Statistical percolation threshold formulation indicator to estimate the fracture network connectivity. Can be used for upscaling of permeabilities. Available in FracaFlow sofware

1. UPSCALING IMPROVEMENT FOR HETEROGENEOUS
FRACTURED RESERVOIR USING A GEOSTATISTICAL
CONNECTIVITY INDEX
M. DELORME, B. ATFEH, V. ALLKEN and B. BOURBIAUX
Reservoir Engineering Department, IFP, France
.
ABSTRACT
The connectivity of a fracture network is a crucial parameter to understand how
fractures affect the reservoir fluid flow response. We have developed two methods
to estimate the equivalent permeability of a fracture network for any cell of a
fractured field model. Whereas the numerical method is CPU-intensive but
applicable for any fracture network, the continuum analytical method is more
rapid but only valid for well-connected fractured media. An innovating
connectivity analysis based on geo-statistical laws used in fractured reservoir
description is proposed. It consists using a simple fracture network connectivity
index to choose the appropriate up-scaling method regarding both the physical
exactness/accuracy of solutions and the computation cost. It is formulated in
terms of a statistical constitutive fracture sets data function. This explicit index
is validated through the comparison of equivalent fracture permeability results
computed with either discrete or continuum models. Finally, the paper underlines
how that index can be implemented within a combined analytical/numerical up-
scaling methodology for setting up reliable flow models of heterogeneous
fractured reservoirs with minimized computational resources.
INTRODUCTION
Characterizing fractured reservoirs and modelling their flow behaviour are
necessary not only in petroleum engineering but also for various earth sciences
applications. Porous fractured reservoirs have a low matrix permeability such
that fractures are the main fluid-conducting medium. Fluids are stored mainly in
the matrix porosity and transferred to producing wells via that conductive fracture
network. Typical workflows for simulating and predicting the flow behaviour of
such reservoirs goes through a characterization of fractures and the construction
of comprehensive Discrete Fracture Networks (DFN).

2. M. DELORME et al
Fracture characterization is based on all conventional reservoir data sources,
including seismic, core analysis, well logs and well tests. Fracture sets can be
detected then parameterized in terms of density, orientation, length and
conductivity distributions. Subsequently, fracture sets parameters can constitute
the input data for the construction of representative DFN that can be used for
flow-property calculation.
As large and small fractures play different roles in fluid flow, they need to be
distinguished. Faults that are often larger than the grid cells size are modelled as a
large-scale discrete fault network conditioned by a deterministic seismic-derived
fault map. Small-scale fracture models and joint sets are generated randomly in
space using fracture set data as input parameters. This modelling methodology is
described in Cacas et al (1990) and implemented within FracaFlow , an IFP
software. DFN models are generated for two main reservoir simulation
objectives. First, they integrate fracture-related data or observations into a
representative network that can be further qualified or calibrated in terms of
hydraulic conductivities from well test data matching at a fracture size resolution
scale, for instance. Secondly, calibrated DFN are used for computing the
equivalent fracture permeabilities for subsequent fullfield scale simulations at a
hectometric cell size resolution. With such a methodology and tools, the geologist
can integrate observations and understanding of the genetic episodes of fracturing
into parameterized fracture models.
In this paper we assume that fracture-related data are already interpreted and are
available. That is, in each reservoir cell, fracture set density, orientation, size and
conductivity are characterized from geostatistical distribution laws or models.
Starting from this quantitative fracture model, we herein introduce a fracture
connectivity analysis method to optimize the reliability and computation
efficiency of available techniques for assigning fracture permeability values to
full-field model cells.
EQUIVALENT FRACTURED MEDIA PERMEABILITY TENSOR
Full-field fractured reservoir numerical models consider that flow is taking place
within two continua, the fracture medium characterized by equivalent or effective
fracture permeabilities specified as cell input parameters, and the matrix medium
represented as an array of identical matrix blocks with size and shape also
specified as cell input parameters.
Herein, fracture flows are modelled using Poiseuille's law (equation 1), with an
equivalent hydraulic aperture, eh, defined from the ratio between a conductivity,
c, and an average mechanical aperture, e. Whereas e is derived from the fracture
network porosity, c is the actual flow parameter that relates the flow rate and the
pressure gradient within an individual fracture. eh and e are equal for ideal
fractures with parallel walls but do not coincide in practice because of the
roughness of fracture walls.
2
1 eh 1 c
u=− ∇P = − ∇P (1)
12 12.e
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3. GEOSTATISTICAL CONNECTIVITY INDEX
with µ the fluid viscosity, ∇P the pressure gradient along the fracture and u the
fluid velocity within the fracture.
Numerical Upscaling
Various computation methods have recently been implemented to determine the
equivalent permeability tensor of a DFN at the reservoir scale. Whereas some
methods involve a discretization of each fracture into finite elements (Koudine et
al., 1998), we adopt a minimum discretization of the DFN based on fracture
intersections in each sedimentary layer (Bourbiaux et al., 1997) in order to reduce
computation costs. Equivalent permeabilities are then determined as the solutions
of a steady-state single-phase Darcy flow problem involving the transmissivities
between fracture nodes. This approach will be considered as our reference
solution method for all DFNs under consideration in this paper. Even though it is
based on a minimum number of fracture nodes, the numerical up-scaling
procedure remains too CPU-intensive when the density of fractures is high, and
cannot in practice be repeated for each cell of a multi-million-cell full-field flow
model. To overcome that limitation, an analytical approach was implemented as
an alternative solution to the fracture permeability up-scaling problem.
Analytical Upscaling
Starting from the approach of Oda (1985), which considers the fracture network
as a flow continuum, we formulated an analytical expression of the equivalent
fracture permeability based on the parameters underlying the DFN realization,
instead of the actual DFN geometry.
Basic assumptions are a linear pressure gradient in the fracture network and
complete connectivity of the fracture network intersecting the studied cell. This
assumption is also valid for faults and well-connected joints.
Under these assumptions, the local velocities u f in each fracture are defined as :
1 cf
uf = − . . N f .∇ P (2)
12.e
f
with N f the unitary projection matrix of the cell-scale pressure gradient, ∇P ,
on the fracture f plane.
Then the identification of the averaged local fluid velocities with the cell scale
Darcy velocity, U , leads to the equivalent permeability tensor that is:
−1 1 nb _ set nb _ fracS c f ,s
K f = −µ.U.∇P = . Vf,s . .N f ,s
Vcell s =1 f =1 12.ef,s (3)
where nb_set is the number of fracture sets, nb_fracs is the number of fractures in
set S, Vcell is the cell volume, V f,s is the fracture volume.
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4. M. DELORME et al
This analytical up-scaling method may be applied either to the individual
fractures of the DFN, or by considering the density and direction of the
constitutive fracture sets in the reservoir cell. This latter "statistical" approach
was adopted as it saves considerable computation resources that are spent in the
former "discrete" approach in generating the DFN, quantifying its geometry, and
inverting for the transmissivity matrix.
Unfortunately the continuum models are not always applicable, in particular if the
fracture network is not well connected.
Analysis
The CPU performances of the two above-described methods are very different
since the analytical method consists of the integration of statistical laws whereas
the numerical method requires a lot of network-density-dependent computations
(Figure 1). These performances were compared for series of DFN differing in the
number of constitutive sets, and in the parameters of those sets, such as the mean
value and dispersion of fracture orientations and lengths. Herein, results are given
only for three series of DFN involving two fracture sets with respective average
fracture lengths equal to 4m and 6m. These series differ in the angle θ ij made by
the mean azimuths of the two constitutive sets. The DFN realizations of a given
series differ only in their fracture density.
Figure 1: Comparison of the analytical and numerical methods for upscaling fracture permeability,
both in terms of accuracy and computation cost. A relative error on the analytical results is determined
with respect to the reference numerical method.
Relative errors between the analytical and numerical methods are more important
for low fracture densities; that is when the numerical method has a high CPU-
efficiency. However, the fracture density threshold above which differences are
acceptable varies with the nature of the fracture model, i.e. with the parameters of
the constitutive fracture sets (Figure 1). A simple criterion is needed for selecting
either the analytical or the numerical method. Indeed, determining the upscaled
fracture permeabilities of a full-field flow model requires adopting the analytical
method whenever it is applicable, i.e. in the presence of any well-connected
fracture network, and applying the numerical method to the remaining low-
connectivity-fracture networks.
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5. GEOSTATISTICAL CONNECTIVITY INDEX
CONNECTIVITY INDEX
As already underlined by Berkowitz (1995), we believe that the geometrical DFN
connectivity is the predominant feature affecting flow and that it can be
quantified as the average number of intersections per fracture of the DFN. The
proposed criteria is then assumed to depend exclusively on geometrical
parameters (orientation, length and density geostatistical laws), thus avoiding the
CPU-intensive generation of representative DFN.
Definition
Although the methodology can be applied to 3D networks, we restrict our
analysis to multi-2D networks with fractures that are orthogonal to the considered
layer and crossing their entire thickness. The reason for that restriction lies in that
petroleum accumulations are most often found in sedimentary basins where the
structure of deposition beds drives the fracturing process, with a propagation of
fractures orthogonally to the bed boundaries. Fracture parameters vary within the
reservoir space but a random distribution of fractures is assumed at a given
reservoir location, i.e. within a given reservoir cell. For diffuse, stochastically-
defined fracture networks, the maximal length of fractures is assumed less than
the characteristic cell length.
Hereafter, a fracture network connectivity index is formulated from the statistical
parameters of the fracture sets at the considered reservoir location.
For each fracture set, denoted by subscript S (S=A or B), geometrical fracture
properties are defined by independent orientation and length probability density
laws, respectively ρ a, s and ρ l , s , with Li,S the mean length , and d s the fracture
density defined as the total length of fractures per area unit. To perform our
demonstration, we split each fracture set into different groups of fractures
according to their orientation. That is, fractures of S having an azimuth
θ ± dθ / 2 belong to the group Gi , s . Their density, d i , s , is then defined by:
i
d i , s = d s × ρ a , s (θ i ) dθ (4)
Considering that fractures are randomly distributed in space, the statistical
elementary surface occupied by a discrete fracture Fi , A of group Gi , A equals:
Sei = Li , A .( d i , A ) −1 (5)
We can express the cumulative length of fractures of set B and azimuth
θ j making the angle θ ij = θ j − θ i with respect to the fracture Fi , A of group
Gi , A within Sei , as:
Lc j , B = d j , B .Se i (6)
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6. M. DELORME et al
Figure 2: Definition of the intersection area, between a discrete fracture Fi , A of group Gi , A and
any fracture of group G j , B , with Sei the elementary surface occupied by Fi , A .
The intersection probability between Fi , A and any fracture of group G j , B can
then be formulated as the intersection area (Figure 2) divided by Sei :
Li , A .Lc j , B sin(θ ij ) (7)
Pij = = d j , B .Li , A sin(θ ij )
Sei
Taking into account all the fractures of Gi , A and considering that fractures
belonging to the same group cannot intersect, the total number of intersections
between fractures of groups Gi , A and G j , B per surface area unit equals:
d i, A
Nc A, B = Pij . (8)
Li , A
If we generalize Equation 8 for a distribution of orientations in each fracture set
and a number, nbset, of fracture sets, we can introduce the dimensionless
analytical connectivity index, Ic, defined as the average number of intersections
divided by the total number of fractures, that is:
nbset nbset nbset nbset −1
di
Ic = Nc ii + Nc ij . (9)
i =1 i =1 j = i +1 i =1 Li
with Ncii and Ncij respectively the number of intersections between fractures of
same set i and fractures of different sets i and j:
2Π 2Π
Ncii = d i .
2
ρ a ,i (θ )ρ a ,i ( β ) sin( β − θ ) dβ dθ
θ = 0 β >θ
2Π 2Π
Ncij = d i d j ρ a ,i (θ )ρ a , j ( β ) sin( β − θ ) dβ dθ
θ =0 β =0
Analysis
In order to validate the fracture connectivity definition, we generated many DFNs
with one or several fracture sets characterized by various values of their density,
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7. GEOSTATISTICAL CONNECTIVITY INDEX
orientation, length statistical laws. The connectivity index of each network was
quantified from the previously-stated index, Ic, and compared with the actual
number of intersections observed in the DFN realization. That extensive study
also led to a valuable data base for analyzing the evolution of the connectivity
with the main fracture sets parameters, like density.
An example is shown below for fracture networks made up of two sets with fixed
parameters, except for their respective orientations. Figure 3 shows the excellent
consistency between the fracture connectivity index function and the actual
connectivity (i.e. the average number of intersections per network fracture)
calculated on the generated networks for different orientations and densities.
Figure 3: Consistency between Ic and the actual connectivity of various networks
Hereafter, we consider again the same two fracture sets for three different angles
θij, but the density is now the variable parameter for DFN realization. The number
of fracture clusters, defined as isolated groups of connected fractures (with more
than two fractures) within the network, is drawn as a function of the connectivity
index (Figure 4). One observes three curves corresponding to the three series of
networks characterized by fixed parameters except for the density. Whatever the
considered series, the number of clusters increases with the fracture connectivity
index and reaches a maximum for a fixed index value, close to 1. Above that
value, the number of clusters decreases as they start to connect, until a single
cluster containing all fractures is obtained. That complete connectivity of the
network corresponds again to a fixed value of the connectivity index, close to 3.
Figure 4: Number of fracture clusters as a function of the connectivity index, Ic: for all the fracture
networks, i.e. whatever the angle in the present example, the maximum number of clusters is observed
for Ic close to 1, then all clusters lump together until a single cluster is obtained for an Ic value close
to 3.
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Such an analysis was also performed for multiple other series of networks
characterized by various distributions of fracture length and orientation of the
constitutive sets and it led to the same observations regarding the Ic index
concept and threshold values:
- the connectivity index, Ic, quantifies the average number of fracture
intersections per fracture of a given fracture network;
- the fracture network forms a single cluster for an Ic value close to 3.
As illustrated in the following section, these two Ic values constitute threshold
values characterizing the flow behaviour of the network.
Application: Flow Interpretation of Ic
The equivalent permeability of previous fracture networks is now analyzed as a
function of their respective connectivity index.
The equivalent horizontal fracture permeability, KH, was computed using the
reference numerical method described before. KH was defined as the geometrical
mean, K K , of the principal permeabilities, K1 and K2, of the 2D fracture
1 2
permeability tensor at the model cell scale. The analytical model was also applied
in order to investigate the range of applicability of that method with respect to the
connectivity index value of the network.
Figure 5: KH results as a function of the fracture network density and as a function of Ic values
Consistently with the previous connectivity analysis, non-zero fracture network
permeabilities are obtained for Ic values higher than 1: Ic=1 then characterizes
the percolation threshold, i.e. the minimum connectivity for the network to
become conductive at the cell scale. For such networks, at least one fracture
cluster connects the opposite faces of the cell, with each fracture of that cluster
having 2 intersections. The distance between fracture intersections is then in the
order of fracture size.
Linear evolutions of the permeability with Ic are also observed for Ic values
higher than 3: Ic=3 corresponds to a second permeability threshold, related to the
existence of a unique fracture cluster at the cell scale. At that stage, any increase
of the fracture density, or of Ic, results in a proportional increase of the fracture
network permeability because any added fracture then contributes to the network
permeability in proportion to the additional length of fracture, i.e. density,
brought to the single constitutive cluster of the network.
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9. GEOSTATISTICAL CONNECTIVITY INDEX
The derivative of the upscaled permeability with respect to the network density is
also represented as a function of Ic in Figure 6, considering both numerical and
analytical upscaling methods. Again, the threshold Ic value for continuum flow
behaviour is apparent. The superposition of analytical slopes to numerical ones
confirms that the analytical method is applicable for networks having a fracture
connectivity index higher than 3, i.e. for networks behaving like flow continua.
Figure 6: Evolution with Ic of the partial derivative of KH with respect to the fracture network density
To end with, the connectivity index of a fracture network as defined herein
appears as a reliable parameter for qualifying its flow behaviour and selecting the
proper upscaling method.
HYBRID UPSCALING FOR FULL-FIELD FLOW MODELS
Previous results give the way to proposing a hybrid methodology for upscaling
the fracture permeability of full-field models. That methodology is simply based
on the fracture connectivity index values determined locally in each cell:
- Ic<1: no effective fracture permeability computation is required;
- 1<Ic<3: a numerical effective fracture permeability computation method
is required;
- Ic > 3: an analytical continuum approach is applicable.
Such a permeability diagnosis can be made a priori from the fracture network
parameters, without having to simulate that network.
This way, the computation time of full-field up-scaling is optimized in a reliable
way and may be drastically reduced, especially in the case of heterogeneous
fracture distributions over the field.
That methodology is applicable to any reservoir where the 3D fracture network
can be modelled as multi-2D networks. A generalization of that Ic formula would
be required for purely-3D fracture networks, such as the ones found in non-
sedimentary geological frameworks, for which a multi-2D modelling is no more
adapted and convenient.
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10. M. DELORME et al
CONCLUSIONS
The assumption that the connectivity of a fracture network is the predominant
driving feature of its flow properties motivates the developments and outcome of
this paper.
A geometrical analysis of synthetic fracture networks confirmed the relevance of
this assumption. A connectivity index was established from the statistical
parameters of fracture sets as might be characterized by the geologist for a given
field case. This index was shown to represent the actual connectivity of the
simulated fracture networks. Furthermore, it was shown to be an efficiently
computed and reliable predictor of the up-scaled network permeability behaviour,
through the use of two threshold values.
These results open the way to an optimized up-scaling methodology of
heterogeneous fractured reservoirs. A fast and reliable connectivity analysis of
the fracture networks drives the choice between numerical and analytical
upscaling methods. Thus, the methodology takes advantage of both the accuracy
of the numerical method for low-connectivity fracture networks, and the CPU
efficiency of the analytical method whenever it is applicable.
Improvements of that upscaling methodology would concern the development of
an extended analytical model, that could be applied to low-connectivity networks,
to further increase the upscaling performance for field models with ever-
increasing resolution or size.
ACKNOWLEDGEMENTS
The authors would like to acknowledge Dr. R. Basquet, presently in Statoil
Company, for the discussions we had to elaborate the connectivity index.
REFERENCES
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