1
A007 REBUILDING AND RECALIBRATING AN
EXISTING RESERVOIR MODEL
MICKAËLE LE RAVALEC-DUPIN, LIN YING HU, OUAËL MEGHIRBI AND FRÉDÉRIC ROGGERO
Institut Français du Pétrole, 1 & 4 avenue de Bois Préau, 92852 Rueil-Malmaison Cedex France
Abstract
The increasing computer power and the recent developments in history-matching can motivate
the re-examination of previously built reservoir models. To save engineer and CPU times, we
develop four distinct algorithms, which allow for rebuilding an existing reservoir model without
restarting the reservoir study from scratch. These algorithms involve techniques such as
optimization, relaxation, Wiener filtering or sequential reconstruction. Basically, they are used to
identify a random function and a set of random numbers. Given the random function, the random
numbers yield a realization, which is pretty close to the existing reservoir model. Once the
random numbers are known, the existing reservoir model can be submitted to a new historymatching process to improve the data fit or to account for newly collected data.
Introduction
A reservoir model is a grid populated by reservoir properties such as permeability. The available
field data do not provide a deterministic description of the reservoir, but are used to infer a
random function whose realizations are likely reservoir models. The drawn realizations usually
honor neither the static data (well logs, cores), nor the dynamic data (pressures, flow rates…).
Static data can be easily accounted for using conditional kriging. Integrating the dynamic data
entails a time-consuming history-matching procedure: an initial reservoir model is successively
modified to improve the match.
To take advantage of the increasing computer power and of the recent developments in historymatching, reservoir engineers may wish to re-examine a previously built reservoir model. The
idea is to update and refine the existing reservoir model by integrating newly collected data. To
do this, reservoir engineers must be capable of rebuilding the existing reservoir model: they have
to know the random function inferred from the data, the random numbers used to generate the
reservoir model, and the different steps in the history-matching procedure. Unfortunately, this
information is not always available.
We supply reservoir engineers with tools, which extract the random numbers from the existing
reservoir model. A preliminary statistical analysis is performed to infer an appropriate random
function. Then, four algorithms are proposed to identify random numbers capable of reproducing
the existing reservoir model. The first one is based upon optimization: it can be used whatever
the considered geostatistical simulator. The second and the third algorithms are designed for
simulators involving a convolution product. The final algorithm is suitable for sequential
(Gaussian, indicator or multipoint) simulators. Once the random numbers are known, the existing
reservoir model can be rebuilt and submitted to a new history-matching process. This capability
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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is demonstrated by rebuilding existing continuous and facies reservoir models, which are then
globally and locally modified.
A few words about geostatistics
Many simulation techniques have been proposed to generate multiple realizations or pictures of
the spatial distribution of an attribute such as permeability or porosity. Before proceeding
further, we revisit two geostatistical simulation techniques suitable for pixel-based realizations.
The basic inputs of geostatistical simulators are a seed, a mean, a variance and a covariance
model. The seed allows for populating each grid block with a random number. The whole set of
random numbers is noticed z. The mean, variance and covariance characterize a random function
S.
Sequential simulation
The sequential simulation approach reduces the problem of generating a N-dimensional random
vector into a series of N univariate generation problems. In this case, the random numbers of
interest are uniform deviates.
First, we define a random path visiting each block of the grid only once. At each grid block, we
estimate, by kriging, the probability distribution of the studied reservoir attribute conditioned to
the values simulated at the previously visited grid blocks. Then, we simulate an attribute value
from that conditional probability distribution using the uniform number attributed to the
considered grid block. This simulation process is repeated until all grid blocks are visited.
In practice, this sequential approach is often used for Gaussian simulation, Indicator simulation
(e.g., Goovaerts, 1997) and simulation based on multipoint statistics (Strebelle, 2002).
FFT-Moving Average simulation
The Fast Fourier Transform – Moving Average (FFT-MA) algorithm (Le Ravalec et al., 2000)
produces Gaussian realizations with stationary covariance functions. In this case, the random
numbers of interest are normal deviates.
Based upon the moving average framework, a Gaussian random field y can be written as:
y = y0 + f 1z
(1
where y0 is the mean of y and z is a Gaussian white noise. Function f results from the
decomposition of the covariance function as a convolution product: C=f 1f. As determining f and
calculating the convolution product f 1z may be difficult, the problem is translated into the
spectral domain using discrete fast Fourier transforms, which makes the calculations much easier
and fast.
Gradual deformation
The gradual deformation method is a geostatistical parameterization technique allowing to
perturb a realization from a few parameters, called deformation parameters, while preserving the
spatial covariance model (Hu, 2000). The basic gradual deformation relation applies to Gaussian
white noises:
z(ρ ) = z1 cos(π ρ) + z2 sin(π ρ).
(2
If z1 and z2 are two independent Gaussian white noises, z(ρ) is also a Gaussian white noise.
Varying deformation parameter ρ yields a continuous chain of Gaussian white noises. As the
deformation rule is periodic, ρ ranges from –1 to 1. For ρ = 0, z is the same as z1 , when ρ = ½, z
3
is the same as z2 . Providing z to a geostatistical simulator yields a Gaussian realization y=S(z).
Smooth variations in ρ induce smooth variations in y. Whatever deformation parameter ρ, z(ρ) is
a Gaussian white noise. As a result, y(ρ) has the same covariance model as realizations y1 =S(z1 )
and y2 =S(z2 ). All of the z components can be modified simultaneously resulting in a global
deformation of y. An alternative is to change only some of the z components, which induces a
local deformation of y. Although we focus on Gaussian white noises, the gradual deformation
rule also applies to any set of random numbers, which can be transformed to a Gaussian white
noise.
These properties make the gradual deformation method very suitable to refine the calibration of a
reservoir model provided the Gaussian white noise used to generate the starting reservoir model
is known.
Rebuilding existing reservoir models
Given an existing reservoir model yres, the problem consists in determining a random function S
and a set of random numbers z so that the function, when applied to the random numbers,
provides a realization y=S(z) similar to the reservoir model. A preliminary statistical analysis can
give the random S function. Thus, we presume that this random function is known and we give
much attention to the estimation of the random numbers. Four distinct algorithms are introduced
for solving such a problem.
Optimization
A very simple and general approach is based upon optimization. We define an objective function
to measure the mismatch between a starting realization y and the existing reservoir model:
J (z ) =
(
1
i
y i (z ) − y res
∑
2 i
)
2
(3
Identifying random numbers yielding a realization similar to the existing reservoir model
consists in determining a set of random numbers, which minimizes the objective function. The
starting set of random numbers is iteratively modified until the objective function reduces to a
low-enough level.
To illustrate the application of the optimization rebuilding procedure, we consider the world map
shown in Figure 1 as an existing reservoir model. It is discretized over a grid of 360x151 blocks.
For a given simulation process, we attempt to identify a set of normal deviates capable of
producing a realization as close as possible to this world map. First, we assume that the world
map can be generated from the truncated Gaussian method (Matheron et al., 1987). This method
calls for a Gaussian realization. When truncated with respect to thresholds proportional to facies
volume fractions, it becomes an indicator realization. For the studied world map, the volume
fractions of the sea and continental facies are 66% and 34%, respectively. The variogram of the
underlying Gaussian realization is supposed to be stable with an exponent of 1.4. It is also
considered as isotropic with a correlation length of 40 grid blocks. The minimization of the
previously mentioned objective function leads to the “Gaussian white noise” depicted in Figure
1. Clearly, it is not a Gaussian white noise: the deviates are not independent, their mean is not
zero and their variance is not one. It actually means that the selected simulation process and the
selected geostatistical parameters may be not appropriate. However, the essential point is that we
obtain a set of random numbers, which allows for rebuilding pretty well the world map (Figure
1, bottom left).
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Figure 1. Left, top: existing reservoir model. Left, bottom: reservoir model rebuilt from the
estimated random numbers. Right, top: estimated random numbers. Right, bottom: histogram of
the estimated random numbers.
Relaxation
In the following two sections, we focus on geostatistical simulations involving a convolution
product as shown in Eq. 1. For simplicity, the y0 mean is set to 0. Thus, the problem boils down
to the estimation of a Gaussian white noise z so that yres ≈ f 1z. As mentioned above, f results
from the decomposition of the covariance function.
We first tackle this problem by referring to the relaxation techniques (Press et al., 1992)
originally developed to solve linear systems of equations such as Ax = B. Basically, an initial
guess of x is computed from some unspecified method before being successively improved.
The exact solution z res of the problem is unknown. Instead, we consider an initial approximation,
noticed z. Thus, the error is simply given by ∆z = z res − z . Unfortunately, the error is just as
unknown as the exact solution itself. However, a computable measure of how well z
approximates z res is the residual ∆y = y res − y .If we rewrite the original problem as:
yres = f 1zres,
(4
we can subtract y = f 1z from the above equation. We find the residual equation:
∆y = f ∗ ∆z
(5
which says that the error satisfies the same set of equations as the unknown zres when y res is
replaced by the residual ∆y .
At this point, we split function f as the sum of a function g and a dirac δ such as δ (0) = a, a being
a constant. g is the same as f everywhere except in 0: g(0)=f(0)- δ (0). Thus, the residual equation
is reformulated as:
∆y = (g + δ) ∗ ∆z
(6
In the frequency space, this relation becomes:
∆Y = G∆Z + a∆Z
(7
where ∆Y , ∆Z and G are the Fourier transforms of ∆y , ∆z and g respectively. This results in
an iteration scheme:
5
∆Y (i ) = G ∆Z (i ) + a∆Z (i +1)
∆Z (i +1) =
(
1
∆Y (i ) − G ∆Z (i )
a
)
(8
We note Z(i) the Fourier transform of the current approximation and Z(i+1) the Fourier transform
of the new updated approximation. The backward Fourier transform of Z(i+1) gives the correction
to add to Z(i). This iteration sweeps are continued until we obtain satisfactory convergence to the
solution.
Wiener filtering
An alternative is the Wiener filtering (Press et al., 1992). Let us come back to Eq. 4. In the
frequency space, it is rewritten as:
Yres = F.Zres,
(9
where Yres, Zres and F are the Fourier transforms of yres, zres and f respectively. Applying the
(
Wiener filtering to compute Zres consists in replacing F-1 by F / F
2
)
+ ε . ε is a constant much
smaller than 1. |F| stands for the F norm and F denotes its complex conjugate. Thus, the
existing reservoir model can be rebuilt from the following Gaussian white noise:
z res = TF
⎛
−1 ⎜
⎜
⎝F
F
2
⎞
Yres ⎟
⎟
+ε
⎠
(10
An example is depicted in Figure 2. Starting from the existing one million grid block reservoir
model shown on the top, left, we use the Wiener filtering technique to estimate the underlying
Gaussian white noise (top, right). The variogram is assumed to be exponential and isotropic with
a correlation length of 300 grid blocks. We observe that the estimated Gaussian white noise
(Figure 2, top right) is not exactly a Gaussian white noise, although the selected variogram was
exactly the same as the one used to generate the existing reservoir model. It exhibits stripes and
its variance is clearly different from 1. In addition, the realization generated from this Gaussian
white noise looks different from the existing reservoir model (Figure 2, bottom left), at least on
the boundaries over a width of one correlation length.
To avoid these boundary perturbations, we also apply the Wiener filtering to the same existing
reservoir model, but extended over one correlation length everywhere as shown in Figure 3 (top,
left). The extra grid blocks located out of the reservoir model are set to the values of the closest
reservoir grid blocks. The results are significantly improved. The estimated Gaussian white noise
is formed of independent normal deviates (Figure 3, top right). Its mean and variance are 0 and
1, respectively. Last, the realization (Figure 3, bottom left) rebuilt from this Gaussian white noise
is very similar to the existing reservoir model, even if there are still some slight differences on
the borders.
Based on this one million grid block example (Figure 3), we performed a few comparison tests to
estimate the efficiency the three previous rebuilding techniques. We showed that the
optimization and relaxation approaches need hours to converge to an acceptable Gaussian white
noise while the Wiener filtering requires only 25 seconds on a standard PC.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Figure 2. Left, top: existing reservoir model. Left, bottom: reservoir model rebuilt from the
estimated random numbers. Right, top: estimated random numbers. Right, bottom: histogram of
the estimated random numbers.
Figure 3. Left, top: same existing reservoir model as in Figure 2, but extended over one
correlation length. Left, bottom: reservoir model rebuilt from the estimated random numbers.
Right, top: estimated random numbers. Right, bottom: histogram of the estimated random
numbers.
Sequential reconstruction
In this last section, we suggest a sequential rebuilding process to determine the set of random
numbers capable of leading to an existing reservoir model. This process can be applied whatever
the simulation process used to simulate the existing reservoir model.
The rebuilding process is just the opposite of the simulation one. First, the random path
established to visit all of the grid blocks is frozen: the reconstruction path has to be identical to
the simulation path.
7
At each grid block, we estimate the probability distribution conditioned to the values of the
previously visited grid blocks. Then, we transform the reservoir value attributed to this grid
block to a uniform deviate following the identified conditional probability distribution. This
uniform deviate can also be turned into a normal deviate. This rebuilding process is repeated
until all grid blocks are visited.
This sequential rebuilding approach is suitable whatever the considered sequential simulation
algorithm.
Deformation of existing reservoir models
Once the random function and the random numbers have been identified, a realization similar to
the existing reservoir model can be generated. The next step consists in coming back to historymatching to refine and update the reservoir model. As the random numbers are known, they can
be modified using the gradual deformation method (Eq. 2). It induces a global deformation of the
reservoir model when all random numbers are modified (Figure 4) or a local deformation of the
reservoir model when only the random numbers populating a given domain are modified (Figure
5).
1
4
2
5
3
6
Figure 4. Global gradual deformation of the world map (from 1 to 6). The whole map is
modified.
1
4
2
5
3
6
Figure 5. Local gradual deformation of the world map (from 1 to 6). Deformation is centred on
Indonesia.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Conclusions
The following main conclusions can be drawn from this study:
-
Four distinct algorithms were presented to rebuild an existing reservoir model. Given a
inferred random function, they entail to the extraction of random numbers from the existing
reservoir model.
-
The first algorithm is an optimization procedure: it can be used whatever the considered
geostatistical simulator. The second and the third algorithms are designed for simulators
involving a convolution product. They are based upon relaxation and Wiener filtering,
respectively. The fourth algorithm involves a sequential rebuilding process. It allows for
extracting random numbers, which are used to rebuild the existing reservoir model from a
sequential (Gaussian, indicator or multipoint) simulation algorithm. It does not require that
the existing reservoir model was actually simulated from a sequential approach.
-
The Wiener filtering technique turned out to be of interest when applied to an enlarged
reservoir model. Extending the reservoir model allows for reducing drastically the boundary
effects. In addition, numerical tests pointed out the efficiency of this approach in terms of
CPU times.
-
Once the random function and the random numbers are known, the existing reservoir model
can be rebuilt and submitted to a history-matching process. Applying the gradual
deformation method to the random numbers results in a global or a local deformation.
Acknowledgements
This work has been performed within the framework of the CONDOR II joint industry project.
The authors thank the participating companies for their support: Groupement Berkine, BHPBilliton, Eni-Agip, Gaz de France, Petrobras and Total.
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