Arild Buland

Bayesian linearized AVO inversion

A new linearized AVO inversion technique is developed in a Bayesian framework. The objective is to obtain posterior distributions for P‐wave velocity, S‐wave velocity, and density. Distributions for other elastic parameters can also be assessed—for example, acoustic impedance, shear impedance, and P‐wave to S‐wave velocity ratio. The inversion algorithm is based on the convolutional model and a linearized weak contrast approximation of the Zoeppritz equation. The solution is represented by a Gaussian posterior distribution with explicit expressions for the posterior expectation and covariance; hence, exact prediction intervals for the inverted parameters can be computed under the specified model. The explicit analytical form of the posterior distribution provides a computationally fast inversion method. Tests on synthetic data show that all inverted parameters were almost perfectly retrieved when the noise approached zero. With realistic noise levels, acoustic impedance was the best determined parameter, ...

Bayesian lithology/fluid prediction and simulation on the basis of a Markov-chain prior model

A technique for lithology/fluid (LF) prediction and simulation from prestack seismic data is developed in a Bayesian framework. The objective is to determine the LF classes along 1D profiles through a reservoir target zone. A stationary Markov-chain prior model is used to model vertical continuity of LF classes along the profile. The likelihood relates the LF classes to the elastic properties and to the seismic data, and it introduces vertical correlation because the seismic data are band-limited. An approximation of the likelihood model provides an approximate posterior model that is a Markov chain. The approximate posterior can be assessed by an exact and efficient recursive algorithm. The LF inversion approach is evaluated on a synthetic 1D profile that is inspired by a North Sea sandstone reservoir. With a realistic wavelet-colored noise model and a S/N ratio of three in the seismic data, the results are reliable. The LF classes and the interfaces between zones are largely correct. The prediction unce...

Bayesian lithology and fluid prediction from seismic prestack data

A fast Bayesian inversion method for 3D lithology and fluid prediction from prestack seismic data, and a corresponding feasibility analysis were developed and tested on a real data set. The objective of the inversion is to find the probabilities for different lithology-fluid classes from seismic data and geologic knowledge. The method combines stochastic rock physics relations between the elastic parameters and the different lithology-fluid classes with the results from a fast Bayesian seismic simultaneous inversion from seismic data to elastic parameters. A method for feasibility analysis predicts the expected modification of the prior probabilities to posterior probabilities for the different lithology-fluid classes. The feasibility analysis can be carried out before the seismic data are analyzed. Both the feasibility method and the seismic lithology-fluid probability inversion were applied to a prospect offshore Norway. The analysis improves the probability for gas sand from 0.1 to about 0.2–0.4 with s...

Rapid spatially coupled AVO inversion in the Fourier domain

Spatial coupling of the model parameters in an inversion problem provides lateral consistency and robust solutions. We have defined the inversion problem in a Bayesian framework, where the solution is represented by a posterior distribution obtained from a prior distribution and a likelihood model for the recorded data. The spatial coupling of the model parameters is imposed via the prior distribution by a spatial correlation function. In the Fourier domain, the spatially correlated model parameters can be decoupled, and the inversion problem can be solved independently for each frequency component. For a spatial model parameter represented on n grid nodes, the computing time for the inversion in the Fourier domain follows a linear function of the number of grid nodes, while the computing time for the fast Fourier transform follows an n log n function. We have developed a 3D linearized amplitude variation with offset (AVO) inversion method with spatially coupled model parameters, where the objective is to...

Bayesian time-lapse inversion

A new, fast inversion approach for time-lapse seismic data is developed where the uncertainty of the inversion results is an integral part of the solution. The inversion method estimates changes in the elastic material properties of a reservoir because of production of hydrocarbons, including uncertainty bounds on these estimates. The changes in elastic properties then can be related to changes in hydrocarbon saturation and reservoir pressure by using rock-physics relations. The inversion operates directly on the difference between a repeat survey and a baseline survey. This is advantageous with respect to the uncertainty calculation, because an estimate of the seismic uncertainty can be obtained directly from the difference data in zones not affected by production. The method is formulated in a Bayesian setting, and the solution is represented by explicit expressions for the posterior expectation and the covariance of the elastic parameter changes. The explicit analytical form of the posterior distributi...

Bayesian wavelet estimation from seismic and well data

A Bayesian method for wavelet estimation from seismic and well data is developed. The method works both on stacked data and on prestack data in form of angle gathers. The seismic forward model is based on the convolutional model, where the reflectivity is calculated from the well logs. Possible misties between the seismic traveltimes and the time axis of the well logs, errors in the log measurements, and seismic noise are included in the model. The estimated wavelets are given as probability density functions such that uncertainties of the wavelets are an integral part of the solution. The solution is not analytically obtainable and is therefore computed by Markov-chain Monte Carlo simulation. An example from Sleipner field shows that the estimated wavelet has higher amplitude compared to wavelet estimation where well log errors are neglected, and the uncertainty of the estimated wavelet is lower.

AVO inversion of Troll Field data

A stratigraphic elastic inversion scheme has been applied to a data set from the Troll East Field, offshore Norway. The objective of the present work is to obtain estimates of the P- and S-wave velocities and densities of the subsurface. The inversion is carried out on {tau} {minus} p transformed common depth-point (CMP) gathers. The forward modeling is performed by convolving a wavelet with the reflectivity that includes water-bottom multiples, transmission effects, and absorption and array effects. A damped Gauss-Newton algorithm is used to minimize a least-squares misfit function. Inversion results show good correlation between the estimated V{sub P}/V{sub S} ratios and the lithologies in the wells. The V{sub P}/V{sub S} ratio is estimated to 2.1--3.0 for shale and 1.6--2.0 for sandstones. In the reservoir, the V{sub P}/V{sub S} ratio is estimated to 1.55 in the gas sand and to 1.62 below the fluid contact.

Improved resolution in Bayesian lithology/fluid inversion from prestack seismic data and well observations: Part 2 — Real case study

We have performed lithology/fluid inversion based on prestack seismic data and well observations from a gas reservoir offshore Norway. The prior profile Markov random field model captures horizontal continuity and vertical sequencing of the lithology/fluid variables. The prior model is also locally adjusted for spatially varying lithology/fluid proportions. The likelihood model is inferred from basic seismic theory and observations in wells. An approximate posterior model is defined, which can be simulated from by an extremely computer-efficient algorithm. The lithology/fluid inversion results are compared to manual interpretations and evaluated by cross validation in one well. Moreover, inversions based on simplified prior models are developed for comparative reasons. Both lithology/fluid realizations and predictions look geologically reasonable. The results seem to reflect general reservoir experience and information provided by the prestack seismic data and well observations. The lithology/fluid propor...

Bayesian Monte Carlo method for seismic predrill prospect assessment

Predrill assessment of the probability that a potential drilling spot holds hydrocarbons (HC) is of vital importance to any oil company. Of equally great value is the assessment of hydrocarbon volumes and distributions. We have developed a methodology that uses seismic data to find the probability that a vertical earth profile contains hydrocarbons and the probability distribution of hydrocarbon volumes. The method combines linearized amplitude variation with offset (AVO) inversion and stochastic rock models and predicts the joint probability distribution of the combined lithology and fluid for the entire profile. We use a Bayesian approach and find the solution of the inverse problem by Markov chain Monte Carlo simulation. The stochastic simulation benefits from a new and tailored simulation algorithm. The computational cost of finding the full joint probability distribution is relatively high and implies that the method is best suited to the investigation of a few potential drilling spots. We applied the method to a case with well control and to two locations in a prospect: one in the center and one at the outskirts. At the well location, we identify the two reservoir zones and obtain volumes that fit the log data. At the prospect, we obtain significant increases in HC probability and volume in the center, whereas there are decreases at the outskirts. Despite the large noise components in the data, the risked volumes in the center changed by a factor of three. We have designed an algorithm for computing the joint distribution of lithology, fluid, and elastic parameters for a full vertical profile. As opposed to what can be done with pointwise approaches, this allows us to calculate success probability and HC volumes.

The impact of common-offset migration on porosity estimation by AVO inversion

The impact of prestack time migration on porosity estimation has been tested on a 2-D seismic line from the Valhall/Hod area in the North Sea. Porosity is estimated in the Cretaceous chalk section in a two‐step procedure. First, P-wave and S-wave velocity and density are estimated by amplitude variation with offset (AVO) inversion. These parameters are then linked to porosity through a petrophysical rock data base based on core plug analysis. The porosity is estimated both from unmigrated and prestack migrated seismic data. For the migrated data set, a standard prestack Kirchhoff time migration is used, followed by simple angle and amplitude corrections. Compared to modern high‐cost, true amplitude migration methods, this approach is faster and more practical. The test line is structurally fairly simple, with a maximum dip of 5°; but the results differ significantly, depending on whether migration is applied prior to the inversion. The maximum difference in estimated porosity is of the order of 10% (about...