A geostatistical framework for incorporating seismic tomography auxiliary data into hydraulic conductivity

A geostatistical approach is presented for the inclusion of seismic tomography data and sonic log data into the estimation of hydraulic conductivity. The procedure accounts for the errors in seismic tomography inversion and for the correlation of such errors, The proposed methodology consists of two steps: (1) the cross-variogram inference is carried out using only the data at the wellbore (both hydraulic and seismic); (2) a co-kriging procedure takes the cross-well data into account to interpolate between boreholes. No postulated a-priori relationship is needed. In order to illustrate the methodology a synthetic data set is generated on the basis of evidence from published case studies and simplified physical considerations. The numerical experiments show that the choice of the excitation frequency is critical. A trade-off exists between the need for a high-resolution survey (asking for higher freqencies) and the need for a good correlation between hydraulic conductivity and seismic properties (asking for frequencies below the squirt frequency of the medium). In the simulation using seismic data with the best excitation frequency (1 kHz in this case), the mean squared error of the hydraulic conductivity estimate is two-thirds lower than using hydraulic data alone. It is important to note also that only a part of the interwell region is adequately sampled by the tomographic experiment, Such a region can be readily identified by calculating the energy of the quasi-null space, through singular value decomposition of the tomographic matrix, In planning this type of experiments, it is necessary to carefully verify case by case whether the adopted range of high frequencies does not prevent the seismic energy from propagating effectively from sources to receivers. (C) 1998 Elsevier Science B.V. All rights reserved.