Modeling Channel Forms and Related Sedimentary Objects Using a Boundary Representation Based on Non-uniform Rational B-Splines

Abstract

This paper aims at providing a flexible and compact volumetric object model capable of representing many sedimentary structures at different scales. Geobodies are defined by a boundary representation; each bounding surface is constructed as a parametric deformable surface. A three-dimensional sedimentary object with a compact parametrization which allows for representing various geometries and provides a curvilinear framework for modeling internal heterogeneities is proposed. This representation is based on non-uniform rational basis splineswhich smoothly interpolate between a set of points. The three-dimensional models of geobodies are generated using a small number of parameters, and hence can be easily modified. This can be done by a point and click user interaction for manual editing or by a Monte-Carlo sampling for stochastic simulation. Each elementary shape is controlled by deformation rules and has connection constraints with associated objects to maintain geometric consistency through editing. The boundary representations of the different sedimentary structures are used to construct hexahedral conformal grids to perform petrophysical property simulations following the particular three-dimensional parametric space of each object. Finally these properties can be upscaled, according to erosion rules, to a global grid that represents the global depositional environment.

Appendix: Construction of an Oxbow Lake

Oxbow lakes are formed during the lateral migration of a channel when a meander becomes much curved and the two concave banks at the extremity of the meander become very close. The meander cutoff occurs when the neck of land between the two concave banks is eroded due to the channel migration or due to strong current during a flooding event. The meander is then abandoned and a shortcut in the channel path is created where two concave banks are connected (Fig. 19).

Fig. 19

Principle of the formation of an oxbow lake during the lateral migration of a meander

In the proposed approach, this process is taken into account during the lateral channel migration by detecting the self-intersections in the control point net of the top surface of the channel.

1.1 Segment Intersection

Considering two non-parallel segments belonging, for instance, to the control mesh of a channel top surface. Their intersections can be computed by determining two scalar values a and b such that

(8)

where $\mathbf {p}_1$ is the origin of the first segment, $\mathbf {p}_2$ the origin of the second segment, and $\mathbf {v}_1$ and $\mathbf {v}_2$ the vectors corresponding to the segments.

Taking the cross-product of Eq. (8) with $\mathbf {v}_1$ and $\mathbf {v}_2$, respectively, the following expressions come for a and b

$$\begin{aligned} \left\{ \begin{array}{l} a \; (\mathbf {v}_1\times \mathbf {v}_2) = (\mathbf {p}_2 - \mathbf {p}_1)\times \mathbf {v}_2 \\ b \; (\mathbf {v}_2\times \mathbf {v}_1) = (\mathbf {p}_1 - \mathbf {p}_2)\times \mathbf {v}_1. \\ \end{array} \right. \end{aligned}$$

(9)

If $0\le a \le 1$ and $0\le b \le 1$, the two segments intersect and the coordinates of the intersection points are given by $\mathbf {p}_{\mathrm{inter}} = \mathbf {p}_1 + a \mathbf {v}_1$.

1.2 Construction of an Oxbow Lake

The construction of an oxbow lake from a NURBS channel is divided in five steps:

• 1. Detection of the self-intersection for each side of the channel top surface (Fig. 20a). The indexes of the channel sections behind and ahead of the intersecting segments are memorized.

• 2. Sorting and reduction of the number of intersections. This operation may be needed to keep only one intersection, see below.

• 3. Increasing knot multiplicity at the parametric coordinate corresponding to the projection of the intersection point on the NURBS. This operation is used to decrease the continuity of the NURBS while preserving its shape.

• 4. Selection of the sections before and after the intersection point (Fig. 20b).

• 5. Lateral closure of the oxbow lake sections and connection of the cutoff sections to maintain channel continuity (Fig. 20c)

Fig. 20

Construction of an oxbow. During the process of a channel lateral migration, self-intersections can occur and are associated with the formation of an oxbow lake. a Self-intersections are detected (red points). b Sections before and after the self-intersection are detected (red lines). c Control point between sections are detected and suppressed to form the oxbow lake and the new channel path

Fig. 21

Sorting of the different cases of self-intersections during the NURBS channel migration. a Unique intersection. b Sections associated with an intersection (in green) are included between the sections corresponding to another intersection (in red). c The first section associated with an intersection (in red) is included between the sections corresponding to another intersections (in green). d The second section associated with an intersection (in red) is included between the sections corresponding to another intersections (in green)

Fig. 22

Result of the oxbow lake formation during the channel lateral migration

The proposed intersection sorting is essential to obtain consistent final geometry. For each intersection, a couple of channel section indexes are determined. Their number has to be reduced so that only one intersection by meander remains. Four possible configurations have been determined. If there is only one intersection in the meander, then it is directly stored in the final intersection list (Fig. 21a). If two sections (in green in Fig. 21b) are included into another one (in red in Fig. 21b), only the largest one is kept in the final list. If the first element of a couple of sections linked with a self-intersection (in red in Fig. 21c) is included in the space defined by another couple (in green in Fig. 21c), the second couple of intersections in kept in the final list of intersections (Fig. 21c). If the second element of a couple of sections linked with a self-intersection (in red in Fig. 21d) is included in the space defined by another couple (in green in Fig. 21d), the first couple of intersections is kept in the final list of intersections (Fig. 21d).

Once the final intersection list has been determined, the control points between two sections corresponding to an intersection are removed from the channel (Fig. 20c). The result of the channel lateral migration process associated with the construction of oxbow lakes is shown in Fig. 22.