A Parallel, Multiscale Approach to Reservoir
Modeling
Omer Inanc Tureyen and Jef Caers
Department of Petroleum Engineering
Stanford University
1
Abstract
With the advance of CPU power, numerical reservoir models have
become an essential part of most reservoir engineering applications.
These models are used for predicting future performances or deter-
mining optimal locations of infill wells. Hence in order to accurately
predict, these reservoir models must be conditioned to all available
data. The challenge in data integration for numerical reservoir mod-
els lies in the fact that each data has its own resolution and area of
coverage. The most common data for reservoir characterization are;
well-log/core data, seismic data and production data. The challenge
is that each data set has its own resolution and area of coverage.
Most current approaches to data integration are hierarchical. Fine
scale models are used for integrating well-log/core and seismic data
while coarse models are used to integrate mostly production data. The
drawback of such a hierarchical approach is such that once the scale is
changed, data conditioning, maintained in the previous scale, is lost.
In this paper, we review a general algorithm as a solution to the
multi-scale data integration. Instead of proceeding in a hierarchical
fashion, a fine model and a coarse model is kept in parallel through
out the entire characterization process. The link between the fine
scale and the coarse scale is provided by non-uniform upscaling. An
optimization procedure determines the optimal gridding parameters
that provide the smallest possible mismatch between fine and coarse
scale reservoir models.
A synthetic example application is given and demonstration of
the methodology. The upgridding is accomplish by a static gridding
algorithm, 3DDEGA. This algorithm aims at preserving geology by
1
minimizing heterogeneity within a coarse grid block. The coarse grids
are provided in a corner-point geometry fashion, hence this allows for
accurate description of the reservoir with fewer number of grid blocks.
1 Introduction
Reservoir modelling and prediction calls for the integration of various data
sources into a single reservoir model. Such data sources can be divided into
several groups of which the most important ones are:
• Geological interpretation of reservoir architecture at all scales ranging
from major faults to facies and bedding configurations. Such infor-
mation is often qualitative in nature, yet may constrain the reservoir
model at all scales. In geostatistics such information can be quantified
through variogram or through 3D training images.
• Well-log and core measurements. Such information is often the most
direct type of information, however is only telling of the near well-bore
reservoir heterogeneity and provides information at a foot scale. In
geostatistics, this type of information is often treated as hard data.
• 3D seismic surveys. This information is probably most exhaustive, yet
often at a scale larger than the reservoir modelling scale. Seismic is
known to act as a low-pass filter, resulting in seismic images that lack
important fine-scale heterogeneities. In geostatistics this type of data
is treated as soft data.
2
• Reservoir dynamic data, most particularly from pressure and flow mea-
surements, or increasingly common, 4D seismic. The scale at which
this information informs the reservoir is largely unknown beyond the
fact that is quite coarse, its scale varies spatially, depending on well
configurations and time, depending on depletion strategies.
In building reservoir models, it should be recognized that each piece of
information has its own characteristic and scale, yet no single source of in-
formation may determine the reservoir model uniquely. While some redun-
dancy of information may be present, it is generally considered that each of
the above four data sources will contribute to modelling the reservoir.
The current practice of reservoir modelling consists of modelling the reser-
voir first using the static data (sources one to three), then only using dynamic
information (source four), see Figure-1. In the static reservoir modelling
stage, the reservoir is modelled on the ”geostatistical” scale. In the vertical
direction, the size of the geostatistical grid cell is typically equal to the scale
of the well-log and core data (about 1 ft), while in the horizontal direction
the grid cell size is often related to the scale at which seismic surveys (100ft
typically).
Two important problems exist with this approach:
• A missing scale problem occurs: the core data at a 0.1ft×0.1ft×1ft
scale is represented by the geostatistical cell of 100ft×100ft×1ft which
implies an intrinsic upscaling of cell properties. This may constitute a
serious problem in strongly heterogeneous system.
• Flow simulation cannot be performed at this scale as flow simulators
3
cannot be evaluated on grid dimensions larger than a few 100.000 grid
blocks. Hence the million cell geostatistical model needs to be upscaled
if any production data needs to be integrated.
We will not deal with the missing scale problem but with the latter prob-
lem only. The common approach to history matching is to history match
on the coarse or upscaled reservoir model, either manually or using some
gradient-based method. Performing history matching in such fashion has the
following problems:
• Any fine scale reservoir information (core or well-log) may be lost when
the coarse scale model is perturbed.
• Important fine and coarse scale geological information may be de-
stroyed while history matching. Particularly when the history match-
ing method does not take into account statistics such as variogram or
multiple-point statistics that are imported in the fine scale geological
model in order to honor the prior geological data.
In many practical settings however, a history match can still be achieved,
even at the cost of destruction of any seismic data conditioning or geological
realism. The cost however is often predictivity of the resulting reservoir mod-
els. For this reason, presuming geology while history matching has deserved
little attention, since often obtaining history match has become a goal on its
own.
This shortcoming in the work flow of building the reservoir model first
static, then dynamic has been recognized by several authors (amongst which
4
the paper by Tran T.T. and Behrens (1999) is pioneering). An improved work
flow has been proposed, shown in Figure-2. Instead of simply history match-
ing the coarse scale model, a posterior downscaling of the history matched
coarse model is performed. The downscaling step allows to re-integrate the
fine-scale information lost during the history matched procedure. The down-
scaled realization can then be upscaled to any desired grid dimensions. The
main problem with this approach is that the final upscaled reservoir model
(Figure-2) need not match the production history due to the fact that the
downscaling procedure introduces noise/error in the match. The downscal-
ing procedure is purely geostatistical, production data is not utilized at this
stage. In fact one may argue that the approach in Figure-2 is still a sequen-
tial hierarchical approach to reservoir modelling. Upscaling, downscaling
and history matching ”operations” are performed in a sequential fashion,
with the last performed ”operation” potentially destroying the achievements
of its predecessors.
In this paper we propose a ”parallel” or ”joint” approach to the problem
of reservoir model building. Instead of going through a sequential procedure
of scale-change (down or up) followed by history matching, we propose to
integrate upscaling into the history matching loop. Instead of perturbing the
coarse scale model, we propose to perturb the fine scale reservoir model. A
simple loop is created of first creating a fine scale realization, then upscaling
it, then evaluating flow on the coarse scale realization. The flow results of the
coarse model are used to perturb the fine scale realization. This sequence of
steps is repeated until a history match is achieved. This approach is not new
and has been successfully applied by the authors Caers (2001), Mezghani and
5
Roggero (2001) and Tureyen and Caers (2002) and also others.
In this paper we show however that this approach has a fundamental and
potentially dangerous flaw. When perturbing a fine scale geostatistical model
using flow results from another, namely coarse scale realization, it is implic-
itly implied that the flow results evaluated on coarse and fine are similar,
hence no or few upscaling errors occur. Indeed, any perturbation of reservoir
parameters linked to the fine scale should be based on the calculation of a
mismatch function that is correctly reflecting fine scale parameter changes.
This is not necessarily true in the presence of upscaling errors. The danger
therefore lies in the fact that the user may assume no upscaling errors exist
(while in reality upscaling errors could be quite severe) and still achieve a
successful history match. Recall that production data may not necessarily
provide a strong constraint to the reservoir model when few wells are avail-
able, hence history matching on poorly upscaled reservoir models is feasible.
The same argument was used above to explain the current state-of-the-art
approach in history matching by perturbing coarse scale realizations. The
result is, as above, a model that matches history but has lost important fine
geological or seismic data, hence lost prediction power or accuracy.
The main contribution of this paper is to correct this flaw by not only
closing the loop between history matching and upscaling but also by reducing
upscaling errors through gridding optimization, while history matching. This
will amount to reducing the upscaling errors between fine and coarse such
that all relevant fine scale geological data is maintained in the coarse scale
model throughout the history matching procedure.
6
The further outline of this paper can now be defined clearly: we will first
review the work of Tureyen and Caers (2002) and show a counter example of
what can go wrong in such a procedure. Next we introduce a fairly general
optimization framework for reducing upscaling errors between fine and coarse
scale realizations. Our methodology or work flow is generic in many ways.
We will therefore use various existing geostatistical, upscaling and history
matching procedures and refrain from using a case study to demonstrate our
point of generality.
2 Review: Basic Parallel Modelling
The objective of the parallel modelling approach for reservoir characterization
is basically to avoid the problem of “choosing a scale” by working on multiple
scales jointly. Unlike the hierarchical modelling approach given in section-
1, the parallel modelling scheme gives the flexibility to update each scale
throughout the entire characterization process.
To provide a clear understanding some of the notation that will be used
through out this paper is first introduced:
z : {z (u), ∀ u∈ Reservoir} the reservoir property at grid block
u=(x,y,z) e.g. permeability;
z (r) : a perturbation of the fine scale reservoir model z . The
magnitude of perturbation is parameterized using
some parameters r ;
z up : the uniquely determined upscaled reservoir model,
upscaled from z ;
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z up(r) : the model upscaled from z (r);
FSM (z ) : the flow simulation model evaluated on z ;
RP(z ) : the flow response when FSM is performed on z ;
S θ(z ) : The upscaling method S evaluated on the fine scale model
z . θ are upscaling or upgridding parameters belonging
to the upscaling method S ;
D : reservoir production data to be matched.
Note at this point that we make full abstraction of the parametrization of
the problem z (r). r can be a single parameter such as in, the gradual defor-
mation method (see Roggero and Hu (1998)), the probability perturbation
method (see Caers (2003)), or r may be parametrization using sensitivity co-
efficients (see Landa (1997) and Wen and Cullik (1998)). Figure-3 schemat-
ically illustrates the main algorithm of the parallel approach on a history
matching problem. First a fine scale geostatistical model (z (u) that honors
the seismic and hard data) is generated. In history matching, some initial
starting model is perturbed using a perturbation scheme. Perturbations are
often parametrized by a set of parameters r that change the reservoir model
z into a perturbed model z (r). Note that we attach such perturbation to
the fine scale, not the coarse scale model. The next step is to upscale z (r)
to a coarser model z up(r), non-uniformly through the following relationship:
z up(r) = Sθ(z (r)) (1)
Here S represents the upscaling technique applied on z (r) and θ repre-
sent the upgridding parameters regarding the upscaling technique S. In the
8
method of Tureyen and Caers (2002), Figure-3, both S and θ are determined
prior to the history matching and remain fixed while history matching.
Once the coarse model is determined, the flow response is obtained by
evaluating the full flow simulation (FSM ) on the coarse model through the
following relationship:
RPz up(r) = FSM(z up(r)) (2)
Finally the r parameters are optimized so that the following objective
function is minimized.
minr
O(r) = min ‖ RPz up(r)−D ‖ (3)
At this point it should be noted that the fine scale model is perturbed
(with the r parameters) with respect to the flow response of the coarse scale
model. Such an approach makes upscaling a part of the history matching
process and is actually the basis of parallel modelling. Some of the advantages
of such an approach can be listed as follows:
• Fine scale and coarse scale data are integrated at the same time at their
relative scales. The well log/core and the seismic data are honored at
the fine scale and the production data is honored on the coarse scale,
• Relevant geology is maintained at all scales. This is accomplished on
the coarse grid through the use of either non-uniform or unstructured
gridding,
• No posterior upscaling is required since it is the non-uniformly gridded
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model that matches the history. This is a result of the fact that full
flow simulations are performed on the coarse model,
• It is a generic method and serves as an alternative approach to reservoir
characterization.
2.1 Example
To make a better understanding of the proposed approach by Tureyen and
Caers (2002), an example will be briefly reviewed in this section. Figure-4
gives the reference permeability field along with its corresponding water cut
curve and sample statistics regarding the generation of the reference field.
This field represents a cross-section of a reservoir which is modelled with 50
grid blocks in the horizontal and 50 grid blocks in the vertical directions. The
flow scheme takes place with an injector and a producer under fixed bottom
hole pressures (5500psi for the injector and 4500psi for the producer). The
production history is for 500 days.
In this example the perturbation was accomplished through a single pa-
rameter gradual deformation method, Roggero and Hu (1998). Sequential
Gaussian Simulation (Deutsch and Journel (1998)) was used to generate the
reference permeability field and to generate realizations for the gradual defor-
mation method. The upgridding technique was adopted from Durlofsky and
Milliken (1997), a single phase upscaling technique (Tran (1995)) was used
to upscale the models once they were upgrided and finally a two-phase fi-
nite difference simulation model was chosen as the full flow simulation model
(FSM ).
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The results of the example given by Tureyen and Caers (2002) are given
in Figure-5. Figure-5a gives the flow responses of 20 realizations conditioned
only to hard data (no history matching has been performed). The relatively
wide scatter of the flow responses are obvious. Most importantly it should be
noted that the flow responses for these 20 realizations were obtained by eval-
uating the full flow simulation model on the fine scale. Figure-5b illustrates
the results in which the proposed algorithm has been applied to 20 realiza-
tions. It is clear in this case that all realizations match the history up to 500
days along with relatively accurate future predictions. Most importantly all
full flow simulations were conducted on coarse models. As mentioned ear-
lier the immediate end result of the proposed algorithm are models that are
coarse, non-uniformly gridded and match the history. Hence the realizations
in Figure-5b are coarse, non-uniformly gridded and match the history.
3 Problems With Parallel Modelling
Although the parallel modelling proposed by Tureyen and Caers (2002) offers
an effective method to reservoir characterization, there are some limitations
to this approach. These can be summarized as follows:
• As mentioned in Section-2, during the entire characterization process
there are two models that are kept in parallel; the fine scale and the
coarse scale. Geostatistics is performed on the fine scale whereas full
flow simulations are performed on the coarse scale. Because of this
reason it is guaranteed that the coarse model will match the history
(as shown in the example given in Section-2.1). However at the end
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of the process the corresponding fine scale model, where all hard data
and seismic data are honored, might not match the history.
• The level of coarsening can play an important role during the process.
Tureyen and Caers (2002) upscaled from a 50×50 fine scale model to
a 25×25 coarse scale model. However the decision regarding the level
of coarsening is fixed prior to characterization. Taking into account
that the fine scale reservoir is perturbed and may significantly change
during the history matching phase, the level of coarsening may not be
known prior to history matching.
3.1 Example Case
To demonstrate the importance of the upscaling level, we present an example
similar to that given in Section-2.1. Most of the outline of the example is
similar to that given in the previous section, hence only the results will be
discussed.
In this example the 50×50 fine grid is upscaled to non-realistic grid di-
mensions of 5×5 on the coarse scale. Figure-6a illustrates this case. It is
clear that the permeability field (Figure-6a) is not representative of the fine
scale geology. However when we consider Figure-6b (which shows the flow
responses of 5 realizations with grid dimensions of 5×5 at the end of the
history matching process) it is clear that models with such grid dimensions
can match the history equally well as the ones shown in the example given
in Section-2.1. Recall that the history matching procedure reduces the mis-
match between field and simulated data on the coarse grid, hence it is possible
12
that the reservoir can be perturbed till history match on the coarse scale,
regardless of the upscaling errors.
As a result, when performing flow simulation on Figure-7a, a serious
mismatch between field and simulated data is observed. This might have the
following consequences:
• The fine scale models are the closest representation of the actual geo-
logical variability. Despite history matching the coarse scale model, the
fine scale model still show considerable variability in the flow response,
Figure-(7b), hence geological variability has not been drastically re-
duced. In other words, a ”poor” history matching procedure does not
necessarily reduce geological uncertainty, despite the fact that a history
match is obtained.
• Fine scale models are often used for well placement optimization prob-
lems (Guyaguler (2001)) hence a history match on the fine scale may
be desirable.
4 Proposed Solution: Gridding Optimization
The essential shortcoming in the method of Tureyen and Caers (2002) lies in
the fact that upscaling errors are not accounted for. The idea of perturbing
the fine scale geological model based on the flow results of the coarse model
would only make sense if the upscaling errors are minimal. Otherwise, if
upscaling errors are not negligible, the fine scale models will not match the
history. In this paper we propose a method that aims at improving the
13
correction between the fine scale and the coarse scale response by reducing
the upscaling errors, while history matching.
First, we proceed by defining additional notation similar to the ones given
in Section-2.
FSM ∗ : a flow simulator that is a fast approximation of FSM ;
RP* : the flow response when FSM ∗ is performed on z ;
Figure-8 illustrates the proposed method schematically. The work flow
starts by constructing a fine scale geostatistical model. However, instead of
directly proceeding with the upscaling and upgridding phase (proposed in
Section-2) a gridding optimization is performed to ensure that the upscaling
errors between the flow responses of the fine scale and the coarse scale models
are minimized. In other words we can define an ε parameter such that;
ε = ‖FSM(zup(r))− FSM(z (r))‖ (4)
or
ε = ‖FSM(Sθ(z (r)))− FSM(z (r))‖ (5)
at the end of the gridding optimization the ε parameter is minimized. It is
clear that in all cases we will not have access to the information FSM (z (r))
(which is the flow response of the fine scale model when the full flow simu-
lation is performed), if we had, there would not be a necessity for upscaling
in the first place. Hence the challenge is to reduce the upscaling errors ε
without knowing FSM (z (r)).
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To achieve this we introduce a fast approximation to the flow model,
FSM∗. The proposed approach for the optimization is such that, instead of
optimizing on the error ε directly, we introduce an approximate error ε∗, for
any given r , given by:
ε∗(θ, S) = ‖FSM∗(Sθ(z (r)))− FSM∗(z (r))‖ (6)
The error ε∗ is evidently a function of the upscaling method S and its
parameters θ. ε∗ can be minimized by finding an optimal set S and θ. Mini-
mizing ε∗ therefore consists of reducing the mismatch between the production
data FSM∗(z (r)) evaluated on the fine scale using the approximated flow
simulator FSM∗ and the production data. FSM∗(z up(r)), is the same flow
evaluated on the coarse scale. Note two important points:
1. To minimize ε∗ the actual flow response D is not used, neither does
one have to use the same boundary conditions as in the reservoir.
2. In order to find an optimal θ and S we need to evaluate FSM∗ one
single time on the fine scale realization z (r) to obtain a reference fast
flow response. Multiple flow simulations are required on z up(r) to
find the optimal θ and S. Hence the CPU time spent in solving the
optimization problem of finding S and θ will be small compared to
running FSM on z up(r).
The assumption made is the following: Once ε∗ has been made small
enough ε has also been reduced, although probably not by the same amount
as ε∗. In other words the objective function defined on FSM∗ namely ε∗ is
monotonically varying with the objective function defined on FSM namely ε.
15
The ranking of models provided by FSM is the same as the ranking provided
by FSM∗: the errors ε and ε∗ need not be the same in absolute magnitude.
A similar approach related to the problem of uncertainty quantification is
taken in Ballin and Journel (1993).
As a result the overall proposed method consists of two optimization
levels: the outer level concerns history matching, the inner level consists of
a grid optimization.
5 Example
In this section we present a synthetic case study for the approach presented in
the previous section. A different upscaling/upgridding approach will be used
in this case. In their example Tureyen and Caers (2002) used the method of
Durlofsky and Milliken (1997), which was a flow based gridding technique.
In this example we use a static based upgridding method described in the
following subsection.
5.1 3DDEGA
3DDEGA (see Garcia (1990)) is an effective and efficient gridding algorithm,
where the objective is to automatically generate coarse grids that fit the ge-
ological heterogeneities. The output of the algorithm are quadrilateral or
hexahedric grids that can be a direct input into commercial numerical reser-
voir simulators through a corner-point geometry description of grid blocks.
The main idea behind the 3DDEGA algorithm is to generate coarse grid
blocks that are as homogeneous as possible in terms of an input variable
16
(permeability map, porosity map, facies map, etc.) that is obtained from an
underlying fine scale model. The grid edges are assigned elastic properties
that allow them to expand or shrink to make the coarse blocks homogeneous.
In order to accomplish this, the algorithm goes through three major steps,
which are:
• Determine the grid block heterogeneities, which is also accomplished in
three steps;
– Retrieve the fine scale grid cells within a coarse grid block,
– Compute the same statistics of the fine grid cells within each
coarse grid block
– Determine a heterogeneity index for the coarse block from the
internal block statistics.
• Update the grid-edge elasticity coefficients (that are a function of the
heterogeneity index),
• Compute the new grid-vertex locations that minimize the heterogeneity
which is defined by the heterogeneity index.
Figure-9 illustrates how the 3DDEGA algorithm responds to a unique
example of fine scale heterogeneity. The fine scale contains a box with high
permeability. In the coarse scale the algorithm adjusts itself to refine around
the region of high permeability and coarsen through out in the other regions.
It is also important to note how the grid blocks are deviated from orthogo-
nality to better preserve the high permeability region ”as it is” on the coarse
grid.
17
To capture heterogeneity quantitatively a block coefficient of heterogene-
ity (heterogeneity index) is defined as follows:
βB =1
∏Nd=1i=1 (1 + µi)− 1
Nd=1∏
i=1
(1 + µi
(σi
B
σimax
)ωi)− 1
(VB
Vnorm
)ωV
(7)
where,
• i = 1,· · ·,Nd, refers to the input variables (fine scale input variables
that are within the limits of the coarse grid block boundaries),
• σimax is the maximum of all internal block variances (or equivalent ex-
pression for a categorical variable) for data variable i,
• µi and ωi (both positive) are a weight and a power assigned to data
variable i,
• VB is block B’s volume,
• ωV is a power assigned to the block volume term,
• Vnorm is such that V ωVnorm=max(V ωV
min,V ωVmax)
However if there is only a single variable then Equation-7 reduces to:
βB =
(σi
B
σimax
)ωi (VB
Vnorm
)ωV
(8)
To obtain a better understanding the procedure will be explained through
Figure-10. For a given coarse grid block (for example grid block B1 in Figure-
10) the inter block variance (σ1B)is calculated (variance of the fine scale input
variable within the limits of coarse grid block B1). Once this is obtained the
18
block coefficient of heterogeneity can be determined for both blocks B1 and
B2. The elasticity of the edge that neighbors blocks B1 and B2 can now be
determined from the block coefficients of heterogeneity of both blocks (the
elasticity is a function of the average block coefficients of heterogeneity).
Given the elasticity, the optimal location of the grid vertices can be deter-
mined such that the block coefficient of heterogeneity will be minimized.
The second term in Equation-8 is used for controlling the coarse grid vol-
umes. If a number of very small and very large grid blocks exist neighboring
each other, this term will be large. This term is therefore introduced to
control the quality of the grid by controlling the relative volumes of coarse
grid blocks. ωi and ωV are weights for emphasizing heterogeneity versus em-
phasizing grid conformity. Some more insight into the importance of these
parameters are provided. Figure-11 illustrates the fine scale reference image
and various upscaled coarse models (obtained by evaluating the 3DDEGA
algorithm on the 100×100 reference fine scale image) through which the sen-
sitivities are given. The top row of Figure-11 represent variation in the ωi
parameter, for fixed ωV , and the bottom row represent variation in the ωV
parameter, for fixed ωi.
The ωi parameter emphasizes on geological heterogeneity. As it is clearly
seen from the top row Figure-11 that if ωi=0.0 then no emphasis is given
to geological heterogeneity hence a uniform grid is obtained. As we increase
the ωi parameter more emphasis is given to the regions of higher geological
variability hence the resulting gridding tries to refine these regions as much
as possible. However for large ωi (Figure-11, ωi=1.0 and ωV =0.0) certain
regions are over-refined leading to small grid blocks neighboring large grid
19
blocks. This might not be desirable in some cases especially for the stability
of flow calculations. Hence to achieve a control on the block volumes the ωV
parameter is introduced.
The bottom row of Figure-11 studies the effect of the ωV parameter that
has a global control on the grid block volumes. When a negative value of
this parameter is used (such as ωV =-0.5) the algorithm allows very large grid
blocks to neighbor very small grid blocks (Figure-11, ωi=1.0 and ωV =-0.5).
With increasing ωV , more restriction is put on the coarse grid block volumes,
hence for a high ωV (such as ωV =2.0 in the bottom row of Figure-11) the
grid blocks become almost uniform.
5.2 Synthetic Example
In this section we present the enhanced parallel modelling approach with a
synthetic example. Before giving details regarding the example, we present
the work-flow specific to this example, see Figure-12. The work-flow starts
with constructing the fine scale model. Streamline simulation is performed
on the fine scale where the pseudo water cut curve is obtained. Using the
3DDEGA algorithm, the fine scale model is upgridded to a coarse model.
Streamline simulation is performed on the coarse scale model, and the mis-
match between the pseudo water cut curves of the fine scale and the coarse
scale models are then calculated. Streamline simulation is performed multi-
ple times on the coarse scale model until the 3DDEGA gridding parameters
are optimized and the mismatch is minimized. In addition to ωi and ωV
a third parameter is also introduced in the gridding optimization process,
namely the number of grid blocks in the x direction (ncx)for the coarse scale.
20
The total number of grid blocks on the coarse scale are fixed (ncx×nc
y remain
fixed). Hence once ncx is varied a value for nc
y can be determined. Once the
gridding optimization is completed, then full flow simulation is performed
on the coarse model and the mismatch is calculated between the calculated
data and the observed field data. This procedure is repeated until a history
match is obtained.
As mentioned earlier the proposed algorithm is generic. In the following
example, the perturbation method (single parameter probability perturba-
tion method) was adopted from Caers (2003). Instead of a variogram based
geostatistical technique, in this case we use a training image based algorithm
for generating realizations, where we model facies instead of continuous per-
meability values. The upgridding is performed by the 3DDEGA algorithm
while the upscaling is achieved by arithmetic averaging of individual values.
The full simulation model and the approximate fast simulation models are
once again taken as a finite difference simulator and a streamline simulator
respectively.
The reference permeability field and its corresponding water cut curve is
given in Figure-13. A constant permeability value of 10000 md is assigned to
the sand facies and 100 md is assigned to the mud facies. The given reference
model is a quarter-five spot pattern with 100×100 grid blocks in the x and
y directions. Each grid block is of 20ft×20ft with a total of 2000 ft in the
x and y directions. The depth is taken as 200 ft. The production history is
taken to be 500 days and the permeability values at the well locations are
treated as conditioning hard data.
Figure-14 illustrates the results for 30 realizations not constrained to
21
water-cut data. As expected the flow responses of the fine scale models con-
ditioned only to hard data give a relatively wide scatter. As a result of the
proposed algorithm, optimally gridded coarse scale models match the history
well, and provide accurate future predictions for the same well configuration.
If grid optimization had not been performed during the characterization
process, then the results given in Figure-15 would be obtained. Even though
a history match is obtained on the coarse scale, the fine scale responses still
present a wide scatter. When performing grid optimization (see Figure-16),
the upscaling errors are minimized and the fine scale models also become
representative of the history.
6 Conclusions
An alternative approach to reservoir characterization is presented in this
paper. The following conclusions are obtained:
• Introducing upscaling/upgridding within the history matching process
is an effective technique for history matching non-uniformly gridded
models directly. However caution needs to be taken such that the
resulting fine scale models need not match history.
• Introducing a second level of optimization for optimizing the gridding
during the upscaling phase results in a better understanding of the
upgridding parameters, furthermore improvements are observed on the
fine scale matches where the full flow simulations are performed only
on the coarse scale models.
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• With the parallel modelling approach, all data are integrated jointly at
their relative scales.
• It is important to note that optimizing gridding parameters while his-
tory matching is different from choosing for an optimal or adequate
set of gridding/upscaling parameters prior to history matching. An
optimal set of parameters for a particular fine scale realization is not
necessarily optimal for another realization, even when both have sim-
ilar geological heterogeneity. The gridding itself should therefore be a
variable of the entire data integration process.
23
References
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Caers, J. (2003). Geostatistical History Matching Under Training-Image
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24
Landa, J.L., H. R. (1997). A Procedure to Integrate Well Test Data, Reser-
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26
7 List of Figures
Figure-1 Hierarchical approach to reservoir characterization.
Figure-2 A traditional approach to history matching.
Figure-3 Basic Parallel Modelling for Reservoir Characterization.
Figure-4 a - Reference field, b - Corresponding water cut curve, c - Sample
statistics regarding the generation of the reference field.
Figure-5 a - Flow responses of 20 realizations (fine scale) conditioned only
to hard data, b - Flow responses of 20 realizations (coarse scale) when
the proposed algorithm is applied.
Figure-6 a - History matched realization with grid dimensions of 5×5, b -
Flow responses of 5 realizations with grid dimensions of 5×5.
Figure-7 a - Fine scale realization corresponding to the history matched
5×5 realization given in Figure-6a, b - Flow responses of 5 fine scale
realizations.
Figure-8 Schematical diagram of the parallel approach with the gridding
optimization.
Figure-9 Typical Gridding Provided by 3DDEGA.
Figure-10 Schematics illustrating various aspects of the block coefficient of
heterogeneity.
Figure-11 Sensitivity results of the ωi and ωV parameters on gridding.
27
Figure-12 The parallel modelling approach specific to the synthetic exam-
ple.
Figure-13 The reference permeability field and its corresponding flow re-
sponse.
Figure-14 Flow responses of 30 non-history matched realizations and 30
history matched realizations.
Figure-15 Comparison of fine scale and coarse scale flow responses in the
case where grid optimization has not been performed.
Figure-16 Comparison of fine scale and coarse scale flow responses in the
case where grid optimization has been performed.
28
8 Figures
Figure 1: Hierarchical approach to reservoir characterization
29
Figure 2: A traditional approach to history matching
30
Figure 3: Basic Parallel Modelling for Reservoir Characterization
31
Figure 4: a - Reference field, b - Corresponding water cut curve, c - Sample
statistics regarding the generation of the reference field.
32
Figure 5: a - Flow responses of 20 realizations (fine scale) conditioned only
to hard data, b - Flow responses of 20 realizations (coarse scale) when the
proposed algorithm is applied.
33
Figure 6: a - History matched realization with grid dimensions of 5×5, b -
Flow responses of 5 realizations with grid dimensions of 5×5.
34
Figure 7: a - Fine scale realization corresponding to the history matched 5×5
realization given in Figure-6a, b - Flow responses of 5 fine scale realizations.
35
Figure 8: Schematical diagram of the parallel approach with the gridding
optimization.
36
Figure 9: Typical Gridding Provided by 3DDEGA
37
Figure 10: Schematics illustrating various aspects of the block coefficient of
heterogeneity
38
Figure 11: Sensitivity results of the ωi and ωV parameters on gridding.
39
Figure 12: The parallel modelling approach specific to the synthetic example.
40
Figure 13: The reference permeability field and its corresponding flow re-
sponse.
41
Figure 14: Flow responses of 30 non-history matched realizations and 30
history matched realizations.
42
Figure 15: Comparison of fine scale and coarse scale flow responses in the
case where grid optimization has not been performed.
43
Figure 16: Comparison of fine scale and coarse scale flow responses in the
case where grid optimization has been performed.
44