TPG4150 Reservoir Recovery Techniques 2003

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Dual Porosity, Dual Permeability Formulation for Fractured Reser-voir Simulation

K. Uleberg and J. Kleppe, Norwegian University of Science and Technology (NTNU), TrondheimRUTH Seminar, Stavanger 1996

Abstract

This study reviews key physical mechanisms andcalculation methods for modelling of fluid flow inNorth Sea fractured reservoirs. The main matrix-fracture fluid exchange mechanisms described aregravity drainage, capillary imbibition and mo-lecular diffusion. Important issues such as capil-lary continuity between matrix blocks, reinfiltra-tion of fluids from higher to lower blocks and ef-fect of block shape on flow processes are also ad-dressed.

Simulation studies of water-flooding in fracturedreservoirs are reported for the purpose of identi-fying the effects of gravity and capillary forces onoil recovery. Included are studies of effects ofcapillary continuity and degree of wetting. Theresults show that for intermediately wetted sys-tems, such as the Ekofisk reservoir, capillary con-tinuity has a major effect on oil recovery.

Laboratory processes involving high pressure gasinjection in fractured systems have been studiedby compositional simulation. The results showthat changes in interfacial tension caused by diffu-sion, may have dramatic effects on oil recovery.

Computational aspects of fluid exchange proc-esses are discussed, including conventional dualporosity formulation, use of matrix-fracture trans-fer functions, and detailed numerical calculation.The only solution to more representative model-ling of flow in fractured reservoirs is more de-tailed calculations. A multiple grid concept is pro-posed which may drastically increase the detail ofthe simulation.

Introduction

Based on the theory of fluid flow in fractured po-rous media developed in the 1960's by Barrenblattet al.1, Warren and Root2 introduced the concept

of dual-porosity models into petroleum reservoirengineering. Their idealized model of a highlyinterconnected set of fractures which is suppliedby fluids from numerous small matrix blocks, isshown in Fig. 1. Kazemi et al.3 were the first toincorporate the dual-porosity concept into a nu-merical model, with application to fluid flow on alarge scale.

Fig. 1 Idealization of a fractured system (fromWarren and Root2)

Since that time, numerical modelling of naturallyfractured reservoirs using dual-porosity modelshas been the subject of numerous investigations.In the dual-porosity and dual-porosity/dual-permeability formulations most commonly used tomodel fractured reservoirs, proper representationof imbibition and gravity drainage is difficult. Insome formulations, attempts have been made torepresent correct behavior by employing a gravityterm, and assuming a simplified fluid distributionin the matrix.4-6 Several authors7-10 have made useof capillary pressure pseudofunctions for the ma-trix and/or the fracture that employ matrix fluiddistributions obtained through some type of his-tory matching with a fine-grid model of a singlematrix block. Others6,11-12 have refined the matrixblocks into multiple blocks. The more recent ofthese publications on dual-porosity modelling dealwith enhancements of the representation of grav-ity effects in the matrix-fracture transfer calcula-tion.

Combined Gas/Water Injection Subprogram

In the North Sea, the Ekofisk Field has been sub-ject to several simulation studies, such as Phillips'study of water imbibition13 and Petrofina's study,14

which includes an evaluation of the effect of cap-illary continuity on recovery.

Current state-of-art in the area of fractured reser-voir simulation, with emphasis on gas/oil gravitydrainage in terms of block-to-block processes, isreviewed by Fung.15 He classifies the currentmethods into four groups: 1) gravity-segregated,2) subdomain, 3) pseudofunction, and 4) dual-permeability models. Fung proposes a new ap-proach for calculation of pseudo capillary pressurepressures, either a priori if vertical equilibriumcan be assumed, or by fine grid simulation if not.

Although several publications have discussed re-imbibition of oil from higher matrix blocks intolower matrix blocks, all models reviewed byFung15 neglect capillary continuity (Fig. 2), exceptthe dual-permeability model, where the matrix isassumed to be completely continuous.

Fig. 2 Effect of vertical capillary continuity onsaturation distribution (from Fung15)

None of the models treats the reinfiltration phe-nomenon properly (Fig. 3). Finally, the impor-tance of the effect of gas-gas and gas-liquid diffu-sion on recovery from fractured reservoirs duringa gas/oil drainage process has not been exten-sively treated in the literature.

It is obvious that the current computational proce-dures are insufficient to represent the physicalflow phenomena taking place between fractureand matrix in highly fractured reservoirs such asin the North Sea. Clearly, additional details in

Fig. 3 Reinfiltration of fluids from higher to lowermatrix blocks (from Fung15)

the description of the flow processes in the mod-els are required.

Simulation of fractured reservoirs using the dual-porosity (and dual-permeability) approach in-volves discretization of the solution domain intotwo continua called the matrix and the fracture.The original idealized model of Warren and Root2

assumes that the matrix acts essentially as asource or sink to the fracture, which is the primaryconduit for fluid flow. In multiphase flow situa-tions, however, complex gravity and capillarypressure driven fluid exchange between fracturesand matrix is occurring, and these processes arenot well understood.

Especially the gas-oil gravity-drainage process infractured reservoirs will require improved model-ling in order to obtain efficient field-scale simula-tion. The gravity segregation in a dual-porositymedium is highly affected by capillary continuitybetween matrix blocks across fractures, and by theprocess of oil reinfiltration from fractures to ma-trix blocks. These mechanisms are not adequatelyrepresented in today's simulation models and sev-eral research groups are working on the prob-lem.15-19 Luan20 presented a systematic study and acomprehensive discussion of the fundamentalphysical involved in gravity drainage of fracturedreservoirs.

The computational side of fractured reservoirmodelling is particularly difficult due to the largescale differences. Important flow processes aretaking place at a scale much smaller than the grid

Dual Porosity, Dual Permeability Formulation for Fractured Reservoir Simulation

blocks normally employed in dual porosity mod-els. Uleberg21 has made a study of application oflocal grid refining techniques in dual-porosityformulations for detailed computation of matrix-fracture fluid exchange.

Physical Characteristics andFluid Flow Mechanisms

Capillary ContinuityThe concept of capillary continuity between ma-trix blocks in fractured reservoirs is now widelyaccepted. A schematic comparison of capillary-gravity dominated saturation distributions for adiscontinuous system and a system with capillarycontact between matrix blocks is shown in Fig. 2.However, the discontinuous concept is still beingused in most commercial dual-porosity models tohandle block to block interactions.

Festøy and van Golf-Racht22 presented simulationresults where the fracture system allowed forvarious degrees of matrix to matrix contact. Theirresults show dramatically higher recoveries forcapillary continuous systems compared to discon-tinuous ones. They suggested that the matrix isbetter described as tortuously continuous.

Luan´s20 discussion on this subject concludes thatthe end effects may be important in a drainageprocess in fractured reservoirs. The saturationdistribution at the endface of the blocks is de-pendent on the wetting conditions and the proper-ties of the fractured medium. Experimental stud-ies23,24 show that the end effects (caused by satu-ration discontinuity) can be reduced by increasedcontact areas between blocks (applying overbur-den pressure in the laboratory).

ReinfiltrationAn important aspect in gas-oil gravity drainage offractured reservoirs is the process of reinfiltration.When drained oil from an upper matrix block en-ters into a matrix block underneath, the process iscalled reinfiltration. Several publications17,18 haveshown that the reinfiltration is a function of bothcapillary forces and gravity, therefore the termreinfiltration should be used instead of the muchused word reimbibition, since the latter could im-ply the capillary effect only.

The flow from one block to another (reinfiltration)is either achieved by 1) film flow across contactpoints or 2) by liquid bridges. This liquid trans-missibility across the fracture is therefore an im-portant parameter for calculating the rate of drain-age of a stack of matrix blocks. Fig. 3 illustratesthe contact points and the liquid bridges.

Experimental results18 have shown that the trans-missibility across a fracture is very sensitive to thefracture aperture, but not so sensitive to the num-ber of contact points or contact area. The reinfil-tration mechanism is also time dependent, sinceliquid bridging provides the main transmissibilityin the initial stage of the gravity drainage process.Later the oil saturation in the fractures will bevery low and the main liquid transmissibility fromblock to block is due to film flow. This final pe-riod is of long duration and is very important forthe overall recovery. The reinfiltration process isnot adequately modelled in the reservoir simula-tors used to predict gravity-drainage oil recoveryin fractured reservoirs.

Da Silva and Meyer16 conducted a simulationstudy of reinfiltration, and concluded that thismay be an important mechanism for systems ofcapillary continuity and large fracture angles.

DiffusionOil may be recovered by diffusion during gravitydrainage in fractured reservoirs. Methods for es-timating the amount and rate of this recovery insuch reservoir processes are in early stages of de-velopment and poorly tested. It is very limitedpublished data against which theories and predic-tion methods can be tested adequately. However,according to Orr,25 the scaling efforts based onfundamental physics must be continued, speciallyin fractured reservoir problems, including work ondiffusion/dispersion as well as gravity, viscousand capillary forces. An interesting effect may bedue to interfacial tension gradients caused by dif-fusion of gas into the oil. The interfacial tensioninduced capillary pressure gradients may result inunexpected saturation profiles.26-27 Hua et al.27

conducted a series of simulations, and were ableto reproduce the experimental results. However,the effects of diffusion on overall recovery isprobably very small and can for most systems beneglected for practical purposes.

Matrix Block Shape and SizeThe shape and size of matrix blocks will stronglyaffect the matrix-fracture fluid exchange process.Torsæter and Silseth28 conducted water imbibitionexperiments on chalk and sandstone cores of dif-ferent shapes and sizes, and typical results arepresented in Fig. 4. Obviously, within a gridblockof a dual-porosity simulation model, matrixblocks of varying shape and size will exist.29 Oneimprovement that could be incorporated in suchmodels, would be the inclusion of some form ofdistribution of size and shape, as indicated in Fig.5. The computation of fluid exchange betweenmatrix and fracture in a matrix block would thenbe the sum of computed exchange from a numberof different geometries.

Combined Gas/Water Injection Subprogram

Fig. 4 Effect of size and shape on imbibition oilrecovery (from Torsæter and Silseth28)

Fig. 5 Frequency distribution of representativematrix blocks (from Torsæter et al.29)

Co-Current and Counter-Current FlowSeveral authors have reported simulation studiesfor the purpose of matching results of laboratoryexperiments.30-35 Such matching of experimentalresults is not trivial, as reported by Kvalheim23

and by Beckner et al.24 The outcome of the simu-lations is very sensitive to the shape and magni-tude of the capillary pressure curve.

A major cause of the difficulties reproducing ex-perimental results is probably that the capillarypressure and relative permeability curves used inthe simulations were measured at flow conditionsdifferent from those of the experiments. An im-portant issue to be addressed in fractured reservoirsimulations is the one of co-current vs. counter-current flow. As reported by Bourbiaux andKalaydjian,36 the experimental imbibition resultsare very much affected by the boundary condi-tions imposed.

Water Flooding SimulationStudies

Matrix-fracture fluid exchange in a fractured res-ervoir is controlled mainly by a combination ofcapillary and gravity forces. The shape and size of

matrix blocks, and the inclination of fractures willaffect recovery of oil by water flooding. Severalsimulation studies37-42 investigating the effects ofthese factors have been conducted. In the follow-ing, the most important results will be presented.

Most of the studies in the past dealing with oildisplacement by water in a fractured reservoirpertain to strong water-wet matrix where capil-lary imbibition dominates the process of fluidtransfer between the matrix block and the fracture.Few studies have been reported dealing with wa-ter flooding in intermediate wet systems whereboth capillary and gravity forces play key roles.In certain situations, the role played by gravityforces may be so dominant that the limited re-covery due to spontaneous imbibition in case ofan intermediate wet rock is significantly im-proved.43

Block to block processes for a gas-oil system hasbeen intensively studied by many authors.15,22,44

Festøy and van Golf-Racht22 showed that for agas-oil system, matrix to matrix contact area in avertical stack of blocks significantly improves therecovery. No such study for oil-water system hasbeen reported in the literature.

Pratap40 conducted fine grid simulations using asimilar geometry in order to understand in detailthe block to block processes involved in oil dis-placement by water in a fractured reservoir. Geo-Quest´s Eclipse 100 model was used for thesimulations. The study makes use of oil-waterrelative permeability and capillary pressures forthe Ekofisk Field, as reported by Thomas et al.45

Results of the study show that a significant frac-tion of oil expelled from the down-dip matrixblock into the separating fracture reinfiltratesinto the block above. This reentering of oil intothe matrix block above is a result of the change inoil pressure in the separating fracture. In the earlyphase of production, the oil pressure in the sepa-rating fracture is less than that of the matrix gridcells above as well as below. Thus, oil will flowto the separating fracture from both the matrixblocks. The oil discharged to the separating frac-ture flows laterally to the vertical fracture. Thereit joins the mainstream of oil produced by sponta-neous imbibition of water from the lateral sides ofthe blocks as they are surrounded by water in thevertical fracture. After some time, the oil pressureof the separating fracture exceeds the oil pressureof the matrix grid cells of the up-dip block incontact with the fracture. This results in signifi-cant fraction of oil in the separating fracture (pro-duced from lower block) to flow into the upperblock. This block to block process which starts atthe edge blocks marks the beginning of the latephase. As time progresses, more and more inner

Dual Porosity, Dual Permeability Formulation for Fractured Reservoir Simulation

matrix grid blocks undergo the process of oil re-drainage. This late phase production continues fora long time.

The study also examines the effect of matrix tomatrix contact on oil recovery from a verticalstack of blocks. The result shows that in absenceof capillary continuity between the blocks, the oilrecovery is low. The presence of matrix to matrixcontact increases the ultimate recovery from thestack dramatically. However, the oil recoveryrate is higher at early times if the contact areabetween the matrix blocks exceeds 5 %. With amatrix to matrix contact of only 1 %, a long timeis required to obtain the ultimate recovery. Someof these results are shown in Figs. 6-7.

The position of the matrix to matrix contact withrespect to the vertical axis of the cylindricalblocks also affects the recovery time. For exam-ple, a 10 % contact at the edge of the matrix blockgives faster ultimate recovery as compared to thesame contact located at the center. The study ona gas-oil system by Festøy and van Golf-Racht22

Fig. 6 Effect of capillary continuity on oil recov-ery for water wet (I) and mixed wet (II) systems(from Pratap30)

Fig 7 Effect of degree of capillary contact on oilrecovery (from Pratap30)showed that with a contact of 25 % between thematrix blocks, the time to obtain the ultimate re-covery is close to that with 100 % contact, andthe present study for an oil-water system confirmstheir findings. It also shows that 30% contact be-tween matrix blocks is sufficiently effective todrain the oil in a reasonable time.

A comparison has been made with a case ofstrong water wet matrix block (Fig. 7). The resultsshow that oil redrainage does not take place if therock is strongly water wet, as the oil phase pres-sure of separating fracture never exceeds that ofthe upper matrix block.

The process of oil redrainage from the lower ma-trix block to the upper matrix block in case of oil-water system has not previously been reported inthe literature. Neither is the finding that the posi-tion of the contact influences the time to reachultimate recovery. The effect of vertical capillarycontinuity and its effect on oil recovery for in-termediate wet rock has been extensively studiedfor the first time.

High Pressure Gas Injection

Despite the efficiency of waterflooding in frac-tured reservoirs, considerable oil will be left be-hind due to relatively high residual saturations.This residual oil may be a target for high pressuregas injection.

The recovery mechanisms involved in high pres-sure gas injections in fractured reservoirs arecomplex and not fully understood. They includeviscous displacement, gas gravity drainage, diffu-sion, swelling and vaporization/stripping of theoil. Interfacial tension gradients caused by diffu-sion may also play an important role on the over-all recovery. Viscous displacement normally playsa minor role, except perhaps in the near vicinity ofthe wells where the pressure gradients are large.

Combined Gas/Water Injection Subprogram

Contrary to conventional reservoirs, diffusionmay play an important role in fractured reservoirs.The injection gas has a tendency to flow in thefractured system, resulting in relatively largecomposition gradients between fracture gas andmatrix hydrocarbon fluids. Thus, there is a poten-tial for transport by molecular diffusion. This isespecially the case in reservoirs with a high de-gree of fracturing (small matrix block sizes). Dif-fusion is difficult to model correctly in a simula-tor. The diffusion flux for a given phase is oftenmodelled as

Jip = Td (fScp)Dip DCip,

where Td is the diffusion transmissibility, Dip isthe effective diffusion coefficients, and the prod-uct fScp represents the fraction of the gridblockinterface where diffusion takes place. The satura-tion term, Scp, is normally chosen as the minimumsaturation of the adjacent grid blocks. Using thisformalism, a problem arises when gas is injectedin an undersaturated reservoir. The minimumcontact saturation, Scp, between the fracture andmatrix grid block will be zero, and no diffusionbetween the two media will be calculated. Withmatrix block heights lower than the capillary entryheight, no mass transfer between fracture and ma-trix systems will occur. In some simulators thisproblem is circumvented by choosing Scp as themaximum saturation between neighbouring gridblocks or by introducing gas-liquid diffusion. Thisdoes not give a physically correct description of adiffusion process in general, and is therefore notrecommended.

Physically, the mechanism for gas entering fromthe fracture to an undersaturated oil would requirean ultra-thin contact zone at the fracture/matrixinterface. In this zone a small amount of equilib-rium gas will exist, and diffusion transfer betweenfracture and matrix can occur via this zone, asgas-gas diffusion between fracture gas and equi-librium gas in the two-phase zone, and as liquid-liquid diffusion between undersaturated matrix oiland saturated oil in contact zone.

Fig. 8 shows the system layout of an experimentperformed at Reslab by Øyno and Whitson,46

where methane was injected around cores filledwith a highly undersaturated oil. The cores wereinitially filled with Ekofisk oil, and methane gaswas then injected into the annulus system.

Fig. 8 Laboratory setup (from Øyno and Whit-son46)

The experiment was simulated using a fully im-plicit compositional simulator (SSI's COMP4model) that accounts for both molecular diffusionand dynamic IFT-scaled capillary pressure. Theeffective matrix block height of the laboratorysystem was lower than the capillary entry pres-sure, so without any modifications to the systemset-up, no oil production could be observed in thesimulations. When a thin (1 mm) two phase zonewas introduced (Fig. 9), mass transfer by diffusioncould take place. Fig. 10 shows the experimentalresults and the simulated best-fit using a thin two-phase zone.

Dual Porosity, Dual Permeability Formulation for Fractured Reservoir Simulation

Fig. 9 Grid system used for modelling of labora-tory experiment (from Øyno et al.49)

Fig. 10 Experimental and modelled oil recoveryvs. time (from Øyno et al.49)

The simulated results indicate that the recoverycan roughly be divided into three productionstages

1. primary swelling of the oil2. secondary swelling and vaporization3. final vaporization of the oil.

The initial stage is dominated by swelling of theoil inside the core, due to liquid-liquid diffusionbetween the undersaturated oil inside the core andthe saturated oil at the outer surface of the core.The light components of the oil (mostly methane),diffuse into the core, while the intermediate oilcomponents from the core diffuse to the outercontact zone. During this stage there is some vis-

cous flow from the center of the core to the frac-ture, due to swelling of the oil and interfacial ten-sion gradients. The oil produced to the fractures isvaporized by the injection gas, and no free oil isobserved.

The first stage ended when some of the oil withinthe core first became saturated. This marks thebeginning of the second stage, where a free gassaturation advances toward the center of the core.As the gas front advances, the gas-gas diffusionwill play a more dominant role on the recoveryprocess. During this stage, mainly light-intermediate, and the intermediate components ofthe oil are vaporised.

When the light-intermediate and most of the in-termediate components have been recovered, thethird stage begins. During the last stage, mostlyheavy-intermediate and heavy components arevaporized. This stage is slow compared to the firsttwo stages, but a large additional recovery may beachieved.

It is worth noticing that there was no significantIFT reduction during production and therefore a

Fig. 11 IFT profile at core center as function oftime - with diffusion (from Øyno et al.49)negligible production by gravity drainage. TheIFT changes due to diffusion may in other caseshave a drastic effect on the recovery.47-48 Fig. 11shows the interfacial tension profile in a core afterhigh pressure gas injection around a core filledwith recombined reservoir oil.49

In this experiment the frontal IFT is reduced dras-tically as gas advances down the core. Composi-tional changes due to gas diffusion cause the IFTbehind the gas front to increase again. The result-ing IFT gradient is so strong that oil is suckedupward against gravity, resulting in the somewhatstrange saturation profile given in Fig. 12.

Combined Gas/Water Injection Subprogram

If diffusion and IFT scaling of capillary pressureare not included in the simulator, these IFT effectswill not be accounted for. Fig. 13 shows the satu-ration profile for the same experiment when diffu-

Fig. 12 Saturation profile at core center as func-tion of time - with diffusion (from Øyno et al.49)

Fig. 13 Saturation profile at core center as func-tion of time - no diffusion (from Øyno et al.49)sion is not included. The IFT gradients are not sopronounced as in the diffusion case, and thereforenot strong enough for the oil to imbibe upwards.This results in a smaller oil saturation behind theadvancing gas-oil front, see Fig. 14.

It was observed that the IFT at the gas front forsome cases (mainly depending upon initial gas-oilratio), were almost vanishing. This resulted in al-most 100 % recovery by gravity drainage. We feelthat the dynamic composition variations, and theinfluence on IFT's, are important issues that needto be addressed in more detail.

Fig. 14 IFT profile at core center as function oftime - no diffusion (from Øyno et al.49)

Mathematical ModelDevelopment

Current state of the art has been reviewed byFung.15 Current models are insufficient for propermodelling of most fluid exchange processes be-tween matrix blocks and fractures. In North Seafractured reservoirs, matrix blocks sizes are typi-cally much less than 1 m3. Thus, a normal gridblock employed in reservoir simulations wouldcontain several tens of thousands of matrixblocks. Obviously, some form of detailed de-scription of the flow processes must be includedin the models.

The dual-porosity and the dual-porosity/dual-permeability formulations are the most commonapproaches used to represent a large number ofindividual matrix blocks in larger computationalblocks. In these models, the processes of water/oilimbibition and gas/oil drainage have caused par-ticular difficulties. Attempts to represent correctbehavior by modifications of the gravity term orby use of capillary pressure pseudofunctions havegenerally not been successful. Others have refinedthe matrix blocks into multiple blocks.

Fig. 15 presents the multiple grid concept for im-proving the fractured reservoir simulation. Theupper grid system represents the coarse grid nor-mally used in dual-porosity simulators. Insideeach grid block, a large number of individual ma-trix blocks exist. It is, of course, not possible to doindividual computation on each matrix block.However, assuming that the large scale gridblocks are chosen so that all the matrix blocksinside exhibit similar behavior (for practical pur-poses), we may do individual computation on one

Dual Porosity, Dual Permeability Formulation for Fractured Reservoir Simulation

representative matrix block inside each large gridblock, and multiply the results with the number ofgrid blocks present. In principle, as previouslysuggested, a distribution of different matrix blockgeometries could be included in the matrix blockdescriptions, so that individual computations areconducted on a number of block groups. The ma-trix block would be locally gridded in 1, 2 or 3dimensions, depending on the situation, and thebehavior simulated using local time steps for fixedboundary conditions during each large time stepof the coarse model. This system would be wellsuited for parallel computing using massive par-allel computers.

Fig. 16 presents another improvement suggestedfor improvement of the fluid flow representationin dual-porosity models. It is proposed that theconventional Warren and Root model with dis-continuous matrix blocks is replaced by a modelwhere all matrix blocks have capillary contact

Fig. 15 Multiple grid concept for fractured reser-voir simulation

Fig. 16 Improved Warren and Root model

with adjacent blocks. This is of particular impor-tance in the vertical direction.

The method has so far been developed and testedfor single and two phase flow50-52 and is currentlybeing extended to a three phase model. The finegrid calculations together with the iterations be-tween the two grid systems require substantiallymore computing time. However, the method iswell suited for parallelization, since each innerblock solutions can be computed separately oncethe boundary conditions from the coarse grid aregiven. So far, the computational speed has beenimproved by a factor of 10 by vectorization andparallelization (Fig. 17).

Fig. 17 Computational speed improvements byvectorization and parallelization.

Combined Gas/Water Injection Subprogram

Conclusions

1. Based on a systematic study of physicalmechanisms and parameters affecting flow infractured reservoirs, it is concluded that allmajor flow mechanisms and flow processesmust be incorporated in a model for fracturedreservoirs. These include gravity, capillaryforces, gravity drainage, diffusion, capillarycontinuity, and reinfiltration.

2. The key to improved modelling of fracturedreservoirs is to include sufficient detail in thecalculation so that all these flow mechanismare represented. A multiple grid concept forthis purpose is recommended.

3. Simulation studies of water flooding of frac-tured reservoirs show that capillary continuitybetween matrix blocks is of high importancefor mixed wetted systems.

4. High pressure gas injection may yield high oilrecoveries due to reduction in interfacial ten-sion caused by diffusion.

5. Literature review shows that current modelsbased on the dual-porosity, dual-permeabilityconcept using simple fluid transfer terms be-tween fractures and matrix blocks, are not suf-ficiently accurate or representative in mostcases.

References

1. Barrenblatt, G.D. et al.: "Basic Concepts in theTheory of Homogeneous Liquids in FissuredRocks," J. of Applied Math. (1960) 24, no. 5,1286-1303 (USSR).

2. Warren, J.E. and Root, P.J.: "The Behavior ofNaturally Fractured Reservoirs," SPEJ (Sept.1963) 245-255.

3. Kazemi, H., Merrill, L.S., Porterfield, K.L.,and Zeman, P.R.: "Numerical Simulation ofWater-Oil Flow in Naturally Fractured Reser-voirs," SPEJ (Dec. 1976) 317-326.

4. Sonier, F., Souillard, P., and Blaskovich, F. T.:"Numerical Simulation of Naturally FracturedReservoirs," SPERE (Nov. 1988) 1114-22.

5. Litvak, B. L.: "Simulation and Characteriza-tion of Naturally Fractured Reservoirs," Proc.,Reservoir Characterization Technical Confer-ence, Dallas, Academic Press, New York,(1985) 561-83.

6. Gilman, J.R. and Kazemi, H.: "Improved Cal-culations for Viscous and Gravity Displace-ment in Matrix Blocks in Dual-PorositySimulators," JPT (Jan. 1988) 60-70; Trans.,AIME, 285.

7. Thomas, L.K., Dixon, T.N., and Pierson, R.G.:"Fractured Reservoir Simulation," SPEJ (Feb.1983) 42-54.

8. Dean, R.H. and Lo, L.L.: "Simulations ofNaturally Fractured Reservoirs," SPERE (May1988) 638-48.

9. Rossen, R.H. and Shen, E.I.: "Simulation ofGas/Oil Drainage and Water/Oil Imbibition inNaturally Fractured Reservoirs," SPERE (Nov.1989) 464-70; Trans., AIME, 287.

10. Coats, K.H.: "Implicit Compositional Simula-tion of Single-Porosity and Dual-Porosity Res-ervoirs," paper SPE 18427 presented at the1989 SPE Reservoir Simulation Symposium,Houston, Feb. 6-8.

11. Gilman, J.R.: "An efficient Finite-DifferenceMethod for Simulating Phase Segregation inthe Matrix Blocks in Double Porosity Reser-voirs," SPERE (July 1986) 403-13.

12. Saidi, A.M.: "Simulation of Naturally Frac-tured Reservoirs," paper SPE 12270 presentedat the 1983 SPE Reservoir Simulation Sympo-sium, San Francisco, Nov. 16-18.

13. Dangerfield, J.A. and Brown, D.A.: "The Eko-fisk Field," North Sea Oil and Gas ReservoirsSeminar I, J. Kleppe et al. (eds.), Graham andTrotman, London (1986).

14. daSilva, F.: "Primary and Enhanced Recover-ies of the Ekofisk Field - A Single and DualPorosity Numerical Simulation Study," paperSPE 19840 presented at the 1989 SPE AnnualTechnical Conference and Exhibition, SanAntonio, Oct. 8-11.

15. Fung, L.S.-K.: "Simulation Of Block-to-BlockProcesses in Naturally Fractured Reservoirs,"SPERE (Nov. 1991) 477-484

16. da Silva, F.V. and Meyer, B.: "Improved For-mulation for Gravity Segregation in NaturallyFractured Reservoirs," paper presented at the6th European IOR-Symposium in Stavanger,May 21-23, 1991.

17. Firoozabadi, A. and Markeset, T.: "An Ex-perimental Study of the Gas-Liquid Transmis-sibility in Fractured Porous Media," paper SPE24919 presented at the 1992 SPE AnnualTechnical Conference and Exhibition, Wash-ington DC, Oct. 4-7.

18. Barkve, T., Firoozabadi, A.: "Analysis of Re-infiltration in Fractured Porous Media," paperSPE 24900 presented at the 1992 SPE AnnualTechnical Conference and Exhibition, Wash-ington DC, Oct. 4-7.

19. Kazemi, H., Gilman, J.R. and Elsharkaway,A.M.: "Analytical and Numerical Solution ofOil Recovery From Fractured Reservoirs WithEmpirical Transfer Function," SPERE (May1992) 219-27.

20. Luan, Z.: "Oil Recovery Mechanisms in Natu-rally Fractured Reservoirs," RUTH ProjectReport, NTH, Nov. 1992.

21. Uleberg, K.: "Improved Methods for Simula-tion of Fractured Reservoirs," SivilingeniørThesis, NTH, May 1993.

Dual Porosity, Dual Permeability Formulation for Fractured Reservoir Simulation

22. Festøy, S. and van Golf-Racht, T.: "Gas Grav-ity Drainage in Fractured Reservoirs Througha New Dual-Continuum Approach," SPERE(Aug. 1989) 271-278.

23. Suffridge, F.E. and Renner, T.A.: "Diffusionand Gravity Drainage Tests to Support the De-velopment of a Dual Porosity Simulator," pa-per presented at the 6th European IOR-Symposium in Stavanger, May 21-23, 1991.

24. Catalan, L. and Dullien, F.A.L.:"Applicationsof Mixed-Wet Pastes in Gravity Drainage Ex-periments," JCPT (May 1992) 28-30.

25. Orr, F.M. Jr.: "Use of Gas for Improved OilRecovery," presentation at the 1992 RUTH-seminar, Stavanger, Dec. 7-8.

26. Schechter, D.S., Dengen, Z., and Orr, F.M. Jr.:"Capillary Imbibition and Gravity Segregationin Low-IFT Systems" paper SPE 22594 pre-sented at 1991 SPE Annual Technical Confer-ence and Exhibition, Dallas, Oct. 6-9.

27. Hu, H., Whitson, C.H. and Qi, Y.: "A Study ofRecovery Mechanisms in a Nitrogen DiffusionExperiment," paper presented at the 6th Euro-pean Symposium on Improved Oil Recovery,Stavanger, May 21-23, 1991.

28. Torsæter, O. and Silseth, J.: "The Effects ofSample Shape and Boundary Conditions onCapillary Imbibition" paper presented at the1985 Chalk Symposium, Stavanger, May 21-22.

29. Torsæter, O., Kleppe, J. and van Golf-Racht,T.: "Multi-Phase Flow in Fractured Reser-voirs", in Fundamentals of Transport Pheno-mena in Porous Media , J. Bear and M.Y. Co-rapcioglu (eds.), NATO ASI Series, NijhoffPublishers, Dordrecht (1986)

30. Kleppe, J. and Morse, R.A.: "Oil ProductionFrom fractured Reservoirs by Water Dis-placement" paper SPE 5084 presented at 1974SPE Annual Technical Conference and Exhi-bition, Houston, Oct. 6-9.

31. Kazemi, H. and Merill, L.S.: "NumericalSimulation of Water Imbibition in FracturedCores" SPEJ (June 1979) 175-182.

32. Kvalheim, B.: "Kapillær oppsuging i kalkstein:Numerisk simulering av laboratorieforsøk,"Sivilingeniør Thesis, NTH, Dec. 1984.

33. Beckner, B.J. et al.: "Imbibition-DominatedMatrix-Fracture Fluid Transfer in Dual Poros-ity Simulators" paper SPE 16981 presented at1987 SPE Annual Technical Conference andExhibition, Dallas, Sept. 27-30.

34. Luan, Z.: "Discussion of Analytical and Nu-merical Solution of Waterflooding in FracturedReservoirs With Emperical Transfer Func-tions," SPERE (July, 1992).

35. Uleberg, K., Torsæter, O. and Kleppe, J.: "AReview of Fluid Flow Mechanisms for Model-ling of North Sea Fractured Reservoirs," Proc.,1993 RUTH Seminar, Stavanger, Oct. 13-14.

36. Bourbiaux, B.J. and Kalaydjian, F.J.: "Experi-mental Study of Cocurrent and CountercurrentFlows in Natural Porous Media," SPERE (Aug.1990) 361-368.

37. Kløv, T.: "Et studium av fluidstrøm mellommatriseblokk og sprekk i et oppsprukket reser-voar," Project Report, NTH, June 1994

38. Skaar, I.: "Toporøsitetssimulering," ProjectReport, NTH, June 1994.

39. Kløv, T.: "Et studium av fysikalske prosesser inaturlig oppsprukkete reservoarer ved bruk avsimuleringsmodeller," Sivilingeniør Thesis,NTH, Dec. 1994.

40. Pratap, M.: "A study of the effect of verticalcapillary contact between matrix blocks in afractured reservoir under water injection,"M.Sc. Thesis, NTH, Dec. 1994.

41. Ese, A. M.: "Effect of gravity and capillarypressure on recovery of oil in intermediate wetreservoirs," Project Report, NTH, June 1995.

42. Fougner, C.: Detailed simulations of matrix-fracture fluid exchange in fractured reservoirsunder various processes," Project Report,NTH, June 1995.

43. Kyte, J. R.: "A Centrifuge Method to PredictMatrix-Block Recovery in Fractured Reser-voirs," SPEJ (June 1970) 164-170

44. Saidi, A. M., Tehrani, D. H., and Wit, K.:"Mathematical Simulation of Fractured Reser-voir Performance Based on Physical ModelExperiments," paper PD10(3), Proc., 10thWorld Petroleum Congress, Bucharest (1979)

45. Thomas, L. K., Thomas, N. D., and Pierson, R.G.: "Fractured Reservoir Simulation," SPEJ(Feb. 1983) 42-54

46. Øyno L. and Whitson C. H.: "ExperimentalStudy of Gas Injection for Secondary and Ter-tiary Recovery in Fractured Chalk Reservoirs,"Proc., 1993 RUTH Seminar, Stavanger, Oct.13-14.

47. Morel, D.D., Bourbiaux, B., Latil, M., andThiebot, B.: "Diffusion Effects in Gas FloodedLight Oil Fractured Reservoirs," paper SPE20516 presented at the 1990 SPE AnnualTechnical Conference and Exhibition, NewOrleans, Sept. 23-26.

48. Hu, H., Whitson, C.H., and Yuanchang, Q.: "AStudy of Recovery Mechanisms in a NitrogenDiffusion Experiment," Proc., Sixth EuropeanSymposium on Improved Oil Recovery,Stavanger, May 21-23 (1991) 543-553.

49. Øyno L., Uleberg K and Whitson C.H.: "DryGas Injection in Fractured Chalk Reservoirs -An Experimental Approach," paper presentedat the 1995 International Symposium of theSociety of Core Analysts, San Fransisco, Sept.12-14.

50. Gimmestad, O.: "Beregning av strømning i to-porøsitets reservoarsystemer," Project Report,NTH, June 1994.

Combined Gas/Water Injection Subprogram

51. Gimmestad, O. and Uleberg, K.: "The Pro-gramming of Fracsim - A Fast Fractured Res-ervoir Simulator," Project Report, NTH, Aug.1994.

52. Gimmestad, O.: "Utvikling av effektive rutinerfor lokal beregning av strømning i oppspruk-kete reservoarer," Sivilingeniør Thesis, NTH,Dec. 1994.