A Bayesian model is developed to estimate porosity, fluid saturation, and pore pressure in reservoirs using seismic and electromagnetic (EM) data. Within the Bayesian framework, unknown reservoir parameters at each pixel in space are considered as random variables and the co-located geophysical properties (seismic P- and S-wave velocity, density, and electrical conductivity), inverted from seismic and EM measurements, are considered as data. Rock-physics models are derived from borehole logs and are considered as random functions between the reservoir parameters and the geophysical properties. Using Markov chain Monte Carlo (MCMC) methods, many samples of each unknown variable are obtained from the Bayesian model, which subsequently are used to infer the unknown variable (reservoir parameter) as well as its uncertainty. A study, based on borehole data from a site in the Troll field, shows that the developed method is more effective for reservoir parameter estimation than traditional regression methods.
Joint inversion of 2D or 3D seismic and EM data for reservoir parameter estimation is computationally expensive (Hoversten et al., 2005). One alternative to this inversion is a two-step process: (1) inverting seismic and EM data separately to produce seismic P- and S-wave velocity, density, and electrical conductivity, and (2) transforming the inverted data to reservoir properties using rock-physics models. Since the relationships between reservoir parameters and geophysical properties are non-unique and subject to uncertainty, traditional deterministic rock-physics models or regression methods are often ineffective and may lead to biases in reservoir parameter estimation. However, in recent years, efforts have been made to incorporate uncertainty into reservoir parameter estimation by using stochastic rockphysics models. Avseth et al. (2001) developed an integrated method to map occurrence probabilities of different lithofacies and fluid properties from seismic amplitude variations with offset (AVO) data, for data collected from a North Sea site. They first defined seismic lithofacies and then used them as the link for tying fluid properties to seismic AVO data using statistical and rock-physics models. The success of the method relies heavily on the existence of seismic lithofacies and distinction in fluid properties among those facies. The method is site-specific and requires considerable geological, geophysical, and sedimentological information. Bachrach et al. (2004) presented a method for quantitative estimation of reservoir parameters (porosity, water saturation, and effective stress) using seismic data. They considered the reservoir estimation given seismic data (seismic P- and Swave velocity, density, or any function of the three variables) as a joint estimation problem within a Bayesian framework. The reservoir parameters in the Bayesian framework are considered as random variables, and the known geophysical attributes are considered as data. The rock-physics relations between the unknown reservoir parameters and the known geophysical data are used to define likelihood functions. The random variables, given data, likelihood functions, and other prior information together define a joint posterior probability distribution of all the unknown variables. The maximum aposterior probability (MAP) and traditional Monte Carlo methods are used to find marginal posterior probability distribution from the joint posterior distribution function. .
