1
A031
GEO-MODELS PRODUCTION FORECAST WITH A NON
LINEAR N-DIMENSIONAL INTERPOLATOR
Emmanuel FETEL, Jean-Laurent MALLET and Jean-Charles Voillemont
Nancy School of Geology - Gocad Research Group
BP 40, Avenue du Doyen Marcel Roubault
54501 Vandoeuvre-les-Nancy, France
Abstract
One of the most challenging problem in reservoir modelling is to handle the uncertainty on the reservoir flow
performance. Common uncertainty analysis approaches use a large number of equiprobable model, however
among them only a limited number are considered for detailed flow simulation. This paper proposes an approach based on a n-dimensional response surface, to forecast the reservoir flow performance on non-simulated
model. This approach is based on the Discrete Smooth Interpolation Algorithm describe in [7]. This algorithm
designed to work in an n-dimensional space is particularly convenient because uncertainty on the data and
contradictory data can be taken into account. The approach has been tested on realistic model and results are
consistent and some time even better than classical techniques.
1
Introduction
Uncertainty exists at each level of the modelling of a reservoir, starting with measurements of raw
data and their interpretation, to the specifications (physical process, fluid properties, etc.) of the
flow model. To account for these uncertainty, a common approach is to generate a large number of
“ equiprobable ” models using geo-statistical techniques. All these realizations are then expected to
represent a sampling of all the possible geological scenarios based on the available data. Such an
approach is efficient for assessing the range of a static reservoir property such as the connectivity or
the oil in place. However due to time-consuming calculations and computer limitations it can not be
applied for dynamic measurement of the reservoir production. In practice, only a limited number of
models (about 4 or 5) are considered for detailed flow simulations. However it could be interesting to
forecast the production performance of the other non-simulated model.
Several papers have been published in the literature on this topic (Hird and Dubrule, 1995 [4],
Deutsch and Srinivasan, 1996 [2], Corre et al., 2000 [1], Friedmann et al., 2001 [3] and Kabir et al.,
2002 [6]). All these methods characterize the available models with one or more criteria (also called
attributes). These criteria could be for example static parameter such as all the input parameters of the
geo-statistical simulations, volumetric or connectivity measurements, or dynamic parameters deduced
from some fluid-flow simulations. They define a n-dimensional space, called attributes space, where
each reservoir model can be represented as an unique point. For each point in this space, the studied
reservoir performance parameter can be estimated with a “ Response Surface ”, a simple mathematical
function linking the attributes and the performance parameter. The response surface is used, then, for
reservoir performance prediction, uncertainty analysis or for choosing new attributes if the response
is not relevant. This approach is sketched in Fig. (1). So far, the most popular methods used to build
a response surface are the multivariate linear or non-linear regressions. These methods are simple,
fast and provide as a result an equation, easy to use afterwards; they are, however also some major
9th European Conference on the Mathematics of Oil Recovery - Cannes, France, 30 August - 2 September
2004
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Figure 1: General scheme for oil-production forecast.
drawbacks with these classical methods: first, the response function is inherently biased, since it depends strongly on the regression model chosen initially, next it is not possible to represents a complex
response and finally the least square solver involved by these techniques may be sensitive to outliers.
This paper proposed to use an interpolation method based on the Discrete Smooth Interpolation
algorithm proposed by (Mallet, 1997 [7]) and adapted to a n-dimensional case. Such method overcomes the major weaknesses of regression algorithm : it allows to model complex response surfaces
and excludes the bias induced by the initial choice of a model observed when using a regression based
method.
2
Interpolation method
By definition, any interpolation method is a mathematical procedure which estimates values of a
function at positions between given values.
In practice, Polynomial and Spline interpolations and Kriging are the most used interpolation
techniques to model a response surface. The two first methods compute mathematical functions
(polynomial and piecewise cubic polynomial) which strictly match the initial data point. Kriging
is a geostatistical interpolation technique that considers both the distance and the degree of variation between known data points when estimating values in unknown areas. Kriging is an exact and
non-biased linear estimator, the estimated values are computed as a linear combination of known values. The weights are determined from a variogram that measures the degree of spatial variation of
the variable. These methods are efficient for small data sets and well spaced data points. However
as the interpolation function strictly match the data point, the response function may present high
oscillations and can not handle contradictory or uncertain data.
3
2.1
Discrete Smooth Interpolation
The Discrete Smooth Interpolation (DSI), introduced by J.L. Mallet in [7], is a method designed for
interpolating a function ϕ(alpha), here the response function, at each node α of a mesh covering the
domain of definition of this function in the attributes space.
According to the notations introduced in [7], the set of all the nodes of the mesh is noted Ω and
the DSI method is designed to find an optimal solution honoring a set of constraints C = {c1 , c2 , ...}.
The main classical DSI constraints are :
• The minimum roughness constraint : It ensures that ϕ varies smoothly when we move from one node to
the others.
• The control node constraint : It consists in specifying that the value ϕ(α) at a node α must be equal to a
given value.
• The control point constraint : It specifies the value of ϕ at location different from Ω.
Any DSI constraint is assumed to have the following general form
Ac (α) · ϕ(α) = bc
(1)
α∈Ω
where ϕ(α) is the function to be determined while Ac (α) and bc are given weighting coefficients
defining the particular constraint c. Note that, introducing the following column matrices ϕ and Ac ,
each constraint c ∈ C can also be expressed in a matrix form :
⎡
⎡
⎤
ϕ(1)
⎢
⎥
..
⎢
⎥
.
⎢
⎥
⎢
ϕ = ⎢ ϕ(α) ⎥
⎥
⎢
⎥
..
⎣
⎦
.
ϕ(M )
⎤
Ac (1)
⎢
⎥
..
⎢
⎥
.
⎢
⎥
⎢
Ac = ⎢ Ac (α) ⎥
⎥
⎢
⎥
..
⎣
⎦
.
Ac (M )
and :
Atc · ϕ = bc
(2)
The DSI method consists in solving the set of linear equations corresponding to the set of all the DSI
constraints:
Atc · ϕ = bc
1
1
..
.
t
Ac · ϕ = b c
N
N
(3)
In practice, this system is solved in a least square sense by minimizing the following quadratic criterion :
c · Atc · ϕ − bc 2
J(ϕ) =
(4)
c∈C
where c is a certainty coefficient used to balance the different constraints.
For a complete presentation of the DSI method and the notion of DSI constraint one can refer to
(Mallet, 1997) [7].
2.2
The Proposed Approach
The DSI method and the DSI constraints have been intensively used to solve 3D problems as the construction of geological surfaces or the interpolation of a given property. However The DSI method is
very general and was designed to interpolate a function ϕ(α), independently of the embedding space
9th European Conference on the Mathematics of Oil Recovery - Cannes, France, 30 August - 2 September
2004
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where the nodes of Ω are located (here the attributes’ space).
Consider a n-dimensional hypercube covering the domain of interest in the attributes space. This
hypercube is assumed to be discretizes by a regular grid whose nodes constitute the set Ω where
the function ϕ(α) must be interpolated. Note that such a grid is a structured object with implicit
geometry and topology, making, thus, this object easy to use and with low memory requirement. At
each node α ∈ Ω, the response surface is represented as a property ϕ(α). Within any cubic cell of
the grid, the response surface values are assumed to be computed as a barycentric interpolation from
the surrounding grid node values. Each of the few selected reservoir models for which a production
parameter has been computed with a regular flow simulator, is set as control points for the function
ϕ(α) to be interpolated on Ω.
This approach allow all the capability of the DSI-method to be used to interpolate the response
surface on Ω. Note that, thanks to the certainty coefficient c , uncertainties on the initial data can be
taken into account during the interpolation.
2.3
Reduction of the Attributes Space
Whatever the method used, the attributes characterizing the models must be chosen in a consistent
way. In particular, the following possible problems must be fixed before modeling the response function:
• Most of the time, the attributes are heterogeneous with very different orders of magnitude. This may
have a detrimental effect on the metric of the attributes space: in this case, the variables having the
largest order of variation will completely override the effect of the other variables.
• It is unavoidable that some of these attributes may be redundant (e.g. correlated): in this case, redundant
variables will also tend to override the effect of the other components.
An excellent technique to avoid these problems is to compute a principal component analysis (PCA)
on the raw data. This method returns a set of non correlated factors, which defines a new attributes
space where the data points are properly spread. Moreover, one can select only the factors which
explain most of the data variation and thus reducing the number of attributes entering for computing
the response surface. In practice keeping the first principal components explaining 80% of the data
variance provides excellent results.
3
Applications
This paper presents two applications, the first one uses a simple mathematical function while the
second is based on a synthetic reservoir data set.
3.1
Mathematical Function
A classical technique to test the efficiency of an interpolation or approximation method consists in
sampling a mathematical function and compared the result of the interpolation of these samples with
the actual values of the function.
The variables x1 , x2 , x3 , x4 and x5 ) are assumed to represent reservoir attribute values. The function F represents the relation between a reservoir production parameter and the attributes :
π
F (x1 , x2 , x3 , x4 , x5 ) = cos(x1 . ) − x1 .x2 − 0.2x3 + x4 + 2x5 − 1
2
(5)
5
The goal of the study is to retrieve this relationship using the proposed approach based on the DSI
algorithm using a set of 50 randomly sampling data points. Among these points 20 points were
selected as a training set for the interpolation. The other 30 points will be the ”testing set” on which
we will compare the ”truth” to the estimation. This provides an estimation of the prediction error.
The proposed approach is also compared to classical regression methods. For this purpose, three
regression models have been generated : a “classic” linear model, a quadratic model and a quadratic
model including the interaction terms.
Figure 2: Mathematical model, ”truth” results (x-axis) vs. estimation results (y-axis) for different response
surfaces a) DSI-based method, b) multivariate linear regression, c) multivariate quadratic regression and d)
multivariate quadratic regression with interaction terms.
Method 2
errors
averageerror
maximumerror
DSI
11,037
4.8%
28%
Linear Reg.
28,679
7.8%
33%
Quadratic Reg.
10,224
7.7%
53%
Quad. Reg. with interaction terms
11,033
8.6%
54%
Table 1: Comparison of estimation error between the DSI-based method and the ”classical” regression
methods.
In Fig. (2), it appears that the DSI-based method predicts the most accurate response surface with a
correlation coefficient of 0.82 between the estimation and the real values. This is confirmed by the
comparison of the estimation error (see Table (1)).
9th European Conference on the Mathematics of Oil Recovery - Cannes, France, 30 August - 2 September
2004
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3.2
Synthetic Model
3.2.1
Model description
A simple 2D grid (100 × 100) with two wells (one injector and one producer) located in opposite corners has been used. A set of 25 realizations of permeability field were generated with unconditioned
Sequential Gaussian Simulation (SGS). The simulations were constraint with 5 types of variograms
and a unique triangular distribution histogram, with value between 150 mD and 750 mD and a mean
at 500 mD. The porosity property is considered constant equal to 20% for all the realizations. For
each model, streamline simulations were performed to model a tracer injection. The first arrival time
of tracer at the producer has been selected as reservoir production parameter to be predicted by the
response function for this study.
Seven models were chosen as a training set and the others as a testing set. The objective of this
study is to predict from the training set the first time arrival of tracer on all the available models.
Finally the results are compared also to the ”truth” values.
3.2.2
Model Attributes
To characterize the reservoir models, the following attributes were computed :
• The time-of-flight swept pore volume, introduced by (Idrobo, 2000 [5]) at 12 and 31 days. The TimeOf-Flight (TOF) is the time for a neutral particle to reach a given distance following a streamline. This
variable is widely used in streamline simulation. The attribute corresponds to the pore volume of the
reservoir part with a TOF to the producer less or equal to a given time;
• The TOF values between the injector well and the producer well, are a good indicator for the reservoir
flow performance. The smallest value of this TOF property and the associated swept volume as defined
before have been selected as attributes;
• Each variogram used for modelling the different permeability fields has been assigned to an index (1 to
5). This index has been selected as an attribute.
Other attributes like the permeability connectivity or volumetric measurement as the oil-in-place are
not relevant for this study. In fact, the unconditioned-SGS carried out for modelling the permeability
field uses a unique distribution histogram. Therefore, for all the models, these attributes have the same
values. They are thus inefficient to separate the models in the attribute space and inconvenient for any
reservoir production prediction method.
3.2.3
First Arrival Time of Tracer Estimation Results
The comparison of the DSI-based approach with a classic linear regression is shown in Fig. (3). As
one can see, for all the models, the DSI-based approach provides a good estimation of the time of
first arrival of tracer. The average error on this data set is equal to 7% and the maximal error between
the truth and the DSI based estimation is equal to 20%. On the other hand the linear regression
technique provide a estimation with an average error of 20% and a maximal error of 68%. The
quadratic regression and the quadratic regression with interaction terms provides even worst results
with an average error respectively equal to 26% and 30%.
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Figure 3: Synthetic Model : Comparison between the “true” results (x-axis) and the DSI-based approach
estimation and linear regression estimation (right). The average estimation error are respectively equal to 7%
and 17%.
Figure 4: Synthetic Model: Cross Validation of the response surface build with the DSI-based approach, the
line represent the ”ideal” curve.
3.2.4
Cross Validation
The quality of the response surface computed with the DSI-based approach has also been tested with
cross-validation techniques (see Fig. (4)). The result are quite good. Except for a small number of
outliers, the points are fairly matching the ”ideal” curve. The presence of outliers in not surprising
as they correspond to extreme values. When they are removed from the set to be validated, the
interpolation does not take them into account, so the new interpolated value reflects only the values
of the other control points.
3.3
Concluding Remarks
The DSI-based approach seems to provide interesting results for modelling a surface response. The
final estimation error lies, generally, in the range of ±7% with a maximal estimation error of ±20%
for the extreme points ,which is quite acceptable.
From a computation cost point of view, the method is quite fast. It takes only 1 minutes for solving
a problem with 2.5 × 104 grid points. More intensive tests must be performed, but it seems that the
algorithm may not converge in a reasonable time for grids greater than 106 points in total size. For
example, for a regular grid with 10 points along each directions, this limits the number of dimension
to 6. If one wants to use a larger number of dimension, reducing the attribute space with a PCA
method is more than advised.
9th European Conference on the Mathematics of Oil Recovery - Cannes, France, 30 August - 2 September
2004
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4
Conclusions and Future Works
In this paper an original approach have been proposed to assess and forecast the oil-production for
a reservoir. This method uses the characterization of geostatistical models in a N-dimensional space
in which the DSI algorithm can be used efficiently to compute a surface response of a particular
reservoir production performance parameter. Moreover this method is particularly convenient because
uncertainty on the data and contradictory results can be taken into account.
The approach has been tested on realistic model and first results appear to be similar and even
better results than common techniques. The Discrete Smooth Interpolation shows results with an
error of ±7% on the response surface in area of interest. Although more tests are required to fully
validate the method.
The method has been presented for the oil-production forecast. But, of course, such a method
can be used in any case when a response surface has to be computed, and more generally when an
non-linear relationship exists between an input and an output. This work will certainly be used in
other various project such as the seismic facies recognition, the computation of non-linear principal
components or the modelling of a fracture network.
Acknowledgement
The authors would like to thanks J. Amiotte (ChevronTexaco) for his valuable input and the work previously performed. This research works was carried out performed in the frame of the gOcad research
project. The companies and universities members of the gOcad consortium are hereby acknowledged.
Authors would like, especially, thank ChevronTexaco for supporting the Ph.D of Emmanuel Fetel.
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