Mathematical Geosciences

, Volume 49, Issue 8, pp 965–994 | Cite as

Reconstruction of Channelized Systems Through a Conditioned Reverse Migration Method

  • Marion N. Parquer
  • Pauline Collon
  • Guillaume Caumon
Article
  • 215 Downloads

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.

Keywords

Stochastic simulation Fluvial Turbidite Oxbow lake Point bar Channel evolution Reservoir 

1 Introduction

Deciphering the fundamental parameters that control the morphology and evolution of meandering systems has been the focus of many researchers for decades (Covault et al. 2016; Jobe et al. 2016; Leopold and Langbein 1966; Seminara 2010). These systems can be found in aerial or submarine environments (Leopold and Langbein 1966; Wynn et al. 2007). In the sedimentary record, channelized systems can also represent interesting reservoirs for natural resources, such as water and hydrocarbons (Fitterman et al. 1991; García-Gil et al. 2015; Mayall et al. 2006), or for storage of gas (e.g., carbon sequestration, Issautier et al. 2014).
Fig. 1

Channel lateral migration (called migration if perpendicular to flow direction, progradation if in the direction of the flow) and vertical migration (called aggradation): a section view and b map view

Turbidite and fluvial systems both migrate in the available space (canyon or floodplain) (Jobe et al. 2016; Wynn et al. 2007). This migration goes along with a perpetual reworking of sediments due to erosion, transport and deposition. Labourdette and Bez (2010) and Peakall et al. (2000) describe a vertical migration (aggradation, Fig. 1a) and two horizontal migrations, one directed downstream in the direction of flow, and one directed perpendicularly to the flow direction (Fig. 1b). The latter is called a lateral migration and contributes to the extension of channel meander loops at small scale and, at a larger scale, to the meander belt. The relative expression of these directions of migration depends on various external and internal factors (e.g., amount of sediment, confinement degree, nature of sediment, paleo-slope, maturity of the system and structural controls) and makes each system unique (McHargue et al. 2011; Wynn et al. 2007).

In addition to gradual migration, the main path of the channel can change abruptly because of avulsions or cut-offs (Hajek and Wolinsky 2012; Slingerland and Smith 2004; Stouthamer and Berendsen 2001). A cut-off occurs when the sinuosity of the channel becomes so high that the channel pinches itself. The trajectory is naturally simplified by the cutting of meander loops. After these disruptive events, paleo-channels and oxbow lakes will be filled with fine-grained material (Issautier et al. 2014; Veeken 2006). Therefore, paleo-channels can be observed on stratal slices (Miall 2014; Posamentier and Kolla 2003). The remnants of oxbow lakes after erosion can provide important clues about paleo-trajectories of the channel. In subsurface studies, well data can also inform us about channel facies positions. Thus, the typical subsurface data that are available to constrain the modeling of channelized systems are composed by borehole data, the last channel path and the observable oxbow lakes.

The long-term evolution of the main channel leads to a complex internal architecture of shale and sand deposits. This complexity, due mainly to the perpetual reworking of the sediments, causes heterogeneities at different scales in the final deposit (Jackson and Muggeridge 2000; Willis and Tang 2010). For example, various types of point bars are deposited along both sides of the channel and distributed around the meander shape (e.g., counter-point-bars, downstream-migrating fluvial point bars and eddy-accretion deposits) (Ghinassi et al. 2016). Such volumes constitute the main reservoir rocks for fluids hosted in channelized sediments. However, these volumes can be compartmentalized due to the presence of shale drapes deposited on the scour surfaces. Understanding reservoir behavior can be achieved by flow simulation. This implies a good reconstruction of the accretion point bars and of the surrounding objects, such as shale drapes.

Many studies have been conducted on the spatial evolution of channels (Peakall et al. 2000; Posamentier and Kolla 2003; Wynn et al. 2007). Most of them focused on fluvial systems because they are easily observable and because their day-to-day evolution directly impacts human activities. Thus, forecasting the future evolution of a channel given a geological and hydrodynamic context has been a wide research topic for decades. Most of the proposed modeling approaches (Edwards and Smith 2002; Sun et al. 1996) are physically-based methods seeking to reproduce the physical processes that generate channelized systems. Many of them are rooted in the seminal works of Ikeda et al. (1981) or Langbein and Leopold (1966) that rely on a hydrodynamic model of the flow inside the channel. This kind of model is calibrated with various parameters such as curvature, channel width values and sedimentary discharge. However, despite the similarity of these forecasts to natural patterns, long-term simulation produces systems that are statistically distinguishable from the natural channels (Howard and Hemberger 1991). In order to stabilize the long-term behavior of these models, Frascati and Lanzoni (2009) worked on a flow field taking into account other morphological features, such as various morpho-dynamic flow regimes. This has permitted researchers to reproduce the full range of morpho-dynamic regimes that can be found inside a channel of variable meander geometries. Although interesting forecasts of channel evolution are given by these methods, their application can be complex and restricted. For example, their parameterization calls for specific types of data such as paleo-hydrodynamic flow and paleo-flow bank erodibility, which are difficult to assess after deposition (Camporeale et al. 2007). Another limit is that these methods give a deterministic solution that does not cover the reservoir architecture uncertainties. To overcome this limit, Lopez (2003) and Pyrcz et al. (2009) introduced stochasticity in their models. Finally, due to the focus on the reproduction of geological processes, data conditioning is difficult, except for well data, which can be managed thanks to attraction or repulsion fields.

Descriptive approaches (Deutsch and Tran 2002; Labourdette et al. 2006; Teles et al. 1998; Viseur 2004), which focus more on the result of the migration than on the geological processes responsible for the migration, are based mainly on the channel geometry (e.g., curvature, width, depth). This simplified parametric construction permits researchers to widen the range of applications to poorly known systems in terms of depositional contexts. Although they give good models of channel geometries, they hardly honor seismic and borehole observations in a simple way. In a direct approach, the output model is strongly dependent on the hypothesis made on the first channel path. Thus, it can be quite difficult to fit to a precisely observed mature channel shape with a direct evolution model of the channel path.

However, our aim is to reproduce the sedimentary record deposited by the channelized system. Thus, honoring data such as the last channel path is of high importance. Our approach is based on the idea that a good approximation of geological processes should help fitting to all conditioning data by getting closer to the reality. This paper proposes a modeling tool based on mainly the geometric parameters accessible through seismic images such as channel curvature. Starting from the last observed channel path, our algorithm goes back in time, reproducing the natural migration of a channel and integrating time step by time step the oxbow lakes that left recorded paleo-trajectories. Although most of the developed approaches support only two dimensions (Howard and Hemberger 1991; Ikeda et al. 1981; Langbein and Leopold 1966), the one presented here reproduces the three migration directions of channels, i.e. downstream, lateral and vertical. Our migration model is based on the study of Mississippi maps detailing several trajectories. Thus, the main channel migration steps are reconstructed, opening the way for the reproduction of accretion point bars. In addition to the continuous migration process, the algorithm also aims at reproducing the discrete evolution steps due to brutal events such as meander abandonment. Absolute values of abandonment ages can be given by a few methods such as radiocarbon analysis (Holbrook et al. 2006) and satellite image analysis (e.g., optically stimulated luminescence (Rowland et al. 2005)). Most of the time, on seismic images, only relative abandonment ages can be constrained by the analysis of cross-cuttings and intra point bar erosion surfaces (Durkin et al. 2015).

In order to honor the last channel path observed as conditioning data, some works have been conducted on reverse migration modeling of channelized systems, starting from the conditioning data and going back in time. Labourdette (2008) proposed a geometric modeling approach based on the seismic observations of a turbidite system (e.g., final stage, channel belt borders) and on the probable migration of characteristic points of the channel trajectory. The idea of Ruiu et al. (2015) is to reduce the sinuosity of the channel by doing a reverse lateral migration of the channel. The result is a channel that tends to produce a straight line between the initial inflection points. This approach gives interesting stacked geometries for channels which have migrated mainly laterally, but does not reproduce deposits due to potential downstream migration. In the absence of unambiguous subsurface images, Rongier et al. (2017a) simulate the last channel path with a formal grammar method using probability distributions of various geometrical parameters (e.g., meander amplitude, meander wavelength, channel width and channel main orientation). Then, a reverse migration step is generated either with a Sequential Gaussian Simulation (SGS) of migration offset constrained by the curvature of the channel or with a Multiple Point Simulation process. This method reproduces the meandering specificities observed in natural deposits such as lateral and down-system migrations (Posamentier and Kolla 2003), aggradation (Peakall et al. 2000), diminution of the sinuosity by natural retro-migration of the channel (Nakajima et al. 2009), abrupt migration (Abreu et al. 2003; Maier et al. 2012) and avulsion (Armitage et al. 2012). However, none of these methods decomposes the migration into the two main directions, lateral and downstream, as advocated, for instance, by Jobe et al. (2016). Moreover, they do not honor oxbow lakes, which are essential remnants of the evolution of the system.

In this paper, a statistical study of the lateral and downstream migrations of the Mississippi river located in the United States over centuries is proposed (Sect. 2). The aim is to determine how the channel migration could be correlated with morphological parameters. Indeed, natural migration is the result of the combination of so many internal and external factors that this irreversible natural process is necessarily simplified in the numerical modeling. Then, an inverse modeling method conditioning to oxbow lakes and reproducing the migration patterns as observed in the studied database is presented (Sect. 3) and applied on a satellite image of the Tangnara river located in Russia (Sect. 4). As the available data never sample the channelized system in its entirety, the method proposes stochastic realizations. Moreover, the stochasticity helps to assess architectural uncertainties (e.g., location of barriers to flow).

2 Natural Migration of a Channel

2.1 Migration Case Study: Mississippi River

At human time scale, the Mississippi river is a good example of an evolutive meandering channelized system. The US Army Corps of Engineers has dated and mapped the last evolutionary steps of the river (Fig. 2). This work has been achieved thanks to historic archives for the four youngest paths since 1765 and with aerial photographs of the remnants of older trajectories (Fisk 1944). Almost twenty stages of migration are observed. This database gives the opportunity to study the link between river geometry and river migration. Most of the previous studies demonstrated a maximum of lateral offset when the ratio of curvature radius to channel width equals three (Furbish 1988; Hickin and Nanson 1975; Hooke 1984). However, channel width can vary locally and is not directly observable on Mississippi maps except for the last channel path of 1944. For the other complete paths, the width should be locally estimated. Otherwise, our aim is to provide a modeling tool also usable on seismic images. On such images, the noise and the limited resolution of these data can make the assessment of river width quite difficult. Thus, we choose not to consider the channel width.

Therefore, we studied the influence of the river curvature on the lateral and downstream migrations by analyzing the four last observed trajectories of the river. Between one thousand and two thousand points were picked along 1200 km of each trajectory of the channel belt, and offsets of migration were computed between pairs of successive channels in terms of evolution through time in the reverse way.
Fig. 2

Picking the four last trajectories of the Mississippi in 1944 (in white), 1880 (in green), 1820 (in red) and 1765 (in blue) on an extract of map (http://lmvmapping.erdc.usace.army.mil/)

For the study, the channel trajectory is divided into meanders, and meanders are divided into half-meanders (Fig. 3). A meander is composed of a pair of opposite loops (Leopold et al. 1964). A half-meander is delimited by inflection points. In order to avoid very small half-meanders, two points, at minimum, are required between two successive inflection points to define a half-meander. Obviously, the number of points between two inflection points depends on the centerline interpretation. Inside each half-meander, the direction of reverse downstream migration and the direction of reverse lateral migration are defined relatively to the segments connecting the inflection points (dashed grey in Fig. 3). Between two successive channels, the downstream and lateral offsets are given by projecting the points of the younger channel onto the older channel along the normal direction of the younger channel path. Then, downstream and lateral offsets are deduced by projection of this vector, respectively, on the downstream vector and on the lateral vector. The curvature computed on each point \(p_{i}\) of the youngest channel is defined between two consecutive segments as the inverse of the radius of the common circle of the three points defining these two segments.
Fig. 3

Points of the youngest channel are projected onto the older channel along the normal direction of each point. Lateral and downstream reverse migration offsets are deduced by projection of the normal vector onto the lateral or downstream vector

Fig. 4

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

The global analysis of the results does not demonstrate a clear correlation between the channel curvature and its natural migration offset (Fig. 4a–c). The lateral and the downstream migrations show a majority of migration patterns with downstream and lateral positive offsets (Fig. 4d). The occurrences of patterns with a negative lateral or downstream migration are lower. The same kind of results have been observed on forward migration rates (Appendix A).
Fig. 5

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

2.2 A Curvature-Dependent Migration

The analysis of migration offsets for each half-meander yields more interesting results. Locally, the considered curvature values are the curvature values divided by the maximal observed curvature of each half-meander, hereafter called normalized curvature. Some of the results are presented in Fig. 5. Inside a half-meander, a light trend can generally be observed between the normalized curvature and the downstream migration offset, and between the normalized curvature and the lateral migration offset. Nonetheless, the trends observed are not constant and can be contradictory for two half-meanders of the same river age. In Fig. 5a, b, f, the downstream migration clearly decreases when normalized curvature increases, while the lateral migration follows an opposite trend. The observed correlation is completely inverse in Fig. 5d. It is interesting to point out that the half-meanders of Fig. 5 present various migration patterns (e.g., lateral migration in Fig. 5a, f, downstream migration in Fig. 5b and a natural retro-lateral migration in Fig. 5d). The maximal values of lateral and downstream migrations vary with the considered half-meander.

Some half-meanders (Fig. 5c, e) do not present any correlation between the normalized curvature and any of the two horizontal migrations. This is not surprising, considering that the related meanders do not show any visual evidence of correlation. Indeed, the meander of Fig. 5c does not present any significant migration between the two mapped steps. On the contrary, the meander of Fig. 5e has quickly migrated between the two mapped steps and shows development in the reverse direction of multiple loops in 1820, whereas only one was observed in 1880. Some intermediate steps are missing that would help us to understand the link between the normalized curvature and the migration offsets between the two mapped stages. This variety of meander migration speeds has been described in many river studies (Brice 1974).

These results have to be tempered due to the uncertainty of river mapping in ancient times, the potential influences of human activities on the river trajectory and of our manual interpretation of the maps. Moreover, the mapped stages are not representative of the gradual migration of the river. It is, however, still interesting to point out that the same results have been observed on forward migration rates (Appendix A). These plots permit us to deduce a model of half-meander curvature-dependent migration. In the following development, all values obtained by sampling are indicated in lowercase and all values deduced from the equations or channelized system analysis are indicated in uppercase. Vectors are indicated in bold.
Fig. 6

Schemes of the correlations at the point \(p_{i}\) for three example values of \(S_\mathrm{L}\) and \(S_\mathrm{D}\): a between curvature and downstream offset \(O_\mathrm{D}(p_{i})\) for Eq. (1); b between curvature and lateral offset \(O_\mathrm{L}(p_{i})\) for Eq. (2)

Because the maximal offset of lateral or downstream migration varies according to the considered half-meander, a sampling of these maximal values \(o_\mathrm{D}\) and \(o_\mathrm{L}\) from probability distributions is proposed. A positive or negative weighting by the local curvature is then applied to reproduce the diversity of half-meander behaviors: the curvature weight w is uniformly sampled from the set of discrete values \(\{-1;1\}\). The horizontal offsets of migration are then smoothed according to the local curvature \(c(p_{i})\) at the point \(p_{i}\) normalized by \(c_\mathrm{M}\), the local maximal curvature. The smoothing parameters \(s_\mathrm{D}\) and \(s_\mathrm{L}\) are sampled from a uniform distribution defined between 0 and two times the maximal drawn offset, \(o_\mathrm{D}\) or \(o_\mathrm{L}\).

If w is positive, the reverse downstream migration offset \(O_\mathrm{D}\) decreases when curvature increases (Fig. 6) as
$$\begin{aligned} O_\mathrm{D}(p_{i}) = \left\{ \begin{array}{ll} o_\mathrm{D}-s_\mathrm{D}\times \frac{c(p_{i})}{c_\mathrm{M}}&{}\quad \hbox {if } w > 0\\ o_\mathrm{D}-s_\mathrm{D}\times \left( 1-\frac{c(p_{i})}{c_\mathrm{M}}\right) &{}\quad \hbox {if }w < 0 \end{array}.\right. \end{aligned}$$
(1)
Similarly, the reverse lateral migration offset \(O_\mathrm{L}\) increases when curvature increases (Fig. 6) as
$$\begin{aligned} O_\mathrm{L}(p_{i}) = \left\{ \begin{array}{ll} o_\mathrm{L}-s_\mathrm{L}\times \left( 1-\frac{c(p_{i})}{c_\mathrm{M}}\right) &{} \quad \hbox {if }w > 0\\ o_\mathrm{L}-s_\mathrm{L}\times \frac{c(p_{i})}{c_\mathrm{M}}&{}\quad \hbox {if }w < 0 \end{array} .\right. \end{aligned}$$
(2)
A negative w value reverses the influence of the curvature (Fig. 6).

3 Reverse Migration Modeling

The study made on the Mississippi system shows the positive and negative influences of the channel curvature on its migration at the half-meander scale. This observation can be used as a constraint for migration modeling. A complete reconstruction of channel belts is proposed with a method that necessitates only information that is deducible from seismic horizon slices or satellite images (Fig. 7, Appendix B). These observations include the geometry of the last channel path and, when available, the meander loops abandoned during channel evolution and the age of their abandonment. For practicality, age is considered as discrete time steps and a maximal number of reverse migration steps is used as an ending condition for the algorithm.
Fig. 7

The reverse migration method accepts data that are available from seismic images such as picking of the main channel path and of its oxbow lakes, estimated minimum and maximum abandonment ages of these abandoned meanders, probabilistic distribution of migration offsets and a number N of migration time steps. The outputs correspond to different stochastic three-dimensional architectures of the reservoir. The algorithm is detailed in Appendix B [Google Earth \(63^{\circ }3439.69^{\prime \prime }\hbox { N}\)\(123^{\circ }3502.08^{\prime \prime }\hbox { E}\)]

3.1 Stochastic Reverse Migration

3.1.1 Curvature-Dependence Modeling

The reverse migration starts from an initial channel path. This trajectory corresponds to the most recent channel position and to the first step of the reverse migration. Inflection points of the channel path permit us to identify meanders and half-meanders (Fig. 8).
Fig. 8

Definition of the three migration directions at a given time step thanks to the inflection points of the channel from synthetic data

Probability distributions of horizontal and vertical migration offsets are needed to reproduce the variability of migration patterns. In the following examples, Gaussian distributions have been chosen.

Along the channel path, for each half-meander, the downstream and lateral horizontal reverse migration directions (\(\mathbf D \) and \(\mathbf L \), respectively) are defined for the current time step (Fig. 8). \(\mathbf D \) is parallel to the line connecting consecutive inflection points and \(\mathbf L \) is perpendicular and oriented toward the center of the meander loop. For each half-meander, a value of maximal horizontal reverse migration offset \(o_\mathrm{D}\) is sampled from the probability distribution of horizontal offsets (Fig. 9). This value is assigned to the downstream reverse migration \(\mathbf D \) and corresponds to the maximal downstream offset of migration in this half-meander for this step. Similarly, another sampling from the horizontal offsets of migration allows us to assign the maximal value \(o_\mathrm{L}\) to the lateral reverse migration L and represents the maximal offset of lateral migration. Values of lateral \(s_\mathrm{L}\) and downstream \(s_\mathrm{D}\) curvature smoothing parameters are also sampled from their corresponding distributions and define the minimal offset values. Then, a value of curvature weight w is drawn from the corresponding distribution and applied to the half-meander (Fig. 10). Curvature is computed on each point of the half-meander and local maxima are deduced for each half-meander. Then, the offset values are computed for each point of the half-meander following the equations deduced from the Mississippi study [Eqs. (1), (2)]. The translated points of the half-meander correspond to all points between both inflection points and to the inflection point located the most upstream of the channel path. This ensures the continuity of the reverse migrated channel path at inflection points. At an inflection point \(p_{i}\), the local curvature \(c(p_{i})\) is null: depending on the sampled value w, \(O_\mathrm{L}\) and \(O_\mathrm{D}\) can be equal to their maximal or minimal values. However, the strict decomposition of the channel path into half-meanders imposes only one translation to each inflection point.
Fig. 9

Sampling of the horizontal and vertical offset migration values in the corresponding probability distributions for each half-meander on synthetic data

Fig. 10

Influence of the channel curvature on the offset of migration with a sampled value of smoothing equal to \(o_\mathrm{D}\) in the downstream direction and \(o_\mathrm{L}\) in the lateral direction on synthetic data. a Positive curvature weighting of the horizontal migration; the downstream offset is maximal for lowest curvatures. b Negative curvature weighting of the horizontal migration; the lateral offset is maximal for lowest curvatures

The translated points correspond to the discretization of the channel path centerline. This centerline is wrapped by non-uniform rational B-splines (NURBS) (Ruiu et al. 2016 and Appendix C) to generate an interpolated volume reproducing the channel shape. Once wrapped, the top surface of the channel is defined by a control mesh made by two external lines and one central control line. Furthermore, the offset translation is applied on the control nodes of the NURBS and not on the interpolated shape. This, combined with the strict discretization of the channel path into half-meanders ensures the continuity of the reverse-migrated paths.

As the Mississippi database presents essentially maps, except some cross sections which are too sparse to permit an exhaustive study, the reverse vertical migration has not been studied. Several authors relate the vertical aggradation \(O_\mathrm{V}\) to be mainly influenced by external factors such as the confinement degree, the sedimentary charge, the accommodation space and, to a lesser point, by morphological parameters such as the sinuosity of the channel (Labourdette and Bez 2010; Peakall et al. 2000; Wynn et al. 2007). Thus, \(O_\mathrm{V}\) could be constrained by layer thickness trend or distality, if available. When such information is not available, for each step of the sequential reverse migration, the vertical offset value \(o_\mathrm{V}\) can be sampled at each time step globally for all half-meanders along the channel path from some input probability distribution (Fig. 7) as
$$\begin{aligned} O_\mathrm{V}(\mathbf {p_{i}}) = o_\mathrm{V}, \end{aligned}$$
(3)
and then assigned to all points of the channel path.

3.1.2 Model Discussion

This approach allows us to reproduce a wide range of meander migration patterns (Fig. 11). Indeed, the decoupling of the channel horizontal migration in the downstream and lateral directions permits us to reproduce pure lateral migration (Fig. 11a), pure downstream migration (Fig. 11b) or an intermediate between both (Fig. 11c). A natural backward migration of meanders can also be simulated when the offset probability distribution includes negative values (Fig. 11d).

The sampling of the horizontal offsets is done at each time step and for each half-meander in order to reproduce the observed variability of migration offset between two half-meanders of the same age (Fig. 11a). It would be possible to correlate spatially the offsets of migration of the same period if a specific migration rate is known. It would also be possible to correlate the offsets for a specific half-meander if an erosional rate has been observed in this specific area.

All these configurations have been observed on the Mississippi maps. In these examples, the reverse migrated channel is located upstream compared to the initial channel, when the offset of downstream migration is positive. As the channel reversely migrates, the channel becomes straighter and less sinuous. Indeed, the reverse migration tends to smooth the channel trajectory toward the straight line defined by the inflection points. This diminution of sinuosity is consistent with older channels which are less mature (Brice 1974).
Fig. 11

Variability of the migration patterns applied to synthetic data. a Pure lateral reverse migration in 10 time steps. b Pure reverse downstream migration in 10 time steps. c Reverse migration of a channel in 15 time steps. The reconstructed path migrates upstream and becomes less sinuous with the increasing age value. d Natural reverse migration in 10 time steps with probability distribution accepting negative offsets of lateral and downstream migration

Between each reverse migration step, the list of inflection points, and so the list of half-meanders, is updated. This permits a complete migration of the system, including inflection points, and a consideration of meander births or extinctions. Working at half-meander scale allows us, whatever the sinuosity of the channel, to avoid most of the potential self-intersections all along the reverse process. Indeed, reverse migrating each half-meander independently permits us to evolve toward the line connecting two successive inflection points. In this direction, the probability of crossing another meander loop is almost null.

Most of the migration modeling methods (Rongier et al. 2017b; Ruiu et al. 2015; Viseur 2004) determine the direction of migration along the normal vectors of each point and its offset in a global way. However, the Mississippi river, as many natural channelized systems, presents meanders of various geometries (Brice 1974). Working between each pair of inflection points allows us to reproduce this diversity in terms of channel geometry.

3.2 Conditioning to Paleo-Path Indicators

3.2.1 Oxbow Lake Abandonment Age and Geometry Interpretations

In addition to the last channel path, other channel belt structures can be observed on satellite or seismic images. These objects can be used to constrain the modeling. This corresponds, for example, to oxbow lakes filled by fine particles following their abandonment. For each oxbow lake observed in a meander belt, the absolute abandonment age is difficult to evaluate. Therefore, we use a probability distribution function for relative abandonment ages.

The distance to the middle of the meander belt and the potential erosion by younger oxbow lakes permit us to estimate minimal and maximal values of relative abandonment ages. The assigned abandonment age a is sampled from a probability distribution defined between these two extreme values (Fig. 12).

As the natural migration of the channel can erode oxbow lake deposits, the complete geometry of the oxbow lake at the moment of its abandonment may not be preserved in the sedimentary record. An interpretation must then be performed to extrapolate a probable shape before its integration in the reverse migration algorithm (Fig. 13).
Fig. 12

Abandonment ages a are sampled for each oxbow lake composing the synthetic data inside a discrete probability distribution \(U(A_{\min }, A_{\max })\) with minimal \(A_{\min }\) and maximal \(A_{\max }\) deduced from geometric rules. Dashed parts of the oxbow lake traces correspond to interpreted extremities that have been eroded by system migration after abandonment of this oxbow lake

Fig. 13

Interpretation of eroded parts in the trace given in input in order to help a good reconstitution of the paleo-paths [Google Earth \(63^{\circ }1225.33''\hbox { N}\)\(122^{\circ }5953.73''\hbox { E}\)]

3.2.2 Integration into the Main Channel Path

Each oxbow lake is defined by its centerline (Fig. 14) and wrapped by NURBS (Appendix C) in the same way as the main channel path. When the iterative time step of the algorithm reaches the value of the assigned abandonment age \(a-1\) of an oxbow lake, this oxbow lake is a candidate for integration into the main channel path for the next step (Fig. 7). In order to prepare the integration, the algorithm detects the channel sections closest to the oxbow lake extremities (Fig. 14b). Then, the distances between each of the detected channel sections and the oxbow lake extremities are computed to check whether they are in the authorized range of distances. Such verification is needed to allow the integration of the complete observed oxbow lake geometry. As a rule of thumb, the distance must range between one and three channel widths (Fig. 14c). If the distance control is successful, the two oxbow lake extremities are connected to the detected sections of the main channel path during the next time step corresponding to the age a of the oxbow lake abandonment (Fig. 14d). This oxbow lake integration is achieved by connecting the control polygons of both objects.
Fig. 14

Reverse modeling of meander cut-off through visualization of the structure of the channel [illustration with the Linde river (Russia)]: a Satellite image of a river and its oxbow lake. b Digitization of the oxbow lake (in black) and of the main channel (in blue) paths. c Detection of the channel closest sections and check of the distances. d At abandonment age a, the extremities of the considered oxbow lake are projected on the main channel path, which is cut between the two projections. The original main channel path is then reconstituted by integration of the oxbow lake path [Google Earth \(65^{\circ }1646.27''\hbox { N}\)\(122^{\circ }2634.65''\hbox { E}\)]

If the distance condition is not respected, the oxbow lake is not integrated. Indeed, for an easy and smooth integration of the oxbow lake, its orientation should fit the orientation of the channel closest sections. The segment that links its two extremities should be approximately parallel to the main channel path (Fig. 14), whereas the integration will not be possible if these segments are perpendicular (Fig. 15).
Fig. 15

Example on synthetic data of oxbow lake with perpendicular orientation with the main channel path. Visualization of the control polygons (in blue) of the channel and oxbow lake paths, and of the control points (in red). a The research of the closest sections of oxbow lake extremities found the same one, so a correction needs to be performed. b The distance check fails: maximal accepted distance is equal to three channel widths. c Example of integration: if it were authorized, the channel would pinch itself

This direct connection between the extremities of the oxbow lake and the channel justifies the importance of the interpretation of the abandoned meander loop. Naturally, a channel always chooses, indeed, the path of lower energy (Langbein and Leopold 1966). In the case of a neck cut-off, it corresponds to the cutting of the meander loop when its bends are very close. Thus the interpreted oxbow lake should present a neck made by two extremities that are quite close to each other and are also close to the main channel path at the abandonment age. The following migration of the system may erode the neck due to its proximity to the new channel path.
Fig. 16

Attraction and repulsion of oxbow lakes during reverse migration process on synthetic data. Visualization of the control polygons (in blue) of the channel and oxbow lake paths, and of the control points (in red). At the abandonment time step, if the oxbow lake is too far from the main channel path (left) the main channel path is attracted toward the center \(C_{1}\) of the oxbow lake. If the oxbow lake is too close (right) the main channel path is repulsed from the center \(C_{2}\) of the oxbow lake

Fig. 17

Reverse migration on synthetic data containing five oxbow lakes. The initial number of reverse migration time steps is set equal to 10. Horizontal offsets follow a normal distribution N(20, 5). a Map view. b Section views

Unsuccessful distance checking for both extremities happens when oxbow lakes are located far from the main channel or, conversely, in the middle of the channel belt. In the first case, the reverse migration process needs to be attracted by these conditioning data (Fig. 16), which can be located at one border of the channel belt. In the second case, the reverse migration process needs to be repulsed from the area of the considered oxbow lake: until the integration of this abandoned meander, this area must be avoided by the migration process. Otherwise, reverse migration steps could erode an observed abandoned meander before its integration. Such a realization would suppose that the oxbow lake has been successively abandoned and eroded. In this case, this abandoned meander would not be observable on the image, which is not the case. Thus, attraction or repulsion vectors are defined equal to the current horizontal migration offset and oriented toward the targeted oxbow lake or in the opposite direction of the oxbow lake to avoid (Fig. 16). These repulsion or attraction vectors are applied on restricted areas of the channel path. These zones are defined by a circle centered around the neck of the considered oxbow lake and with a radius R equal to half the mean wavelength of the channel path. The mean wavelength of the channel is defined by the mean distance between two successive inflection points. The offset of attraction or repulsion is weighted by the inverse distance to the center of the oxbow lake (Fig. 16). This prevents brutal distortion of the channel path between sections inside and outside of the restriction area.

Finally, if a complex configuration has prevented the integration of an oxbow lake at the wanted time step (Fig. 15), the integration is delayed to the next time step while conserving the relative age order between oxbow lakes. This postponement to posterior steps calls for adapting the ending condition of the algorithm. In our implementation, the number of reverse migration steps can be increased up to three times the target number of steps in order to improve conditioning rate. This value has been arbitrarily defined after testing various percentages. The best would be to let the algorithm run until all oxbow lakes would be integrated (i.e., without constraint on the maximal number of time steps). However, it is impossible up to now, because all configurations are not handled, and because oxbow lake absolute ages may also need to be reconsidered during reverse migration (e.g. Fig. 15). Without such improvements, there is a risk of the algorithm falling into an infinite loop.

The integrated abandoned meander loops extend the channel reverse migration to the complete meander belt and permit us to reproduce the natural reworking of the channel belt (Fig. 17). Conditioning to fairway data is not considered here but could be easily handled. Indeed, a fairway datum can be considered in the algorithm as an indicator of past meander location in the same way as an oxbow lake. Information about the paleo-orientation of the migration can be observed on fairway data through the stacking patterns of point bars and used to orient the meander loop.
Fig. 18

a Satellite image of the Tangnara river (Russia). 24 oxbow lakes can be observed and have been identified by a specific id number. b, c Realizations of the stochastic reverse migration with conditioning to, respectively, 9 and 13 oxbow lakes in 30 time steps (the maximal threshold of steps for this application). e Map of the average occurrences of 100 realizations [Google Earth \(63^{\circ }09'04.57''\hbox { N}\)\(122^{\circ }59'29.12''\hbox { E}\)]

4 Application to a Satellite Image

We tested the reverse migration method on a \(15 \times 4\) km satellite image of the Tangnara river located in Russia (Fig. 18a) [Google Earth \(36^{\circ }12'34.31''\hbox { N}\)\(123^{\circ }00'01.02''\hbox { E}\)]. This river presents 24 oxbow lakes that we digitized and classified by relative abandonment ages (Fig. 20). The probability distributions of abandonment ages were attributed by considering the distance to the channel and relative locations of the other oxbow lakes. A target of 10 reverse migration steps was defined and Gaussian distributions were attributed to horizontal and vertical migration offsets, respectively N(500,100) and N(100,10) in meters. We generated one hundred realizations, of which two are presented in Fig. 18b, c. All realizations have the same parameters. Their differences come from the sampling of migration offsets, curvature weights and abandonment ages. A map of the average occurrences with information about the oxbow lake occurrences of integration and of reverse migration step locations over the hundred realizations is shown in Fig. 18d. It can be noticed that, even with the increased number of reverse migration steps of 300% (i.e. maximal 30 reverse migration time steps), due to the delayed integration of oxbow lakes after ten time steps, some oxbow lakes were not integrated. From one realization to another, the number of successful integrations of oxbow lakes varies between 3 and 15 (Fig. 19a). The same oxbow lakes, for example the 1, 4, 8, 12, 13, 16, 17 and 23 are often left aside. As these are the most ancient oxbow lakes, their successful integration depends on the integration of the youngest abandoned meanders (Fig. 20).
Fig. 19

a Number of conditioned oxbow lakes for each realization among the 24 observed paleo-geometries. b Number of successful integration for each oxbow lake and for 100 realizations

Fig. 20

Probable ages of oxbow lakes versus rate of successful integrations over 100 realizations

The method needs some further development to accept a wider diversity of oxbow lakes. The improvements concern precise geometrical configurations that are arduous: e.g., when the segment of the oxbow lake extremities is perpendicular to the main channel path. Another explanation for the non-integration of some oxbow lakes is a mistaken abandonment age estimation. As a perspective, working with scenarios of abandonment ages relying on observed obvious relative abandonment ages could help in identifying relative oxbow lake chronologies that permit reconstruction of more probable paleo-systems. This calls for a better management of relative ages. Relativity between abandonment ages is easy to guess for two juxtaposed oxbow lakes, but quite arduous for oxbow lakes far away one from another. Study of the natural clustering of oxbow lakes could help in defining relative abandonment ages.

Finally, some oxbow lakes could have been completely eroded by the migration of the system. Indeed, the oxbow lakes that are observable at a given moment do not necessarily retrace the exhaustive history of the channel belt. The simulation of eroded oxbow lakes could help to fit the observed ones and to reproduce the complete channel belt evolution.

5 Conclusions

Most present-day observations of channel belt systems are suitable to retro-simulate channel evolution and to reconstruct possible sedimentary structures and heterogeneities. The types of available data for channel belt modeling motivate to go back in time to reconstitute the whole system from the last channel position. The lack of information about the paleo-hydrodynamics of the river and of the bank erodibility led us to develop a geometrically-based method of stochastic reverse migration that aims at reproducing the natural geological processes. Moreover, the complexity and the irreversibility of the natural processes calls for a simplification and a randomization of the model. A statistical study on the Mississippi river suggests an anti-correlation trend of reverse downstream and lateral migration offsets at the half-meander scale versus the local curvature. This anti-correlation could perhaps be attributed to the energetic compensation between both migration directions of the channel at the half-meander scale. Comparison with correlations observed in other studies about the ratio of curvature radius and channel width could be realized on the last mapped channel path. At this stage, the channel width can be precisely observed. Perfect conditioning to the mapped width could be done using the shape fitting of NURBS objects proposed by Ruiu et al. (2015).

The simulation of the migration in the three spacial dimensions permits us to reproduce many observed patterns of the internal sedimentary architecture in the subsurface. It includes a conditioning to the observable remnants of the system such as oxbow lakes. The stochasticity of the method could help in assessing the impact of the heterogeneities and the associated uncertainties on the subsurface flow, but calls for appropriate spatial discretization (Botella et al. 2016; Labourdette et al. 2006; Massart et al. 2016). As discussed by Pyrcz et al. (2015), the reverse migration may change statistics on the simulated geometric features of the channel (e.g. wavelength). As channels can naturally show complex variations in time and space, this can be seen as a strength, even though it raises questions about how to validate the simulation results.

Oxbow lakes with specific orientations are still difficult to integrate and led to postponing the integration of the older ones. Analysis of their natural clustering could help to assign more accurate abandonment ages and simulation of the eroded ones could facilitate the integration of each oxbow. The development of such a semi-automated tool is one of the short-term perspectives of this work. Another constraint of rotation of the main channel path could be added to overcome this limit and avoid orientation misfit. Finally, the difficulty of the integration of some oxbow lakes can also be due to unexpected behaviors of the natural river, which are hard to anticipate in our geometric approach. Oxbow lakes postponed too many times could also result from a wrong age estimation. Their exclusion could be considered. This could also be further used for invalidating some realizations. An a posteriori study of the migration offsets could be done for that purpose.

One of the main perspectives of this work is to test the method on subsurface data of fluvial or turbidite environments. Conditioning to data other than oxbow lakes could also be considered, such as point bars or borehole data. Point bar data can be assimilated to paleo-channel geometries in the same way as oxbow lakes. Well data would indicate sand or shale facies which could be conditioned through the simulation of oxbow lake geometries inside the channel belt.

Notes

Acknowledgements

This work was performed in the frame of the RING project at Université de Lorraine (http://ring.georessources.univ-lorraine.fr/). We would like to thank for their support the industrial and academic sponsors of the RING-GOCAD Consortium managed by ASGA. The software corresponding to this paper is available in the GoNURBS plugin of SKUA-Gocad. We also acknowledge Paradigm for the SKUA-Gocad Software and API. The authors thank Michael Pyrcz and an anonymous reviewer for their constructive remarks that helped us to improve this paper.

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Copyright information

© International Association for Mathematical Geosciences 2017

Authors and Affiliations

  1. 1.GeoRessources - UL/CNRS/CREGUVandoeuvre-les-NancyFrance

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