3.- Fast and Efficient Modeling and Conditioning of Naturally Fractured Reservoir Models Using Static and Dynamic Data

8/12/2019 3.- Fast and Efficient Modeling and Conditioning of Naturally Fractured Reservoir Models Using Static and Dynamic Data

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Copyright 2007, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Europec/EAGE Annual Conference andExhibition held in London, United Kingdom, 1114 June 2007.

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Abstract

A large proportion of petroleum reservoirs is known to benaturally fractured with consequences on their flow behaviorhence on reservoir performance. Though the modeling of suchreservoirs has been the purpose of many research works, itremains a challenging task. Too simplistic reservoir models donot allow capturing essential features like large-scale

fracturing trends, or non-linear multivariate relationshipsbetween the equivalent (generally anisotropic) permeability ofthe fracture system, and fracture densities and properties to becharacterized on a directional fracture-set basis. Conversely,too complex reservoir models, intended to be more realistic,require computationally intensive and memory consumingalgorithms. They also involve numerous parameters, a largepart of which cannot be estimated from available data.

In-between, there is a need for reasonably complex modelsand methods to generate them in a consistent way with variousfracturing and dynamic data in order to produce conditionalmodels. This paper presents such an approach, which has beendeveloped as a workflow.

The approach is based on an original conceptual model offracture systems and a notion of scale-dependent effectiveproperties. It is also a two-step modeling approach in whichthe fracture system is first characterized, then converted intoequivalent flow properties for reservoir simulation purposes.Key aspects of the approach include the geostatisticalmodeling of fracture densities, scale-dependent calculation ofequivalent within-layer horizontal permeability tensors basedon spatially periodic discrete fracture networks, analyticalcalculations of vertical inter-layer permeabilities, andconditioning to well-test permeabilities by using steady-stateflow-based evaluation of reservoir model responses. All theseaspects rely on innovative and CPU-time efficient methods.

They are introduced and illustrated by case-study results.

Introduction

Three main reasons explain the tremendous research work onnaturally fractured reservoirs.

1. They represent a large proportion of the world'shydrocarbon reserves.

2. Fracturing is critical for oil recovery.3. The geology and flow behavior of naturally fractured

reservoirs are highly complex and require uneasy andtime-demanding modeling approaches.

Typically, once directional fracture-sets, i.e. fractureorientations and types, have been identified from cores orborehole images, the reliability of naturally fractured reservoirmodels relies on the following critical steps.

1. Calculation of fracture densities along wells for eachdirectional fracture-set.

2. Full-field modeling of the spatial distribution offracture densities for each directional fracture-set.

3. Calculation of (scale-dependent) equivalent flowproperties, for the overall fracture system including allfracture-sets, everywhere within the reservoir model.

4. Conditioning of the equivalent flow property models(mainly permeability) to dynamic data yet preservingthe consistency of the underlying fracture model.

Fracture densities or spacings are generally computedalong wells using moving-window averaging. The fracturedensity (also denoted FD hereafter) may be expressedindifferently as a number of fractures per unit lengthperpendicular to some averaged fracture plane, a cumulativelength of fractures per unit area, or a cumulative surface offractures per unit volume. The calculation must take intoaccount the dispersion of orientations of the directional

fracture-set, the well-path direction, and possibly the boreholediameter (Narr 1996). A relevant observation scale must alsobe selected, as related to a moving-window size, large andsmall-scale fracture densities having a different meaning anduse (Garcia et al.2005).

Fracture densities being generally known at a few sparsewell locations, various approaches have been proposed in theliterature to supplement well data with better knowninformation that can explain fracturing. Some approaches areempirical and deterministic, as the one by Ericsson et al.(1996) to relate the fracture density to geological andstructural attributes. They lead to qualitative more thanquantitative fracture models. Others try to make use of

geomechanical models, either directly (Mac 2006) or using a

SPE 107525

Fast and Efficient Modeling and Conditioning of Naturally Fractured Reservoir ModelsUsing Static and Dynamic DataM. Garcia, SPE, FSS Intl., and F. Gouth, SPE, and O. Gosselin, SPE, Total


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geostatistical approach (Gauthier et al. 2002a, Colin 2001,Heffer et al. 1999). Though fracturing should be related togeomechanical (strain and stress) conditions, fractureorientations, types and densities generally result fromsuccessive poorly known tectonic episodes, which are difficultto identify hence to model. Relating fracture densities toseismic attributes has also been the purpose of research work

(Zheng 2006, Zeidouni & van Kruijsdijk 2006, Pearce 2003,Gauthier et al. 2002c). If valuable information may beexpected from seismic data, the availability of good qualityseismic data must come along with specific processingalgorithms to extract relevant attributes. In practice, all thoseconditions are seldom met. It follows that a more general andsatisfactory approach should be able to take into account allavailable information that potentially explains fracturing. Thespatial distribution of fracture densities being necessarilyuncertain, a probabilistic approach should also be preferred,that relies on multivariate statistical analysis, for relatingfracture density to explicative variables, and on geostatisticsfor addressing the spatial variability issue. Such an approach

has been proposed by Gauthier et al. (2002a, 2002b and2002c) where it proved efficient on different case-studies. It isthe one presented later in this article.

The simulation of flows requires that equivalent flowproperties be assigned to naturally fractured reservoir models.Required properties include the equivalent permeability of thefracture system (in percolation conditions), matrix to fracturetransfer parameters (e.g. shape factor), and multiphaseproperties (capillary pressures, relative permeabilities). Onlythe equivalent permeability, which primarily controls flowsand can be directly confronted to dynamic data, is addressed inthis paper.

The difficulty with equivalent permeability calculation

resides in the discrete nature of fractures, the multi-dimensionality of often anisotropic equivalent permeabilities(tensor form), and the multivariate relationship between theequivalent permeability and the numerous fracture parametersthat control the connectivity and conductivity of the fracturesystem (see Bruines 2003 for sensitivity analysis results). Theequivalent permeability of fractured porous media has been anintensive research field for a long time. Empirical and physicalapproaches can be distinguished. The former call for laws tocalculate equivalent properties from other supposedly moreeasy to obtain variables. For example, power-laws are used torelate, under percolation conditions, equivalent permeability(value or tensor) to porosity, fracture density or strain (Masihiet al.2005, Suzuki 2005, Hefferet al.1999, Sardaet al.1999,Bernabe 1995). Such laws may be useful in some situationsbut remain very approximate and uneasy to calibrate. Adifferent empirical approach is proposed by Oda (1985), andlater by Brown and Bruhn (1998), to derive directly blockpermeability tensors from known orientations and propertiesof within-block fractures from different directional fracture-sets. This approach assumes fully crossing fractures in alldirections, however, hence small enough blocks compared tofracture dimensions. Questionable corrections are proposed ifthe previous assumption is not met.

Regarding physical approaches, they rely on simulatedflow responses of discrete fracture networks (DFNs) and theirpossible fluid exchanges with the matrix. They are commonly

used to evaluate equivalent block permeabilities fromstochastic simulations of two or three-dimensional DFNs ofvarious complexity in terms of fracture shape, spatialdistribution of fractures, and probability distribution offracture properties. So calculated block permeabilities areclosely related to the geometry, orientation and sizes of blocks.It also depends on the boundary conditions used to simulate

the flow response of the within-block system of fractures(Kfoury 2004, Pouya & Courtois 2002, Bourbiaux et al.1997).The flow simulation may be carried out in the discrete fracturesystem (Bourbiaux et al.1997), or through a fine grid modelof the fractured medium (Araujo et al.2004). Applied to 3Dcases, the approach tends to be CPU-time and memoryconsuming.

Conditioning to dynamic data raises a number of specificissues, owing to the tensor form of equivalent permeabilitiesand the fact that they are not directly modeled but derivedfrom numerous fracture-set parameters through complex non-linear multivariate relationships. Issues include evaluatingflow responses or large-scale average properties (forward

modeling), and calibrating model parameters to matchdynamic data. Without getting into details, spatial and non-spatial model parameters are generally distinguished, as wellas short-scale and large-scale dynamic data, thus leading tomultiple-step and iterative approaches. Calibration of modelparameters is often hand operated (Araujo et al. 2004,Gauthieret al.2002b and 2002c, Heffer et al.1999), seldomautomatic (Suzuki et al.2005).

Whatever the conditioning approach, evaluating the flowresponse of a naturally fractured reservoir model remains anarduous task. Even average well-test permeabilities need flow-based evaluations of models (Sarda 2001), the strong andspatially varying anisotropy of permeability making

inappropriate other elsewhere successful forward modelingmethods as those based on power averaging, multiple-pointproxy or well-test response approximations (Gautier andNoetinger 2004, Srinivasan and Caers 2000, Sagar 1993).

This paper presents an integrated approach that has beendeveloped as a workflow for modeling naturally fracturedreservoirs (Figure 1). The approach relies on an originalconceptual model of fracture systems. Emphasis has beengiven to reach a limited but relevant model complexity, thusallowing easy but consistent flow characterization andconditioning to static and dynamic data. Three key steps of theapproach are detailed and illustrated. They are all based oninnovative and CPU-time efficient methods.

Geostatistical modeling of fracture densities to honor wellfracturing data and observed spatial trends.

Scale-dependent calculation of full permeability tensors,based on spatially periodic discrete fracture networks forhorizontal within-layer permeabilities, and analytical so-lutions for vertical interlayer permeabilities.

Calibration of reservoir models using steady-state flow-based evaluation of equivalent well-test permeabilities.


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Conceptual model of fracture systems and notion ofscale-dependent effective properties

The discrete nature of fracture systems, associated withpercolation and scale effect issues, makes it extremely difficultto evaluate equivalent flow properties without simulating insome aspects their flow behavior. As an alternative to fully

randomized 3D DFN, a conceptual model has been devised tocapture, with a minimum complexity, relevant fracture-systemfeatures consequential to flow. It is expected from theconceptual model a fast evaluation of the following equivalentflow properties everywhere or over any reservoir region.

1. Within-layer connectivity (percolating part of thesystem) and related shape factor (matrix/fracturetransfer).

2. Within-layer horizontal permeability tensor.3. Vertical inter-layer permeability.Instead of block properties, that are associated with and

depend on a grid definition, an original notion of scale-dependent effective propertiesis considered here. This notion

consists in looking at the equivalent property, as being relatedto a statistically homogeneous property field (i.e. stationaryand ergodic), based on local fracture-system characteristicsevaluated at a desired observation scale.

As an example, suppose that we want to estimate theequivalent permeability of a fracture system from a spatiallyvarying fracture density known at a particular observationscale, all other fracture properties being constant. Given acalculation point, the scale-dependent effective permeability atthat point is nothing but the effective permeability that wouldhave a fictitious statistically homogeneous domain, where thefracture density is the one applying to the calculation point atthe observation scale. By construction, this effective

permeability is independent of any block definition and flowcondition. This notion will be further illustrated later, afterhaving introduced spatially periodic fracture systems.

Back to the conceptual model, a naturally fracturedreservoir is modeled as a stack of fractured layers also calledmechanical units. Fractures are supposed perpendicular tounits (i.e. vertical for an horizontal unit), simple verticalconductivity corrections applying to non perpendicularfractures. Each mechanical unit contains its own fracturenetwork, which may have different characteristics than thoseof the fracture network of the unit above or below.

A fracture network is characterized by a number ofdirectional fracture-sets, each fracture-set corresponding to anorientation class, one or more fracture types (e.g. sealed vs.open, joint vs. shear), and specific geometric and flowproperties (e.g. length, conductivity). A directional fracture-setis also assigned to one mechanical unit, or several provided itis identically defined in all of them with similar spatial andnon-spatial properties (i.e. same fracture orientations, types,densities and properties).

The fracture system being so defined, horizontal andvertical permeabilities, as corresponding to flows parallel andperpendicular to units, can be distinguished. Horizontal within-unit permeabilities can be seen as in-

herited from 2D fracture networks.

Vertically, inter-unit, instead of within-unit, permeabil-ities must be evaluated as required for flow simulation

purposes. They depend on the fracture density and thefracture properties of the different fracture-sets present ineach unit, on the unit thicknesses but also on the propor-tions of persistent and bedding-terminated fractures.

Such a decoupling only assumes that each unit is uniformlyfractured vertically (geomechanically homogeneous). At anylocation, fracture spacings and properties do not change with

depth. Figure 2 summarizes this conceptual model and therelevant model parameters for the calculation of within-unitand inter-unit permeabilities.

Spatial modeling of fracture densities

Spatial modeling of fracture densities (FD) requires thatdirectional fracture-sets be first defined from fracturesobserved from cores or interpreted from borehole images. Thiscalls for fracture classification based on orientation and type.Each fracture-set is also attached to geometric and flowproperties, and to one or several equally-fractured mechanicalunits (see previous section). If a fracture-set generally

originates from one particular tectonic episode, its propertiesmay result from subsequent compression or reactivationevents, or depend on present day stress conditions (Guiton2001).

Given a fracture-set, a realistic model of the spatialdistribution of fracture density is expected to honor fracturedensity data, as (suitably) calculated along wells, but also toreproduce areal large-scale spatial trends as obtained fromexplicative (correlated) geomechanical, structural, seismic orother geological attributes. To be valid, explicative variablesmust show monotonic (increasing or decreasing) trends toallow extrapolating them beyond sampled intervals, i.e. rangesof values as seen from well locations. Valid trends must also

have a physical meaning or interpretation (e.g. increased FDwith decreasing distance to faults). For the sake of easymodeling, it is required that all explicative attributes beexhaustively known everywhere over the reservoir.

To address the multivariate data integration problem, andto allow quantifying spatial uncertainty, multivariate analysisand geostatistical conditional simulation techniques are used.The aim of multivariate analysis is to relate the fracturedensity to available explicative attributes so as to derive asingle function of the explicative attributes that capitalizes atbest all relevant spatial trends. Discriminant analysis ispreferred over other techniques, non-linear correlation beingcommonly observed between fracture density and explicativevariables (Figure 3).

Based on a class decomposition of the fracture density (e.g.small, medium and high FD), discriminant analysis isperformed to find, in the space of explicative variables, thedirection that minimizes the intra-class dispersion (Figure 4).This direction is given by the first discriminant analysiscomponent, denoted C1, as a linear function of the explicativevariables. Knowing C1, a bivariate relationship can beestablished between FD and C1. Non-linear S-shapescatterplots are generally obtained with two plateaus for thesmall and highFDvalues (Figure 5). Scatterplot smoothing iscarried out to generate a bivariate histogram model that fullydefines the relationship between FD and C1 (Hrdle andMarron 1995, Xu and Journel 1995). The interpretation of S-


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shape scatterplots is straightforward. By looking at C1 as afracturing index, theFDis zero or almost zero until fracturingconditions are met. ThenFDtends to increase with C1until toreach fracture saturation conditions with a maximumFDvaluethat cannot be exceeded. The vertical dispersion of pointsreflects the prediction power of C1. The smaller thedispersion, the more precise is C1 to predict FD. The

dispersion may vary however with C1 (heteroscedasticresidual variance), hence the prediction power.

To take into account the complex nonlinear relationshipbetween FD and C1, FD is simulated using a sequentialindicator simulation method. The integration ofC1is done bycokriging through a Bayesian formalism in which C1 isconverted into soft (probability-like) data (Gooverts 1997,7.3, Zhu and Journel 1992). Modeling tasks can be reducedby considering a Markov approximation (Schmaryan andJournel 1999), which leads to a collocated cokrigingsimulation approach, with the cross-variogram (between FDindicators and soft probabilities) being written as a function ofone of the two auto-variograms. Figure 6 gives an overview of

the simulation approach. All modeling steps, from FDcalculation along wells to geostatistical simulation, must berepeated for all fracture sets.

An illustration on a case-study involving different seismic,structural and geological attributes can be found in Gauthieretal.(2002c). Other case-studies are also presented in Gauthieret al.(2002a and 2002b).

Calculation of equivalent flow properties

Simulation of multiphase flow in naturally fractured reservoirscalls for specific flow models. Several options of increasingcomplexity can be considered.

1. Single porosity-permeability model: the only requiredmodel parameters are flow properties of the fracturesystem (permeability, porosity, capillary pressures,relative permeabilities).

2. Dual porosity and single permeability model: inaddition to the previous parameters, shape factors,characterizing matrix to fracture flow transfers, andmatrix capillary pressures must be defined.

3. Dual porosity and permeability model: matrixpermeability and relative permeabilities are alsorequired.

We focus here on the way relevant equivalentpermeability fields of fracture systems can rapidly becomputed as required for conditioning naturally fracturedreservoir models to dynamic data. The calculation of capillarypressure and relative permeability functions is beyond thescope of this paper. Nor are discussed shape factor aspects.

Tackling equivalent permeability of fracture systems, thefollowing flow simulation aspects must be kept in mind.

1. Due to preferential fracture orientations, equivalentpermeability tensors are needed to capture localanisotropy that may spatially vary with fracturedensity.

2. Within layers, equivalent permeability componentsshould be best directly computed at block interfaces,thus avoiding uneasy block tensor averaging.

3. Interlayer permeabilities are also required vertically atblock interfaces to take into account bedding-terminated and persistent fractures.

As already mentioned in introduction, the complexity offracture systems makes it risky to evaluate equivalentpermeabilities without somehow appraising flow responses.Conversely, using a 3D DFN for flow simulation purposes

raises the question of how detailed should be the DFN tobecome realistic enough, yet making model calibrationpossible without multiplying the number of poorly knownmodel parameters. Practically, it appears that even the mostcomplex DFNs remain a poor approximation of actual fracturesystems, and so are also flow simulations. More important isthe capability of the model to reproduce essential features ofthe fracture system as related to spatially varying connectivityand permeability anisotropy.

To do so, a scale-dependent effective approach has beendeveloped to evaluate equivalent permeability tensorsindependently of any block definition and flow boundarycondition. In this approach, the calculation of within-layer and

inter-layer equivalent permeabilities is decoupled. Besidescomputation speed, decoupling can be justified by the need ofhaving equivalent permeabilities computed at different blockinterfaces horizontally and vertically, as previously argued.

Based on the conceptual fracture-system model alreadypresented and depicted in Figure 2, within-unit and inter-unitscale-dependent effective permeabilities are calculated asfollows.

Within-unit equivalent permeability. Spatiallyperiodic 2D DFNs are generated from which 2Dsymmetric positive-definite permeability tensors canbe derived.

Inter-unit equivalent permeability. Analytical flowsolutions, taking into account bedding-terminated andpersistent fractures, have been developed from whichinter-unit permeability components can be obtained.

Both calculation methods are innovative, and only requirethat fracture densities and fracture properties be evaluated at auser-defined observation scale. Details about the methods andillustrative results are given hereafter.

Horizontal within- unit permeability tensor

At local scales, fracture systems can be considered as self-repeating. Similar patterns can be identified that suggestspatial periodicity. Assuming a spatially periodic fracturesystem, effective permeability tensors can be derivedindependently of any block definition and flow condition.Only an observation scale must be defined, which determinesthe underlying spatial periodicity. This observation scale maybe related to a block size, not a geometry, or be smaller if lessfractures are enough to represent properly the fracture-systemconnectivity. Conceptually, the aim of spatially periodic DFNis not to cover the whole reservoir domain with the sameperiodic fracture system. Instead, it consists in generating,from local fracture densities and fracture properties, realisticenough fracture systems with good spatial and statisticalproperties to allow deriving scale-dependent effectivepermeabilities.

The so called elementary-patch based (EPB) method hasbeen devised to generate spatially periodic fracture systems.


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Given an elementary-patch (or EP), the method can generatewithin the EP a spatially periodic fracture system thatreproduces input fracture densities, orientations and lengths,and can be seen as an equivalent statistically-homogeneousfracture system. In 2D, an EP is nothing but a polygon with aneven number of parallel edges and good geometrical (paving)properties. Rectangular EPs are generally preferred. The EP

size is controlled by the desire observation scale. Spatiallyperiodic fracture systems generated by the EPB method areperiodic in the sense that any ingoing fracture through an EPedge is associated with an outgoing fracture through theopposite EP edge (Figure 7). The method can generate varioustypes of spatially periodic fracture systems as shown in Figure9. The principle of the EPB method is beyond the scope of thispaper and will be the purpose of a forthcoming publication.

The fracture system being spatially periodic, any steady-state linear flow conditions (i.e. uniform hydraulic potentialgradient) necessarily leads to spatially periodic flow rates andpotential differences (Figure 8). By simulating the flows fortwo different (best perpendicular) gradient directions, a scale-

dependent effective permeability tensor is obtained, which issymmetric positive-definite provided the fracture system is atleast partly percolating. Details about the calculation are givenin Appendix A. Effective permeability tensor results areillustrated in Figure 9. Figure 10 shows the effectivepermeability tensor field of a three-fracture-set system ascomputed using the EPB method from geostatisticalrealizations of FD. It can be noted the spatially varyingdirections of anisotropy associated with contrasted maximumpermeabilities and minimum to maximum permeability ratios.

Vertical inter-unit permeability

According to the conceptual model of fracture system

previously introduced and depicted in Figure 2, a mechanicalunit decomposition of the reservoir domain is considered, eachmechanical unit being supposed uniformly fractured vertically(geomechanically homogeneous). Within each unit, fracturesare also supposed to be fully crossing, or to cover at least theupper or lower half thickness of the unit.

Looking at the inter-unit vertical permeability kv betweentwo units A and B, each fracture-set ican contribute to verticalflows in one of the following ways.

1. The fracture-set is only present in one unit: allfractures from this fracture-set are bedding-terminated.

2. The fracture-set is present in both units: a proportionPiof fractures is persistent, a proportion 1 Pi beingbedding-terminated.

Providing some simplifying assumptions, the vertical inter-unit permeability can be computed analytically based on theprevious decomposition in persistent and bedding-terminatedfractures, and average flow and geometric fracture properties.The simplest and often good enough approach is to supposethat flows through persistent and bedding-terminated fracturesare decoupled (not interacting). The vertical inter-unitpermeability can then be written as:

bedv

pervv kkk += (1)

where pervk andbedvk are the contributions to the vertical

inter-unit permeability of persistent and bedding-terminatedfractures, respectively.

Fracture planes being considered, linear flows apply topersistent fractures (Figure 11), from which the expression of

pervk can be easily derived. We find:

==i

iii

i

periv

perv FDkePkk , (2)

In this equation, the sum is over all fracture-sets ipresentin both units A and B, kei and FDi being the fractureconductivity and the fracture density of fracture-set i,respectively.

The contribution of bedding-terminated fractures is lessstraightforward to obtain. The analytical calculation requiresthat point connections be considered between connectedfractures. For a two-fracture system with a single-pointconnection (see Figure 12), an analytical flow solution can beestablished. The calculation assumes average central locations

for the connection points, at the middle of each fracture edge.Local radial-circular flows are also assumed around theconnection points. This is made possible by consideringequivalent semi-cylindrical borehole-like connections. Foreach fracture, the borehole radius is chosen so as to preservethe area of the connection zone (see Figure 13). Theconnection area is function of the thicknesses and orientationsof fractures. Unless explicitly defined, the fracture thickness(or aperture) can be approximately drawn from the fractureconductivity ke (input model parameter) using the Poiseuilleplane equation for (laminar) flows, C being some roughnessparameter generally set to 0:

( )3 112 keCe += (3)

Within each fracture, the flow problem to solve is the onedepicted in Figure 14. Using the superposition principle,borehole imagesare needed to take into account the specificboundary conditions that apply to each fracture. The analyticalflow solution can be derived from Schneebelis solution for asingle borehole row (see Marsily 1986). By equating the flowrates in the two fractures (conservation equation) and thehydraulic potential at the connection point, the followingexpression is found for the equivalent vertical permeability ofthe so defined two-fracture systems:

( )

i

BAi

j

Bjj

i

Aii

bedvL

HHFD

ke

HLR

ke

HLRk ji +

+= ),,0,(),,0,(

2,

(4)

In this equation, Ri (resp. Rj) is the equivalent boreholeradius in fracture i(resp.j), and the functioncan be seen asa geometric factor depending on the fracture length and height(see Appendix B).

A generalization to any number of fracture-sets withmultiple-point connections can be obtained in different wayswith various degrees of complexity or approximation. Thesimplest one is to reformulate the multiple fracture-set

problem as an equivalent two-fracture-set problem associated


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with appropriate average flow and geometric fractureproperties. Details about this calculation will be presented in asubsequent paper dedicated to the calculation of equivalentfracture-system flow properties.

As an illustration, the vertical inter-unit permeability hasbeen computed for increasing values of the proportion ofpersistent fractures. The calculation has been repeated for a

two and three-fracture-set system. All fracture-sets have samelength (L = 10 m), same conductivity (ke = 100 md.m) andsame fracture density (FD= 1 or 0.25 m-1). The proportion ofpersistent fractures is also the same for all fracture-sets. Thetwo units are 5 m thick. Results are presented in Figure 15. Itcan be seen how the inter-unit vertical permeability (kv)increases with the proportion of persistent fractures to reach amaximum (kvmax) for a proportion of 1. The smaller the fracturedensity, the more contrasted is the kv/kvmaxratio. This is due tothe smaller number of intersection points along the fractures.

Uncertainty issue about equivalent permeability

Two sources of uncertainty are associated with the equivalent

permeability of a fracture system.1. Parameter uncertainty about the naturally fractured

reservoir model.2. Calculation uncertainty due to possible non-

uniqueness of the equivalent permeability solution.Model parameters include spatial and non-spatial

parameters. Are considered as spatial parameters, those that doindeed vary spatially and are sufficiently known to allowspatial modeling. Main spatial parameters are fracturefrequencies, as random functions, and unit thicknesses asobtained from a usually deterministic geomodel. If they can berelated to fracture frequencies, fracture conductivities may alsobe made spatial. All other parameters are generally non-

spatial. They may be associated with a single (average) scalarvalue or with a probability distribution function (pdf).Particularly important is the fracture orientation of adirectional fracture-set, whose pdf can directly be inferredfrom available fracture data. Model calibration againstdynamic data then requires different techniques for spatial andnon spatial parameters. They are further discussed in the nextsection.

Calculation uncertainty is a consequence of the non-uniqueness of the local (horizontal) effective permeability-tensor solution when the number of within-EP fractures issmall (as in Figure 9.b for example). From one DFNrealization to another, significantly different effectivepermeability tensors may be obtained. Figure 16 illustrates thistype of uncertainty, which increases with decreasing fracturedensity. It follows an underdetermined fracture-system model,the available model parameters being not always capable ofcompletely defining local permeability. Conditioning todynamic data then is all but possible without being able tocontrol everywhere the equivalent permeability of the fracturesystem.

To do so, an additional parameter is required that allowsspecifying, where needed, the permeability level that locallyprevails, given local fracture-set characteristics (i.e. fracturedensities and properties). This calls for a measure to rankpossible permeability-tensor candidates according to apermeability level. Such a measure can be the tensor

determinant (kmin kmax = square of average mean),provided the tensor is positive definite (kmin 0), i.e. thefracture system is connected in all directions. Another moregeneral measure is the matrix trace (kmin + kmax). Aconnectivity index can also be used to relate the (effective)permeability-tensor to a scalar degree of connectivity. Anadditional advantage then is the potential for rapidly

identifying fracture density conditions that give rise to non-uniquely defined permeabilities. Then, and only then,uncertainty about the effective permeability tensor should bequantified and a connectivity index should be exploited.

Conditioning to dynamic data

Spatial modeling of the equivalent permeability of a fracturesystem is justified when fracture permeability is much greaterthan matrix permeability (generally 10 times greater or more).It is then essential to make the equivalent permeability fieldconsistent with dynamic data, i.e. well-test interpretedpermeabilities and production data.

Conditioning of naturally fractured reservoir models todynamic data raises, however, a number of issues.

1. Equivalent permeability tensors must be spatiallymodeled, instead of mere scalar permeabilities.

2. Unless the equivalent permeability tensor is directlymade a (multidimensional) random function, which isa contemplated research avenue, conditioning todynamic data must be done through fracture-relatedmodel parameters.

3. The equivalent permeability tensor then appears as acomplex non-linear multivariate function of fracture-related model parameters. The latter must be definedfor different fracture-sets that contribute all together tothe equivalent permeability.

4. Some of the model parameters are spatial (e.g. fracturedensities), whereas others are non-spatial (e.g.geometric or flow properties of fractures).

5. Conditioning to dynamic data must preserve modelconsistency with other fracture-related (static) data.This includes well data and spatial statistics aboutfracture densities.

It follows that equivalent permeability tensor fields are notsimilarly influenced by all fracture-related model parameters.Nor are the model parameters similarly characterized by alldynamic data, certain types of dynamic data providing moreinformation than other types about some model parameters.Model parameters and dynamic data should therefore behierarchized to be better related each other, thus allowing moreefficient conditioning of naturally fracture reservoir models todynamic data.

Different optimization methods are also required to processspatial and non-spatial parameters. The latter can benefit fromgradient-based or other conventional optimization methods.This is not true with spatial parameters, which require specificmethods. Multivariate versions of stochastic geostatisticalinversion methods should be preferred. Potential methodsinclude the probability perturbation method (Hoffman andCaers 2004), the gradual deformation method (Hu et al.2001),

and the sequential self-calibration method (Wen et al.

2000,Gmez-Hernndez et al.1998). In all cases, flow responses of


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naturally fractured reservoir models must be evaluated andconfronted to dynamic data.

Inversion and flow evaluation of reservoir models are twoimportant research fields. We only focus here on two limitedaspects. One is about the hierarchical organization of bothfracture-related model parameters and dynamic data in a multi-step conditioning approach. The other one bears on the

evaluation of well-test interpreted permeabilities on naturallyfractured and more generally heterogeneous reservoir models.A novel approach is presented. It relies on steady-state flowevaluations of reservoir models. Well-test permeabilities arederived that can be seen as effective-gradient based averagingof fine scale permeabilities.

Multi-step conditioning approach

As previously discussed (see Uncertainty issue aboutequivalent permeability), spatial and non-spatial fracture-related model parameters are all uncertain. The uncertainty cantake different forms.

1. Spatial parameters give rise to spatial uncertainty. If

their values are imprecisely known far from wells,their spatial repartition must be consistent with large-scale spatial trends, as observed from correlatedexplicative attributes (see Spatial modeling of fracturedensities), and must reproduce spatial statistics.

2. Whether they are associated with a single (average)scalar value or with a pdf, range constraints apply tonon-spatial parameters. Scalar parameters can varybetween minimum and maximum values, whereasrandomized model-parameters have their pdfparameters so constrained.

Information about model parameters comes from static anddynamic data. Static data are mainly related to fracture

densities. Dynamic data are expected to provide informationabout the equivalent fracture-system permeability. They aretherefore linked to all model parameters. A distinction can bemade, however, between short and large-scale dynamic data.Typically, well-test data represent smaller drainage areas thanproduction data that may reflect large-scale well interferencesor flow events. The former are therefore less sensitive to thespatial continuity of equivalent permeability, or similarly tothe spatial continuity of fracture densities.

In terms of optimization, inner and outer inversion loopscan be considered. In the inner loop, non-spatial parametersare calibrated against short-scale dynamic data using standardoptimization techniques. In the outer loop, spatial parametersmust be optimized to fit large-scale dynamic data. Aspreviously mentioned, the preference should be given tostochastic geostatistical inversion methods, which can takeinto account both static and dynamic data, and can reproducespatial statistics. Figure 17 summarizes such a multi-stepapproach based on a hierarchical organization of modelparameters and conditioning data.

To take into account the spatial uncertainty of spatialmodel parameters in the inner loop, without including them yetin the inversion process, the flow response of the reservoirmodel can be evaluated for several possible realizations of thespatial parameters. Denote spand npthe vectors of spatial and

non-spatial model parameters, respectively, and ( )spnpO

the objective function to minimize, based on thethrealizationofsp. A more general objective function can be defined as:

( ) ( )=

spnpnp OO (5)

where the sum is over a reasonable number ofequiprobable realizations ofsp.

Evaluation of well-test interpreted permeability

By definition, all dynamic data result from reservoir flowproblems. Conditioning to dynamic data then usually requiresthat the reservoir model be evaluated in one of the twofollowing ways.

1. The flow problem is simulated on the reservoir modeland the flow response is confronted to theexperimental one (e.g. curves of pressure or rateversus time).

2. Average flow properties (generally permeabilities) aresomehow calculated by using homogenizationtechniques or by interpreting the simulated transientflow response of the reservoir model. Comparison isthen made between average properties calculated onthe model and from the dynamic data.

As already discussed (see Introduction), homogenizationmethods are inappropriate when they are used to evaluateequivalent permeability tensor fields instead of scalarpermeability fields. Regarding the numerical simulation ofactual transient flow problems, they may be time-consuming,numerically difficult to simulate, and uneasy to exploit foroptimization purposes (e.g. problem of curve comparison).

An alternative consists in relating interpreted dynamic datato simple flow problems that can easily and automatically beinterpreted. As an illustration of this type of approach, theevaluation of well-test interpreted permeabilities from meresteady-state flow simulation is presented here.

Buildup or drawdown well-tests involve transient flowresponses around wells. The analysis is usually carried outusing the Theis method (or Jacob's semi-logarithmicapproximation), which allows deriving from late-time pressuredata an equivalent (average) k.h(horizontal permeability timesreservoir thickness) representative of a drainage area aroundthe well. Knowing the (average) thickness of the tested unit(s),an equivalent horizontal permeability kcan be obtained.

The interpreted part of the pressure curve assumes that thefollowing flow conditions are met.

Flows not influenced by partial penetration of well,non-vertical well-path, nearby boundary conditions, orflow exchanges between tested units.

Homogenization of permeabilities, or more preciselyof transmissivities (k.h product), with a relativelyconstant average permeability (or k.h) within thedrainage area.

Specific to naturally fractured reservoirs, instanta-neous pressure equilibrium in matrix and fractures.

So interpreted well-test permeability is related to a localaverage permeability around the tested well. Consequently,when evaluating a heterogeneous reservoir model, anequivalent well-test permeability must be derived around the

well. This equivalent permeability is necessarily flow


8/20

8 SPE 107525

condition dependent and must reflect the somehow planar andradial (single source) nature of flows. This supposes to knowand ideally to be able to locate the drainage volume or area ofthe test around the well. It also requires a relevant upscalingmethod to evaluate the equivalent permeability of the drainagevolume under radial flow conditions.

An approach has been developed to evaluate well-test

interpreted permeabilities from steady-state around-well flowsimulations. The latter requires a full permeability tensorsimulator to take into account properly the around-wellanisotropy of permeability. The approach relies on the threefollowing aspects (see Figure 18).

1. The shape and orientation of the drainage areadepends on the local anisotropy of permeability. Thisanisotropy can easily be assessed from fracturedensities and properties locally known around thetested well. The so defined drainage area is taken asthe simulation domain for simulating the steady-stateflow problem (Figure 18.a).

2. Appropriate boundary conditions must apply to the

previously defined simulation domain in order toreproduce consistent around-well flow paths similar tothose generated by the well-test (Figure 18.b).

3. Flow-based averaging of permeability is carried outusing the so-called effective-gradient based averagingmethod. The calculation can be done for any closedsurface located within the simulation domain andcontaining the source well. By repeating it fordifferent drainage radii (rd), a curve of well-test k.hversus rdis obtained, which can directly be comparedwith the actual well-test interpreted k.h(Figure 18.c).

The effective-gradient based average permeability iscalculated as follows, wheredenotes a closed surface aroundthe well:

( )

( ) ( ) ( )

=

dq

dq

knK

1 (6)

The terms in this equation have the following meaning. ( ) ( ) ( ) ( ) = Kn .q is the absolute value of flux

throughat location. K() is the local permeability tensor at location. n() is the normal vector to the surface at location.

() is the hydraulic potential.Details about the determination of boundary conditions andthe effective-gradient based averaging method are beyond thescope of this paper. They will be addressed in the forthcomingpaper already mentioned.

Conclusion

Modeling of naturally fractured reservoirs cannot be carriedout without considering a multi-step approach. Fracturing andexplicative (geomechanical, seismic, structural or geological)information must be integrated to evaluate spatial and non-spatial model parameters on a directional fracture-set basis.

These model parameters must in turn be translated into

equivalent flow properties, which have to be consistent withdynamic data representative of the overall fractured medium.

It follows that a multi-step approach means a hierarchicalorganization and modeling of reservoir model parameters,associated with a hierarchical organization of conditioning(static and dynamic) data.

Such an approach has been presented and discussed. It

involves several critical modeling steps that require relevantand fast solutions:

1. Spatial modeling of fracture densities.2. Calculation of equivalent flow properties and

especially of equivalent permeability tensor fields offracture systems.

3. Conditioning to dynamic data, which calls for theevaluation of flow responses of reservoir models.

New notions and methods have been developed asalternatives to existing solutions that appear poorlyoperational, whether they are too data-specific, too engineer orcomputer-time consuming, or simply inefficient.

The new solutions cover the following aspects.

1. A geostatistical approach for modeling the spatialdistribution of fracture densities from variousfracturing and explicative information. The approachhas already been applied to different case-studieswhere it proved efficient and helpful.

2. A conceptual model of fracture systems that allowsdecoupling the calculation of within-layer and inter-layer permeabilities.

3. Notions of scale-dependent effective properties andspatially periodic fracture systems that lead to rigorouscalculations of within-layer effective permeabilitytensors of fracture systems, independently of anygridblock definition and flow boundary conditions.

4. Analytical solutions for the inter-layer verticalpermeability that take into account fracture densitiesand properties, and proportions of persistent andbedding-terminated fractures.

5. Notions of equivalent simple flow problems andeffective-gradient based permeability averaging thatallow fast and appropriate evaluation of flow-responses of reservoir models, thus leading to easycomparison with dynamic data.

Details and illustrations have been given about the newnotions and methods. Forthcoming publications will furtherpresent some of them.

Ongoing work focuses on inversion aspects to develop andapply stochastic geostatistical inversion methods to spatialmodel parameters of naturally fractured reservoirs. It isexpected from the new notions and methods that they facilitatethis challenging goal.

Nomenclature

A, B = mechanical unit indicesC = roughness parameterDx, Dy = EPs dimensions parallel to ix, iy resp. [L]e = fracture thickness or aperture [L]F//, F = fluxes parallel, perpendicular to G [LT

-1]G = uniform unit-potential gradients [ML-2T-2]G1, G2 = unit-potential gradients // to ix, iy resp.


9/20

SPE 107525 9

H = fracture height [L]h = investigated height (well tests) [L]ix, iy = two perpendicular vectors (EP axes)K = permeability tensork = (horizontal) permeability [L2]ke = fracture conductivity [L3]k.h = transmissivity (well test) [L3]

kmin = minimum effective permeability [L2]kmax = maximum effective permeability [L2]kv = vertical permeability [L

2]kvmax = maximum vertical permeability [L

2]L = fracture length [L]n = vector normal to surfacenp = vector of non-spatial model parametersO = objective function (dynamic conditioning)P = proportion of persistent fractureQx,1, Qy,1 = flow-rates generated by G1 through the

boundary edges of EP parallel to ixand iy resp. [L3T-1]Qx,2, Qy,2 = flow-rates generated by G2 through the

boundary edges of EP parallel to ixand iyresp. [L3T-1]

Qx, Qy = flow-rates generated by G through the boundary edges of EP parallel to ixand iy

resp. [L3T-1]q = flux through surfaceR = equivalent borehole radius [L]rd = drainage radius [L1]sp = vector of spatial model parametersx, z = coordinates [L1] = location on surface =closed surface around the well

= geometric factor (function) = hydraulic potential [ML-1T-2] = angle between Gand ix1, 2 = main directions of anisotropySubscripts

i, j = fracture set indices = realization indexSuperscripts

bed =bedding-terminatedper =persistentAcronyms

pdf = probability density functionC1 = first discriminant analysis component

DFN =discrete fracture networkEP = elementary patchEPB =elementary patch basedFD = fracture density [L-1]

AcknowledgementWe thank Total SA for its financial support and for authorizingthe publication of this paper. We are also grateful to BertrandGauthier, Yann Lagalaye, David Foulon, Sylvie Delisle,Grard Massonnat and Laure Moen-Maurel who contributed,in different ways during the last four years, to make this workpossible. Jaime Gmez-Hernndez is also acknowledged for

his valuable contribution to the development of a fullpermeability-tensor flow simulator.

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Appendix A: Calculation principle of the effectivepermeability tensor of spatially periodic fracturesystemsThe calculation method is presented for a spatially periodicfracture system generated within a 2D rectangular EP. Withoutloss of generality, the EP is supposed to be parallel to the xand y-directions ixand iy(see Figure 19).

The effective permeability tensor is obtained byconsidering the (supposedly infinite) spatially periodic fracturesystem as being embedded in a uniform hydraulic potentialgradient field. The fracture system being spatially periodic, soare the flow rates and the hydraulic potential differences insuch steady-state linear flow conditions (see Figure 8).

Looking at the EP, each pair of opposite boundary fracture-points can be seen as associated with a flow-line (Figure 19).The flow-line is conductive if the flow rate through thefracture-points is non-zero for some gradient direction.Conductive flow lines can then be interpreted as associatedwith boundary fracture-points connected to the fracture systemand hence participating to flows.

A two-step approach is needed to calculate the effectivepermeability tensor of a spatially periodic fracture system.

1. The principal directions of anisotropy must be found.They are the ones along which a uniform potentialgradient generates flows in the same direction (nocross-flow).

2. Knowing the principal directions of anisotropy, themaximum and minimum equivalent permeabilitiesalong these two directions are derived.

Both steps require that flow rates be simulated on thespatially periodic fracture system for two different hydraulicpotential gradient directions. Let us consider two constant unitpotential-gradients G1and G2parallel to EPs main directions(see Figure 19). Each gradient potential generates flow-ratesalong the flow-lines. Denote: Qx,1 and Qy,1 the global flow-rates generated by G1

through the boundary edges of EP parallel to ix and iy,

respectively. These global flow-rates are nothing but thesum of flow-line rates (see Figure 19), i.e.,

( )

( )

=

=

b

by

a

ax

QQ

QQ

11,

11,

G

G

Qx,2and Qy,2those generated by G2.Any constant unit potential-gradient G(), that makes an

anglewith the direction ix, can be written as:

( ) ( ) ( )

( ) ( ) 21 sincos

sincos

GG

iiG

+=

+= yx


11/20

SPE 107525 11

G() generates global flow-rates Qx() and Qy() throughthe boundary edges of EP parallel to ixand iy. According to thesuperposition principle, these flow-rates can be written as:

( )

( )

sincos

sincos

2,1,

2,1,

yyy

xxx

QQQ

QQQ

+=

+=

Fluxes parallel and perpendicular to the potential gradientcan be derived from the global flow-rates. These fluxes arerequired to identify the principal directions of anisotropy andto calculate the maximum and minimum permeabilities.Denote F//() and F() these two fluxes, which are alsofunctions of. We find:

( ) ( ) ( )

cossincossin

cossin

2,1,21,22,

//

+++=

+=

y

y

x

x

y

y

x

x

y

y

x

x

D

Q

D

Q

D

Q

D

Q

D

Q

D

QF

( ) ( ) ( )

y

y

x

x

y

y

x

x

y

y

x

x

y

y

x

x

y

y

x

x

y

y

x

x

D

Q

D

Q

D

Q

D

Q

D

Q

D

Q

D

Q

D

Q

D

Q

D

Q

D

Q

D

QF

2,1,

1,2,2,1,

1,2,22,21,

2sin2cos

cossinsincos

sincos

+

+

+=

+=

=

It can be shown that the cross-flow fluxes Qx,1

/Dx and

Qy,2/Dy are equal, G1 and G2 being both unit andperpendicular. This property is directly related to thesymmetry of the equivalent (effective) permeability tensor. Itfollows that:

( ) 2sin2cos 1,2,2,1,

+

+=

y

y

x

x

y

y

x

x

D

Q

D

Q

D

Q

D

QF

The principal directions of anisotropy are the ones thatmake null the perpendicular fluxF(), i.e.,

2/

arctan

2

1

12

2,1,2,1,1

+=

+=

x

x

y

y

y

y

x

x

D

Q

D

Q

D

Q

D

Q

By applying the principal directions of anisotropy1and2to F//(), the minimum and maximum permeability values areobtained.

Appendix B: Geometric factor for the calculation ofequivalent permeability of bedding-terminatedfractures

The functionin (4) can be approximated as follows:

( )

( ) ( ) ( )

( ) ( )

+

+

+

+

=

L

x

L

nHz

L

nHH

L

x

L

nHz

L

nHH

L

x

L

z

L

HHLzx

n

n

n

2cos

22coshln1

22coshln

2cos

22coshln1

22coshln1

2cos

2coshln1

2coshln,,,

0

1

where n0= Max(5L/H, 5z/H),xin [-L/2;L/2],zin [0;H].


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SPE 107525 12

Figure 1: Overview of the workflow for modeling naturally fractured reservoirs.

Calculationpoint

Well direction

direction

khmaxDir. ofkhmax

khmin/ khmax kvtop

FSet 1

FSet 2

FSet 3

6. Calculation of equivalent K and

block size fields (dual model)

7. Conditioning to dynamic data

2. Definition of fracture-sets (FSet)

Stereo-plotTarget units

1. Construction of the

mechanical-unit grid

3. Moving-window calculation of along-

well fracture densities (FD)

8. Output to a flow simulator

Structural

Geomechanical

Seismic

Geological or other

FDdata Vs. Explicative variablesBivariate histogram C1map

FD

C1

One best correlated

component C1

Multivariate analysis

4. Identification and characterization of spatial trends

Flow responses of

fractured model

Vs.

Measured or

interpreted actual

reservoir flow responses

5. Geostatistical conditional

simulation ofF D

TRXTRYTRZ

SIGMAV


13/20

SPE 107525 13

Figure 2: Conceptual model of fracture system and model parameters for the calculation of within-unit horizontal permeability tensor and ver-tical inter-unit permeability.

Figure 3: Typical non-linear monotonic relationships between fracture-density and explicative variables.

Figure 4: Use of discriminant analysis to find, in the space of explicative variables, the direction minimizing the intra-class dispersion (orsimilarly maximizing the inter-class dispersion).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Geomechanical Opening N20

N20

Frac.

Freq.

(Nb/ft)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Seismic Coherence

N20

Frac.

Freq.

(Nb

/ft)

V1

V2

C1

High FD

Medium FD

Small FD

Hk

BHk

UnitB

UnitA

Persistent fracture

Bedding-terminated

fractures in unit B

Bedding-terminated

fractures in unit A

Top of unit A

Bottom of unit B

BAvk

,

HA

HB

Fracture-set related parameters

FD = fracture density (nb / m perpendicularly to some average fracture plane) = fracture orientation

L = fracture length (m) ke = fracture permeability times thickness (md.m) = possible relationship parameter between keandFD PA,B = probability (proportion) of persistent fracturesOthers

HA,HB= unit thicknesses

AHk

BAvk

,


14/20

14 SPE 107525

Figure 5: Non-linear (S-shape) scatterplot ofFDvs. C 1and related smoothed bivariate histogram.

Figure 6: Geostatistical simulation of fracture density (FD) on a fracture-set basis. All simulations honor the fracturing data and tend to repro-duce the input statistical (bivariate histogram, spatial correlation or variograms) and the spatial fracturing trends as provided by the C1 map.

Figure 7: Spatially periodic fracture system based on fracture corre-spondences between opposite EP edges.

Figure 8: The spatially periodic fracture system being embedded in aconstant potential-gradient field G (steady-state linear flow condition),spatially periodic flow rates and potential differences are obtained.

FD

C1 0 30 0 6 0 0 90 0 1 2 00 1 50 0 1 8 00 2 1 00 2 4 00 2 70 0 30 0 00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

|h|

(|h|)

Bivariate histogramC1map

FDdata

Spatial statistics

Conditional simulations of FD

Input data and

statistical models

Elementary-patch(EP)

Two by twofracture

correspondence

FDRel. freq.

C1

A priori probability density

function (histogram) ofF D

corresponding to C1= -1.5

0

0.1

0.2

0 1 2FD

Pdf

C1= -1.5

G

(uniform potential gradient)


Two by twofracture

correspondencePi

Pi

Pj

PjQi

Qi= -Qi

Qj= -Qj

QjPotential difference:

ii= =jj


15/20

SPE 107525 15

a) Numerous short fractures (N60, N20 or N120). b) A few long fractures (N60, N20 or N120).

c) Mixed network of short and long fractures (N60, N20 or N120). d) Numerous long fractures (N20, N30 or N40).

Figure 9: Simple examples of spatially periodic fracture networks generated by the EBP method. All networks involve three fracture-sets, eachwith constant fracture orientation, conductivity and length. Only fracture segments connected to others are displayed. The gray ellipses showthe permeability anisotropy.

Dir of kmax: N30kmax: 1300 mdkmin: 20 md

Dir of kmax: N60kmax: 34 mdkmin: 4.7 md

Dir of kmax: N45kmax: 77 mdkmin: 57 md

Dir of kmax: N40kmax: 16 mdkmin: 8.6 md


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16 SPE 107525

a) Direction of maximum within-unit permeability (azimuth in degree). b) Maximum within-unit permeability (md).

c) Minimum to maximum within-unit permeability ratio (log-scale). d) Vertical inter-unit permeability with the overlying unit (md).

Figure 10: Effective permeability tensor field of a three-fracture-set system as computed by the EPB method from FDrealizations.

Figure 11: Flow through a single persistent fracture. Figure 12: Flow through two bedding-terminated fractures with a cen-tral single-point connection.

Bottom of fracture at potentialB

LayerB

LayerA

Top of fracture at potentialT

Linearflow

HA

HB

Fracture from setjwith bottom at potentialB

LayerB

LayerA

Fracture from set iwith top at potentialT

Single-pointflow connectionHA

HB


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SPE 107525 17

Figure 13: Equivalent semi-cylindrical borehole-like connection between two connected fractures.

Figure 14: Image-borehole layout equivalent to the boundary conditions applying to a single (top) fracture. Layout = infinite series of regularlydistributed infinite-length rows with regularly distributed boreholes. Borehole spacing along a row = L (L = fracture length), row spacing = 2.H(H= fracture height). Boreholes from a same row have same rate. Boreholes from two successive rows have opposite rates. Q= vertical-flowrate crossing the fracture.

Fracture from set i

Fracture from setj

ej

ei

i,j]0;/2]ej/|sini,j|

ei/|sini,j|

a) Intersecting parallel-plate fractures. b) Parallelogram-shape of the connection zone.

c) Equivalent cylindrical borehole preserving the area of the connection (exchange) zone.

Parallel-platefracture from set i

Radius = ej/|sini,j|/

ei

Q

-Q

Q Q

-Q

Q

-Q

Q

-Q

Q

-Q

H

L

-Q

-L/2 L/2

(x,H) =T

borehole row n= 0

borehole row n= -1

borehole row n= 1

borehole row n= 2

x

(x,z)

(x,-H) =T


18/20

18 SPE 107525

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.2 0.4 0.6 0.8 1

Proportion of persistent fractures

Inter-unitkv/kv

max

3 Fsets, FD = 1 m-1

2 FSets, FD = 1 m-1

3 FSets, FD = 0.25 m-1

2 FSets, FD = 0.25 m-1

Figure 15: Effect of the proportion of persistent fractures on inter-unit vertical permeability. Two fracture-sets (striking N20 and N120) andthree fracture-sets (striking N20, N60 and N120) are considered. All fracture-sets have same length (L = 10 m), same conductivity (ke= 100md.m) and same fracture density (FD= 1 or 0.25 m

-1). The proportion of persistent fractures is also taken the same for all fracture-sets. The

two units are 5 m thick. For each calculation case, the inter-unit vertical permeability (kv) is normalized by the maximum inter-unit permeabil-ity (kvmax) corresponding to fully persistent fractures (proportion of persistent fractures = 1).

Figure 16: Illustration of the non-uniqueness of equivalent permeability tensor solutions. Sensitivity analysis on the fracture density (FD) ofone fracture-set as the only non-constant parameter in a two-fracture-set fracture system.

Dir. of kmax() kmax(md) kmin(md)

0.1 0.10.1

66

51

79

54

32

23


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SPE 107525 19

Figure 17: Hierarchical organization of fracture-related model parameters and conditioning data in a multi-step conditioning approach. Frac-ture densities (FD) are supposed to be the only spatial model parameters. If fracture conductivity or any other fracture property is linked toFD, it must be taken into account in the optimization step ofFD.

Static data(fracturing + geomodel)

Spatial and multivariatestatistics

FDfields(on a fracture-set basis)

Fracture properties(on a fracture-set basis)

Equivalent permeabilityfields

Compliance withshort-scale dynamic data

YES

NO

NO

Optimize non-spatialfracture properties

YESStop

Compliance withlarge-scale dynamic data

Optimize spatialfracture frequencies


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20 SPE 107525

Figure 18: Steady-state flow based evaluation of k.h well-tests.

Figure 19: Flow-lines associated with opposite boundary fracture-points for a spatially periodic fracture system embedded in a uniform poten-

tial gradient field. The flow-lines are nothing but a convenient representation of the spatially periodic flow conditions.

a) Calculation of an appropriate simulation domain taking into

account the local around-well anisotropy of permeability.

b) Full permeability tensor simulation of around-well flow using appropri-

ate boundary conditions to avoid adverse boundary condition e ects.

Permeability tensor field(dir. ofkmax, kmax, kminand

kv, vertical inter-unit perme-ability)

Simulation domain

Simulatedheads andflow rates

c) Effective-gradient based calculation of k.h (permeability times thick-

ness) versus rd (equivalent radius of drainage) around the well.

Test rd

Model k.h Stabilizationzone


Flow-line associated withopposite fracture-points

PbandPbPb

Pb

Pa

Pa

Qb(G)

Qa(G)

G1// ix

G(uniform unitpotential gradient)

G2// iy

ix

i

O

Dx

Dy

Documents

3.- Fast and Efficient Modeling and Conditioning of Naturally Fractured Reservoir Models Using Static and Dynamic Data