Geostatistical Inversion Methods

Posterior probability distribution = Built based on a priori information and forward modeling theory Stochastic seismic inversion = Viewed as a probabilistic estimation process Bayesian inversion = Treats subsurface model parameters as random variables governed by probability distributions Sequential Gaussian Simulation (SGS) = Used in stochastic seismic inversion for sampling posterior distributions

Populating subsurface model with acoustic impedance data = Obtained from well logs as initial constraints Defining a random path through (x, y) trace locations = Determining the order of inversion at each trace location Performing local stochastic optimization at each trace location = Includes generating trial Acoustic Impedance sequences using SGS Retaining the best Acoustic Impedance sequence = Characterized by the smallest synthetic-seismic error

The described algorithm was slow = Due to unrestricted search space requiring a large number of trials Difficulty in converging to an answer = Resulted in significant computational demands and inefficiencies

Acoustic impedance = A property represented in subsurface model parameters Vp/Vs ratio = A property represented in subsurface model parameters Litho-Fluidal Classes = A property represented in subsurface model parameters Seismic data (d) = Usually represented by partial angle stacks

Likelihood function = Quantifies the probability of observing seismic data given a set of model parameters Prior information = Obtained from geological knowledge, well logs, or previous studies A-priori model = Required for any seismic inversion algorithm, represents beliefs about parameters before observing seismic data Secondary data (d) = Correspond to seismic data usually represented by partial angle stacks

Bayesian geostatistical inversion = Obtaining a consistent model of facies and elastic properties Stochastic simulation = Implementing prediction of additional parameters using joint probability distributions Generalized geostatistical inversion scheme = Solving for facies and continuous properties simultaneously to reproduce reservoir heterogeneity Facies-based inversion = Providing a more geologically realistic approach to modeling subsurface heterogeneity

Volumes of facies, petrophysical, and engineering properties = Consistent volumes reflecting desired heterogeneity and geological realism Multiple realizations = Quantitative estimates of uncertainties for each desired property Reservoir engineering dynamic models = Utilizing obtained results for production history matching and forecasting Geostatistical inversion = Integrating multidisciplinary data to provide probabilistic characterizations of the subsurface

Elastic parameterization with Gaussian approximation = Development of efficient algorithms for geostatistical inversion Facies-based parametrization = Providing a more flexible approach compared to Gaussian approximation Stochastic simulation methods = Offering an approximate solution by generating multiple realizations of the subsurface model Bayesian approach = Providing estimates of subsurface properties along with valuable insights into uncertainty

Geostatistical inversion = Conditioned geological modeling Rock Physics Typing = Describing property distributions of deposits Stochastic simulation process = Generating random 3D model realizations Facies-based seismic inversion = Utilizing a-priori facies model specification

Estimating uncertainty by calculating standard deviation across realizations = Bayesian geostatistical inversion Facilitating efficient algorithms for Bayesian inversion = Gaussian approximation for prior and likelihood models Producing a series of model realizations = Markov chain Monte Carlo simulations Deriving probabilistic posterior facies model from observed seismic data = Stochastic Facies-based seismic inversion

Elastic properties of reservoir with fixed saturation = Single-mode or Gaussian distribution A-priori model for highly porous reservoir saturated with various fluids = Complex multi-modal distribution Joint probability distributions of elastic parameters for each facies = Facies-based seismic inversion algorithms Mean P-impedance, mean S-impedance, and mean density in 3D models = Estimated from generated realizations

Developing highly efficient algorithms for Bayesian inversion = Gaussian approximation for prior and likelihood models Representing likelihood of each facies occurring at each location in subsurface = Probabilistic posterior facies model Updating a-priori model using available seismic and well data = Conditioned geological modeling Generating consistent realizations adhering to both a-priori model and seismic data = Stochastic simulation process

3D model of mean elastic parameters and their uncertainty = Estimated from generated realizations Probabilistic posterior facies model based on observed seismic data and a-priori facies model = Result of facies-based seismic inversion A-priori model of Facies or Litho-Fluidal Classes in probabilistic form = Specifying joint prior probabilities of Facies for each cell of the model Joint probability distributions of elastic parameters for each facies = Specification in facies-based seismic inversion algorithms

Kriging algorithm = Produces overly smooth models Stochastic simulation idea = Generated multiple realistic answers Seismic inversion = Incorporates seismic data constraints Sequential Gaussian Simulation-based approach = Reduces modeling uncertainty

Early 1950’s = Application of Kriging algorithm for reservoir parameters 1994 = Introduction of stochastic simulation idea by Journel Present = Utilization of Sequential Gaussian Simulation-based approach

Smoothness in final models = Stochastic simulation idea Modeling uncertainty = Incorporation of seismic data constraints Computational efficiency = Sequential Gaussian Simulation-based approach

Synthetic seismogram = Artificial data matched against real seismic data Geostatistical inversion = Parametrization of model using elastic properties Facies-based geostatistical inversion = Inversion process considering rock types Real seismic data = Data used to make models more plausible

Geostatistical Inversion for elastic parametrization of the model = Determining model properties based on elasticity Seismic inversion as geostatistical modelling process = Utilizing seismic data for modeling Stochastic simulation idea by Journel = Generating multiple realistic models Sequential Gaussian Simulation-based approach to stochastic seismic inversion = Reducing uncertainty through simulated seismic data

Tendency of Kriging to produce smooth models = Unrealistic spatial variability representations Differences between individual stochastic simulation realizations = Increased geological plausibility but higher variability Challenges related to computational efficiency in early implementations = Large number of simulations required for each location

Geostatistical inversion methods = Imply a strong statistical relationship between model parameters corresponding to adjacent locations A-priori model = Defined by specifying a background model of elastic properties and a variogram model Likelihood function = Defined based on misfit between synthetic and real seismic data for partial angle stacks Random realizations of the subsurface model = Represent plausible scenarios of the subsurface considering uncertainties in input parameters and geological features

Sequential trace-by-trace inversion methods = Do not explicitly incorporate information about spatial correlation into the definition of the posterior probability distribution Gaussian approximation for prior probability distributions of elastic parameters = Commonly used in many inversion algorithms Stochastic methods applied to solve geostatistical inversion problems = Represent results of inversion as a set of random realizations to quantify uncertainty associated with estimated subsurface properties Modifications of geostatistical inversion algorithms involving three elastic parameters = Parameterize subsurface model using P-Impedance, S-Impedance, and Density

High dimensionality in inversion algorithms = Poses significant challenges for practical implementation due to increased computational complexity and memory requirements Structural framework guided by seismic horizons and faults = Used to calculate parameters of the a-priori model between wells Background model utilized in inversion process = Incorporates low-frequency components of P-Impedance and S-Impedance sequences obtained through interpolation, filtering, and smoothing procedures 500 independent 3D realizations of the model generated in inversion process = Represent plausible scenarios of the subsurface considering uncertainties in input parameters and geological features

Resolution of a single realization aligns more closely with well log resolution = Than the seismic frequency bandwidth Averaged P-Impedance model across all realizations = Estimation of the mean (most expected) posterior model of P-impedance given seismic and well data Estimation of the normalization factor = Not explicitly mentioned in the text but related to ensuring probability distributions sum to 1 Inversion results represented by a set of random realizations = Quantify uncertainty associated with estimated subsurface properties