-signed to nearly uniformly distribute computation and mem-ory storage to all processors. We show that the inversion ap-proach can estimate porosity and fluid saturation distributionsand identify the location of hydrocarbon by fitting both EMand seismic data misfits.

ACKNOWLEDGMENTS

The authors thank M. Zaslavsky and V. Druskin from Schlumberger-Doll Research for providing the 3D electromagnetic forwardmodeling code and J. Liu from Schlumberger for his contribu-tions on the 3D electromagnetic inversion code.


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