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A001 NEPTUNE PROJECT – MODELLING AND
SIMULATION OF CARBONATE
ENVIRONMENTS
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Bruno Leflon & Gérard Massonnat
2
1
Gocad Research Group, Rue du Doyen Marcel Roubault – BP 40, 54501 Vandoeuvre-Lès-Nancy, France
2
Total CSTJF, 64018 Pau Cedex, France
Abstract
Neptune is a software developed to provide realistic and consistent stratigraphic grid and facies
proportions from well data and geology knowledge. It is based on the relationship between the
facies and the bathymetry of deposit. Such a relation is generally known, approximately, by
sedimentologists and is case study dependant. It is defined as bathymetry distribution of
probability for each facies. A method has been developed to compute automatically and for each
well path, a curve of bathymetry deduced from observed facies log and given distributions of
bathymetry. Extrapolation of bathymetry inside the reservoir grid is made using accommodation
concept. Accommodation is computed at well locations and extrapolated in a geo-chronological
model which is constrained by wells correlations assumed to be provided by a geologist. At any
location in the subsurface, using accommodation, paleo bathymetry can be obtained, and facies
proportions can be computed from this paleo-bathymetry. From the bathymetry (=paleotopography), exposition of sediments to prevailing winds is deduced to identify high energy
facies areas. When building a reservoir model, any stratigraphic grid can then be “painted” with
the above properties, in order to rebuild the stratigraphy in the grid to better fit the isogeological-time surfaces of the geo-chronological model. In practice, using a geo-chronological
model yields very realistic results even in the case of severely faulted or folded reservoirs.
Introduction
Stochastic simulations of reservoir heterogeneity are often limited by the following hypotheses
[3]:
1) Spatial stationarity. Such a hypothesis may be valid at a given scale but not at a
different scale. For categorical random functions (used in facies simulation), stationarity
means that prior proportions of facies are constant in the reservoir. In that case, the
facies proportions are often set equal to the facies proportions observed along the well
paths. To avoid non-realistic results induced by stationarity, additional information,
such as, for example, seismic attributes, must be included in the processing. However,
even in the case where seismic attributes are known, other problems may occur, as the
one induced by the difference of resolution between the seismic image and the model or
as the lack of correlation between a continuous variable (seismic attribute) and a
categorical variable (facies). Although rarely done in practice, a better solution would
be to integrate geological information, such as the concept of accommodation in
sequential stratigraphy sequence that can be used to avoid the use of the stationarity
hypothesis (e.g., see [3]).
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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2) Constant sedimentation rate throughout geological time. Such a hypothesis is often
made implicitly through the geometry of the reservoir grid which can be rectilinear or
curvilinear (stratigraphic). In both cases, for any geological time, it is generally assumed
that the sedimentation rate is constant as far as the thickness of grid cell is itself
constant. In others words, it is assumed that each layer corresponds to a homogeneous
time interval. This hypothesis is generally false. It would be better to use a stratigraphic
grid whose layers’ boundaries would correspond to constant geological times (see also
[2]).
Based on these two remarks, a new methodology called “Neptune”, was developed to compute
prior facies proportions based on the accommodation concept and to create a time-layered grid.
This accommodation concept is generally used in forward modelling. It is often deduced from
global sea level changes (eustatism), tectonic knowledge and not on well data. As a consequence
the results rarely fit well observations. To avoid such an inconsistency, it is proposed in this
work to estimate the accommodation directly along the well path in order to provide a consistent
bathymetry model.
Neptune principles
Definitions
As shown in figure 1 and as explained below, several concepts can be associated with
accommodation, each of these concepts being a function of the paleo-coordinates (x,y) on a
paleo-map and of geological time t:
z
x,y
Fig. 1: Cross-section of the sedimentary basin at geological time t showing the variables
associated with accommodation
1) bathymetry b(x,y,t), also called water depth, is the distance between the surface of the
water and the interface water / sediment at time t. It is deduced from observed facies
along well paths. Note that even though b(x,y,t) could be directly extrapolated in a 3d
grid, this is not recommended because it induces too high uncertainty especially in area
where there is not enough data.
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2) sediment thickness e(x,y,t) is the distance at time t between the water / sediment
interface and an arbitrarily chosen reference horizon (substratum). Note that, at any
given location (x,y), e(x,y,t) is a non decreasing function of the geological time t.
3) eustatic sea level B(x,y,t) is the distance between the surface of water and the centre of
the earth. It can be written as B(t) because, at the reservoir scale, it is independent of the
location (x,y).
4) altitude of the horizon H(x,y,t) is the distance between the reference horizon
(substratum) and the centre of the earth. The derivative of this altitude relative to
geological time ( ∂ H ) is called subsidence when H(x,y,t) decreases ( ∂ H <0) and uplift
∂t
∂t
when H(x,y,t) increases ( ∂ H
∂t
>0).
5) accommodation A(x,y,t), also called relative sea level, is the distance between the
reference horizon and the surface of water.
Extrapolation of accommodation
Using the above definitions, it comes
A(x,y,t) = b(x,y,t) + e(x,y,t)
(1)
A(x,y,t) = B(t) – H(x,y,t)
(2)
The relative subsidence or potential of accommodation s(x,y,t) is then defined as the variation of
accommodation between a reference time t0 and the time t. From equation (2), it comes
s(x,y,t) = A(x,y,t) – A(x,y,t0) = B(t) – B(t0) – (H(x,y,t) – H(x,y,t0))
(3)
If it is assumed that, in small regions at the surface of earth, subsidence or uplift do not depend
on x or y, then s(x,y,t) ≈ s*(t). Moreover, from equation (1)
s*(t) = b(x,y,t) – b(x,y,t0) + e(x,y,t) – e(x,y,t0)
(4)
Using equation (4), for each iso-time t, the potential of accommodation can be easily computed
along well paths, where paleo-bathymetry and thickness are observed. By convention, the
reference time t0 is generally assumed to correspond to the bottom of the reservoir. As s*(t) does
not depend on the location (x,y) and depends only on geological time t, it can be observed that
s*(t) should be the same for every well. An average value of the values of s*(t) observed along
the well paths, (called common factor cf(t)) is computed for each time t (known at wells through
correlation diagrams) and is extrapolated in a geo-chronological grid (a grid whose parametric
coordinates are x, y and t). The difference between this average value cf(t) and the value s(x,y,t)
observed at wells is called residual and is noted r(x,y,t). There are several reasons for residuals
not to be null: for example, the imprecision of bathymetry is one reason, differential subsidence
in the studied area is another one. Depending on the cause of residuals, it can decided to neglect
them (s(x,y,t) = s*(t) = cf(t)), or to interpolate them piecewise continuously in each fault block
and to add the result to the common factor : s(x,y,t) = cf(t) + r(x,y,t).
From accommodation to bathymetry
Once the potential of accommodation s(x,y,t) is known everywhere, thanks to equation (4), the
bathymetry b(x,y,t) can be easily deduced in the grid. Moreover, the thickness e(x,y,t) can be
computed from seismic data or by interpolating the values observed along well paths. Note that
e(x,y,t) – e(x,y,t0) is the thickness of sediments between the two iso-times. At that point, a
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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reference time t0 must be chosen and the bathymetry b(x,y,t0) at such a reference time t0 must be
given as a paleo-bathymetry map (landscape at the reference time t0).
b(x,y,t) = s(x,y,t) + b(x,y,t0) + e(x,y,t) – e(x,y,t0)
(5)
Workflow
Computing bathymetry at wells
The main data used in Neptune are facies observed along well paths. In the case where these
facies can be associated to ranges of bathymetry, a bathymetry curve is deduced along these well
paths. A naïve way to compute such a curve is to set a point at the average value of bathymetry
for each facies layer and to connect the points. However, in practice the curves obtained with
such a simplistic approach are rough (Fig. 2.a). and this is why it was decided to use the Discrete
Smooth Interpolation (D.S.I.) to get more realistic curves.
bathymetry facies
bathymetry facies
bathymetry facies
main
sequence
limits
0
25
(a)
0
25
(b)
0
25
(c)
Fig. 2: Creation of the bathymetry curve. (a) the initial curve with one data point set at the
average value for each facies layer. (b) the same curve smoothed by DSI interpolation. (c) an
interpolated curve obtained after densification of the initial curve (higher resolution).
The D.S.I. method ([1], chapter 1) is an interpolator that accounts for a series of linear constraint.
Each DSI constraint is assumed to be defined as
∑ A (α ) ⋅ ϕ (α ) ≡ b
α∈Ω
c
c
(6)
where Ω is the set of all the nodes α (discrete location) of the model, Ac(α) and bc are given
coefficients which are specific to the constraint c, φ(α) is the value of the property to be
interpolated at a node α and the symbol ≡ represents indifferently one of the symbols =, ≈ or ≤.
Note that several constraints can be set at the same time and, in the case where some constraints
are inconsistent, they are then respected in a least square sense.
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By default, a constraint specifying that the function φ(α) must be smooth at any node α is
installed in order to ensure a pseudo-continuity of the interpolated properties.
To interpolate the bathymetry, two types of DSI constraints are used: “box constraints” keeping
bathymetry between the given minimum and maximum associated with the facies, and “control
node” specifying that the value of bathymetry should be, preferably, close to the mode of the
distribution (if a modal distribution is given). Note that control node constraint ensures that the
resulting curve reflects variations of facies.
Creating geo-chronological space
arbitrary time scale
arbitrary time scale
The accommodation can only be interpolated along iso-time surfaces: this makes it necessary to
work in a 3D space (x,y,t). The geological time t is discretized between t0 (the bottom of the
reservoir) and t1 (the top of the reservoir) in such a way that reservoir heterogeneities are well
represented. Correlations between wells are then drawn with respect of that discretization (see
Fig. 3). Note that the shape of the accommodation curve along well paths may be used by the
geologist to set these correlations: since this property is well correlated along iso-time surfaces, a
peek of accommodation on a well should be represented on other wells too (Fig. 3).
a. wells before transformation
b. wells after transformation
Fig. 3: Two correlations are drawn between wells. The dark curve is the interpolated bathymetry,
the light curve is accommodation potential. After each correlation, the accommodation potential
is recomputed using bathymetry and the thickness of sediment of each time layer. Note that
accommodation peeks are good markers for well correlations.
Extrapolating properties
Once correlations are drawn, the thickness for each geological time layer is known along well
paths and is interpolated in a geo-chronological grid ([2]). In the same time, the variations of
accommodation layer by layer (=accommodation potential) are easily extrapolated along x and y
(see Eq. 4). Using cell-thickness and accommodation potential, variations of bathymetry from
one layer to another are deduced. If bathymetry is known at a given time (for example b(x,y,t0)
on reservoir bottom), the bathymetry b(x,y,t) can be computed everywhere (see Eq. 5)(Fig. 4).
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Fig. 4: Creation of the facies proportions. Cell thickness (a) and potential of accommodation (b)
are extrapolated from wells correlations. Cell thickness can also be constrained by seismic data
(reservoir thickness, internal layer thicknesses…). A paleo-bathymetry map is given for a given
layer (here the bottom). From these three data, 3d paleo-bathymetry is deduced. Then facies
proportions are computed from bathymetry using the initial given distributions.
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At the beginning of the workflow, distributions of bathymetry probabilities were defined for
each facies. They are then used to compute a facies probabilities grid from bathymetry b(x,y,t).
Moreover, if paleo-directions of prevailing winds are known, high energy zones can be deduced
from the paleo-morphology and facies proportions are changed consequently.
Transferring data from geo-chronological space to geographical space
The previous grid was defined in paleo-geographical coordinates x and y. Those coordinates can
be either actual geographical coordinates (little deformed reservoirs) or a parameterization of the
reservoir volume. Vertically, the previous grid was in time, however the cell thickness is also
known. The proposed method is to build a grid in the geographical space having the same
number of horizontal layers than the geo-chronological model. That grid is painted with the
properties of the chronological grid layer to layer and using the values of the coordinates X and
Y. Then the geographical grid is reshaped using the thickness property (Fig. 5).
Usual geographical space
Geo-chronological space
Fig. 5 From geo-chronological space to geographical space. The final grid is reshaped using the
thickness data.
Perspectives
Accommodation is theoretically easy to extrapolate. However, since it is computed from an
estimated bathymetry, there will always be residuals that should not be taken into account.
Moreover cell thickness is extrapolated independently of bathymetry or accommodation, and is
taken to compute 3d bathymetry. Incertitude in thickness increases incertitude in bathymetry.
About thickness, a post-processing is generally done on thickness to make the sum on a column
equal to the observed seismic thickness and to make thickness null if the bathymetry is negative
(at emersion, there is no sedimentation). That modification of thickness should have an impact
on the bathymetry; we enter here in a loop.
Further developments on this work are to improve the D.S.I method in order to interpolate
together both the cell-thickness and the bathymetry. Equation 5 shows that relations between
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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accommodation, bathymetry and thickness are linear equations. Since the accommodation should
be constant on the same layer, the equation can be transformed into the following D.S.I.
constraint:
φ0(i,j,k+1) – φ0(i,j,k) + φ1(i,j,k+1) – φ0(i+p,j+q,k+1) + φ0(i+p,j+q,k) – φ1(i+p,j+q,k+1) ≈ 0
(7)
where φ0(i,j,k) is the bathymetry of cell (i,j) at the top of the layer k and φ1(i+p,j+q,k+1) is the
thickness of layer k at cell (i,j). In other words, that constraint means that the potential of
accommodation in cell (i,j) must be nearly equal to the potential of accommodation in cell
(i+p,j+q).
Using D.S.I., albeit is possible to fix the thickness equal to null if bathymetry is negative or to set
the sum of thickness in a column of the grid equal to a given value. As it is taken as constraints
in the interpolator, artefacts due to post-processing should be avoided.
Conclusion
The major points of the present method are that the stratigraphic grids are directly built in the
geo-chronological space which is the most naturalistic space to work in and that the well data are
directly used as soft constrains in the forward model. Both points provide very realistic models
where heterogeneities due to stratigraphic process are represented.
First versions of Neptune have been tested on several real case studies in Total, and the results
are always very persuasive, since they are geologically consistent and robust. Studies are in
progress to investigate the uncertainties produced by the Neptune approach. The first results
show that the main uncertainty is on the input map of the paleo-bathymetry. To take those types
of uncertainties into consideration, it is planed to introduce a multi-realization approach.
Acknowledgements
This research works was performed in the frame of the gOcad research project. The companies and universities
members of the gOcad consortium are hereby acknowledged. The authors address a special thanks to Total for their
financial support provided for this PhD and to Jean-Laurent Mallet for his precious advices.
Bibliography
[1]
Mallet, J-L. (2002). Geomodeling. Oxford University Press, 1st edition.
[2]
Mallet, J-L. (2004). Space-time mathematical framework for sedimentary geology.
Mathematical Geology, 36(1):1-32.
[3]
Massonnat, G. (1999). Breaking of a paradigm: geology can provide 3d complex
probability fields for stochastic facies modelling. In SPE Annual Technical Conference and
Exhibition, Houston, Texas.