1
A013 HISTORY MATCHING WITH RESPECT TO
RESERVOIR STRUCTURE
1
2
SIGURD IVAR AANONSEN ; ODDVAR LIA ; AND OLE JAKOB ARNTZEN
2
1
Centre for Integrated Research, University of Bergen, Allégt. 41, N-5007 Bergen, Norway
2
Statoil Research Centre, N-7005 Trondheim, Norway
Abstract
A practical method for history-matching with respect to geological fault properties (fault
displacement, average dip and strike, smear-gouge-ratio, shale-smear-factor, etc.) is presented.
The method is based on coupling commercial software for fault modelling with a reservoir
simulator. The history-matching and control of these tools are performed using standard
optimization routines in MATLAB. Both large seismic faults as well as small subseismic faults
can be modelled. For the large faults the simulation model grid is automatically deformed in
each iteration of the history-matching process. The sealing effect is based on an advanced fault
seal model and is included as transmissibility multipliers across individual cell interfaces. The
subseismic faults may either be included in the same way as the large faults, or as permeability
modifiers applied to the existing grid. The methodology provides a good link between geological
modelling (sedimentology, structural geology) and fluid flow simulation. The method is applied
to reservoir models of typical shallow marine and fluvial depositional systems.
Introduction
A large number of publications on history-matching have appeared during the last three decades.
However, they almost exclusively concentrate on the estimation of petrophysical properties, like
permeability and porosity. Very few papers have considered the reservoir structure. If faults are
considered, it is mostly limited to the adjustment of a single transmissibility multiplier for each
fault. In a real case, however, the transmissibility will vary over the fault and depend on factors
like fault throw, fault permeability, shale smear, etc. In this paper, the parameterization of
seismic and subseismic faults used in a commercial fault modelling tool[1] has been evaluated for
history matching. This tool can perform stochastic simulation of faults and fractures as well as
structural uncertainty modelling and also add faults to simulation grids[2]. History-matching with
respect to structural geological fault parameters is performed on reservoir models of shallow
marine and fluvial systems.
Fault Modelling
Two types of faults have been considered: Small subseismic faults and larger seismic faults. The
subseismic faults are generated stochastically, and their effect is normally incorporated in the
flow model as modifications to the grid cell permeabilities. Sealing effects of the subseismic
faults are based on a fault permeability, which may be given explicitly or calculated from the
neighbouring cell permeabilities. The larger faults are modelled explicitly through a parametric
description. These faults are denoted pfm faults. The simulated faults will automatically be
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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incorporated in the reservoir simulator grid by splitting of grid nodes and modification of layer
depths in a region around the faults. The size of this region depends on the fault throw. When
the fault displacement is used as a history-matching parameter, the grid is thus automatically
adjusted in each iteration. For faults incorporated in the grid, transmissibility multipliers are
calculated for each individual cell surface based on e.g., shale smear effects. Shale smear is
included in two different ways, which may be combined: i) As a fault permeability or a fault
permeability multiplier, which is given as a function of the smear-gouge-ratio (SGR)[3] or ii) as a
complete blocking of flow in a “tongue” up or down the fault zone with a length depending on
the thickness of the shale layer. The final fault permeability associated with each cell surface
(from which the transmissibility multiplier is calculated) is obtained by multiplying the initial
fault permeability with the SGR factor (SGRF) and the blocking factor taking the values 0 or 1.
The length of the “shale-smear tongue” is related to the thickness, t, of the shale facies layer by
the formula:
L = bt a .
(1)
SGRF is generally specified as a tabulated function of SGR. Here, the following parameterization
was used for history-matching purposes:
log(SGRF ) = log( F 0) − C ( SGR − SGR0) .
(2)
Any of the parameters F0, SGR0, C, a, and b may in principle be adjusted during the historymatching process. Here, only SGR0 and b were used. In the following, the parameter b will be
denoted just SSF (Shale-Smear-Factor).
History Matching
The mis-match between simulated and measured production history is measured using a standard
least-squares objective function weighted by a measurement error (standard deviation). Root
Mean Square (RMS) is defined as the square root of the objective function divided by the number
of measurements. That is, RMS = 1 if the deviation between simulated and measured data is
identical to the measurement error. The minimization of the objective function is performed
using standard optimization routines in MATLAB, more specifically, the routines “fminsearch”
and “fminunc”[5]. The routine fminsearch uses a simplex method which does not require
gradients, while fminunc uses a gradient method based on the BFGS quasi-Newton method. The
gradients are calculated numerically, using multiple simulations. To stabilize the MATLAB
optimization algorithms, the parameters were normalised according to their initial value and a
typical variation. For the fault permeability factor, a logarithmic transformation was used. For
the other parameters we used a linear transformation. For details on the optimisation algorithms
we refer to the MATLAB documentation[5]. The objective function is evaluated for a given set of
parameter values by first running the fault modelling tool to generate updated fault properties
(grid and transmissibility multipliers), and then a commercial reservoir simulator to generate
production profiles.
The fault modelling tool and the reservoir simulator are called
automatically in each iteration from the MATLAB objective function routine.
History-matching parameters for seismic (pfm) faults are fault displacement and the sealing
parameters described above. History matching of subseismic faults is based on a similar
procedure as described by Landa[4], where parameters of the stochastic fault model are varied
while keeping the seed constant. That is, applying the stochastic modelling software as a purely
deterministic tool. This approach may be applied to parameters for which the resulting fault
realization depends continuously on the input parameters as long as the seed is kept constant, e.g.
global fault parameters which are modelled by Gaussian random fields. For the fault modelling
3
Fig. 1. Model geometry
Fig. 2. Fault model. Seismic (pfm) faults are shown as
planes; subseismic faults as lines.
tool applied, this applies to average displacement, strike and dip. Also the fractal dimension
describing the distribution between smaller and larger subseismic faults[1] may be used as a
history-matching parameter. The distribution of faults, however, is based on a marked point
process implying that a parameter such as the number of subseismic faults cannot be used in a
matching procedure as described here.
The rationale behind such an approach is that in a real situation, the true solution is not known,
and some sort of uncertainty analysis should be performed. Liu and Oliver[6] give a good
overview of different methods for assessing uncertainty in reservoir models. They showed that
the randomized maximum likelihood method (RML) as proposed by Oliver et al.[7], although
being exact only for linear models, produced distributions that were quite similar to the correct
distributions also for a highly non-linear reservoir fluid flow problem with Gaussian statistics.
They also showed that this is the only practical alternative that provides an acceptable
assessment of uncertainty. In the RML method, the uncertainty is assessed by running multiple
history-matching runs with measured data and initial conditions drawn from the data pdf and
prior pdf for the history-matching parameters, respectively[7]. When these distributions are
known, the main objective of the history-matching process reduces to that of generating a
parameter realization that matches a given set of data from a given initial condition as efficiently
as possible.
Examples
The examples are based on a synthetic reservoir model populated with a fault model and
petrophysical properties based on two different depositional environments: Case 1 (shallow
marine) and Case 2 (fluvial). The reservoir, which is initially oil-filled, is produced by injecting
water in two wells downflank and producing from three wells upflank (Fig. 1). The fault
configurations are based on a predefined set of 50 subseismic faults and 17 larger seismic faults
(Fig. 2). The reservoir dimensions are 1000x1500x30m. The volume was discretized using a
regular grid with 20x30x90 cells. Well control is based on keeping the producers at a given
bottom-hole pressure target combined with 100% voidage injection. History data to be matched
are oil production rate and water cut. One set of parameters was chosen to represent the true
values, and the production history is defined as the result of a simulation with this parameter set.
Measurement errors are set to 10 Sm3/d for oil production and 0.02 for water cut.
Figs. 3-4 and Table 1 show the results of an initial history-matching run with four parameters,
three pfm fault parameters and one parameter for subseismic faults. The parameters for the pfm
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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40
35
RMS
30
25
20
15
10
5
0
1
11
21
31
41
51
61
No of simulation runs
Fig. 3. HM run Case 1 with 4 parameters.
x
Parameters:
Initial
Final
“True”
PFMF
1.0
2.2
1.75
SSF
1.0
2.0
5.0
SGR0
-0.1
-0.1
-0.2
FPF
0.1
0.2
0.01
Table 1. HM run Case 1 with 4 parameters.
x
x
x
x
x
x
x
x
x
x
x
Fig. 4. Result of 4-parameter history-matching run Case 1, fluvial.
faults are: Throw multiplier (PFMF), Shale-Smear-Factor b (SSF), and reference point for SGRcurve (SGR0). For the subseismic faults: Fault-permeability-factor (FPF). The objective
function minimization is performed with the simplex method. Fig. 3 shows RMS development
vs. the number of simulation runs, i.e., the number of objective function evaluations. Fig. 4
shows initial and final production profiles compared with the “true” case. Note that an RMS
value of around 5 is quite satisfactory. This is because a measurement error of 10 Sm3/d
corresponds to a relatively small relative error with the typical rates obtained here.
From Table 1 it is seen that the solution converges to a local minimum different from the true
solution. The displacement of the pfm faults is too large, and this is compensated with lower
values for all the fault seal parameters. To obtain more insight to this problem, a sensitivity
5
Fig. 5. Variation in RMS vs. HM parameters, Case1.
study was performed where the four parameters were systematically varied and the resulting
RMS-values calculated. The result is shown in Fig. 5. It is seen on the left plot that there is a
“valley” of low RMS-values. Also there are some “wiggles” in the objective function with
variations in SSF. As a consequence, a local optimisation method may converge into a local
minimum in this valley. A global minimum could possibly have been found using a global
optimisation method. However, this would require a very large number of iterations, and the
question is whether it is worth the additional cost, or even desirable if the history-matching is
done as a part of a randomized maximum likelihood approach as described above.
History-matching examples with eight parameters for the two cases are summarized in Figs. 6-8,
and Table 2. Again it is seen that multiple solutions exist, all with similar final RMS. Notice that
the gradient method is very efficient. Even with eight parameters, the algorithm converges after
only one iteration for Case 1 and after four iterations for Case 2. The examples indicate that the
most important means of obtaining a match is to find a reasonable value for the “overall”
transmissibility of the reservoir, and that such a value can be found very efficiently with the
gradient method. In these examples a single set of parameters have been used for all the pfm
faults, and similar for the subseismic faults. The number of parameters would increase
significantly if different values were used for each individual fault or fault group. However, it is
a reasonable assumption that geological parameters, which we have used here, are globally more
valid than for instance the traditional transmissibility multipliers, which are the same over the
entire fault plane.
In these and other examples not presented here, incorrect values for most of the parameters
typically can be accounted for by some extra modifications of one single parameter. Also, very
different solutions, all matching history, can be found if different starting points are chosen. Fig.
5 also illustrates a typical property of the objective function which have been observed in all
cases run: The relative sensitivity with respect to the different parameters depends heavily on the
location in parameter space where the sensitivities are evaluated. That is, a traditional sensitivity
study varying one parameter at the time may lead to completely wrong conclusions with respect
to which parameters being most important and most effective for history-matching.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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CASE
Method
No of simulation runs
No of iterations
CASE 1 (SHALLOW MARINE)
RMS
35
30
SIMPLEX
25
GRA DIENT
20
15
RMS
rmax
fdim
strikeexp
dipexp
FPF
SSF
SGR0
PFMF
“True”
10
5
0
1
11
21
31
41
51
61
71
81
91
No of s im ulation runs
Fig. 6. Case 1 with 8 parameters. Comparison between
Simplex method and Gradient method.
RMS
rmax
fdim
strikeexp
dipexp
FPF
SSF
SGR0
PFMF
0
50
1.75
30
60
0.01
5
-0.2
1.75
1 (shallow marine)
simplex Gradient
178
124
100
10
Initial values
33.14
33.14
10
10
1
1
0
0
90
90
0.1
0.1
1
1
-0.15
-0.15
1
1
Final values
5.31
3.15
23
10
1
1
-22
0
90
90
0.0044
0.098
1.06
3.35
-0.26
-0.185
1.38
1.85
2 (fluvial)
simplex gradient
178
97
99
7
8.34
10
1.00
0
90
1.0E-01
1.0
0.00
0.50
8.34
10
1.00
0
90
1.0E-01
1.0
0.00
0.50
0.37
23
1.00
-15
48
2.8E-03
0.4
-0.23
1.58
0.96
22
0.90
0
89
3.1E-02
2.4
-0.23
1.74
Table 2. Results from HM runs with 8 parameters.
Conclusions
It has been demonstrated that fault parameters applied in a commercial fault modelling tool can
be used to match a reservoir simulation model to dynamic data in an automatic loop. The loop is
controlled by standard optimization routines in MATLAB. In each iteration of the loop, the
simulation grid and permeabilities/transmissibilities are adjusted by the fault modelling tool and
applied in a commercial reservoir simulator to obtain production performance.
With the advanced fault sealing model, “basic” geological fault properties can be adjusted as
opposed to the traditional method, where the history matching parameters typically are one fault
transmissibility multiplier per fault. Since these geological parameters are more likely to be
similar for different faults in the same reservoir, this method should be more efficient than the
traditional method with respect to obtaining a match.
Since the parameters being adjusted are geological parameters, the results of the historymatching process can be implemented directly in the geological model. Upscaling errors should
then be taken into account. In this project, the geomodel grid was used also for the flow
simulations, so upscaling was not an issue.
For the cases studied, a good match depends to a large degree on having a correct overall
communication, but not so much on the values of the individual fault parameters. There is thus a
path of acceptable parameter values crossing through the parameter space. Choosing a particular
solution along this path requires the use of prior geological information. Alternatively, this path
may be characterized through multiple history-matching runs.
The gradient method was much more efficient than the simplex method even if the first used
numerical gradients. This is in accordance with general experience, see e.g. Fletcher[8].
Seismic faults have been history-matched with respect to displacement, and in this process the
grid was adjusted in each iteration. These adjustments consist of moving the depth nodes
7
Fig. 7. Production profiles Case 2, gradient method. Initial and final solution compared with the “true” solution.
according to the change in fault displacement as modelled by the fault modelling tool. Other
modifications of fault geometry, which require moving the grid pillars, have not been
considered.
Acknowledgments
This work has been financed by Statoil, and the authors want to thank Statoil for permission to
publish this paper. We also want to thank the Norwegian Computing Centre for valuable
software support, Jan Tveranger (CIPR) who made the geomodels and Signe Ottesen (Statoil) for
help on various issues related to fault modelling.
References
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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1.8
1.6
1.4
1.2
SIMPLEX
GRA DIENT
0
100
35
30
25
strike
10
9
8
7
6
5
4
3
2
1
0
fdim
RMS
8
1
0.8
0.6
0.4
0.2
0
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SIMPLEX
GRA DIENT
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150
No of s im ulations
ssfb
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fpf
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0.001
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0
0
SIMPLEX
-0.05
GRA DIENT
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pfmf
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-0.15
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4
0
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50
100
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No of s im ulations
SIMPLEX
GRADIENT
2
40
TRUE
1.5
20
GRA DIENT
10
0
0
0
50
100
150
No of s im ulations
200
30
SIMPLEX
TRUE
200
TRUE
GRA DIENT
50
0.5
50
100
150
No of s im ulations
7
6
60
-0.25
0
SIMPLEX
2.5
1
-0.3
9
8
3
-0.2
200
1
0
TRUE
50
100
150
No of s im ulations
50
100
150
No of s im ulations
3
2
rmax
dip
0
GRA DIENT
0
TRUE
10
0.1
10
0
sgr0
200
SIMPLEX
SIMPLEX
GRA DIENT
GRA DIENT
5
1
50
40
30
20
SIMPLEX
0
0
80
70
60
15
10
TRUE
No of s im ulation runs
100
90
20
200
0
50
100
150
No of s im ulations
200
Fig. 8. Results Case 2 (fluvial). Comparison between Simplex method and Gradient method.
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