Reconstruction of Channelized Systems Through a Conditioned Reverse Migration Method

Abstract

Geological heterogeneities directly control underground flow. In channelized sedimentary environments, their determination is often underconstrained: it may be possible to observe the most recent channel path and the abandoned meanders on seismic or satellite images, but smaller-scale structures are generally below image resolution. In this paper, reconstruction of channelized systems is proposed with a stochastic inverse simulation reproducing the reverse migration of the system. Maps of the recent trajectories of the Mississippi river were studied to define appropriate relationships between simulation parameters. Measurements of curvature and migration vectors showed (i) no significant correlation between curvature and migration offset and (ii) correlation trends of downstream and lateral migration offsets versus the curvature at half-meander scale. The proposed reverse migration method uses these trends to build possible paleo-trajectories of the river starting from the last stage of the sequence observed from present-day (satellite or seismic) data. As abandoned meanders provide clues about the paleo-locations of the river, they are integrated time step by time step during the reverse simulation process. We applied the method to a satellite image of a fluvial system. Each of the different resulting geometries of the system honored most of the available observations and presented meandering patterns similar to the observed ones.

Appendices

Appendix A Forward Migration Analysis of the Mississippi River

Migration rates have also been studied on the Mississippi river between pairs of paleo-trajectories in the forward direction of the paleo-flow. The lateral and downstream offsets have been computed in the same way as in the reverse direction but with inverted vectors. The results are close to the ones deduced from the reverse direction: no clear global correlation can be observed between curvature and forward migration. Correlation trends can be observed at the half-meander scale. These smooth correlations vary depending on the considered half-meander (Figs. 21, 22).

Fig. 21

Horizontal offset values versus curvature on four successive paths considered in the forward direction. a curvature versus downstream migration offsets, b curvature versus lateral migration offsets, c curvature versus sum of the downstream and lateral migration offsets and d downstream versus lateral migration offsets

Fig. 22

Plots of values of offset versus normalized curvature for five meanders of the four last Mississippi trajectories considered in the forward direction. They summarize the main trends observed for normalized curvature and downstream migration in red, and normalized curvature and lateral migration in orange. If some trends can be locally observed, the very low values of some correlation coefficients demonstrate the absence of general rules and the high variability of migration patterns

Appendix B Algorithm of the Reverse Migration Method

The proposed algorithm for channelized system reconstruction requires as input data the number of wanted realizations, the number of targeted reverse steps, the horizontal and vertical distributions of migration offsets, the final channel path centerline, the list of identified oxbow lake centerlines with their corresponding estimated minimal and maximal values of abandonment period. It iterates on the wanted number of stochastic realizations, on the wanted number of the reverse migration time steps and on the number of half-meanders found, to reverse migrate the main channel path while conditioning to paleo-geometries observed. The Algorithm 1 is illustrated in Fig. 7.

Appendix C Flexibility in Interactive Channel Modeling Brought by Non-uniform Rational B-Splines

Non-uniform rational B-splines (NURBS) are parametric surfaces defined by a network of control points and an interpolation function (Piegl and Tiller 1997). The interpolation function is a continuous polynomial function whose degree is directly related to the number of interpolated points (Bézier 1983). NURBS objects present a curvilinear framework that permits to choose the grid resolution without shape deformation. These surfaces have been combined to define volumetric objects independent from the support (Ruiu et al. 2016). Thus, channels, levees, point bars and lobes can be modeled with NURBS (Fig. 23) allowing us to reproduce channelized systems in both fluvial and turbidite contexts.

Fig. 23

Modeling of a channel using 5 NURBS surfaces. Red lines compose the control polygon and the blue points are the control points: a map view and b cross view

In many steps of the workflow, the number of control points varies in an inhomogeneous way. This is the case at each connection of two objects (e.g., the integration of an oxbow lake) or at each local deformation of the NURBS object (e.g., the reverse migration process). This decreases the handling of the NURBS object and the computational efficiency (Fig. 24a). Therefore, the algorithm of Tiller (1992) has been implemented to determine the smallest control polygon while minimizing the deformation imposed to the NURBS (Fig. 24b).

Fig. 24

Re-parameterization during reverse migration process permits us to avoid the collapse of sections affecting the channel geometry: a reverse migration without re-parameterization of the control polygon at each step and b reverse migration with re-parameterization of the control polygon at each step