Conditioning Spectral Simulation Method By Horizontal Well Data

Spectral simulation is a relatively new geostatistical approach to 3D probabilistic reservoir property simulation. In spectral method simulated property is considered as a realization of a stochastic field, and well logs as realizations of stochastic processes. Well logs are decomposed into Fourier series of coefficients w.r.t. some L2 basis. Coefficients among different wells are grouped according to the basis function, each group representing samples of 2D stochastic fields (surfaces) of coefficients. For each group stochastic surfaces of coefficients are simulated, based on obtained samples and full 3D stochastic field is reconstructed as sum of Fourier series at each lateral point.

One of the features of the spectral method is conditioning simulation results (i.e. reproducing hard data) only on data along vertical wells, which is considered as a limitation in practical applications when reservoirs with large number of horizontal wells are modeled. Hard data on non-vertical wells impose different type of conditioning on simulated stochastic fields of coefficients. In order to satisfy the new type of conditions, generalization of kriging and new type of conditioning of stationary fields, based on this generalization, is proposed. The new type of conditioning is proved to modify simulated surface-coefficients such that conditions imposed on resulting 3D stochastic field on any finite set of points (including points on trajectories of horizontal wells) can be satisfied while preserving statistical parameters of the stochastic field.

Numerical algorithms are provided for analytical derivations, which are confirmed by illustrative simulation experiment for a simple one-dimensional model. The new algorithm is implemented in experimental software and demonstrated to be scalable by conducting conditional simulation for real-field geophysical parameter on a full-scale reservoir model. The results are compared to those of more traditional methods and shown to be more adequate from geological point of view and better reproduce statistical parameters of well data.