The Use of the Conventional Well Model toPredict the E¤ect of Discrete FractureNetwork Flow on Reservoir Flow

Performance

Joe VoelkerDepartment of Petroleum Engineering, Stanford University

Spring, 2004

Abstract

This report describes the e¢ cacy, and conditions of applicability,of the well model, within �nite di¤erence reservoir �ow simulation,to simulate the e¤ect of discrete fracture network �ow, on reservoirperformance.The well, or source, model is critical to feasibly history match pro-

duction data within an 11-well study area in a large, Middle Eastern,carbonate reservoir that contains enigmatic, abnormally high �ow ca-pacity, geologic features called "super-k." Characterization of super-kis desired for future placement of water injection and production wells.Premature well abandonment has become a problem for the operator,due to the inability to mitigate the high injection water conductivityof super-k, and associated early water breakthrough.The history matching study attempts to predict primarily the

spatial distribution, orientation, lengths, and transmissibilities, givensparse geologic data, of discrete fracture �ow networks, since theymay comprise a predominant component of super-k. The well modelis used to simulate �ow in the fracture networks.The well model o¤ers several advantages over fracture discretiza-

tion and dual porosity models, particularly for �ow simulations per-formed as part of an optimization algorithm. Without the compu-

1

tational and gridding burdens of fracture discretization, and the poorresolution of a coarse dual grid, the well model provides a means bywhich discrete fracture �ow simulation, given fracture �ow networkmodels of �ne resolution, may be performed with conventional �owsimulators, on coarse grids.The construction of an e¤ective "fracture well" is developed, specif-

ically with regard to well connection density and location. The ef-�cacy and appropriate conditions of usage of the fracture well arediscussed analytically, and demonstrated with 3-D �ow simulation ex-amples. 2-D �ow simulation for a synthetic case futher demonstratesthe capabilities of the fracture well model.

2

Contents

Contents 3

List of Figures 4

1 Introduction 6

2 Brief review of the history matching study 102.1 Study area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Reservoir management of super-k . . . . . . . . . . . . . . . . 122.3 Geology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Construction of a Combined Facies / Discrete Fracture Reser-

voir Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 The use of sources to model discrete fracture �ow 173.1 The conventional �ow simulation well model . . . . . . . . . . 183.2 Fracture connection transmissibility . . . . . . . . . . . . . . . 22

4 Discrete fracture �ow networks and the connections used todescribe them 234.1 Representation of the fracture �ow network . . . . . . . . . . . 244.2 Terminal connections . . . . . . . . . . . . . . . . . . . . . . . 254.3 Intersection connections . . . . . . . . . . . . . . . . . . . . . 284.4 Near-well and near-neighbor connections . . . . . . . . . . . . 294.5 Conditioning fracture �ow networks to super-k �ow intervals . 30

5 Elementary analytical considerations for the application ofthe well model 325.1 Multiphase e¤ects . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Gravity e¤ect . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3 Highly heterogeneous fracture permeability . . . . . . . . . . . 355.4 The simpli�ed well model . . . . . . . . . . . . . . . . . . . . 35

5.4.1 Ignoring non-linear viscous �ow pressure drop . . . . . 365.4.2 Constructing fracture �ow network transmissibilities . . 365.4.3 Ignoring low-�ux connections . . . . . . . . . . . . . . 37

3

6 Sensitivity study of the fracture �ow network model and thesource model 446.1 Facies model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Fracture �ow network realizations . . . . . . . . . . . . . . . . 466.3 Production predictions . . . . . . . . . . . . . . . . . . . . . . 47

7 A model for the time varying behavior of super-k 507.1 2D synthetic example . . . . . . . . . . . . . . . . . . . . . . . 55

7.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . 557.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.1.3 Relative permeability e¤ect . . . . . . . . . . . . . . . 61

7.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

8 Future work 62

Nomenclature 64

APPENDICES 65

A An object-based discrete fracture �ow network model withconnection mapping 65A.1 Model input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67A.2 Model output . . . . . . . . . . . . . . . . . . . . . . . . . . . 70A.3 Intersections of planes . . . . . . . . . . . . . . . . . . . . . . 70A.4 Examples of transmissibility connection placement . . . . . . . 71

References 74

List of Figures

1 Study area location[3] . . . . . . . . . . . . . . . . . . . . . . 102 Study area well locations . . . . . . . . . . . . . . . . . . . . . 113 Flowmeter results compared to porosity and permeability [3] . 144 Elements of super-k [3] . . . . . . . . . . . . . . . . . . . . . 155 Reservoir and �ow simulation grids . . . . . . . . . . . . . . . 166 Comparison of �ow simulation methods for discrete fractures . 187 Comparison of �ow simulation methods for discrete fractures . 22

4

8 Fracture connection transmissibility in terms of matrix trans-missibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9 Hierarchy of fracture �ow network development[6] . . . . . . . 2610 Mapping transmisibility connections from a fracture �ow net-

work model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711 Near-neighber and near-well connections . . . . . . . . . . . . 2912 Placement of fracture connections at well blocks . . . . . . . 3113 Calibration of source transmissibilities . . . . . . . . . . . . . 3814 Fracture pressure compared to reservoir pressure . . . . . . . 4015 The e¤ect of non-equal source transmissibilities . . . . . . . . 4216 The e¤ect of heterogeneity in permeability . . . . . . . . . . . 4317 Fixed facies model . . . . . . . . . . . . . . . . . . . . . . . . 4618 Fracture �ow network regions . . . . . . . . . . . . . . . . . . 4719 Initial fracture �ow network model . . . . . . . . . . . . . . . 4820 A realization from the �rst outer iteration . . . . . . . . . . . 4921 A realization from the second outer iteration . . . . . . . . . . 4922 A realization from the third outer iteration . . . . . . . . . . . 5023 A realization from the fourth outer iteration . . . . . . . . . . 5024 Top view of realization from third outer iteration . . . . . . . 5125 Sensitivity results for four wells . . . . . . . . . . . . . . . . . 5226 Sensitivity results for four wells . . . . . . . . . . . . . . . . . 5327 Super-k �ux decrease with time . . . . . . . . . . . . . . . . . 5428 Synthetic 2D case . . . . . . . . . . . . . . . . . . . . . . . . . 5629 Synthetic 2D case boundary conditions . . . . . . . . . . . . . 5730 Prediction of super-k �ux decrease with time . . . . . . . . . . 5931 Prediction of super-k �ux decrease with time . . . . . . . . . . 6032 Relative permeability for synthetic case . . . . . . . . . . . . . 6133 DISCFRAC realization . . . . . . . . . . . . . . . . . . . . . . 6634 ECLIPSE well output �le . . . . . . . . . . . . . . . . . . . . 6735 DISCFRAC parameter �le . . . . . . . . . . . . . . . . . . . . 6836 Region de�nition �le . . . . . . . . . . . . . . . . . . . . . . . 6837 Super-k data �le . . . . . . . . . . . . . . . . . . . . . . . . . 6938 Production and injection well data �le . . . . . . . . . . . . . 6939 Connections at grid borders . . . . . . . . . . . . . . . . . . . 7140 Near-well and near-neighbor connections . . . . . . . . . . . . 7241 Fracture network generated from an intersection . . . . . . . . 73

5

1 Introduction

This report elaborates on an important component of a previous report[1],which describes a history matching study in a small area within the GhawarField, Saudi Arabia. The subject, use of the well model to simulate discretefracture network �ow, warranted further discussion and analysis, given itscritical role in the study, and its potential for application in other reservoirsin which discrete fractures signi�cantly a¤ect reservoir performance.The use of the fracture well model is apparently not extensive, given

that no references as to its use were found. The reasons for this are notevident, since, under certain conditions, the use of the conventional wellmodel is appropriate, as will be demonstrated in this report. Furthermore,the availability of the well model in all conventional reservoir �ow simula-tors prompts, at least, a cursory trial, given the alternative of discretizingfractures, or adopting a dual grid, which was designed for fracture systemsdescribed more appropriately with a continuum model.The apparent rarity of in-situ (as opposed to hydraulically induced for

well stimulation) discrete fracture problems, as well as the prevailing andoverwhelming lack of data to su¢ ciently constrain discrete fracture reservoirmodels, a severe obstacle hindering this history matching study, may explainthe dearth in the literature. While the latter consideration is probablylargely responsible, the former should not be, as fractured rock most certainlyprevails in many reservoirs, particularly carbonate, and the continuummodel,as seen clearly in surface outcrops of the gamut of sedimentary deposits, isnot always more appropriate than the discrete model. However, the lack ofdata may render the problem as practically unsolvable, and thereby perhapscontribute to a tendency to ignore the treatment of discrete fractures in mosthistory matching problems.Nonetheless, despite the lack of feasibility in the face of sparse data, mod-

eling of �ow in discrete fractures is, in some cases, imposed by necessity, asin the case of performance prediction in reservoirs a¤ected by super-k. Al-though no single geologic model of super-k has emerged as preeminent atGhawar, the role of open mode discrete fracture networks, provided su¢ -cient geologic constraint does not nullify the probability of their existence,certainly must be considered as likely objects which can possess both thenecessary high magnitudes of conductivity which characterize super-k, andsigni�cant lateral extent. The latter quality, required to resolve observed,massive, unmitigated hydraulic conductivity between injection and produc-

6

tion wells under large spacing (1 km spacing in the case of Ghawar), is espe-cially important. It may be di¢ cult to propose an alternative facies super-kmodel, laterally extensive, connected, with uniformly high permeability, in ahighly heterogeneous facies deposition, as in the near shore carbonate depo-sition that is Ghawar.The association of open-mode discrete fracture networks with injection

wells operating at high pressures, is also compelling, in that, fracture porepressures most certainly are over-predicted in fractured rock, and in-situ, dis-connected fracture networks are susceptible to connection under conditions ofabnormally high pore pressure. Although the very high matrix permeabilityat the Ghawar study area might decrease this tendency, very high conduc-tivities in the fractures may prevail instead. Indeed, the fact that mineralprecipitation has not diminished the matrix perm at the Ghawar study areato below darcy levels, may suggest that fracture permeability enjoys the samelack of diminution.Given that it may be necessary to treat discrete fracture �ow networks

as a signi�cant component of a �ow simulation model, whether or not thismay be the case at the Ghawar study area, this report discusses the develop-ment of one option for doing so. The well model is discussed and developedwith a combination of analytical, elementary proposals for a practical im-plementation, used in the history matching study. Otherwise, conditionsunder which they are not valid are presented. It is intended that a workingmodel, applicable not only to reservoirs like Ghawar, but more generally,be proposed.Also included in the report is a brief review of the history matching study

(Sec. 2). Much of this review is taken from earlier reports; it is includedhere for convenience and because it is the overall project of which discretefracture network modeling is only one aspect.This report also presents the discrete fracture �ow network model used in

the history matching study, a stochastic, grid-less, object-based model whichgenerates GOCAD object �les and fracture network transmissibility connec-tions for input as ECLIPSE well speci�cations. This model is discussed inAppendix A. Although it is designed to take advantage of the source modelfor fracture �ow network modeling, it is not directly related to the subjectof this paper. In fact, any fracture �ow network model that can transformfracture coordinates into �ow simulation block coordinates, can utilize thesource model. The model presented in Appendix A has some desirable fea-tures which place transmissibility connections in the most pertinent locations

7

along the network.A �nal introductory remark may be warranted regarding the incorpora-

tion of the well model as a discrete fracture �ow model. It has been ourexperience that those introduced to the idea, initially conclude that a proxyintended for conventional wells is being applied to discrete fractures. How-ever, the well model is not a construct that is peculiar to oil and gas reservoir�ow simulators. It is more fundamental: a source function in the governingmass balance equation. That a discrete fracture network may be treated asa source function, a spatial distribution of mass �ux densities not describedby the �ux term in the governing equation, is believable, if not evident.The source function is not restricted; it may assume any geometry, for

example a surface, as with fractures, or a curve, as with wells. The geometrywill a¤ect the computational performance of the simulator; however, no otherrestriction is imposed by commercial simulators on the spatial geometry ofthe source function.The fact that these sources are hydraulically constrained is the peculiar

aspect to reservoir engineering. It is merely coincidence that the identicalconstraint holds for sources used in modeling production and injection wells.Therefore, although we do so consistently in this report, in the interest ofnot straying too far from familiar nomenclature, referring to the fracture�ow model as a well model, rather than as a source model, detracts from itsfundamental nature.A similarly non-intuitive idea is that of modeling �ow in wells, not with

sources, but with blocks. The lack of computational feasibility does notdetract from the e¢ cacy of modeling wells by discretizing them. The reasonsources are used instead is that the hydraulic constraint in wells just happensto be amenable to an analytical treatment, and the reason for this is that,at its simplest, it is a simple function of one parameter, depth. Extend-ing the constraint to include �ow is not di¢ cult, as the �ow properties ofthe well are typically very homogeneous, and can be known very precisely.The analytical treatment eliminates the computing requirement of discretiza-tion. Fracture �ow networks are de�nitely not as homogeneous as wells, andtheir �ow properties cannot be known precisely. However, it is reasonableto propose that fracture �ow networks that are so conductive as to signi�-cantly a¤ect reservoir �ow performance, may approach a homogeneity in �owproperties which may a¤ord an analytical treatment. Therefore, the use ofsources for fracture �ow modeling, although not as e¢ cacious as for wellmodeling, may be, under some conditions, fundamentally more appropriate

8

than discretization.The developers of ECLIPSE, the �ow simulator used in this study, and

all other successful commercial simulators, have coupled the sources to theirhydraulic constraints very e¤ectively. Given the hydraulic constraint infracture networks is identical to that of production and injection wells, onlythe �ow geometries are di¤erent, the addition of a discrete fracture �owmodule would not represent a coding development requiring new paradigms.As evidenced in this study, if the hydraulic constraint is simpli�ed to thatof static equilibrium only, no further coding development is required at all- the �ow simulator may be used without modi�cation. Furthermore, ifthe hydraulic �ow constraint is also imposed, and may be described using apipe friction pressure gradient model, again the �ow simulator may be usedwithout modi�cation.However, it may be argued that nonetheless a separate, at least in name,

module within a commercial �ow simulator, is desired for fracture �ow mod-eling. The justi�cations may be:

1. The availability of a modeling tool will encourage the incorporation ofdiscrete fracture �ow networks into �ow simulators, when appropriate,

2. A friction pressure gradient model applicable to fracture �ow geome-tries will provide for more straightforward application than a pipe �owmodel,

3. A connection location and transmissibility input paradigm which con-forms to planar geometries would provide a more natural user interface.

It is hoped that this report may promote at least some contemplationof incorporation of the source model to discrete fracture network �ow sim-ulation in commercial reservoir simulators. That currently, no feasible �owsimulation approach exists for discrete fracture �ow networks, and that anyfracture �ow network model, no matter how well-conditioned, is useless to areservoir engineer if it cannot be practically incorporated into a �ow simula-tor, conveys a de�nite need for the source model to be examined as a feasibleapproach.

9

2 Brief review of the history matching study

The Ghawar Field and the study area are shown in Fig. 1 and Fig. 2. Thereader is referred to the previous report[1] for details of the history matchingstudy.As a brief summary, the �eld is a giant anticlinal trap, entirely carbonate,

sealed by an overlying anhydrite. Ghawar oil is moderate to low GOR.The operator of the Ghawar �eld, Saudi Aramco, has been injecting waterat the �eld �anks during much of the producing history. The reservoir isundergoing secondary recovery using a peripheral water injection schemeproviding e¤ective reservoir pressure maintenance and oil sweep.

Fig. 1: Study area location[3]

The study area is located on the western �ankof the �eld, and adjoins the �eld�s aquifer.The objective of the history matching study

is the reservoir characterization of a geo-logic/hydraulic, poorly understood, feature called"super-k." The signature of super-k is abnormallyhigh production rates from thin zones. Currently,no other diagnostic is used to indicate super-k, asit has not been characterized successfully beyonda production signal.Super-k may form high conductivity conduits

from water injection wells to adjacent productionwells, causing early water breakthrough. Produc-ing wells in this class often must be prematurelyabandoned. On the other hand, super-k enhancesproductivity in areas e¤ectively still under primaryproduction; these so-called "dry areas" have notyet experienced e¤ective water�ooding.The principal production data used in the

study is �owmeter data, as it was determined froma previous study[2] that conventional well perfor-mance data (well �owing pressures, GOR, and wa-ter cut), were not su¢ ciently constraining in thereservoir characterization sought. It was con-cluded that a history match of �owmeter data,in addition to conventional well performance data,was essential in the characterization of super-k.

10

2.1 Study area

The study area was selected by the operator, and satis�es two main criteria:tractable size for the study , and prevalence of super-k production. It wasdesirable that the study area be small, with a reasonably small number ofwells, given the complexity of the geology, the enigmatic nature of super-k,and the extensive production history to be matched.

Fig. 2: Study area well locations

The study area, compris-ing the region in white, is5 mi by 1.8 mi by 250 ftthick. There are 11 wells inthe study area, 9 producingwells and 2 water injectionwells. The wells are drilled onapproximately 250 acre spac-ing (1 km spacing, as is thespacing for the remainder ofthe �eld).Four wells in the study

area, 1 injector and 3 produc-ers, have been identi�ed ashaving super-k zones, and areunderlined in Fig. 2.Three of the newest wells

have little or no productionhistory. The remaining8 wells, having a signi�cantamount of production data,are numbered in Fig. 2.The producing region has

been active for 60 years. The Ghawar Field has been producing since 1951.Peripheral water injection began in the mid-1960�s. Study area developmentbegan in the mid-1970�s; over 25 years of well performance history is availableto characterize the study area.The history matching of well performance, that is, transient shut-in pres-

sures, and oil, gas, and water production rates, has been successfully accom-plished on a �eld-wide basis. Therefore, reservoir �ow capacity, pore volume,and �uid saturations are predicted to an acceptable degree of con�dence.

11

Flow simulations, using the history-matched reservoir model provided by theoperator, showed an acceptable match of well performance.[2]The operator has submitted actual �owmeter data, comprised of 18 sur-

veys, taken irregularly over 25 years, from the two water injection wells andsix production wells.Additionally, static data, well logs from all study area wells and core data

from one producing well, were also made available. Conceptual "data," thegeologic facies model of the study area, has been provided by the operator�sgeologists and conveyed in a published paper[3].The �owmeter, log, and core data, as well as the conceptual geology, were

integrated to characterize the facies distribution within the study area.

2.2 Reservoir management of super-k

The operator de�nes super-k as intervals in which the liquid volume �ux,injection or production, exceeds 500 BLPD/ft. The operating de�nition cur-rently incorporates an additional criterion, that the interval account for morethan 50% of the total well �ow. However, for the purposes of this study,only the �ux de�nition will be retained. The de�nition is general becausesuper-k has otherwise escaped characterization. The benchmark �ux maybe predicted by conventional in�ux performance models, but very often, this�ux, and higher, cannot be predicted with such models.Super-k structures are, potentially, secondary recovery management prob-

lems. Some instances of production well abandonment have been attributedto super-k. High water cut at producing wells that are hydraulically con-nected to injection wells through super-k structures, may be mitigated solelyby production well abandonment. Super-k structures thieve so e¤ectively asto sometimes prevent zonal isolation by cement squeezing.Super-k, however, contributes to primary recovery e¢ ciency in dry areas,

providing high conductivity conduits from high storage, high permeabilityfacies units.The objective of the study is the characterization of the joint distribution

of high permeability beds, and high �ow capacity discrete fracture networks,providing a reservoir model that accurately simulates both primary produc-tion behavior and secondary production behavior. Most valuable will be acharacterization that provides the operator with probable locations of super-k, and therefore pathways for early water breakthrough during injection op-erations. These locations may then be avoided when designing secondary

12

recovery well development. A successful characterization may also providethe operator with in�ll locations in dry areas. These areas bene�t from theoccurrence of super-k.

2.3 Geology

The Ghawar Field area under study is a marine shoal deposit consisting of acomplex dominantly of carbonate sands and carbonate muds deposited bothin more-or-less horizontal beds and in channels.The original geological model and interpretation for this study were gen-

erated by Meyer et al[3]. The high conductivity super-k features in theGhawar Field have to date de�ed simple geologic characterization, despiteextensive �eld development and an extensive stratigraphic and petrophysicalstudy of super-k core data[3]. Currently, these features are identi�ed solelyfrom �owmeter data, as zones of anomalously high �ow rates.The di¢ cult nature of these features is manifested in their lack of strict

correlation to stratigraphy. The high rate zones are known to be high perme-ability, thin layers bounded above and below by thinner, impermeable layers.The rates of �ow from these layers are much higher, however, than can bederived from their computed �ow capacities, including consideration of skindamage or stimulation. Also, the rate-time decline of these zones is muchless than can be attributed to their apparent pore volumes, deduced fromwell log porosities.These thin permeable zones are conjectured to be hydraulically connected

to high conductivity beds or channels under conditions of high interface trans-missibility. The high transmissibility could originate from fractures, faults,or eroded surfaces, any or all of which have breached the bounding imperme-able layers. If such a thin zone is not favorably connected to proli�c beds orchannels, production rates are lower and depletion is more signi�cant. In thiscase, the high rate phenomenon associated with super-k is less likely to bemeasured. Therefore, mere presence of high permeability layers in wells doesnot guarantee super-k �ow rates. Super-k is observed only in those layerswhich are supported hydraulically by other proli�c sources, or are hydrauli-cally connected to high rate injection wells.Fig. 3 is taken from a well near the study area and shows the unpre-

dictable nature of super-k within a well; porosity and permeability cannotin themselves de�ne super-k. Note the occurrence of high porosity, highpermeability zones at this well which do not possess super-k �ow rates.

13

Fig. 3: Flowmeter results compared to porosity and permeability [3]

Although connection to high volume sources is required for super-k toappear during primary production, it is not required for the occurrence ofsuper-k �ows under water injection operations. Here, all that is required isa high �ow capacity connection between injector and producer. In fact, thelower storage connections exacerbate early water breakthrough.Fig. 4 displays conceptual models proposed[3] for super-k structure. The

so-called stratiform structure consists of a thin, permeable facies, usuallygrainstone, bounded above and below by a tight facies, as described previ-ously.The high permeability facies is not restricted to grainstone. However,

14

Fig. 4: Elements of super-k [3]

for brevity, we will refer to this element as grainstone in the remainder ofthe report. Various other facies may form this high permeability element.These facies, when they form a high permeability unit, will be referred to asgrainstone.Although super-k may connect to wells via fractures, it is more likely

the connection to wells is achieved through the stratiform structure. Thefractures, on the other hand, most likely provide a high conductivity conduit,

15

connecting other super-k elements, perhaps in remote regions of the studyarea.

2.4 Construction of a Combined Facies / Discrete Frac-ture Reservoir Model

Fig. 5: Reservoir and �ow simulation grids

The reservoir model to be used for �owmeter his-tory matching, incorporates proposed essentialelements of the super-k network:

� A grainstone which serves as the connec-tion of the network to the well. As de-scribed in Sec. 2.3, this is a thin bed ofgenerally uniform lithology.

� A discrete fracture which serves as theprinciple �ow conduit of the network,

� A high permeability bed which serves asa high transmissibility connection fromthe discrete fracture. This element is re-ferred to as a super-k bed. This elementmay comprise various lithologies, includ-ing tempestite beds, grainstones, sucrosicdolomites, and oolitic limestones.

� Background facies, which comprises thecomplement of the facies described in thethree points above.

The model is composed, therefore, of a faciesmodel, and a discrete fracture network model.The facies model, which is �xed over an opti-mization run, is generated using multiple pointstatistics borrowed from training images[4], andis populated with permeability which is constant for each facies. The mag-nitudes of the permeabilities is such that an adequate history match of wellperformance data is maintained. The 3D discrete fracture model, which isobject -based, stochastic, and perturbed in part by gradual deformation of

16

drawing numbers[5], is incorporated subsequently. It is described in Appen-dix A.Reservoir simulations are conducted on the Cartesian grid shown in Fig. 5

. The �ow simulation grid is also shown in Fig. 5. As mentioned above,both grids are discretized with the same dimensions, 32 x 12 x 60. The �owsimulator is a black oil model.

3 The use of sources to model discrete frac-ture �ow

Conventional �ow simulation of fractures lays signi�cant obstacles within ahistory matching algorithm. Two conventional methods are discretization offractures, and the dual porosity model.The advantages and disadvantages of these two models, with respect to

their inclusion in a history matching algorithm, are shown in Fig. 6.Included also in Fig. 6 is a list of pros and cons for the method used in

this study, the conventional well model, or more appropriately, the use ofsources.The well model is implemented in a �dump�ood�manner. That is, the

�well� is not produced at the surface, but instead is open only to back�owbetween the well connections. A mapping algorithm is used to place wellconnections in �ow simulation blocks intersected by the discrete fracturenetwork, provided a model for the networks exists.A discretized fracture model, given that very �ne blocks may be used

to de�ne the fracture networks, may resolve the geometries of the networksvery well, although at a severe cost of simulation convergence performance,even if unstructured grids are used, simply because of the severe contrastin volumes between the coarse and �ne blocks. This cost currently rendersdiscretization as generally not feasible for this use. Additionally, everyrealization of a discrete fracture model, upon which the �ow simulation gridis built, requires a new �ow simulation grid. Thus, any history matchingalgorithm employing optimization methods, which must, by necessity, runnumerous �ow simulations, will require an automatic gridding module aspart of the algorithm.The dual grid model was developed to simulate �ow through fracture

systems more appropriately de�ned by a continuum model, and therefore is

17

Fig. 6: Comparison of �ow simulation methods for discrete fractures

generally not suitable for application to discrete fractures. A dual grid maybe used to model discrete fracture networks, by selectively assigning highpermeabilities, in the fracture grid, to those blocks which contain fractures.However, since the dimensions of the fracture grid is identical to the matrixgrid, resolution of fracture geometries is poor, or at least limited to the coarsegrid resolution.

3.1 The conventional �ow simulation well model

The conventional well model is represented simply as a set, Sw, of connectiontransmissibilities, Twj , in hydrostatic equilibrium,

Sw =�Tw1 ; T

w2 ; :::; T

wj : �fluid

;

18

where w is a fracture name, and j is the block being connected. Thecondition that the connections are in hydrostatic equilibrium is representedby �fluid, the density of the well �uid. This set is shown pictorially in Fig. 6,as a set of small yellow cubes, with the hydrostatic constraint also included.Another, more general, way of describing Sw is as a set of sources, with

each source j having a strength represented by Twj , and having a commonconstraint represented by w.A further constraint of the set may be added, that is, a function which

maps rate and location within the well, to wellbore friction pressure gradi-ent. This constraint comprises advanced well features in most commercialsimulators, including ECLIPSE.The friction gradient function is not considered in this report. However,

it is recognized that fracture friction pressure gradient is not insigni�cantrelative to matrix �ow pressure gradient, in many instances. We assumeit is negligible in our study area, currently, while reserving the prospect ofincorporating the friction gradient function, using ECLIPSE�s pipe frictionmodel, in future work.The simplicity of representing the fracture as the set Sw is symbolic of

the simplicity in which it is implemented in all conventional �ow simulators.Advantages a¤orded the use of the well model are primarily:

� ease of implementation: adding fractures is as straightforward as addingwells,

� invariance of the �ow simulation grid: no updating of the �ow simula-tion coarse block discretization is required with a new fracture networkmodel realization,

� the geometry of sources is not constrained: the sources in Sw may liealong a curve, as in a production or injection well, in a plane, as in afracture plane, in multiply oriented and connected planes, or any othergeometry in 3D. An example of the lack of geometric constraint isseen in Fig. 6. Note that for representation of the the discrete frac-ture network by well transmissibility connections, only two connectionsare used, although the apparent �ow path of the network includes fourblocks. Although connections in the intermediate blocks may be de-sired under some conditions, they are not required in order for thesource model to deliver from one block with a connection to the other.There are conditions in which omission of intermediate connections

19

does not introduce signi�cant error in predicting the e¤ect of �ow inthe fracture network on reservoir performance. These conditions arediscussed in Sec. 5.

� there is no limit on the total number of sets S�, and multiple setsS� may occupy a single grid block: any number of fracture networksmay be introduced, and multiple networks, none of which are directlyconnected hydraulically, may occupy a single grid block, just as thereis no limit on the number of producing wells which may by input intoa �ow simulator, and multiple producing wells may occupy one block.Of course, there are computational limits on the numbers of sets S�.

� The transmissibility of connections are controlled by the user, and maybe di¤erent from the transmissibility as may be computed from �owgeometry and the block transmissibility. This ability is available inall commercial �ow simulators, to incorporate, for instance, skin e¤ectin production wells, and is desirable due to the variability of fractureconnection transmissibility due to complex, small scale fracture geome-tries, as in transmissibility increases resulting from brecciation.

� generality: the well model approach may be used with any discretefracture reservoir model. The only requirement is that the discretefracture model be enabled to be mapped onto the �ow simulation grid

The last point is emphasized: the well model does not require the speci�cfracture modeling approach discussed in Appendix A, or any other speci�cmodel, but instead requires only that an algorithm be incorporated whichmaps the fracture network model onto �ow simulation block locations, astransmissibility connections. One such model is presented in Appendix A.The well model approach, therefore, does not limit the choice among discretefracture reservoir models.Appendix A presents a gridless, 3D, object-based, stochastic 3D model

of discrete fracture networks. This model also performs the mapping of wellconnections to a �ow simulation grid, given a fracture network realization,the �ow simulation grid, and the location of production and injection wells.The model places connections in simulation blocks at key points along thefracture network, including those near production or injection wells, nearother fracture networks, at fracture network intersections, and at fracturenetwork terminals. The model also establishes intersecting fracture networks

20

as hydraulically connected, given an intersection criterion. Currently, themodel provides the mapping of connections in the ECLIPSE format.The advantages remaining for the discretized fracture model, not found

in the well model, are: multiple, varied, transmissibility connections to asingle coarse grid block from multiple fracture grid blocks, and simulationof the viscous �ow process through the fracture. However, if a well frictionmodel is incorporated into the well model, although not done in this study,the latter advantage is eliminated. The principal drawback of the discretizedfracture model, the requisite generation of a new �ow simulation grid forevery reservoir model realization, may render the remaining advantage overthe well model as minor.An important disadvantage, currently, of the well model is that its in-

corporation into a discrete fracture model is new; the �ow simulation solu-tion convergence behavior may be improved, as was discussed in a previousreport[1].Fig. 7 further compares the fracture models, with respect to the steady

state fracture liquid �ow rate, qfrac, and the dimension of the Jacobian ma-trix, dim J , of the solution system.The �exibility of the well model with respect to controlling transmis-

sibilities is shown in the calculation of qfrac. The transmissibility terms,Tm(matrix transmissibility), and Tmf(matrix-fracture transfer function), arecontrolled through the permeabilities and geometries of the blocks de�ningthe discrete fracture, in both the discretization and dual porosity models.The number of these transmissibility terms in the �ow calculation is deter-mined by the number of blocks de�ning the discrete fracture.The transmissibility governing qfrac in the well model, however, are con-

trolled by the terms Tw, the well connection transmissibility, which is con-trolled directly by the user. Furthermore, the number of terms, for any onefracture, can be as few as two. Therefore, control over qfrac is independent ofthe permeabilities and geometries of the blocks; the reservoir fracture modelmay be independent of the facies model, and no change in the facies modelis required to add fractures and simulate fracture �ow. Also, the fracturede�nition may be as simple as manipulating two transmissibility terms.The dimension of the Jacobian matrix, dim J , determines, in large part,

the CPU cost of the �ow simulation. The Jacobian matrix is increased insize directly by the number of additional discrete fracture blocks, Nf , addedto the original coarse grid, having N blocks, for the discretization method.The dual porosity model, by de�nition, requires a 2N dimensional Jacobian

21

Fig. 7: Comparison of �ow simulation methods for discrete fractures

matrix.The dimension of the Jacobian matrix is increased only by the number of

discrete fractures added to the reservoir model, for the well model approach.Thus, if one discrete fracture is added to the reservoir model, as shown in Fig.7, for example, then dim J is increased only by one, no matter how manyfracture connections are included in the single fracture. The typical bandstructure of the matrix will be changed, however, possibly having an adversea¤ect on the linear solution convergence behavior, although this a¤ect ontotal CPU time may be minor compared to that of signi�cantly increasingthe size of the matrix.

3.2 Fracture connection transmissibility

22

The simplest treatment of the well connection transmissibility, Tw, is toderive it relative to the matrix transmissibility, Tm, as shown in Fig. 8.

Fig. 8: Fracture connection transmissibility in terms of matrixtransmissibility.

Here, Ty is the matrix trans-missibility in the y-direction.This calculation of fracture con-nection transmissibility serves asan order-of-magnitude estimate;connection transmissibilities usedin the �ow simulation may begreater or less than this es-timate depending on whether,for instance, the user wishes tosimulate a fracture which brec-ciates the block and generates alarge connection transmissibility,

or perhaps pinches out in the block with a resulting smaller connection trans-missibility.

4 Discrete fracture �ow networks and the con-nections used to describe them

A limitation of the well model is inherited from it�s one dimensional simplic-ity: it cannot simulate the e¤ects of gravity segregation and capillary �ow inthe fracture. Notwithstanding this limitation, the well model captures theessential elements to describe fracture �ow:

� connection transmissibility,

� hydrostatic continuity,

� viscous friction drop in the fracture through the appropriate selectionof a well friction model.

The last aspect of fracture �ow is not utilized in this study. It is assumedhere that viscous pressure drop in the fracture is insigni�cant, relative to thatfrom, and into, the connections.It must be emphasized that the use of the well model to simulate discrete

fracture �ow is more appropriately applied to a fracture �ow network. A

23

fracture �ow network may be comprised of one fracture, but more commonlythe network is comprised of multiple fractures in hydrostatic equilibrium.A fracture �ow network may assume a geometry that a single discrete

fracture may not, as for example that generated by intersecting fractures, orthat developed over large distances, comprised of many fractures arrangeden echelon.The modeling of individual fractures is undesirable for the purposes of

reservoir simulation, given that the �ow characteristics of fractures often aredetermined at scales much larger than that accommodating individual frac-tures. The conductivity of discrete fractures only becomes important, inits e¤ect on reservoir performance, when many individual fractures amalga-mate into a �ow conduit that extends a distance appreciable relative to wellspacing.

4.1 Representation of the fracture �ow network

A fracture �ow network is the apex of a hierarchical system beginning withen echelon individual joint planes, which are in turn associated with otherfracture system components, including oblique joints, dissolution or compres-sion bands, crushed rock shear regions, brecciated zones, and even faultedzones, in turn forming discrete fracture system groups and complexes, someparts of which form an e¤ective �ow conduit, the �ow network. Thus, the�ow network is actually a subset of the intersecting fracture system complex,since not all components may contribute e¤ectively to the conduit. This hi-erarchy is �rmly established by the scale encompassed within a typical coarse�ow simulation block, as in that of this study, measuring 250m x 250m x 4ft.Fig. 9 presents the elements of the �ow network hierarchy, as observed

in �eld data[6]. Individual joints are seen to comprise only a part of acomplex fracture system, which could alone form an isolated �ow network.Intersecting fracture systems, forming groups and more extensive complexes,may form one or more networks. The complex may become very extensive,yet be entirely contained within the scale of a single column of �ow simulationgrid blocks.A practical discrete fracture �ow network model cannot generate com-

ponents of the network, only the culminating network itself. However, inthe absence of observed fracture distribution maps, or perhaps fracture den-sity maps derived from seismic interpretation, this model is necessarily con-strained by those data which inform the formation of the fracture system,

24

that is, bed lithology and geometry, and the history of stress orientation andmagnitude. In other words, although the �ow network model is updatedin a history matching algorithm by production data, its generation is reliantheavily on geologic data. Often, as in the case of this study, that geologicdata is very sparse.The fracture �ow network model described in Appendix A, generates re-

alizations of �ow networks represented as isolated or intersecting planes. Afracture �ow network in this model, despite actually being comprised of acomplex fracture system, or a series of intersecting fracture systems, is as-sumed to have a gross geometry which is planar in the least, as with an iso-lated fracture system, or at most complex, comprised of intersecting planes,as for intersecting fracture systems.

4.2 Terminal connections

Given the emphasis on �ow, it is further desirable to limit the number of wellconnections that are mapped to the simulation grid, as circumstances maydictate that only a few key connections are pertinent. The �ow at the termi-nal regions of the fracture network, for example, are often the most importantto �ow, and therefore most relevant to the �ow simulation. The terminallocations, by de�nition, are regions of maximum transfer �ux to the matrix.Indeed, a proli�c fracture �ow network is developed only where transmissibil-ity connections conduct �uid at high rates to the matrix. Geologic featuressuch as fracture complexes, even those in hydrostatic equilibrium, are e¤ec-tively nulled if the transmissibility between the fracture and the matrix isnot appreciable at two or more connections. The �ow network is de�nedby the connections of highest fracture-matrix transmissibility, separated bythe greatest distances, and yet de�ning the basic skeleton of the network.The fracture �ow network model generates realizations of objects assumedto have such favorable terminal connections.Fig. 10 presents a 2D example, showing the geometry of a realization of a

fracture model, and its corresponding mapping of terminal connections, rep-resented as squares. Fig. 10 also includes fracture intersection connections,to be described in Sec. 4.3. The connections are colored uniquely accordingto the set in which they belong, that is, the �ow network in which hydrostaticequilibrium is established.High fracture-matrix transfer �ux at terminal connections may arise from

a number of conditions, including high matrix or connection transmissibility,

25

Fig. 9: Hierarchy of fracture �ow network development[6]

26

Fig. 10: Mapping transmisibility connections from a fracture �ow network model

low friction pressure drop, and large reservoir pressure drop over the regionscontaining the network. Each of these conditions require speci�c fractureproperties. The existence of these properties, unfortunately, are highly un-certain, in that they are rarely measured directly and extremely di¢ cult toinfer because of a lack of data. The proposition of their existence is highlydependent on a geologic fracture mechanics model based on data informingthe given lithology under the given stress history of the reservoir, and datainforming the diagenetic history of the reservoir . This data is commonlyvery sparse, if not nonexistent, and therefore these critical fracture propertiesvery uncertain.Nevertheless, an examination of these high �ux conditions is useful for an

elementary understanding of high-conductivity fracture networks.High matrix transmissibility at a terminal source will result when the

fracture terminates in a high permeability facies. Also, the imposition of highsource transmissibility may be warranted, due to brecciation, for example,

27

at the terminal source. The source will conduct high �uid �uxes at theseconnections. The probability of existence of fractures in a given lithology,as well as the tendency of a speci�c lithology for brecciation, is, again, adetermination by application of fracture mechanics to the given reservoir.Fracture geometries which enable low friction pressure drop �ow, rela-

tive to the pressure drop associated with viscous �ow through the matrix,enable high pressure drops across terminal connections, and therefore high�uid �uxes. This condition results from fractures in which permeability isvery high relative to permeability outside the fracture, in the non-fracturedrock. Fracturing does not necessarily imply high conductivity, as manymechanisms leading to fracturing also diminish, rather than increase, rockpermeability. Therefore, it is appropriate to discern, as much as is possible,whether conditions leading to fracture in any speci�c case are amenable tohigh-conductivity fractures.Fracture �ow networks which have su¢ cient lateral extent to breach re-

gions of signi�cantly di¤ering pressure regimes, such as, for example, onewhich spans the reservoir between an injection and production well, may ex-perience large fracture-matrix �ux, due to the pressure drop in the matrixat the terminal connections. The existence of this condition, unlike thetwo previously discussed, is determined more readily, as it depends on welllocations and matrix permeability, and therefore is a reservoir engineeringdetermination.

4.3 Intersection connections

When individual fracture systems intersect, they may or may not form aunique hydrostatic �ow unit, due to the complexity of the fracture systemsand that of the intersection region. However, when they form a network inequilibrium, the intersection region may be important in that the connectiontransmissibility may be enhanced or diminished there. Also, the overallgeometry of the �ow network may be better de�ned with a connection at theintersection region, especially for fracture systems with signi�cantly di¤erentazimuths and plunge. Therefore, transmissibility connections are placed atall e¤ective fracture network intersections. An intersection is determined tobe e¤ective through a random drawing, as explained in Appendix A.

28

Fig. 11: Near-neighber and near-well connections

4.4 Near-well and near-neighbor connections

Points in the network that are near well blocks may also be critical, becausethese are regions where �uid �ux in the reservoir is large. Furthermore, thein�uence of fracture networks is highest when they are near wells.Finally, points in the network that are near neighboring, networks are im-

portant for the following reason: geologically, discrete fracture systems arecomplex failure regions, often not con�ned to narrow and restricted geome-tries, as may be individual fractures. Fracture systems in close proximityto each other may possess damage zones which overlap, although withoutthe development of conductivity to allow substantial inter-fracture �ow, butnevertheless possibly inducing higher fracture-matrix transmissibility thanexisted in the undamaged rock. Fig. 11 shows an example of the placementof these type of connections.

29

4.5 Conditioning fracture �ow networks to super-k �owintervals

The previous sections describe the placement of fracture/well connections,including those near production or injection wells. Given that the objectiveof the study is �owmeter data history matching through discrete fracturenetwork characterization, it follows that intervals in which super-k has beenobserved, are intervals which may be in�uenced by fracture �ow networks.Therefore, �ow network location is conditioned to these intervals. The dis-crete fracture �ow network model described in Appendix A, performs thisconditioning.This is demonstrated in Fig. 12. Here, at an injection well for example,

super-k injection �ow is observed at well blocks z = 50, 51, and 52, where zis the vertical dimension of the reservoir modeling grid, and in the particularcase of the Ghawar study area, also the dimension of the �ow simulation grid(total 60 vertical blocks). The yellow cubes represent fracture �ow networkconnections conditioned to those three well blocks.The implication of this placement of fracture �ow network connections

at the well blocks is not well stimulation. That is, the fracture is not beingsimulated as connecting directly with the injection wellbore, as would a wellstimulation hydraulic fracture. The implication is that the fracture is nearthe well, and more precisely, that it is near the Peaceman[7] radius, ro, ofthe well,

ro = 0:208�x

where �x is the lateral length of the well block. The distance ro isapproximately 160 ft in the �ow simulation grid of this study.Note that the conventional well model precludes the intersection of two

wells. Wells may share a common block, but may not share a commonconnection. Therefore, a fracture may not directly intersect an injectionor producing well in this model. Indeed, fractures are di¢ cult to observeat wells, and no substantiation exists for assuming that super-k networksconnect directly to wellbores via fractures, with any signi�cant frequency.

30

Fig. 12: Placement of fracture connections at well blocks

31

5 Elementary analytical considerations for theapplication of the well model

It is readily shown that the source model, if used with currently available fric-tion pressure drop capability, and with transmissibility connections placed atall coarse blocks intersected by the fracture �ow network, has only the follow-ing shortcomings, when compared to discretizing the fracture �ow network:

1. It is not appropriately applied when multiphase e¤ects are importantin the fracture,

2. It cannot appropriately model �ow in which gravity segregation e¤ectsare signi�cant in the fractures,

3. It is di¢ cult to apply in fractures having highly variable permeabil-ity, where very �ne discretization must be constructed to capture theheterogeneity, and the resulting fracture/matrix transmissibility varia-tion cannot be easily predicted and then resolved with a single sourcetransmissibility.

4. It may not be appropriately applied to cases in which accumulationin the fracture is signi�cant, relative to that in the matrix. However,since discrete fracture systems comprise a miniscule volume relativeto the pore volume of the matrix, for any commercial reservoir, thiscondition is irrelevant and will not be discussed further.

The source model may be used, however, in a �ow simulator in whichfracture �ow networks are discretized, to account for the conditions above. Itcan model speci�cally networks in which the above can be ignored. Similarly,the source model may be used within a dual grid model, to simulate �owin abnormally conductive networks, in which the above conditions are notimportant.If the source model is restricted to usage in conditions other than those

enumerated above, it is conceptually equivalent to fracture discretization,assuming the following:

1. The friction pressure drop facility can be calibrated to the pressuredrop due to viscous �ow through the fracture,

32

2. The source transmissibility can be calibrated to the fracture/matrixtransmissibility

The equivalence between the source model and the discretized model is es-tablished because well �ow friction models can indeed be calibrated to Darcy�ow models, given adequate homogeneity and isotropy in �ow properties,and source transmissibility functions can certainly be adequately calibratedto inter-block transmissibility functions, again under adequate homogeneityand isotropy.

5.1 Multiphase e¤ects

Capillary and relative permeability e¤ects in the fracture are important onlyin that their magnitudes are large when the fracture �ow network is notsigni�cantly more conductive than the surrounding matrix, a condition thatis not of great interest.Per se, capillarity is not important because the critical reservoir processes

it controls, displacement and ultimate recovery, is not important in the frac-ture, due to its miniscule reservoir volume.Nevertheless, the implication of large capillarity, to fracture conductivity

can be seen in the capillary number, Ncap,

Ncap =u�

where u is the total volumetric �ux, the Darcy velocity, � is the meanviscosity of the liquids, and is the interfacial tension between the liquids,and more importantly, by examining the relative permeability function inthe fracture.It may be safely assumed that if Ncap is large enough in the matrix that

capillary e¤ects are insigni�cant relative to viscous e¤ects, then capillarye¤ects may be ignored in the fracture as well. However, often this is not thecase, as capillary e¤ects are commonly very important in the matrix. Thedi¤erence between the capillary number in the fracture, and the capillarynumber in the matrix, can be assumed to lie in u: So, when the capillarynumber is small in both the matrix and the fracture, it simply expresses thatu in the fracture is not signi�cantly larger than in the matrix, a conditionwhich may arise if the fractures are not appreciably more permeable than

33

that of the matrix. In this case, the fracture networks may not stronglya¤ect reservoir performance.Similarly, that highly non-linear relative permeability curves persist from

the matrix to the fracture, suggests liquid-solid interaction is important inthe fracture, nullifying signi�cantly high conductivity �ow. These fracturesare not important in this study.

5.2 Gravity e¤ect

The tendency toward gravity segregation may be examined with the gravitynumber, Ng,

Ng =u�

g(��)kv

H

L

where�� is the density di¤erence, kv is the vertical permeability, H is thevertical thickness of the reservoir or fracture, and L is the length scale overwhich �ow occurs, such as fracture �ow network length. Evidently, a di¤er-ence between the gravity number in the matrix and that in the fracture willmost likely be determined by the di¤erence in the ratio uH

kvL. Furthermore, if

isotropy is assumed in the fracture, then the di¤erence in the aspect ratio HL

will determine the di¤erence in gravity number. Here, it is seen that long,thin fracture �ow networks will be more susceptible to gravity segregationthan the reservoir.Gravity segregation in a fracture network is important only when at-

tempting to history match a �ow rate pro�le along the vertical thickness ofthe fracture system, for example when the fracture a¤ects a production wellwhich has been surveyed with a �owmeter. Then, the pro�le over the inter-val a¤ected by the fracture �ow network, assuming its vertical resolution isadequate, cannot be resolved precisely if segregation is ignored.Typically, this precision in �owmeter history matching is not required.

Indeed, it is most important to discern the a¤ected interval from non-a¤ectedintervals, rather than predicting the �ow pro�le over the a¤ected interval.Speci�cally in the case of this study, super-k �ow generally occurs over smallvertical intervals. The �ow usually contrasts sharply with the �ow outsideof the a¤ected interval. Therefore, the variation of �ow within the a¤ectedinterval is not important, and often cannot be resolved by the �owmeter.Finally, gravity segregation is not important with respect to recovery

within the fracture, since, again, the volume within a fracture system is

34

negligible.

5.3 Highly heterogeneous fracture permeability

This last condition under which the source model is not appropriate, is rarelyconsidered, simply because a fracture �ow network cannot be resolved to sucha �ne scale that would justify the application of any heterogeneity in �owparameters within the simulated network. There currently exists no mea-surement paradigm that would enable the acquisition of variation in fracture�ow properties such as permeability and fracture aperture, for fractures thatcannot be directly measured. Indeed, commonly fracture description is veryuncertain at the largest scales, with the objective in characterizing thembeing to capture the large scale �ow e¤ects. If, on the other hand, thefracture �ow networks are very precisely characterized, including the prop-erties within the networks, then perhaps fracture network discretization iswarranted.It is seen therefore, that the source model is appropriate for the modeling

of fracture network �ow, under the conditions where this �ow signi�cantlya¤ects reservoir performance.

5.4 The simpli�ed well model

It is imperative that �ow simulation models be optimized as much as possiblewith regard to speed and accuracy, when the model is used in a historymatching algorithm. This is due to the need for numerous �ow simulationsin the optimization algorithm.The complete well model, in which all possible connections are included,

that is, all blocks which are intersected by the fracture �ow network aregiven transmissibility connections to the network, and in which a frictionmodel is utilized to compute viscous pressure drop within the network, al-though accurate in its description of �ow through the network, may becomecomputationally burdensome.The computational load is especially severe with irregular geometry, and

large numbers, of transmissibility connections. This is due to the diminu-tion of uniformity of the structure of the Jacobian matrix, brought aboutby the connections. This problem will not be elaborated on more in thispaper, except to say that, fortunately, the prominence of horizontal wells hasprompted work to improve the performance of unconventional well models in

35

�ow simulators, and that any computational gains made will greatly bene�tthe use of sources to model fracture network �ow.Therefore, it is advantageous to discern those conditions in which a sim-

pler well model may be used.One such simple well model, currently being used in this study, employs

the following modi�cations to the complete model:

1. Flux in the fracture network is assumed to vary linearly with pressuregradient,

2. Flux through terminal regions, near production and injection wells, andnear other fracture �ow networks, are assumed much greater than atother regions in the network.

The well model resulting from these assumptions, therefore does not em-ploy a friction model to describe viscous �ow pressure drop within the frac-ture, and does not place transmissibility connections anywhere in the net-work, except in those blocks at the terminus, near production and injectionwells, and near other networks.

5.4.1 Ignoring non-linear viscous �ow pressure drop

A non-linear pressure drop function of �ux in the fracture �ow network iscaused by turbulence, and so ignoring the non-linearity implies the �ow in thenetwork is laminar. Expressed as a backpressure relation, the assumption isequivalent to specifying a backpressure exponent, n, close to 1:

q = (�p)n ; n~=1

The pressure drop due to laminar �ow may be modeled with transmis-sibility connections alone, without an additional, friction pressure drop con-straint.

5.4.2 Constructing fracture �ow network transmissibilities

When �ow in the network is laminar, then the viscous pressure drop throughthe network may be modeled with as few as two source transmissibility con-nections, each located at the terminal points of the network. Fig. 7 presentsthe linear �ux relation between two matrix blocks, for the three fracture

36

models of interest. Equating the �ow terms from the discretization modeland the well model, the following results,

1

Tw1+1

Tw2=X 1

Tmi

This amounts to simply calibrating the fracture �ow network source model(well model) transmissibilities, Tw1 and T

w2 , with the discretized model frac-

ture block transmissibilities, Tmi . Here, it is assumed that only two connec-tion transmissibilities, Tw1 and T

w2 , are being used to characterize the fracture

�ow network for the source model.Fig. 13 demonstrates the elements involved in this calibration, for the case

in which a fracture connects a region near an injection well, and a region neara producing well. A fracture �ow pressure curve, assuming high conductivity(relatively �at pressure gradient), and linear �ow, is shown. Also, the well, orsource, model �ow pressure curve, which is �at, as expected for the exclusionof a friction pressure drop model, is presented. A reservoir �ow simulationgrid is shown at the top of the �gure, in which an injection well and producingwell are placed. Also, two source model connections are placed in the wellblocks.The transmissibility of the reservoir simulation blocks are not consid-

ered here; the transmissibilities, Tmi , are those of the �ne scale blocks ofthe discretized fracture model, which is not shown in the �gure. The frac-ture pressure curve is assumed to be that which would be generated by thediscretized fracture model.Therefore, with the assumption of laminar �ow in the fracture network,

a turbulent friction pressure drop model is not required to model �ow inthe network. Most commercial �ow simulators, like ECLIPSE, for example,have turbulent �ow friction models, otherwise known as pipe friction models.The calibration is practically achieved by production history matching,

and therefore becomes a critical history matching parameter. The calibrationgenerally cannot be accomplished directly, since permeability variation datain fracture �ow networks is rarely obtained.

5.4.3 Ignoring low-�ux connections

Convergent �ow around injection and production wells largely determinesthe pressure distribution in reservoirs, which generally is characterized byrelatively small gradients over most of the reservoir volume, and much larger

37

Fig. 13: Calibration of source transmissibilities

gradients in a small volume containing the well. This follows from thelogarithmic pressure distribution generated by radial �ow.It may be assumed that �ow in highly conductive fracture �ow networks

are likely not to be radial, given the linear, planar geometry of the network,but rather, linear. Under this assumption, the linear �ow equations in Fig. 7

38

are appropriate.Given that highly conductive fracture �ow networks will also have rela-

tively �at �ow pressure gradients, then the question of whether or not fractureregions which are not near well blocks may have large �uxes depends on:

1. The coarseness of the �ow simulation grid,

2. The magnitude of the di¤erence between the source transmissibilitiesat the terminal regions of the fracture network.

Fig. 14 shows the signi�cance of coarse �ow simulation grids, with regardto resolving reservoir pressure near wells. The grid shown at the top ofFig. 14 is approximately the grid spacing of the Ghawar study �ow simula-tor, relative to well spacing in the study area. Note that the majority ofpressure change due to the injection well is enclosed in the well block, andthat the pressure gradient in the adjacent block is relatively �at. Indeed,assuming that the well block pressure, p1, may be computed as the Peace-man pressure[7], occurring at a radius from the center of the injection well,ro = 0:208�x, then simple radial �ow in a homogenous reservoir dictatesthat the pressure of the adjacent block, p2, is such that,

p1 � p2p1 � pmid

~= 0:8

for this particular reservoir �ow simulation grid spacing.where pmid is the reservoir pressure at the midpoint between the injection

well and production well, computed from the radial �ow equation. Generallythe following holds, again from the radial �ow equation in a homogeneousreservoir, for a �ow simulation grid which has a grid density of N blocksbetween wells, where N includes the well blocks, (N = 4 in this example):

p1 � p2p1 � pmid

=1:57

ln[(N � 1)=2] + 1:57

Note in Fig. 14 that both the fracture pressure curve and the well pressurecurve intersect the in�ection point of the reservoir pressure curve. This con-dition, the �at fracture pressure gradient, and the coarse grid, all combineto generate small di¤erences between the fracture pressure and the reser-voir block pressures. Thus, under these conditions, liquid �ow between thefracture and these inter-well matrix blocks is limited.

39

Fig. 14: Fracture pressure compared to reservoir pressure

This set of conditions warrants the absence of source transmissibility con-nections in the inter-well reservoir blocks.The symmetry in Fig. 14 is due to the transmissibilities between the

matrix and fracture, or the source transmissibilities, at the terminal points,

40

being equal, in this example.The equations presented in Fig. 13 show how these conditions arise. The

ratio of �p1 and �p2 is,�p1�p2

=Tw2Tw1

Thus if Tw1 and Tw2 di¤er, then the fracture pressure in the source model,

pf , will be displaced from the block pressures of intermediate blocks, anexample of which is shown in Fig. 15.The scenario generating pf in this case is one in which Tw1 is larger than

Tw2 . Thus, liquid in the fracture, when entering blocks 2 and 3, for example,will tend to exit the fracture, into these two blocks, with �ux magnitudesthat are proportional to (pf � p2) and (pf � p3).These conditions may require a source transmissibility connection be

placed in blocks 2 and 3, depending on the magnitude of the source modelpressure, pf .Note, however, that very large di¤erences in source connection transmis-

sibilities must be excluded from consideration if the fracture �ow network isvery conductive. This is seen from the �ow equation for the fracture,

q =�p

1Tw1+ 1

Tw2

If either Tw1 or Tw2 is small, then q may be small enough that the e¤ectof the fracture �ow network on reservoir performance is small.Assuming di¤erences in fracture geometry between the terminal points

of the network may be neglected, then source connection transmissibilitydi¤erences arise from di¤erences in the permeability of the rock at the ter-minal points. A heterogeneous permeability distribution may result in largedi¤erences in the permeabilities of the terminal regions. However, this het-erogeneity also a¤ects the reservoir pressure gradient, and in a fashion whichdiminishes the e¤ect of the heterogeneity on the source connection transmis-sibilities, as shown in Fig. 16.Here, permeability heterogeneity, for example, higher permeability near

the injection well, result in both, source connection transmissibility increaseat the injection well block, and reservoir transmissibility increase near theinjection well. Since permeability regions near the wells signi�cantly a¤ectthe reservoir pressure gradient, the reservoir pressure curve may be shiftedsigni�cantly, as is the fracture pressure curve. Thus, pressure di¤erences

41

Fig. 15: The e¤ect of non-equal source transmissibilities

between the fracture and the reservoir at inter-well block regions tend to beameliorated, as shown in Fig. 16.Similarly, lower permeability near the injection well will cause both frac-

tures and reservoir curves to shift downward, again decreasing the pressuredi¤erence between the two.

42

Fig. 16: The e¤ect of heterogeneity in permeability

Therefore, if signi�cant source connection transmissibility di¤erences ariseat the terminal regions, due to heterogeneity in reservoir permeability, theomission of source transmissibility connections at inter-well blocks may notintroduce signi�cant error.

43

6 Sensitivity study of the fracture �ow net-work model and the source model

A history matching algorithm is currently being implemented to characterizethe super-k distribution of the Ghawar study area. The algorithm utilizesboth the source model, and the fracture �ow network model described inAppendix A.The algorithm is a stochastic optimization program, with gradual defor-

mation of drawing number sequences[5] used to perturb the parameters. Theparameters consist of characteristics of the fracture �ow network. The algo-rithm optimizes the parameters through minimization of prediction error inproduction data. Speci�cally, the production data of interest is �owmeterdata.Using sensitivity studies, critical optimization parameters are currently

being determined, as the history matching work is still in progress. Currentlythe facies model (see Sec. 2.4) is frozen during the optimization, as it hasbeen determined that super-k �ow performance is more readily predicted byfracture models imbedded into facies models, than by facies models alone[2].Parameters which can be perturbed in the algorithm include:

1. fracture �ow network density, by region,

2. individual network location,

3. network orientation,

4. network length.

5. network vertical thickness,

6. network plunge,

7. network connection transmissibility,

This section presents an example sensitivity study, to demonstrate thevariation in the models generated, and the variation in the �owmeter produc-tion predictions which result from �ow simulations on the reservoir models.The example represents a single run of the algorithm, in which multi-

ple changes of the random seed, and multiple iterations for each random

44

seed are performed. The iterations are performed within a single-parameteroptimization, achieved with the Brent algorithm[9].Although optimization is performed for each random seed, the purpose

of the example is merely to show the sensitivity of production performancepredictions to one or more parameters.Perturbations were made only on the locations of fracture �ow networks,

in this example, during iterations (so-called "inner iterations") for any givenseed. However, orientation, length, vertical thickness, and plunge were ran-domly drawn with each new seed (a change of seed constitutes an "outeriteration"). Fracture density, although varied by region, was �xed for theoptimization, as was fracture network connection transmissibilities.This run consists of 4 outer iterations, each with 5 inner iterations. Each

inner iteration requires the construction of a fracture �ow network realiza-tion, the imbedding of that realization into the �xed facies model, and �owsimulation of the combined realization.Flow simulation predicts 25 years of production history data from the

study area, as well as �owmeter survey data, from eight wells. A total of19 �owmeter surveys were conducted at various times during the producinghistory of the study area. Simulated �owmeter predictions, compared tomeasured results, for one survey from each of the 8 surveyed wells will bepresented here.Each �ow simulation duration was 45-75 minutes. The duration of con-

struction of each reservoir realization is a few seconds.

6.1 Facies model

The facies model is shown in Fig. 17. As described in Sec. 2.4, the model isa realization of a training image-based algorithm[4].The super-k bed is so-called because it may be a necessary constituent

of super-k �ow networks. Previous super-k models have directly associatedfracture �ow networks, and high permeability beds at the terminal regionsof the networks[2]; the current model does not necessarily associate the two,however it may be determined that this association is necessary. If so, thenthe fracture �ow network model will be enabled to condition network locationto lithology.Facies permeabilities are constant, with the following values:

1. super-k bed, 5000 md.,

45

Fig. 17: Fixed facies model

2. grainstone, 2000 md.,

3. background, 300 md.

6.2 Fracture �ow network realizations

One realization from each of the four outer iterations is presented here, toshow the variation brought about from changing the seed.The model speci�es, in this example, 3 regions, each with a �xed pro-

portion, or density, of fracture �ow networks, out of a total �xed number,in this case, 20. The regions vary only laterally, with the speci�ed networkproportion applied to the entire thickness of the reservoir. Generally, how-ever, regions may be de�ned as any volume in the reservoir. The regionproportions are 0.7, 0.2, and 0.1, in this example. No data currently is

46

used to constrain regional proportions, the constraints given here are madeonly as a demonstration of the region capability. Fig. 18 shows the regionproportions.

Fig. 18: Fracture �ow network regions

Fig. 19 through Fig. 23 present the ini-tial model, and four models, one each fromthe set of inner iterations associated witheach of the four outer iterations.Shown within the reservoir strati-

graphic grid, which has a vertical exagger-ation of 200, are planes representing frac-ture �ow networks, and cubes representingtransmissibility connections. The loca-tions of these connections are exported toan ECLIPSE data input �le, as well spec-i�cation data, so as enable the well modelto simulate �ow in the networks. Alsoshown are production and injection wells,represented as curves with cubic connec-tions. See Appendix A for a complete

description of the fracture �ow network model.Note in Fig. 19 that conditioning of networks to two wells occurs in the

model. The well on the west side is an injection well, and on the east, aproducing well. The conditioning can be seen in Fig. 20 through Fig. 23.Finally, Fig. 24 is a plan view of the realization from the third outer

iteration, to show that the distribution of networks honors the regional pro-portions. As expected, the networks are concentrated in the southern half,and in the northwest, and are sparse in the north to northeast.

6.3 Production predictions

Flowmeter simulation predictions for the 8 wells in the study area in which�owmeter surveys were conducted, are shown in Fig. 25 and Fig. 26. Sim-ulation results from 20 realizations, and the initial model, are presented foreach well. The results include those from 2 water injection wells, and 6producing wells.The �owmeter data is presented as a vertical pro�le of liquid �ux, in

B/D/FT. The z axis is measured in grid cells.The optimization objective function was computed in this example, for

47

Fig. 19: Initial fracture �ow network model

simplicity, using the �owmeter data from one well, the west injection well,indicated in the previous section, one of two wells which provide fracture �ownetwork conditioning. The history matching algorithm is currently being

48

Fig. 20: A realization from the �rst outer iteration Fig. 21: A realization from the second outer iteration

modi�ed to enable regional optimization, which will improve the performanceof the algorithm; it is anticipated that without local optimization control,history matching success is very unlikely.Note that good history matching results are obtained in the west injection

well, shown in Fig. 26, as characterized by one or more predictions having arelatively small prediction error.

49

Fig. 22: A realization from the third outer iteration Fig. 23: A realization from the fourth outer iteration

7 A model for the time varying behavior ofsuper-k

Observed super-k behavior at Ghawar, and in particular, in the study area,is, in some instances, time dependent. That is, intervals which exhibit super-k �ux magnitudes in a �owmeter survey, may exhibit signi�cantly di¤erent�ux magnitudes in later, or earlier surveys.

50

Fig. 24: Top view of realization from third outer iteration

An increase in �ux magnitude over time may be explained by an increasein the conductivity of the super-k network, perhaps by dilating or extendingthe fracture �ow network. Alternatively, �ux magnitudes may increase dueto a corresponding decrease in �ux in an adjacent interval in the same well.Flux magnitude decreases in time are more di¢ cult to explain, in that,

with the exception of an adjacent interval, in the same well, becoming more

51

Fig. 25: Sensitivity results for four wells

conductive, a mechanism for conductivity decrease of a super-k network mustbe proposed.Fig. 27 shows examples of signi�cant super-k �ux decrease with time, in

two wells. These are the west water injection well, and the east producing

52

Fig. 26: Sensitivity results for four wells

well.Super-k �ux is observed in the earliest �owmeter pro�le of the west in-

jector, in 1977, in vertical �ow simulation cells 50, 51, and 52, as shown.

53

Fig. 27: Super-k �ux decrease with time

This �ux is then absent in the subsequent surveys, in 1978 and 1993. Cor-respondingly, a super-k �ux is measured in the latest survey in cells 56 and57. This �ux may have arisen because of the decrease in �ux at the lowerinterval, however, the mechanism behind the �ux decrease in that intervalhas been elusive.Note that the later surveys were normalized for di¤ering total well injec-

tion rates. The normalization was performed using the earliest well injectionrate as a base. Normalization allows for a valid comparison of �ux betweenthe surveys.

54

Similarly, super-k �ux is seen to decrease signi�cantly in the east pro-ducer, for surveys taken in 1977 and 1990. Flux in the later survey wasnormalized relative to the �rst survey.Note that the intervals open to injection and production, in both of these

wells, did not change during these time intervals. Therefore, barring unmea-sured changes in interval productivity, due to changes in well skin damage,the changes in �ux recorded by these �owmeter surveys are not wellborerelated.

7.1 2D synthetic example

An investigation to ascertain a possible reservoir performance explanation forsuper-k �ux decrease with time, was conducted by analyzing a 2D syntheticcase, in which individual connection rates in injection and producing wells,as well as in fracture �ow networks, was computed and analyzed.This 2D case is extracted directly from the 3D model, with reservoir

properties, pore volumes of production and injection, and reservoir pressures,nearly identical to that of the 3D model and the actual production andpressure history of the study area.Fig. 28 presents the synthetic case model, a 2D cross-sectional slice from

the east-west center line of the 3D model, which was presented in Fig. 5.Both the �ow simulation and stratigraphic grids are shown.One down-dip injection well, and two up-dip producing wells, are placed,

as shown, in the 2D model. These well locations are similar to actual,relative, locations, of the most signi�cant wells in the study area. Thewestern producer is called the "near producing well," and the eastern, the"far producing well."A simple fracture �ow network is installed, a single curve connecting two

simulation grid blocks, one in a super-k bed, and a well block in the nearproducing well. Note that although the well appears to be dipping downfrom left to right, in the stratigraphic model, it is actually inclined upwardfrom west to east in the �ow simulation model, due to the westerly dip ofthe study area.

7.1.1 Boundary conditions

Fig. 29 presents the production / injection constraints on the wells. Theseconstraints mimic the actual depletion, during primary production for the

55

Fig. 28: Synthetic 2D case

study area, as well as the actual pore volumes injected and produced duringwater injection in the area, during the 60 year production history at Ghawar.The production constraint on the fracture network is necessarily a zero

net �ow, although back�ow, that is, �ow between connections, is allowed.

56

Fig. 29: Synthetic 2D case boundary conditions

7.1.2 Results

Fig. 30 and Fig. 31 presents how time dependency, speci�cally a decrease insuper-k �ux, may occur. The four plots in each �gure are well connection�ow rates vs. time, in the injection well, two producing wells, and the simple

57

fracture network. Note there are only two source transmissibility connectionsin the fracture network.Fig. 30 presents total liquid �ow rates, and Fig. 31 presents water �ow

rates.The multiple curves in the plots represent the multiple connections in

each well. The plots show �ow rates from all connections.Note �rst the e¤ect of the fracture network on the productivity of the

near producer. During primary production, the e¤ect is not apparent, bya comparison between the connection rates of the two producers. However,when water injection begins, the e¤ect of the fracture network on the nearproducer is evident. It is almost instantaneous due to the relative incom-pressibility of the system, which is essentially gas free.During water injection, the fracture network�s connection to a well block

of the near producer, and its increase in the productivity of the well block, isvery evident. This solitary, local productivity increase results in an increasein the variance of connection rates in this well, relative to that existing priorto water injection, and as compared to that in the far producer, which, dueto its location, is not a¤ected by the fracture network.Although there is also a variation in connection rates in the far producing

well, note that the rates are grouped into a few, tight groupings. The post-injection connection rates of the near producer, on the other hand, show agreater spread of rates within any grouping. This is expected well responseto a system with highly variable interval productivity. This response mayarise even when only one interval with an extreme productivity, high or low,is present.Most signi�cantly, unlike the rate vs. time behavior of the injector and

the far producer, the connection rates in the near producer become time de-pendent.Of course, the variation of connection rate in time �rst occurs, in all

wells, as expected, with the onset of water injection. The time dependencyof interest, that of connection rates in the near producer, begins not withinjection, but only after a delay following injection. Note in Fig. 31 that, asindicated by the dashed line, the delay coincides precisely with the arrivalof water to the western connection, the upstream connection, of the fracturenetwork. In fact, prior to this event, the connection rates in both the fracturenetwork and near producer, are constant.Fig. 30 shows that, following the arrival of water at the fracture network,

the back�ow rate in the network steadily declines until, eventually, it is zero.

58

Fig. 30: Prediction of super-k �ux decrease with time

Fig. 31 indicates that this period of decline coincides with increasing waterproduction, and therefore water cut, in the fracture network. Coincidentally,this decline in fracture network back�ow rate precisely mimics a decline inrate from the most productive interval in the near producer.

59

Fig. 31: Prediction of super-k �ux decrease with time

It is concluded, that the fracture network loads up with water and eventu-ally dies, simply due to the �uid density increase concurrent with increasingwater cut, and the dip of the reservoir, and that this decline in back�owrate induces a decline in productivity in the interval a¤ected by the fracture

60

Fig. 32: Relative permeability for synthetic case

network, in the near producer.The subsequent variation in near producer connection rates is such that

as intervals which initially produce at high rates, decline with time, thoseinitially at low rates, are induced to increase with time, so as to maintain aconstant total well rate.Finally, when the fracture network back�ow rate is zero, the e¤ect of the

fracture is nulled completely, and the connection rates in the near producertend to converge to a few tight groupings, similar to that of the una¤ected,far producer.Note that the conductivity of the fracture network does not e¤ect the

injection well, due the distance separating the western end of the fracturenetwork and the injection well.

7.1.3 Relative permeability e¤ect

The production wells both produce at nearly 100% oil for the entire life,negating any relative permeability e¤ect on the time dependent rate result.In fact, the relative permeability e¤ect is nonexistent: a linear relative per-meability curve, presented in Fig. 32, such that total mobility, oil and water,is constant , was used in this run.

61

7.2 Summary

The signi�cance of this result is that it was achieved in the �rst simulationrun; no modi�cations were made to the properties of the 2D extraction model,and no modi�cations were made to the production and injection constraints,so derived to honor that which was actually observed in the study area, inorder to induce this result.The conditions which may actually induce this result at Ghawar evidently

include high permeability reservoir matrix, such that water injection pres-sures are not su¢ ciently high, even at proli�c injection rates, to overcomewater loading of fracture networks. The loading of fractures occurs becausethey �ll with water more quickly than does the surrounding matrix, whileinjection pressures are still low, not high enough to prevent the loading. Ofcourse, after the fracture loads up, as water continues to move in the matrixupdip, eventually injection pressure becomes high enough to �ow water in thefracture again, which will again induce a local productivity increase at thenear producer. This productivity increase is then expected to be permanent,given that the �uid in the fracture is at or near 100% water.In general, this phenomenon may occur in any dipping reservoir, given

injection pressures insu¢ cient, early in the water injection operation, to liftwater in the fracture networks.

8 Future work

As working components are in hand to progress in history matching pro-duction data in the study area, this will be the focus of future work. Thispoint has come about only after proper history matching data was recog-nized, that of the �owmeter pro�les, and only after feasible methods weredeveloped to construct fracture models, and more importantly, to implement�ow simulation for the fracture network components.Further veri�cation of the well model is currently under way, speci�cally

in comparing the results of synthetic cases, such as in Sec. 7, using a discretemodel in which the fracture �ow network is discretized in 1D. This �owsimulation model is being developed in the Supri-B research group.Finally, recent interest has arisen in further geological paradigms for the

formation of super-k. Speci�cally, the formation of horizontal fracture bed-ding, due to the sequential formation of compression solution surfaces, by na-

62

ture horizontal, followed by shearing, which then forms the fractured zones.It is reported that these horizontal fracture zones, formed in a compres-sive environment, are often very conductive to �uids, given the corrugatedmorphology of the solution surfaces, which are then broken upon shearing,leaving conductive shear zones[11]. These structures are abundantly appar-ent in outcrops, and are prevalent in carbonates, as solution surfaces favor�ne grained lithologies.The combination of the horizontal orientation and the high �uid con-

ductivity of these structures, build favorably toward the necessary elementsof super-k: extensive lateral continuity, and high conductivity. Therefore,these structures warrant further investigation, and if appropriate, inclusionin the fracture network model.

63

Nomenclature

g = gravity acceleration

H = reservoir vertical thickness

J = Jacobian matrix

k = permeability

kv = vertical permeability

L = reservoir length

N = number of �ow simulation blocks

Nf = number of discrete fracture �ow simulation blocks

p = reservoir pressure

pf = fracture network pressure

q = liquid �ow rate

qfrac = liquid �ow rate in fracture �ow network

Tm = block transmissibility

Tmy = block transmissibility in Y direction

Tmf = matrix-fracture transfer function

Tw = source connection transmissibility

u = total volumetric �ux

= liquid interfacial tension

� = liquid viscosity, cp

�fluid = well �uid density

�x = lateral block size

64

APPENDICES

A An object-based discrete fracture �ow net-work model with connection mapping

This appendix describes a fracture �ow network model, used in the Ghawarproduction history matching study.Fig. 33 presents aGOCAD rendering of a model realization. The model is

constructed without a grid; the 3D grid shown is a �ow simulation grid, uponwhich the fracture �ow network model is superimposed. The grid correspondsto that used in the study: 32 x 12 x 60. The vertical exaggeration isapproximately 200.The planar objects represent fracture �ow networks, with varying az-

imuths, plunge, lengths, and vertical thicknesses. There are 28 such net-works in this example. One of the networks, colored white, is represented asan intersection of two planes. This network will be examined in more detail.Also shown are production and injection well traces, with well connectionsmarked with small spheres. The well paths and connection locations areplotted as they exist in the �ow simulation gridThe cube markers in Fig. 33 indicate the upper and lower extents of the

vertical intervals over which �ow simulation transmissibility connections areplaced. For example, a pair of markers on the end of a plane indicate thattransmissibility connections are placed at those locations, as well as all gridblocks in a vertical line between the markers. These connection markers areplaced at various points:

1. terminal points of planes,

2. intersection points of planes,

3. points of intersection of planes with the grid boundary,

4. points on the planes near to production or injection wells,

5. points on planes near other fracture �ow network planes.

These connections are shown as they are mapped onto the simulationgrid, and therefore coincide with simulation blocks.

65

Fig. 33: DISCFRAC realization

66

Fig. 34: ECLIPSE well output �le

The planes, in some instances, extend outside the boundary of the �owsimulation grid. Transmissibility connections are, however, placed at pointswhere the planes pierce the boundary, and therefore the e¤ective terminalpoints of the network are at the boundary, and not beyond.An ECLIPSE well �le, shown in Fig. 34 is generated by the model; this

well �le is included into the ECLIPSE general input �le.This feature, generation of a �ow simulation well �le, is a necessary mod-

ule for any fracture network model, if the well model is chosen to simulate�ow through discrete fracture networks. Otherwise, the fracture networkmodel has no other restrictions.

A.1 Model input

The parameter �le, Fig. 35, shows the input for the program DISCFRAC,the discrete fracture �ow network model. The format of the �le follows theGSLIB [8] convention, as does all of the input �les to be described here. Themodel computations will be explained in the context of this parameter �le.DISCFRAC stochastically distributes planes within the volume of the

3D �ow simulation grid, according to a total number of planes speci�ed bythe user, and a regional proportion, also speci�ed by the user. Therefore,within a region de�ned by the user, the number of fractures will equal theregion proportion of the total. The region description, as well as the regionproportion, is speci�ed in the region �le, here named "regions," and shownin Fig. 36. Regions are speci�ed using the �ow simulaion grid, with a region

67

Fig. 35: DISCFRAC parameter �le

Fig. 36: Region de�nition �le

number and proportion given to each grid block.As described in Sec. 4.5, DISCFRAC conditions fracture networks to

production or injection well intervals in which super-k �ow was observed.The super-k data �le, here named �data," and shown in Fig. 37, speci�esthe locations of observed super-k, again using the �ow simulation grid as areference.Production and injection wells are input in the well �le, named "wells,"

in this example, Fig. 38. Note that the well coordinates are input us-ing ECLIPSE format, which is di¤erent from the GEO-EAS format usedin GSLIB. The ECLIPSE format follows the standard �ow simulation for-mat, with X increasing East, Y increasing South, and Z increasing down.The input of the well data using this format allows the user to input wellcoordinates directly from the ECLIPSE input data �le.The �ow simulation grid dimensions are input next. This input de�nes

the volume within which fracture networks will be placed, and allows the

68

Fig. 37: Super-k data �le

Fig. 38: Production and injection well data �le

mapping of the fracture networks connections into the �ow simulator.Flow simulation grid block dimensions are then input, to provide for

proper scale for the later speci�cation of fracture network dimensions. Theunits selected for this input must be the same as that for the fracture dimen-sion inputs.The total number of fractures is input. This number, along with the

regional proportion input, determines the number of fractures placed in eachregion.The next four sets of input specify the ranges of distributions for the

random drawing of length, vertical thickness, azimuth, and plunge of fracture�ow network planes. The drawing currently occurs on uniform distributionswithin these ranges. The units of these ranges must be the same as thatspeci�ed for the �ow simulation grid block dimensions.The tolerances for which connections are placed in proximity to produc-

tion and injection wells, and other fracture network planes, is then input.Fracture network planes which approach wells or other planes within thesetolerances, input in �ow simulation block units, will have fracture connectionsgenerated in the plane, at points nearest to the object of interest.

69

A.2 Model output

Finally, GOCAD output �les are given the root name as input on the lastline of the parameter �le. There are four GOCAD output �les generated foreach realization:

1. A GOCAD object �le containing the plane surfaces of isolated fracture�ow networks, that is, those which did not arise from intersections.This �le is named simply, root.

2. An object �le containing the transmissibility connection markers (thecubes in Fig.33). This �le is named root.wells.

3. An object �le containing the plane surfaces fracture �ow networksformed from intersections. This �le is named root.int

4. An object �le containing the transmissibility connection markers for theintersection fracture �ow networks formed from intersections. This �leis named root.intwells.

A.3 Intersections of planes

Mere intersection of planes does not generate fracture �ow network intersec-tions resulting in a single network in hydrostatic equilibrium. Geologically,two discrete fractures may intersect without forming a single �ow conduit,due to the fact that the complexity of fracture systems persist to extremely�ne scales, including the scale over which the intersection zone occurs. Pre-dicting a probability 1.0 occurrence of hydrostatic equilibrium with everyintersection, does not recognize this persistence of complexity. Therefore,when two planes intersect in the model, the occurrence of hydrostatic equi-librium between the two are randomly drawn from a distribution determinedfrom the geometry of the intersection itself. The probability P that equi-librium occurs is assumed to be proportional to the fraction of intersectionlength to combined planar thickness length,

P =hint

h1 + h2 � hintwhere hint is the length of intersection, and h1 and h2 are the thicknesses

of the two intersecting planes.

70

A.4 Examples of transmissibility connection placement

Fig. 39: Connections at grid bor-ders

Fig. 39 shows a view facing west, of the realization previouslypresented. It is seen that connections are placed where theplanes pierce he �ow simulation grid boundaries, even thoughthe planes may extend outside the grid.Fig. 40 presents examples of connections arising from

planes near wells and planes near each other. Note for ex-ample, the green plane and the blue well, and the two orangeplanes and the two orange wells. An example of a connec-tion arising from nearness to another fracture network is thegreen plane near the purple plane.Fig. 41 presents the fracture �ow network formed by an

intersection. Here it is seen that the intersection thickness islarge relative to the total thickness, so the probability of hy-drostatic equilibrium, described in Sec. A.3, is large, resultingin a single fracture network. Note the connection markers atthe terminal points of the planes. The pair of markers whichdo not coincide with the edge of the plane extending fromleft to right, correspond to piercings of the plane with thegrid boundary. Note �nally the markers coinciding with thegreen well, and the absence of markers at the intersection.This results from the fact that these two sets of markers oc-cupy the same simulation blocks. The absence of markerson the red well is due to the large distance of the well fromthe network.

71

Fig. 40: Near-well and near-neighbor connections

72

Fig. 41: Fracture network generated from an intersection

73

References

[1] Voelker, J.: "Geostatistical Characterization of Superpermeability FromFlow-Meter Data: Application to Ghawar Field," SCRF Report No. 16,May, 2003.

[2] Voelker, J.: "A Characterization of Ghawar Super-k DistributionThrough Flowmeter History Updating of Training Image Based Maps,"SCRF Report No. 15, May, 2002.

[3] Meyer, Franz O., Price, Rex C., and Al-Raimi, Saleh M.: �Stratigraphicand Petrophysical Characteristics of Cored Arab-D Super-k Intervals,Hawiyah Area, Ghawar Field, Saudi Arabia,� GeoArabia, v.5, No.3,355-384.

[4] Strebelle, Sebastian, Journel, Andre G: �Reservoir Modeling UsingMultiple-Point Statistics,�paper SPE 71324 presented at the 2001 SPEAnnual Technical Conference and Exhibition, New Orleans, Louisiana,30 September-3 October.

[5] Hu Ly, Blanc G., and Noetinger B.: "Gradual deformation and iterativecalibration of sequential stochastic simulation," Mathematical Geology,vol. 33, no. 6, 2001

[6] Davatzes, N. and Aydin, A.: "The formation of conjugate normal faultsystems in folded sandstone by sequential jointing and shearing, Water-pocket monocline, Utah," Journal of Geophysical Research, vol 108, no.B10, 2003.

[7] Peaceman, Donald W.: �Interpretation of Well-Block Pressures in Nu-merical Reservoir Simulation,�SPEJ (June,1978) 183-94; Trans., AIME,265.

[8] Deutsch, Clayton V., and Journel, Andre G.: "GSLIB GeostatisticalSoftware Library and User�s Guide," Oxford University Press, 1998, p.21

[9] Brent, Richard P. Algorithms for Minimization Without Derivatives.Prentice Hall, Englewood Cli¤s, NJ, 1973, Chap. 5.

[10] Karimi-Fard, M: " An e¢ cient discrete fracture model applicable forgeneral purpose reservoir simulators," SPE Journel, 2004.

74

[11] Graham, B., Antonellini M., and Aydin A.: "Formation and growth ofnormal faults in carbonates within a compressive environment," GEOL-OGY, January, 2003.

75