P006 NON-UNIFORM UPLAYERING OF HETEROGENEOUS RESERVOIR MODELS
Ahmed Sharif
Roxar, Gamle Forusvei 17, N-4065 Stavanger, Norway
Abstract
A method for non-uniform uplayering of heterogeneous geological models is proposed. The method
generates layering scheme by which coarse scale
simulation models preserve the impact of reservoir heterogeneity on fluid flow and production
response. The uplayering process involves generation of streamline-based dynamic response of
the geological models. The response data were
then analysed and processed to establish a set
of connectivity-based attribute logs. Multivariate analysis and pattern recognition techniques
were finally applied to produce layering scheme
for coarse-scale simulation. The performance of
the proposed method was investigated under a set
of realistic field-scale reservoir models. The results indicate that flow-based non-uniform uplayering can improve the performance of coarse-scale
simulation models, compared to that of the simple
uniform uplayering approach.
1
Introduction
In reservoir characterisation, geologic models are
now routinely built with high resolution grid systems in the order of 106 − 107 cells, whilst production performance and underlying reservoir flow
behaviour are, for practical reasons, modelled with
lower resolution grids in the order of 104 − 105
blocks. This disparity in scale invites the process
of upscaling, by which large-scale properties for
the coarse model are computed. A major challenge to practicing reservoir engineers is to design coarse-scale representation of geologic models that accurately estimates important geological features that may dictate reservoir fluid flow,
whilst keeping the simulation model computationally efficient to enable multiple runs. Hence, the
question of scale and estimation of accurate largescale reservoir properties have attracted much attention. Various methods (from robus flow-based
numerical algorithms to the most exotic analytical approaches) that differ both in complexity and
accuracy were developed to deal with this issue.
The ability of an upscaling process to produce
large-scale properties that capture flow behaviour
of fine-scale models depends upon: 1) proper design and construction of coarse-scale grid models, 2) correct sampling of the fine-scale properties, and 3) application of relevant and accurate
averaging method. This work presents a method
that can be utilised to design coarse-scale simulation grids based on the dynamic behaviour of
fine-scale models. The work focuses exclusively
on uplayering − the design of simulation layering. This is not a particularly farfetched strategy; from depositional perspective, properties of
heterogeneous sedimentary rocks usually exhibit
higher variability in the vertical direction than in
horizontal directions. Geologic models for such
heterogeneous reservoirs are therefore bound to
have a large number of high resolution fine-scale
layers. Besides, with the current computational resources, flow simulation on coarse-scale grids that
have similar resolution in x- and y-directions as
the geologic models is relatively feasible. In the
vertical direction, however, some form of uplayering are often necessary. In practice, representation
of fine-scale layering in reservoir simulation can be
conveniently tackled by one of three approaches:
1) uniform uplayering of fine-scale models based
on a pre-conceived number of simulation layers,
2) preliminary uniform uplayering followed by selective splitting of certain coarse-scale layers, and
3) direct use of the geologic-scale layering. The
first approach, despite being fairly common, can
be potentially erroneous due to the risk of indiscriminately lumping layers with highly distinctive
hydraulic responses. The second one is obviously
an attempt to preserve certain important features
of the geologic model, whereas the last approach
is desirable but computationally prohibitive unless
the geologic models are relatively small.
9th European Conference on the Mathematics of Oil Recovery − Cannes, France, 30 August - 2 September 2004
2
A number of approaches have been proposed for
the design of proper uplayering that on one hand
preserve much of the details of the geologic models whilst on the other hand the simulation model
remains computationally efficient. An attempt to
tackle this problem was probably first presented
by Testerman[1] , who proposed a contrast seeking
statistical algorithm that utilises permeability as
a guide. The basic idea in that approach was to
develop reservoir zonation such that the variance
within individual zones is minimised while retaining maximum variance between zones. Among the
more recent approaches is an algorithm presented
by Stern et al.[2] that was devised to produce an
optimal fine-scale layer grouping based on breakthrough time and flux. Li et al.[3] presented an
optimisation algorithm which generates simulation
layering through minimisation of the discrepancy
between selected properties of the geologic and
simulation models. Durlofsky et al.[4] proposed a
flow-based uplayering technique by which layer averages of the fine-scale flux are employed to guide
grouping of fine-scale layers. Younis et al.[5] proposed a heterogeneity preserving procedure that
first identifies spatially connected important geologic bodies which mainly control reservoir flow
behaviour.
This paper presents an uplayering procedure
that makes use of single-phase flow solution on the
3D fine-scale model and streamlines flow characterisation. Although an essential feature still remains proper choice of an uplayering property (e.g.
permeability, flow rate), neither grouping (or segmentation) of layer averages nor explicit identification of hydraulic bodies within the flow domain is
required. Instead, streamline solutions were used
to aid generation of a set of connectivity-based
attribute logs that reveal regional profiles of the
fieldwide hydraulic characteristics. The streamline
solutions offer an efficient approach to fluid flow
modelling and thereby powerful depiction of dynamic connectivity of the reservoir at fine geologicscale. By making fine-scale total flow rate serve as
an uplayering property, we utilise developments in
feature extraction and pattern recognition to generate coarse-scale layering for simulation models.
Through its reliance on streamline-based regional connectivity description, the approach advocated here reveals feasibility of spatially correlatable regional layering, which is recognisable
from log deflections and variation of response pro-
files brought about by the underlying reservoir heterogeneity. However, it is worthwhile to note that
the pattern recoginition approach identifies presence of appropriate layering due to the similarity implied by response profiles of the fine-scale
model. In plain terms, the pattern recognition algorithm does not inherently grasp the underlying
cause-and-effect relationship between the reservoir
response and rock properties. To deal with this obvious weakness, the utilised algorithm was trained
with model-related background knowledge that supervise it thoughout the uplayering process.
The uplayering obtained by means of the proposed method was tested by carrying out numerical two-phase flow simulations, and the performance of the model was compared with that of its
parent fine-scale solution as well as the solution
obtained via a competing and uniformly uplayered
model. Although no assertions are made here to
have accomplished general solution to the uplayering problem, the approach highlighted here (one
of several currently under investigation) produced
layering design that outperformed the uniformly
layered coarse model.
The outline of this paper proceeds as follows.
First, description of the reservoir model and response measure are introduced in Section 2. Relevant concepts in streamline simulation and the
generation of connectivity-based response logs are
also presented. Section 3 introduces feature extraction method that was applied to pre-process
the response data prior to the pattern recognition
process. The feature extraction operation helps to
’de-noise’ the response data by reducing intercorrelations between response logs, such that the pattern recognition algorithm will focus only on the
intrinsic features present in the data. Section 4
presents the framework of the pattern recognition
technique and the modelling of the uplayering. Finally, the performance of the proposed uplayered
method is demonstrated in Section 5 though numerical waterflood simulations. Further discussion
and conclusions are also highlighted in Section 6.
2
Reservoir Response Measure
The proposed uplayering approach requires flow
modelling on the fine grid and generation of
streamlines. For this purpose, a streamline simulator in RMS was used to solve single-phase flow
equation on the full 3D fine-scale grid with bound-
9th European Conference on the Mathematics of Oil Recovery − Cannes, France, 30 August - 2 September 2004
3
ary conditions defined by injection and production
wells. The reservoir model was populated with
stochastically modelled permeability and porosity.
Fig.[2] shows a snapshot of permeability field with
streamlines. Key fine-scale data having a connection with the matter at hand are flow rate (obtained from pressure via Darcy’s law), time-offlight, streamlines and volumes drained/flooded by
individual producers/injectors. The time-of-flight,
Figure 1: Permeability field and streamlines for finelayered reservoir model. Ten injectors and nine producers were used to generate fine-scale
designated by τ (x), is a central parameter in the
proposed uplayering procedure. By definition[6] ,
s
φ
τ (x) =
dξ
(1)
0 |u|
where u is the total velocity and φ is porosity.
The parameter describes the travel time at which
a neutral tracer along a streamline would move
from zero (at the injection well) along some trajectory, s. In the context of this work, the flexibility of the time-of-flight parameter in tracking
the movement of fluid has been an essential to the
proposed approach. For instance, by selecting a
certain time-of-flight, τ ∗ , the area or volume of
the region enclosed by the envelope τ (x) ≤ τ ∗
can be identified. For an injector producer pair,
the time-of-flight increases from zero (at the injection well) to some maximum (post-breakthough)
value. Hence, for various levels of τ ∗ , the swept
volume can be easily calculated. For this purpose,
let Dk designate pairwise connected region for a
given layer k for which τ (x) is less than or equal
to some prescribed value. The swept volume in
terms of the physical space is provided by
Θ(x − x ) dx
(2)
Vk =
Dk
where Θ is a step function equal to unity within
Dk . Following this solution, the average flow rate
through each layer in Dk was calculated and the
response log for each connected well pair was generated. In the 3D fine-scale grid, the volume elements in Dk were determined by using an indicator
parameter that first selects relevant connect pair
of wells and then filters out cells whose time-offlight exceeds τ ∗ . The choice of τ ∗ is primarily
arbitrary and various values may lead to different
response profiles. A good initial estimate can be
obtained by analysis of the cumulative production
of individual producers connected to a given injection well. Fig.[2] presents profiles of flow rate logs
for 6 connect pairs (out of 30 in the model) in this
study. It may be noted that despite the complex
and seemingly erratic representation of the local
heterogeneity by the individual response logs, the
presence of layering can be nonetheless detected.
Following sections present techniques that systematically detect similarity in response logs.
3
Extraction of Key Response Feature
The response data for the uplayering constitutes
a multivariate data space defined by n logs and
m layers. Let X ∈ Rm×n designate the response matrix in which the elements of the jth
column represent the values of the total flow
rate for layers i = 1, · · · m. Furthermore, let
F = {xj ∈ Rm | j = 1, · · · , n} designate data set
in feature space. Through some functional mapping, we will now apply feature extraction technique that effectively captures essential information hidden in X by means of a p−dimensional
subspace. The feature extraction operation is primarilt intended to ’de-noise’ the data by reducing intercorrelations between response logs, such
that the subsequent pattern recognition algorithm
will focus only on the intrinsic features present
in the data. The functional mapping can be accomplished by means of classical subspace techniques, which map original data along the axes
of some new framework in which a fewer number
of orthogonal vectors systematically absorb most
of variance. An effective mathematical tool in the
9th European Conference on the Mathematics of Oil Recovery − Cannes, France, 30 August - 2 September 2004
4
ual factors. Thus,
the total variance is the sum of
trace(ΣT Σ) = nj=1 σj2 .
Of particular interest here is the column projection XV = U Σ which can be exploited to provide
straightforward analysis of the pattern in the original data through successive partitioning of Σ and
U . Noting that the singular values are ordered,
it follows from the summation in the right hand
side of Eq.[3] that the feature extraction process
can be viewed from the perspective of a partition
along some pth column
Σ
U
)
(4)
X = (U
(V V )T
Σ
0
20
40
60
Layer Number
80
100
Figure 2: Profiles of total flow rate for a set of connected well pairs. The individual logs are scaled by
division with maximum value of each log
context of this work is the singular value decomposition (SVD) technique. The SVD of the response
matrix, with m > n, is the factorisation[7]
X = U ΣV T =
n
σj uj vjT
(3)
j=1
where U ∈ Rm×n is an orthogonal matrix that
contains left singular vectors uj (eigenvectors for
XX T ), V ∈ Rn×n is an orthogonal matrix that
contains right singular vectors uj (eigenvectors for
X T X) and Σ =diag(σj , · · · , σn ), whose entries σi
(arranged in the order σ1 ≥ · · · σn ≥ 0) are the
singular values of X. The magnitude of σj2 (i.e.,
eigenvalues of the covariance matrices) reveals the
proportion of variance described by the individ-
contains the first p singular values, and Σ
where Σ
contains the remaining
n−p singular values. Since
the total variance is nj=1 σj2 , the above partition
admits relationship between the dimension p of the
desired subspace and the cost (in variance) associated with the truncation of the remaining n − p
components. In geological data analysis, there exists a number of approaches[8] that are employed
to decide the dimension of the desired subspace.
A common and simple approach is to specify the
amount of variance loss which can be afforded in
a given analysis. Denoting α by the percentage of
total variance that is lost due to the shrinkage of
the data space, the dimension of the feature subspace was obtained through the following equation
p
2
j=1 σj
(5)
α = 100 1 − n
2
j=1 σj
Fig.[3] presents the percentage variance explained
by retaining p out of n = 28 factors. Using a
variance loss of α = 5%, the feature space was
condensed to 10 components. The resulting reduced model was then passed to the pattern recognition algorithm that eventually generates the desired uplayering.
4
Uplayering by Pattern Recognition
Using the results obtained through the preceding feature extraction procedure, let now
X ∈ Rm×p designate the column projection
of the feature subspace.
Furthermore, let
P = {xi ∈ Rp | i = 1, · · · , m} denote the pattern
space of the data. Uplayering by pattern recognition involves application of cluster analysis that
9th European Conference on the Mathematics of Oil Recovery − Cannes, France, 30 August - 2 September 2004
5
Variance absorbed (%)
100
follows:
α = 5%
80
1. Initialise clustering process by selecting L
cluster centroids.
60
2. Calculate distance to cluster centroids and assign each pattern xi ∈ P to its closest cluster
center.
40
3. Update cluster centroids using the current
cluster memberships.
20
0
4. If a prescribed convergence criterion is not
met, go to step 2.
0
0.2
0.4
0.6
0.8
1.0
p/n
Figure 3: Percentage variance explained vs the number of factors, p, retained out of the total n = 28. Discarding the last 20 components costs 5% loss of total
variance, whereas retaining the first two factors alone
costs 50% of the variance.
generates grouping of pattern data along the layering dimension, i = 1, · · · , m, such that pattern vectors that exhibit similarity tend to fall
into the same class or cluster, whereas those that
are relatively distinct tend to separate into different clusters. The comparison of responses was
accomplished by using distance metric that measures similarity (or rather dissimilarity) between
patterns. In earth sciences, cluster analysis techniques have been widely used in segmentation
of well logs[10,11,13] and zonation of sedimentary
sequences[9,11,12] . In this work, the k-means clustering method[14] , an iterative partitional clustering technique, was utilised to generate a set of
pairwise disjoint clusters or classes that ultimately
define the desired coarse-scale layering. The kmeans operates as an analysis of variance in reverse: it starts with the selection of centers for
a desired number of clusters, L, in P, and returns a set of clusters, C = {C | = 1, · · · , L}, that
partition the pattern data such that some criterion function (e.g., squared error) is optimised.
In Euclidean formulation, the regular (i.e., unsupervised) k−means algorithm is the least-squares
functional
J=
xi − m 2
(6)
xi ∈P
where xi − m is a dissimilarity measure (distortion) between the pattern xi and centroid m
of cluster . The algorithm can be summarised as
Typical convergence criteria are minimal decrease
in dispersion, or that there is no further reassignment of patterns to new cluster centres (i.e., stable
partition).
The regular k−means algorithm, as presented
above, is not directly useful in this context due to
the obvious lack of supervision. A supervised version of the algorithm was therefore applied by incorporating model-related background knowledge
into the clustering procedure. The background
knowledge was presented to the algorithm as constraints that aid initialisation (step 1) and assignment of patterns to their closest cluster centres
(step 2). First, a model-based initialisation was estimated by generating field response markers, illustrated in Fig.[4], that were obtained through lateral correlation of response shape (peaks, troughs,
levels, ramps etc.) attributes[15] .
The shape attributes were in turn defined by
through the first and second derivatives obtained
by means of smooth differentiation filters[16] of
each curve, and the shape attributes were thereby
identified based on interpretation of the slope, inflexion, concavity and curvature along each trace.
The other set of constraints were designed to
supervise step 2. The scheme involved generation
of a framework that has pairwise ’must-link’ and
’cannot-link’ constraints (with an associated cost
of violating each constraint) between patterns in
the dataset[17] . The cost of having a linkage (or
no linkage) was derived based on facies (or permeability rock type) information, stratigraphic constraints and optional preservation of high (or low)
flow zones.
Penalties with graded severity (i.e., proportional
to the ’seriousness’ of the violation) were used to
enforce the imposed constraints. The level of the
9th European Conference on the Mathematics of Oil Recovery − Cannes, France, 30 August - 2 September 2004
6
of violating the model constraints was set-up by
means of the following optimisation function
Peak
J=
Trough
xi − m 2 +
xi ∈P
Small peak
Small trough
Level
Ramp
Large trough?
Medium peak
Positive spike
Figure 4: Illustration of curve shape analysis and
marker identification for cluster initialisation
penalty was selected to reflect how far apart a
pair of pattern vectors are. The cost of violating constraints for patterns that should (or should
not) link was set higher than the cost of violating constraints for patterns that are in turn far
apart. Moreover, grouping of patterns that originate from the same rock type was given a cost
lower than the cost of grouping of patterns that
differ in rock type but fairly similar in response.
Furthermore, stricter penalties were imposed on
violation of the stratigraphic constraints (i.e., adjacency constraints) and preservation of distinctive
high/low flow zones. For instance, the penalty for
grouping a pair of patterns, (xi , xj ) whose layers
are separated by a specified surface was set high,
whereas the presence of a hydraulically distinctive
pattern vector, xi in an otherwise predominantly
homogeneous body entails preservation (i.e., maximum penalty) of xi at the cost of rejecting the
pairwise groupings (xi−1 , xi ) and (xi , xi+1 ).
Using such model-related background information, linkage conditions implied in the constraints
were used to generate the set L with must-link
pairs (xi , xj ) ∈ L and the set L̄ with cannot-link
pairs (xi , xj ) ∈ L̄, with associated weights and
penalties. Following Basu et al.[18,19] , a supervised
version of the k−means algorithm, which combines
the objective function in Eq.[6] along with the cost
π(xi , xj ) +
(xi ,xj )∈L
π̄(xi , xj )
(7)
(xi ,xj )∈L̄
where π and π̄ are functionals that quantify the
overall cost of violating linkage constraints for a
must-link and cannot-link between two patterns
xi and xj .
The above clustering algorithm was employed
to design the uplayering of a 20 layers coarse-scale
flow simulation model. The number of layers was
first chosen based on streamline breakthrough time
analysis of an arbitrary uniform coarse model, see
Fig.[5].
1.0
Rate of change in relative breakthrough
Small trough
Large Peak
Medium peak
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
Number of fine layers in a coarse layer
Figure 5: Change of relative breakthrough time with
the number of fine-scale layers lumped in a coarse layer
By increasing the number of fine-scale layers
assigned to a coarse layer from 2 to 20, a uniform coarse model with 1:5 (i.e., 20 layers) layering contrast was created. Finally, the corresponding non-uniform model was generated by means of
the above described clustering algorithm. The levels of fine-scale layer lumping in this non-uniform
model are {6, 7, 2, 4, 5, 3, 9, 10, 2, 3} for the upper
10 layers and {2, 6, 7, 9, 2, 2, 2, 8, 5, 6} for the lower
10 layers. Here 6, 7, · · · represent number of finescale layers that constitute layers 1, 2, · · · of the
coarse-scale model.
9th European Conference on the Mathematics of Oil Recovery − Cannes, France, 30 August - 2 September 2004
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Numerical Simulation Results
6
Discussion and Conclusions
The uplayering method presented here was intended to examine application of streamline-based
connectivity information in simulation grid designed. In spite of the high level of rock heterogeneity, the performance of the non-uniform uplayering approach has been excellent. Further investigation of this issue is planned and several enhancement of the analysis methodology are currently
under consideration. Nevertheless, generation of
simulation model layering through a quantitative
analysis of streamline-based regional connectivity
response data was found promising.
1.0
Fractional oil rate, qo /qotarget
The performance of the proposed uplayering
method was evaluated using RMSflowsim. A set
of waterflood simulations was carried out using following three simulation grid models: 1) a reference
fine-scale model with 100 layers, 2) a corresponding 20-layers and non-uniformly uplayered coarse
model, and 3) uniformly uplayered 20-layers coarse
simulation model. The coarse-scale permeability
fields were obtained by flow-based diagonal tensor upscaling algorithm that employs sealing (noflow) boundary conditions. The reservoir fluid is
modelled as an undersaturated black oil with viscosity ratio µo /µw = 3, capillary pressure is neglected and relative permeabilities are described
by the Corey-type parameterisation krw = S 3.75
and kro = (1 − S)2.75 with an initial water saturation of Swi = 0.2 and residual oil saturation of
Sor = 0.1. Fig.[6] presents fractional flow of oil
produced versus pore volume water injected for the
three cases. As can be seen, for tD ≤ 1.0, the nonuniform case does not only outperform the uniform
case, but it also adequately reproduces the early
production response of the fine model. This is further illustrated through the comparison of average
reservoir pressure in Fig.[7]. This outperformance
is mainly due to the ability of the non-uniform
model to capture early fluid movements and rapid
advance of injected water along high permeability
streaks. At late post-breakthrough times, however, injection-driven build-up has more or less
stabilised and the gravity force acquired sufficient
time to stabilise the slowly moving water front;
something which the non-uniform model apparently could not satisfactorily reproduce.
Fine Model
Non-uniform
Uniform
0.8
0.6
0.4
0.2
0
0.5
1.0
tD
1.5
2.0
Figure 6: Fraction of target oil rate produced vs
pore volume water injected, tD , for fine-scale reference model (dotted), uniformly uplayered (dashed)
and non-uniformly uplayered model
1.0
Reservoir pressure, p̄/pi
5
Fine Model
Non-uniform
Uniform
0.9
0.8
0
0.5
1.0
1.5
2.0
tD
Figure 7: Average reservoir pressure (as fraction of
initial pressure) vs pore volume water injected, tD , for
fine-scale reference model (dotted), uniformly uplayered (dashed) and non-uniformly uplayered model
9th European Conference on the Mathematics of Oil Recovery − Cannes, France, 30 August - 2 September 2004
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Nomenclature
k
φ
t
tD
qo
qotarget
pi , p̄
τ
Θ
V
u
U , Σ, V
X
x
α
L, L̄
π, π̄
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
Permeability
Porosity
Time
Pore volume injected
Oil production rate
Target oil production rate
Reservoir pressure
Time of flight
Unit function
Volume
Flow rate
SVD matrices
Response matrix
Data vector
Proportion of variance
Pattern linkage sets
Cost function for linkage
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9th European Conference on the Mathematics of Oil Recovery − Cannes, France, 30 August - 2 September 2004