A simple linear interpolation of missing log data between wells enables estimation of a reflectivity series and TDR; however, it does not account for variations in lithology or structure. Several methods have been proposed to estimate missing logs that could be used for a more accurate seismic-well tie. Gardner's equation (Gardner et al., 1974) has been shown to provide a reasonable relationship between sonic and density for a large number of brine saturated rock types. Additionally, the Faust method (Faust, 1953) and Smith method (Smith, 2007) provide empirical relationships between resistivity and sonic logs. Saggaf and Nebrija (2003) note a high interdependence of different log types and apply regularized back-propagation neural networks to estimate missing portions of sonic logs. Each method assumes that specific well logs are collected in every well to carry out the estimation.
An alternative approach is to assume that rock properties do not vary significantly in lateral space, which allows using density and sonic logs from a nearby well to estimate a TDR. Because this assumption does not take into consideration structural or stratigraphic variations in lithology, applying a TDR generated at one well to a nearby well may result in a mis-tie with the seismic data. To account for these variations, the well must be correlated to a common geologic time. Wheeler and Hale (2014) and Wu et al. (2018) use dynamic time warping (DTW) (Berndt and Clifford, 1994; Hale, 2013) to correlate multiple well logs. Shi et al. (2017a) use local similarity scan (LSIM) (Fomel, 2007a) to optimally sort and flatten multiple well logs. Once the well logs are flattened or aligned in geologic time, which is analogous to a stratigraphic correlation, missing well log sections can be estimated from the available data by horizontal interpolation. Bader et al. (2018) flatten well logs from depth to relative geologic time domain and interpolate a missing sonic log using several available sonic logs and several empirical relationships assuming fluid variations have a negligible effect on the well logs.
With a complete well-log suite, including those sonic and density logs estimated by interpolation, we are able to further tie the wells to seismic data. The manual seismic well tie involves matching common reflectors between modeled synthetic and seismic data by stretching and squeezing the synthetic until a desired correlation between the datasets is achieved (White and Simm, 2003). To reduce interpreter bias and improve consistency between multiple seismic well ties, several automatic methods have been proposed. Muñoz and Hale (2012) use DTW to automatically align real and synthetic seismograms; this approach is extended to automatically and simultaneously tie multiple wells to seismic by estimating a synthetic image to tie with the seismic image ensuring lateral consistency of the well ties (Muñoz and Hale, 2015). Further, Wu and Caumon (2017) show that laterally consistent seismic well ties can be achieved by using DTW to correlate synthetic and seismic data that are `flattened' to relative geologic time. An alternative approach to carry out the seismic well tie is LSIM; Herrera et al. (2014) compare DTW with LSIM, showing that both methods can successfully compute a seismic well tie. Their study shows that using DTW can achieve a higher correlation between synthetic and seismic data compared to LSIM; however, the resulting TDR using DTW shows an undesirable oscillatory behavior due to stretching and squeezing.
Once each well is tied to the seismic data, the high spatial coverage of seismic can be utilized to understand lateral variations in log properties. Several methods have been proposed to interpolate log data along local seismic structures. Assuming available log data is properly tied to seismic and conforms to seismic image features, Hale (2010) uses image guided blended neighbor interpolation (Hale, 2009) for seismic guided well log interpolation. Alternatively, Karimi et al. (2017) show that predictive painting (Fomel, 2010) can be used to interpolate log data along seismic structures to generate accurate starting models for post stack inversion. Fomel (2016) presents a fast interpolation algorithm for interpolating scattered data to a regularly sampled grid. Interpolation along seismic structure using well log data generates log property volumes that conform to both well log and seismic datasets. Wu (2017) proposes to compute such a structurally conformable model in the flattened space, where the seismic and well-log data are unfaulted and unfolded.
In this paper, we address limitations brought about by missing well log data as well as challenges associated with achieving consistent seismic well ties and propose a workflow that integrates the data matching techniques, LSIM and predictive painting, to estimate missing logs, tie synthetic seismiograms to seismic, and finally interpolate all available well log data along seismic structures. We use cross-validation with a blind well test to test the consistency of seismic well ties. We apply our method to tie 26 wells and the 3D Teapot Dome seismic dataset.