estimation of rock properties from seismic, em and gravity

EGM 2010 International WorkshopAdding new value to Electromagnetic, Gravity and Magnetic Methods for Exploration

Capri, Italy, April 11-14, 2010

Estimation of rock properties from seismic, EM and gravity well-log measurements

F. Miotti1, G. Bernasconi1, D. Rovetta1, P. Dell’Aversana2 1Politecnico di Milano, 2eni e&p division

Summary Geophysical methods are used to obtain physical properties of the subsurface, by inverting the constitutive equations. If there exists a common set of rock properties (cross-properties) that influence different measurements, their joint utilization can reduce the ambiguities of the interpretation. We select a real test case in a reservoir scenario, and we explore how to determine rock porosity and fluids saturation from P-velocity, conductivity and density of the equivalent compound medium. Introduction The main objective of hydrocarbon exploration is the quantitative characterization of the subsurface properties, and in particular the estimation of the petrophysical parameters of a potential reservoir. This problem is still challenging and subject to a large degree of uncertainty. Seismic still represents the most used and efficient technique, and new imaging and inversion procedures are continuously proposed: one of the current frontiers is the recovery of information at very low frequencies (Cerney et al., 2007). Electromagnetic and gravimetric data provide precious complementary information, and they are increasingly acquired and processed (Chen et al. 2007, Gallardo et al 2003). For example, during the last years, many studies involved the Controlled Source Electromagnetic method (CSEM) as integration of the seismic and gravimetric methods (Eidesmo et al 2002). The key point is that different geophysical measurements sense different rock properties, and their systematic integration can reduce the uncertainty of the geological interpretation (Chen et al. 2009). The analysis of the relations between rock properties and geophysical measurements (Mavko et al., 1998; Schön, 1996) permits to understand the conditioning of the inverse problem and to find the cross-links between different domains (Carcione et al., 2007).

Figure 1. Rock properties and constitutive equations.

On the rock domain side, cross-properties are physical variables, for example porosity and fluids saturation, influencing measurements of different nature, such as seismic, density and electromagnetic (Figure 1). Cross-properties make it possible an efficient data integration (Miotti et al., 2009).



In this paper we use well-log data to explore the potential of the joint utilization of geophysical methods. We describe a Bayesian procedure to estimate porosity and fluids saturation in a reservoir scenario from velocity, conductivity and density measurements of the equivalent compound medium. We derive also a quantitative indication of the reliability of the estimated parameters. The procedure is successfully tested on real data. Estimation of rock properties The starting point is the definition of the constitutive equations for a medium composed of a porous dry matrix filled with fluids. The literature offers a wide variety of models (Schön 2004), so that it is necessary to tune the equations to the particular case. For the seismic data, the equation relates the matrix and fluid bulk modulus, shear modulus, density and porosity, to the measured seismic wave velocity. In this study we use the Raymer model (Raymer, 1980). P-wave velocity Vp of the equivalent medium is,

( ) 1 2fluidsolidp VVV ⋅+⋅−= φφ (1)

with

solid

solid

solidVρ

μ⋅+= 3

4 Ksolid,

fluid

fluid ρKVfluid = ,

gow SSS ⋅+⋅+⋅= gowfluid KKKK , , iisolid KfK ⋅= ∑=

solidN

1i

i

N

1iisolid

solid

f μ⋅=μ ∑=

∑=

⋅ρ=ρsolidN

1iiisolid f , gow SSS ⋅+⋅+⋅= gowfluid ρρρρ .

Φ is the porosity, K is the Bulk modulus, µ is the shear modulus, ρ is the density, fi is the fractional volume of the i-th solid component, and S the fluid saturation. Pedix i refers to the i-th solid phase, o to oil, w to water, and g to gas. For the electromagnetic data, the equation relates the electrical conductivity of the individual specimens and the porosity, to the overall measured conductivity. In this study we use the Complex Index Refractive Method (CRIM; Schon, 2004). For negligible permittivity, the electric conductivity of the compound medium is

( ) (2

2/12/12/1

1

2/1)(1 ⎥⎦

⎤⎢⎣

⎡++⋅+⋅⋅−= ∑

=ggooww

N

iii SSSf

solid

σσσφσφσ )

)

. (2)

For the gravity data, the equation relates the matrix and fluid densities and the matrix porosity, to the equivalent medium density (Mavko, 1998):

( ) ( ggooww

N

ii SSSsolid

ρρρφρφρ ++⋅+⋅⋅= ∑=1i

f-1 . (3)

According to Figure 2, the forward problem can then be phrased as a non linear matrix relation,

)m(gd = , (4) where m is the vector of model parameters (rock properties) and d is the vector of geophysical measurements. Rock parameters are derived by inverting equation (4). We use a probabilistic approach (Tarantola, 2005). In this way we are able to describe, through probability densities, model and data uncertainties, as well as any prior information we have on the particular scenario. We define prior uncertainties with Gaussian probabilities, and we start from an initial model

Tow SSm ],,[ 0000 φ=



Figure 2. Data and model space, reservoir scenario.

The solution is obtained by solving iteratively the following functional,

[ ] ( ) ( )[ ]0kkobsk1

dT

k

11Mk

1d

Tk01k mmGd)m(gCGCGCGmm −−−+−= −−−−

+

with Gk Jacobian matrix at the k-th iteration, Cd data covariance matrix (uncertainity/noise in the data), CM model covariance matrix (uncertainity/prior information on the model parameters), dobs observed data. The algorithm stops when the model update stays below a threshold. The outputs are the estimated porosity, the fluids saturation, and their confidence. Analysis of the Jacobian matrix reveals that porosity is the best resolved parameter, water saturation is the second one, and that there is a strong leakage between the estimation of oil and gas saturation (Miotti, 2009). We show here the inversion of rock properties from a real well-log. Figure 3 is the result: oil targets are confirmed by drilling observations. The estimated petrophysical parameters are also inserted in the “forward” relation to visually check and to validate the procedure (Figure 4). A crucial phase is the selection of the constitutive equations: this can explain the (small) mismatch between the real data and the recalculated one with the inverted parameters.

Figure 3. Well-log real data (blue curves on the left): P velocity, density, and electrical conductivity. Estimated

petrophysical parameters (red curves on the right): porosity, oil saturation, and water saturation.



Figure 4. Original data (blue) and recalculated data (red) from inverted rock properties.

Conclusions Geophysical measurements can be linked to subsurface physical parameters through constitutive equations. The analysis of these equations permits to investigate the parameter observability and the reliability of the interpretation. We have presented a Bayesian approach for the derivation of the petrophysical parameters in a reservoir scenario, from the measured properties of the compound medium, namely P-wave velocity, electrical conductivity and density. Ambiguities are reduced by making use of the cross-properties that link heterogeneous physical domains. Furthermore, the algorithm can be ”tuned” by setting the uncertainity of data and a priori model, and by choosing the more appropriate rock physic model. Acknowledgements We acknowledge eni e&p division for supporting this study, for providing the real data, and for permission to present this work. References Carcione J., Ursin, B., Nordskag, J.I., 2007, "Cross-property relations between electrical conductivity and the

seismic velocity of rocks", Geophysics, 72, 193–204. Cerney B., Bartel, D.C., 2007, “Uncertainties in low-frequency acoustic impedance models”, The Leading Edge,

26, 74-87. Chen J., Dickens, T.A., 2009, “Effects of uncertainty in rock-physics models on reservoir parameter estimation

using seismic amplitude variation with angle and controlled-source electromagnetics data”, Geophysical Prospecting, 57, 61-74.

Chen J., Hoversten, G.M., Vasco, D., Rubin, Y., Hou, Z., 2007, “A Bayesian model for gas saturation estimation using marine seismic AVA and CSEM data”, Geophysics, 72, 85–95.

Eidesmo T., S. Ellingsrud, L.M. MacGregor, S. Constable, M.C. Sinha, S. Johansen, F.N. Kong, and H. Westerdahl, 2002, “Sea bed logging, a new method for remote and direct identification of hydrocarbon filled layers in deepwater areas”, First Break, 20, 144-152.

Gallardo-Delgado L.A., Perez-Flores, M.A., Gomez-Trevino, E., 2003, “A versatile algorithm for joint 3D inversion of gravity and magnetic data”, Geophysics, 68, 949–959.

Hashin, Z., Shtrikman, S., 1963, “A variational approach to the elastic behavior of multiphase materials”, J. Mech. Phys. Solids, 11, 127-140.

Mavko, G., Mukerji, T., Dvorkin, J., 2004, "The Rock Physics Handbook", Cambridge University Press. Miotti F., Bernasconi, G., Rovetta, D., 2009, “Joint inversion of rock properties: a case study”, Proceeding of 71st

EAGE Conference.



Raymer, L. L., Hunt, E.R., Gardner, J.S., 1980, “An improved sonic transit time to porosity transform”, 21st Annual Logging Symposium, Transactions of the Society of Professional Well Log Analysts, Paper 546.

Schön, J.H., 2004, “Physical Properties of Rocks, Fundamentals and Principles of Petrophysics”, Handbook of Geophysical Exploration, Seismic Exploration, Vol. 18, Elsevier.

Tarantola, A., 2005, “Inverse Problem Theory”, SIAM.

estimation of rock properties from seismic, em and gravity

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