Complexity of the Geological Framework and Use of Mathematics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 21

Fig. 1.9 Example of decomposition of a reservoir into a set of hexagonal adjacent 3-D cells adapted
to the specific needs of the flow simulator. The edges of these cells cross neither the horizons nor the
faults and are aligned to constitute a 3-D curvilinear coordinate system ( u , v , t ). There are distortions
of the lengths of horizontal edges between the top and bottom cells of the reservoir (a) Geological
space is transformed into (b) Parametric space (Source: Mallet 2004 ,Fig.1)
As illustrated in Fig. 1.9a , the first generation of flow simulators used in reservoir
engineering is based on a decomposition of the subsurface into a set of adjoining
hexahedral cells whose edges nowhere cross the horizons and the faults. These cells
can be aligned to generate a stratigraphic grid whose edges induce a curvilinear
coordinate system ( u , v , t ) with t oriented in the vertical direction and ( u , v ) parallel
to the bedding (Mallet 2002 ). As a result, each point in the subsurface has an image
in the ( u , v , t ) parametric domain and images of the nodes of the stratigraphic grid in
this parametric space constitute a rectangular grid where bedding is horizontal and
not faulted as illustrated in Fig. 1.9b . The curvilinear coordinates account for the
shapes of the horizons which themselves controlled geological continuity. How-
ever, this original geomodeling approach had two drawbacks: (1) Distortions of
horizontal distances could not be avoided when faults are oblique relative to the
horizons; as shown in Fig. 1.9a : a pair of faults with V shape in the vertical direction
can generate significant distortions of horizontal cell sizes increasing from top to
bottom of a reservoir; and (2) the new generation of flow simulators uses unstruc-
tured grids based on decomposition of the subsurface into polyhedral cells that
cannot be used to compute curvilinear distances.
Mallet ( 2004 ) later developed a mathematical “ge o- chronological” (GeoChron)
model in which the original G-space is replaced by a G-space with the property that
distortions and difficulties related to the curvilinear coordinate ( u , v , t ) system
downward from the be dd ing planes do not occur (Fig. 1.10 ). Mallet's ( 2004 )
GeoChron model in the G-space impr ov es upon the earlier G-based approach by
eliminating its inherent drawbacks. In G-space the rectangular (Cartesian) coordi-
nate ( x , y , z ) system can be used allowing, for example, standard geostatistical
estimation of rock properties. In G-space the ( x , y , z ) system only ap plies in the
immediate vicinity of the folded and unfaulted horizons but in G -space any
curvature-related distortions do not exist. Mallet ( 2004 ) uses the expression