Miguel Bosch

Seismic inversion for reservoir properties combining statistical rock physics and geostatistics: A review

There are various approaches for quantitative estimation of reservoir properties from seismic inversion. A general Bayesian formulation for the inverse problem can be implemented in two different work flows. In the sequential approach, first seismic data are inverted, deterministically or stochastically, into elastic properties; then rock-physics models transform those elastic properties to the reservoir property of interest. The joint or simultaneous work flow accounts for the elastic parameters and the reservoir properties, often in a Bayesian formulation, guaranteeing consistency between the elastic and reservoir properties. Rock physics plays the important role of linking elastic parameters such as impedances and velocities to reservoir properties of interest such as lithologies, porosity, and pore fluids. Geostatistical methods help add constraints of spatial correlation, conditioning to different kinds of data and incorporating subseismic scales of heterogeneities.

Lithologic tomography: From plural geophysical data to lithology estimation

The information provided by different geophysical data sets (gravimetric, magnetic, seismic, etc.) can be used, together with petrophysical and geostatistical information, to estimate the major lithological properties of the rocks within the studied volume. The formalization of this inverse problem requires a joint representation and parameterization of the different media properties in the model. The information relating rock properties together couples the inversion of the plural geophysical data sets and allows one to relate the observations with the lithological parameters of the model. The representation by probability density functions (pdfs) of the different types of information entering the problem is also required and provides the mathematical framework to formulate their combination. The resulting joint posterior pdf is composed of two factors: the joint likelihood function, which is the product of independent likelihood functions associated with each geophysical data set, and the joint prior pdf. The latter decomposes, following a partition of the model parameter space in a primary (lithological) subspace and secondary (physical) subspace, as a marginal pdf over the primary model parameter space and a conditional pdf over the secondary model parameter space. A Markov-chain Monte Carlo method was adapted to sample joint models from the posterior pdf: (1) the method starts with a Markov-chain sampling primary models from the marginal prior pdf, (2) the chain is extended to the joint model space, by sampling from the conditional pdf of the secondary parameters with respect to the primary parameters, and (3) it is modified to sample from the posterior pdf, by applying the Metropolis rule, which uses the evaluation of the joint likelihood function to accept or reject model transitions in the sampling chain. Finally, posterior marginal or posterior conditional pdfs for the model parameters or the model properties can be straightforwardly calculated from the set of joint models sampled by the chain.

Joint inversion of gravity and magnetic data under lithologic constraints

Interpreting exploration data requires combining different types of information to solve the geologic puzzle. It implies bringing together all data components into an image that makes conceptual sense in terms of the geology of the exploration area. The identification of geologic objects and the inference of a spatial description of the lithology—consistent with all available information—are the objectives of the process.

Petrophysical seismic inversion conditioned to well-log data: Methods and application to a gas reservoir

Hydrocarbon reservoirs are characterized by seismic, welllog, and petrophysical information, which is dissimilar in spatial distribution, scale, and relationship to reservoir properties. We combine this diverse information in a unified inverse-problem formulation using a multiproperty, multiscale model, linking properties statistically by petrophysical relationships and conditioningthemtowell-logdata.Twoapproacheshelpus:1Markov-chainMonteCarlosampling,whichgeneratesmanyreservoir realizationsforestimatingmediumpropertiesandposteriormarginalprobabilities,and2optimizationwithaleast-squaresiterativetechniquetoobtainthemostprobablemodelconfiguration. Our petrophysical model, applied to near-vertical-anglestacked seismicdataandwell-logdatafromagasreservoir,includesadeterministic component, based on a combination of Wyllie and Wood relationships calibrated with the well-log data, and a random component, based on the statistical characterization of the deviations of well-log data from the petrophysical transform.At the petrophysical level, the effects of porosity and saturation on acoustic impedance are coupled; conditioning the inversion to well-log data helps resolve this ambiguity. The combination of well logs, petrophysics, and seismic inversion builds on the correspondingstrengthsofeachtypeofinformation,jointlyimproving 1 cross resolution of reservoir properties, 2 vertical resolution of property fields, 3 compliance to the smooth trend of property fields, and 4 agreement with well-log data at well positions.

Joint gravity and magnetic inversion in 3D using Monte Carlo methods

We jointly invert gravity and magnetic data following a Monte Carlo method that provides estimation for a 3D model of the structure and physical properties of the medium. In particular, the model layer geometry and the density and magnetic susceptibility fields within layers are estimated, and their uncertainties are described with posterior probabilities. This method combines the gravity and magnetic data with prior information of the mass density and magnetic susceptibility statistics, and statistical constraints on the model interface positions. The resulting model realizations jointly comply with the observations and the prior statistical information.

A Monte Carlo approach to the joint estimation of reservoir and elastic parameters from seismic amplitudes

Inversion of seismic data and quantification of reservoir properties, such as porosity, lithology, or fluid saturation, are commonly executed in two consecutive steps: a geophysical inversion to estimate the elastic parameters and a petrophysical inversion to estimate the reservoir properties. We combine within an integrated formulation the geophysical and petrophysical components of the problem to estimate the elastic and reservoir properties jointly. We solve the inverse problem following a Monte Carlo sampling approach, which allows us to quantify the uncertainties of the reservoir estimates accounting for the combination of geophysical data uncertainties, the deviations of the elastic properties from the calibrated petrophysical transform, and the nonlinearity of the geophysical and petrophysical relations. We implement this method for the inference of the total porosity and the acoustic impedance in a reservoir area, combining petrophysical and seismic information. In our formulation, the porosity and impedance are related with a statistical model based on the Wyllie transform calibrated to well-log data. We simulate the seismic data using a convolutional model and evaluate the geophysical likelihood of the joint porosity-impedance models. Applying the Monte Carlo sampling method, we generate a large number of realizations that jointly explain the seismic observations and honor the petrophysical information. This approach allows the calculation of marginal probabilities of the model parameters, including medium porosity, impedance, and seismic source wavelet. We show a synthetic validation of the technique and apply the method to data from an eastern Venezuelan hydrocarbon reservoir, satisfactorily predicting the medium stratification and adequate correlation between the seismic inversion and well-log estimates for total porosity and acoustic impedance.

Lithology discrimination from physical rock properties

The estimation of lithology from multiple geophysical survey methods needs to be addressed to develop advanced tomographic methods. An initial requirement for lithology discrimination is that lithology should be discriminable from the media properties physically related to the geophysical observations. To test this condition for different combinations of the most common crustal rocks, we performed several lithology discrimination exercises on rock samples under laboratory conditions. The physical properties included mass density, compressional velocity, shear velocity, electric conductivity, thermal conductivity, and magnetic susceptibility. A categorical description of the sample lithology was followed; hence, the inference consisted of predicting the sample rock category (lithotype) membership. The joint information provided by the physical properties of the rocks allowed us to discriminate the sample lithotype correctly, with an overall success rate of 100% in the most favorable situation and over 85% in the least favorable situation. We obtained successful classification results for a variety of common lithotypes (granite, gabbro, limestone, tuff, marble, basalt, and gneiss) using three common classification methods: clustering analysis, Gaussian classification, and discriminant analysis. Although discrimination was positive with each of these multivariate classification techniques, discriminant analysis showed some advantages for the classification and graphic analysis of the data. These results support our postulate that lithology can be estimated reliably if multiple geophysical observations are considered jointly.

Inversion of traveltime data under a statistical model for seismic velocities and layer interfaces

We invert large-aperture seismic reflection and refraction data from a geologically complex area on the northeast Atlantic margin to jointly estimate seismic velocities and depths of major interfaces. Our approach combines this geophysical data information with prior information on seismic compressional velocities and the structural interpretation of seismic sections. We constrain expected seismic velocities in the prior model using information from well logs from a nearby area. The layered structure and prior positions of the interfaces follow information from the seismic section obtained by processing the short offsets. Instead of using a conventional regularization technique to smooth the interface-velocity model, we describe the spatial correlation of interfaces and velocities with a geostatistical model, using a multivariate Gaussian probability density function. We impose a covariance function on the velocity field in each layer and on each interface in the model to control the smoothness of the solution. The inversion is performed by minimizing an objective function with two terms, one term measuring traveltime residuals and the other measuring the fit to the statistical model. We calculate the posterior uncertainties and evaluate the relative influence of data and the prior model on estimated interface depths and seismic velocities. The method results in the estimation of velocity and interface geometry beneath a basaltic sill system down to 7 km depth. This method aims to enhance the interpretation process by combining multidisciplinary information in a quantitative model-based approach.

P wave velocity tomography of the Venezuelan region from local arrival times

Arrival times from the local seismological network of Venezuela were used to estimate a three-dimensional P wave velocity model for the region between longitude 60°–74° W and latitude 6°–14° N to a depth of more than 80 km. The inversion was carried out by damped least squares, describing the media by homogeneous velocity blocks. The resolved lateral velocity variations in the first layer (0–30 km depth) showed a correlation with the main stratigraphic features of the area, while second layer (30–50 km depth) showed the influence of Moho depth variations through the region, generating a pattern well correlated to the Bouguer Anomaly Map. Lithospheric seismic velocities below the Moho appear to be influenced by the major crustal fault systems. An important low-velocity zone is present below the triple junction of the fault systems of Oca, Bocono, and Moron in northwestern Venezuela. Farther south, a similar low-velocity zone is present below the junction of the Bocono and the Santa Marta fault systems. Those are the two continental corners of the triangular Maracaibo Block. Below 80 km depth (the fourth layer) the low velocity zones show a connected pattern that follows or is adjacent to the crustal fault zones. The presence of subducted Atlantic lithosphere below the Eastern Venezuelan Basin could explain the high-velocity zone at this location. A similar interpretation emerges from the tectonic wedging model, previously proposed to explain the pronounced minimum of the gravity anomaly.

Seismic inversion using a geostatistical, petrophysical, and acoustic model

Interpretation of seismic data for structure and stratigraphy is commonly based on the geological knowledge of the area and the correlation of seismic reflections with well-log data. However, the relation between the seismic image and the well-log data is not straightforward, being often obscured by the wave-propagation phenomenon. Part of the problem is that reflectivities do not measure the interval properties but result from the property contrasts of consecutive strata. Seismic inversion provides insight into the interpretation process, transforming reflection amplitudes into physically meaningful variations in interval properties, which can be directly related to the well-log data after appropriate scale considerations.