Upscaling Improvement for Heterogeneous Fractured Reservoir Using a Geostatistical Connectivity Index delorme atfeh allken bourbiaux fraca fracaflow


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

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Upscaling Improvement for Heterogeneous Fractured Reservoir Using a Geostatistical Connectivity Index delorme atfeh allken bourbiaux fraca fracaflow

  1. 1. UPSCALING IMPROVEMENT FOR HETEROGENEOUSFRACTURED RESERVOIR USING A GEOSTATISTICALCONNECTIVITY INDEXM. DELORME, B. ATFEH, V. ALLKEN and B. BOURBIAUX Reservoir Engineering Department, IFP, France.ABSTRACTThe connectivity of a fracture network is a crucial parameter to understand howfractures affect the reservoir fluid flow response. We have developed two methodsto estimate the equivalent permeability of a fracture network for any cell of afractured field model. Whereas the numerical method is CPU-intensive butapplicable for any fracture network, the continuum analytical method is morerapid but only valid for well-connected fractured media. An innovatingconnectivity analysis based on geo-statistical laws used in fractured reservoirdescription is proposed. It consists using a simple fracture network connectivityindex to choose the appropriate up-scaling method regarding both the physicalexactness/accuracy of solutions and the computation cost. It is formulated interms of a statistical constitutive fracture sets data function. This explicit indexis validated through the comparison of equivalent fracture permeability resultscomputed with either discrete or continuum models. Finally, the paper underlineshow that index can be implemented within a combined analytical/numerical up-scaling methodology for setting up reliable flow models of heterogeneousfractured reservoirs with minimized computational resources.INTRODUCTIONCharacterizing fractured reservoirs and modelling their flow behaviour arenecessary not only in petroleum engineering but also for various earth sciencesapplications. Porous fractured reservoirs have a low matrix permeability suchthat fractures are the main fluid-conducting medium. Fluids are stored mainly inthe matrix porosity and transferred to producing wells via that conductive fracturenetwork. Typical workflows for simulating and predicting the flow behaviour ofsuch reservoirs goes through a characterization of fractures and the constructionof comprehensive Discrete Fracture Networks (DFN).
  2. 2. M. DELORME et alFracture characterization is based on all conventional reservoir data sources,including seismic, core analysis, well logs and well tests. Fracture sets can bedetected then parameterized in terms of density, orientation, length andconductivity distributions. Subsequently, fracture sets parameters can constitutethe input data for the construction of representative DFN that can be used forflow-property calculation.As large and small fractures play different roles in fluid flow, they need to bedistinguished. Faults that are often larger than the grid cells size are modelled as alarge-scale discrete fault network conditioned by a deterministic seismic-derivedfault map. Small-scale fracture models and joint sets are generated randomly inspace using fracture set data as input parameters. This modelling methodology isdescribed in Cacas et al (1990) and implemented within FracaFlow , an IFPsoftware. DFN models are generated for two main reservoir simulationobjectives. First, they integrate fracture-related data or observations into arepresentative network that can be further qualified or calibrated in terms ofhydraulic conductivities from well test data matching at a fracture size resolutionscale, for instance. Secondly, calibrated DFN are used for computing theequivalent fracture permeabilities for subsequent fullfield scale simulations at ahectometric cell size resolution. With such a methodology and tools, the geologistcan integrate observations and understanding of the genetic episodes of fracturinginto parameterized fracture models.In this paper we assume that fracture-related data are already interpreted and areavailable. That is, in each reservoir cell, fracture set density, orientation, size andconductivity are characterized from geostatistical distribution laws or models.Starting from this quantitative fracture model, we herein introduce a fractureconnectivity analysis method to optimize the reliability and computationefficiency of available techniques for assigning fracture permeability values tofull-field model cells.EQUIVALENT FRACTURED MEDIA PERMEABILITY TENSORFull-field fractured reservoir numerical models consider that flow is taking placewithin two continua, the fracture medium characterized by equivalent or effectivefracture permeabilities specified as cell input parameters, and the matrix mediumrepresented as an array of identical matrix blocks with size and shape alsospecified as cell input parameters.Herein, fracture flows are modelled using Poiseuilles law (equation 1), with anequivalent hydraulic aperture, eh, defined from the ratio between a conductivity,c, and an average mechanical aperture, e. Whereas e is derived from the fracturenetwork porosity, c is the actual flow parameter that relates the flow rate and thepressure gradient within an individual fracture. eh and e are equal for idealfractures with parallel walls but do not coincide in practice because of theroughness of fracture walls. 2 1 eh 1 c u=− ∇P = − ∇P (1) 12 12.e GEOSTATS 2008, Santiago, Chile
  3. 3. GEOSTATISTICAL CONNECTIVITY INDEXwith µ the fluid viscosity, ∇P the pressure gradient along the fracture and u thefluid velocity within the fracture.Numerical UpscalingVarious computation methods have recently been implemented to determine theequivalent permeability tensor of a DFN at the reservoir scale. Whereas somemethods involve a discretization of each fracture into finite elements (Koudine etal., 1998), we adopt a minimum discretization of the DFN based on fractureintersections in each sedimentary layer (Bourbiaux et al., 1997) in order to reducecomputation costs. Equivalent permeabilities are then determined as the solutionsof a steady-state single-phase Darcy flow problem involving the transmissivitiesbetween fracture nodes. This approach will be considered as our referencesolution method for all DFNs under consideration in this paper. Even though it isbased on a minimum number of fracture nodes, the numerical up-scalingprocedure remains too CPU-intensive when the density of fractures is high, andcannot in practice be repeated for each cell of a multi-million-cell full-field flowmodel. To overcome that limitation, an analytical approach was implemented asan alternative solution to the fracture permeability up-scaling problem.Analytical UpscalingStarting from the approach of Oda (1985), which considers the fracture networkas a flow continuum, we formulated an analytical expression of the equivalentfracture 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 andcomplete connectivity of the fracture network intersecting the studied cell. Thisassumption 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 fwith 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 scaleDarcy 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 inset S, Vcell is the cell volume, V f,s is the fracture volume.VIII International Geostatistics Congress
  4. 4. M. DELORME et alThis analytical up-scaling method may be applied either to the individualfractures of the DFN, or by considering the density and direction of theconstitutive fracture sets in the reservoir cell. This latter "statistical" approachwas adopted as it saves considerable computation resources that are spent in theformer "discrete" approach in generating the DFN, quantifying its geometry, andinverting for the transmissivity matrix.Unfortunately the continuum models are not always applicable, in particular if thefracture network is not well connected.AnalysisThe CPU performances of the two above-described methods are very differentsince the analytical method consists of the integration of statistical laws whereasthe numerical method requires a lot of network-density-dependent computations(Figure 1). These performances were compared for series of DFN differing in thenumber of constitutive sets, and in the parameters of those sets, such as the meanvalue and dispersion of fracture orientations and lengths. Herein, results are givenonly for three series of DFN involving two fracture sets with respective averagefracture lengths equal to 4m and 6m. These series differ in the angle θ ij made bythe mean azimuths of the two constitutive sets. The DFN realizations of a givenseries 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 determinedwith respect to the reference numerical method.Relative errors between the analytical and numerical methods are more importantfor low fracture densities; that is when the numerical method has a high CPU-efficiency. However, the fracture density threshold above which differences areacceptable varies with the nature of the fracture model, i.e. with the parameters ofthe constitutive fracture sets (Figure 1). A simple criterion is needed for selectingeither the analytical or the numerical method. Indeed, determining the upscaledfracture permeabilities of a full-field flow model requires adopting the analyticalmethod whenever it is applicable, i.e. in the presence of any well-connectedfracture network, and applying the numerical method to the remaining low-connectivity-fracture networks. GEOSTATS 2008, Santiago, Chile
  5. 5. GEOSTATISTICAL CONNECTIVITY INDEXCONNECTIVITY INDEXAs already underlined by Berkowitz (1995), we believe that the geometrical DFNconnectivity is the predominant feature affecting flow and that it can bequantified as the average number of intersections per fracture of the DFN. Theproposed criteria is then assumed to depend exclusively on geometricalparameters (orientation, length and density geostatistical laws), thus avoiding theCPU-intensive generation of representative DFN.DefinitionAlthough the methodology can be applied to 3D networks, we restrict ouranalysis to multi-2D networks with fractures that are orthogonal to the consideredlayer and crossing their entire thickness. The reason for that restriction lies in thatpetroleum accumulations are most often found in sedimentary basins where thestructure of deposition beds drives the fracturing process, with a propagation offractures orthogonally to the bed boundaries. Fracture parameters vary within thereservoir space but a random distribution of fractures is assumed at a givenreservoir location, i.e. within a given reservoir cell. For diffuse, stochastically-defined fracture networks, the maximal length of fractures is assumed less thanthe characteristic cell length.Hereafter, a fracture network connectivity index is formulated from the statisticalparameters of the fracture sets at the considered reservoir location.For each fracture set, denoted by subscript S (S=A or B), geometrical fractureproperties are defined by independent orientation and length probability densitylaws, respectively ρ a, s and ρ l , s , with Li,S the mean length , and d s the fracturedensity defined as the total length of fractures per area unit. To perform ourdemonstration, we split each fracture set into different groups of fracturesaccording to their orientation. That is, fractures of S having an azimuthθ ± 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 statisticalelementary 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 azimuthθ j making the angle θ ij = θ j − θ i with respect to the fracture Fi , A of groupGi , A within Sei , as: Lc j , B = d j , B .Se i (6)VIII International Geostatistics Congress
  6. 6. M. DELORME et alFigure 2: Definition of the intersection area, between a discrete fracture Fi , A of group Gi , A andany 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 canthen 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 ) SeiTaking into account all the fractures of Gi , A and considering that fracturesbelonging to the same group cannot intersect, the total number of intersectionsbetween fractures of groups Gi , A and G j , B per surface area unit equals: d i, A Nc A, B = Pij . (8) Li , AIf we generalize Equation 8 for a distribution of orientations in each fracture setand a number, nbset, of fracture sets, we can introduce the dimensionlessanalytical connectivity index, Ic, defined as the average number of intersectionsdivided 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 Liwith Ncii and Ncij respectively the number of intersections between fractures ofsame set i and fractures of different sets i and j: 2Π 2ΠNcii = d i . 2 ρ a ,i (θ )ρ a ,i ( β ) sin( β − θ ) dβ dθ θ = 0 β >θ 2Π 2ΠNcij = d i d j ρ a ,i (θ )ρ a , j ( β ) sin( β − θ ) dβ dθ θ =0 β =0AnalysisIn order to validate the fracture connectivity definition, we generated many DFNswith one or several fracture sets characterized by various values of their density, GEOSTATS 2008, Santiago, Chile
  7. 7. GEOSTATISTICAL CONNECTIVITY INDEXorientation, length statistical laws. The connectivity index of each network wasquantified from the previously-stated index, Ic, and compared with the actualnumber of intersections observed in the DFN realization. That extensive studyalso led to a valuable data base for analyzing the evolution of the connectivitywith the main fracture sets parameters, like density.An example is shown below for fracture networks made up of two sets with fixedparameters, except for their respective orientations. Figure 3 shows the excellentconsistency between the fracture connectivity index function and the actualconnectivity (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 networksHereafter, we consider again the same two fracture sets for three different anglesθij, but the density is now the variable parameter for DFN realization. The numberof fracture clusters, defined as isolated groups of connected fractures (with morethan two fractures) within the network, is drawn as a function of the connectivityindex (Figure 4). One observes three curves corresponding to the three series ofnetworks characterized by fixed parameters except for the density. Whatever theconsidered series, the number of clusters increases with the fracture connectivityindex and reaches a maximum for a fixed index value, close to 1. Above thatvalue, the number of clusters decreases as they start to connect, until a singlecluster containing all fractures is obtained. That complete connectivity of thenetwork 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 fracturenetworks, i.e. whatever the angle in the present example, the maximum number of clusters is observedfor Ic close to 1, then all clusters lump together until a single cluster is obtained for an Ic value closeto 3.VIII International Geostatistics Congress
  8. 8. M. DELORME et alSuch an analysis was also performed for multiple other series of networkscharacterized by various distributions of fracture length and orientation of theconstitutive sets and it led to the same observations regarding the Ic indexconcept and threshold values:- the connectivity index, Ic, quantifies the average number of fractureintersections 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 thresholdvalues characterizing the flow behaviour of the network.Application: Flow Interpretation of IcThe equivalent permeability of previous fracture networks is now analyzed as afunction of their respective connectivity index.The equivalent horizontal fracture permeability, KH, was computed using thereference numerical method described before. KH was defined as the geometricalmean, K K , of the principal permeabilities, K1 and K2, of the 2D fracture 1 2permeability tensor at the model cell scale. The analytical model was also appliedin order to investigate the range of applicability of that method with respect to theconnectivity index value of the network.Figure 5: KH results as a function of the fracture network density and as a function of Ic valuesConsistently with the previous connectivity analysis, non-zero fracture networkpermeabilities are obtained for Ic values higher than 1: Ic=1 then characterizesthe percolation threshold, i.e. the minimum connectivity for the network tobecome conductive at the cell scale. For such networks, at least one fracturecluster connects the opposite faces of the cell, with each fracture of that clusterhaving 2 intersections. The distance between fracture intersections is then in theorder of fracture size.Linear evolutions of the permeability with Ic are also observed for Ic valueshigher than 3: Ic=3 corresponds to a second permeability threshold, related to theexistence of a unique fracture cluster at the cell scale. At that stage, any increaseof the fracture density, or of Ic, results in a proportional increase of the fracturenetwork permeability because any added fracture then contributes to the networkpermeability in proportion to the additional length of fracture, i.e. density,brought to the single constitutive cluster of the network. GEOSTATS 2008, Santiago, Chile
  9. 9. GEOSTATISTICAL CONNECTIVITY INDEXThe derivative of the upscaled permeability with respect to the network density isalso represented as a function of Ic in Figure 6, considering both numerical andanalytical upscaling methods. Again, the threshold Ic value for continuum flowbehaviour is apparent. The superposition of analytical slopes to numerical onesconfirms that the analytical method is applicable for networks having a fractureconnectivity 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 densityTo end with, the connectivity index of a fracture network as defined hereinappears as a reliable parameter for qualifying its flow behaviour and selecting theproper upscaling method.HYBRID UPSCALING FOR FULL-FIELD FLOW MODELSPrevious results give the way to proposing a hybrid methodology for upscalingthe fracture permeability of full-field models. That methodology is simply basedon 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 networkparameters, without having to simulate that network.This way, the computation time of full-field up-scaling is optimized in a reliableway and may be drastically reduced, especially in the case of heterogeneousfracture distributions over the field.That methodology is applicable to any reservoir where the 3D fracture networkcan be modelled as multi-2D networks. A generalization of that Ic formula wouldbe required for purely-3D fracture networks, such as the ones found in non-sedimentary geological frameworks, for which a multi-2D modelling is no moreadapted and convenient.VIII International Geostatistics Congress
  10. 10. M. DELORME et alCONCLUSIONSThe assumption that the connectivity of a fracture network is the predominantdriving feature of its flow properties motivates the developments and outcome ofthis paper.A geometrical analysis of synthetic fracture networks confirmed the relevance ofthis assumption. A connectivity index was established from the statisticalparameters of fracture sets as might be characterized by the geologist for a givenfield case. This index was shown to represent the actual connectivity of thesimulated fracture networks. Furthermore, it was shown to be an efficientlycomputed 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 ofheterogeneous fractured reservoirs. A fast and reliable connectivity analysis ofthe fracture networks drives the choice between numerical and analyticalupscaling methods. Thus, the methodology takes advantage of both the accuracyof the numerical method for low-connectivity fracture networks, and the CPUefficiency of the analytical method whenever it is applicable.Improvements of that upscaling methodology would concern the development ofan 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.ACKNOWLEDGEMENTSThe authors would like to acknowledge Dr. R. Basquet, presently in StatoilCompany, for the discussions we had to elaborate the connectivity index.REFERENCESBerkowitz, B. (1995), Analysis of Fracture Network Connectivity Using Percolation Theory, Mathematical Geology Vol. 27 No 4. , 467-483Bourbiaux, B.; Cacas, M-C; Sarda, S.; Sabathier, J-C.; (1998), A Fast and Efficient Methodology to Convert Fractured Reservoir Images into a Dual-Porosity Model, SPE No 38907 presented at the SPE Ann. Tech. Conf. and Exh., San Antonio, Tx, Oct. 5-8.Cacas, M.C.; Ledoux, E; de Marsily, G.; Tillie, B.; Barbreau, A.; Durand, E.; Feuga, B.; Peaudecerf, P.; (1990) Modeling fracture flow with a sthochastic discrete fracture network: calibration and validation., Water Resources Research, 26,3,479-489Delorme M. and Bourbiaux B.(2007), Méthode pour estimer la perméabilité dun réseau de fractures à partir d’une analyse de connectivité, Patent Application FR07/04.703,Koudine N., Gonzalez Garcia R., Thovert J.-F. and Adler, (1998) Permeability of Three-Dimensional Fracture Networks, Physical Review E, 57, No 4. , 4466-4479.Oda M. (1985): Permeability tensor for discontinuous Rock Masses, Geotechnique Vol 35, 483-495.Long, J.C.S. and Witherspoon, P.A. (1985), The Relationship of the degree of interconnection to Permeability in Fracture Networks, Journal Of Geophysical Research, Vol 90 NOB4, 3087-3098. GEOSTATS 2008, Santiago, Chile