SCA : A DUAL NETWORK MODEL FOR RELATIVE PERMEABILITY OF BIMODAL ROCKS: APPLICATION IN A VUGGY CARBONATE

Transcription

1 SCA : A DUAL NETWORK MODEL FOR RELATIVE PERMEABILITY OF BIMODAL ROCKS: APPLICATION IN A VUGGY CARBONATE Andrés Motezuma 1,2, Samir Békri 1, Catherine Larohe 1, Olga Vizika 1 (1) Institut Français du Pétrole, 1 et 4 avenue de Bois-Préau, Rueil-Malmaison, Frane (2) Instituto Mexiano del Petróleo, Méxio DF., Méxio. This paper was prepared for presentation at the International Symposium of the Soiety of Core Analysts held in Pau, Frane, September 2003 ABSTRACT Carbonate roks often present bimodal pore size distributions usually attributed to the presene of vugs, large pores inserted in the porous matrix. This vugular porosity an ommuniate through the matrix (separated vugs) or it an form an interonneted pore system (touhing vugs) (Luia, 1999). These different degrees of interonnetivity onfer to the porous medium transport properties very different than the ones haraterizing homogeneous samples. In the present work a numerial simulator is presented, that alulates multiphase flow transport properties in vuggy arbonates taking into aount their speifi pore struture parameters. The omplex real pore spae is represented as a dual- network model that inorporates information on the primary (matrix) and the seondary (vugs, fratures) porosity and aounts for the onnetivity of the seondary porosity. An original methodology is presented to dedue experimentally realisti pore size data to onstrut the network model taking into aount the omplex struture of the bimodal roks. It onsists in ombining merury invasion data and NMR measurements on partially saturated samples to dedue small-sale data haraterizing pore-body and pore-throat size distributions needed to build the 3-D network model. The methodology has been validated on an outrop limestone sample presenting a dual porosity. Based on the experimental data a dual pore-network has been onstruted. It satisfatorily reprodues the apillary pressure urve, the porosity and the permeability determined experimentally. Through alulations performed with this model it is demonstrated that wetting phase relative permeability is strongly affeted by both the vugs and matrix harateristis, while non-wetting phase is mainly ontrolled by the vuggy system. INTRODUCTION Modeling of two or three-phase flow in porous media is of prime importane in improving reservoir simulators preditivity and proess effiieny evaluation. For an aurate predition of the transport properties of a porous medium, small-sale data on the pore spae geometry and topology are needed. Pore-size data are most frequently obtained from

2 merury intrusion or retration experiments. Using these easily obtained data to alulate transport properties is very attrative and hallenging. Merury invasion apillary pressure urves need to be interpreted in order to extrat pore-throat radii distribution. A modeling of the pore spae as well as important assumptions about its geometry and topology are hidden behind this interpretation. Network modeling to interpret merury porosimetry drainage and imbibition urves taking into aount partiular pore geometry, surfae roughness and pore-size orrelation for homogeneous (or single porosity) roks has been reported previously (Tsakiroglou et al., 1997; Kamath et al., 1998; Xu et al., 1999; Larohe et al., 2001). However extension to dual-porosity porous media has been looked at very little (Ioannidis and Chatzis, 2000; Békri et al., 2002). Dual-porosity strutures are haraterized by a primary (or matrix) porosity and a seondary porosity onsisting of vugs or fratures. The interonnetivity of these oexisting networks is a major issue, whih severely affets fluid distributions within the pore spae and onsequently all related petrophysial properties (Ehrlih, 1971). The objetive of this work is to develop a tool to alulate multiphase flow transport properties taking into aount pore struture speifiity and rok/fluid interations. The present paper onsists of two parts. In the first part a new methodology is presented to extrat small-sale struture data by ombining merury porosimetry and NMR on partially saturated samples. In the seond part these data are introdued in the dual-porosity network model and are used to alulate transpor t properties. In this dual-porosity network the onneted vugs are treated in a disrete form (onsidered as forming a network) while the matrix is taken into aount through its average marosopi properties (apillary pressure and relative permeability) alulated with single porosity network modeling. METHODOLOGY TO CHARACTERIZE THE CORE SAMPLE In this part a methodology is desribed to obtain data on the porous medium struture that will be used to onstrut a pore network model representative of a double porosity rok. The method onsists in aquiring NMR spetra of the rok, at various well-determined water saturations, and merury porosimetry data. Combination of these two informations gives onsistent pore-size distributions and pore/pore-throat orrelations as shown below. A arbonate rok with bimodal porosity struture was used to apply this methodology. The total porosity is 33.8% and the permeability 115 md. The sample presents a bimodal pore size distribution. It onsists of a fine-grained matrix and both well-onneted and isolated (ommuniating through the matrix) maropores. Desription of the experimental proedure During a water-air drainage the sample is analyzed by NMR at different levels of water saturation ontrolled by entrifugation at preseleted pressure levels. Initially the sample is 100% saturated with water. Eah stage of saturation Sw orresponds to a apillary pressure P given by the veloity of rotation ω (rpm). At eah pressure level, NMR gives the water distribution that an be assoiated to pores oupied by water. It must be said that these are apparent "pores sizes", beause magnetization may be homogenized by fast moleular diffusion over pore spae regions that do not neessarily orrespond with geometri

3 boundaries of pores. Besides some residual water remaining in big pores is interpreted as water filling smaller pores. This is probably in the origin of the slight shift of NMR signal urves to lower T 2 (Figure 1). When this experiment is finished, the merury inje tion apillary pressure urve is determined on a piee of the used sample. This method being destrutive, it is applied only at the end of the proedure, when all other non-invasive haraterizations are ompleted. Figure 1 shows the NMR spetra obtained on eah level of water saturation Sw(P ). The urves reflet the bimodal harateristi of the sample during the whole drainage proess. Before entrifugation, all the pores are filled with water (S w =1), and the NMR measurement reflets the whole pore-size distribution of the sample. With inreasing ω water is progressively displaed by air. The part of the urves on the right is assoiated to the maropores (relaxation times around 1000 mse), and the left part to the miropores (relaxation times around 70 mse). At ω = 4000 rpm, irreduible water saturation (21%) is reahed. The small peak in the right hand side orresponds evidently to the maropores that are ompletely isolated behind the matrix and do not ontribute to the well-onneted network of vugs. The merury injetion results are presented in Figures 4 and 12. The bimodal harater of the pore struture is again apparent. The interpretation of the above results, as desribed below, will permit to separate the two oexisting systems (matrix-primary porosity / onneted maropores-seondary porosity). The apillary pressure urve of eah porosity system and the pore size distributions obtained in this way will be used in the onstrution of the dual network model. Interpretation Determination of the Capillary Pressure for eah System After porosity alibration, the NMR amplitude signal is equivalent to volume; the total volume of water in the sample for eah value of S w (P ), presented in Figure 1, is given by V T (S w )=Σ V(T 2,S w ) (1) Considering the minimum amplitude between the two peaks as threshold between the maro- and miropores, the respetive volumes an be distinguished. If V M (S w ) is the volume of maropores (onneted, V M and isolated, V Mi ), and V m (S w ) the volume of the miropores, then the volume fration is given by f M (S w ) = V M (S w )/V T (S w ) (V M= V M+ V Mi ) (2a) f m (S w ) = V m (S w )/V T (S w ) (2b) Figure 2 shows the evolution of these frations as a funtion of the total water saturation in the sample. For the initial urve, S w =1, f M is ~0.6 (maropores) and f m ~0.4 (miropores). During the drainage it is seen that the volume fration of the miropores remains lose to the initial value of 0.4 until S w =0.55 (950 rpm), what signifies that during the first drainage steps the water prodution omes only from the maropores. At this stage, the total saturation is used to estimate the volume of the well-onneted maropores; V M =(1-0.55) V T (S w =1). In fat it is observed that starting from this point miropores and isolated

4 maropores are simultaneously emptied. The total water prodution at eah entrifuge step, V prod (S w ), orresponds to V prod (S w )= V Mi (S w ) + V m (S w ) (3) Knowing V m (S w ) from the NMR measurements, the fration of the produed water orresponding to the miropores is given as fm(sw)= Vm(Sw)/ Vprod(Sw) (4) Figure 3 shows these data as a funtion of the water saturation of the sample. For water saturation above 0.55 (before the matrix starts to get invaded) this fration is zero. Then a rapid inrease of f m is followed by a muh slower one roughly orresponding to two straight lines. From the merury injetion measurements, we obtain the differential volume of the wetting phase as a funtion of the saturation of the sample V Hg (S w ). The differential volume that orresponds only to the miroporosity is then alulated by V m (S w )= V Hg (S w ) f m (S w ) (5) The differential volume distribution as a funtion of the pore-throat radius as issued from the merury porosimetry experiment along with the part orresponding to the miropores is shown in Figure 4. The pore-throat radii have been alulated by r = 2γ osθ (6) ( 0 ) P where γ o is the merury surfae tension and θ the ontat angle. The differential volume urve for the maropores (V M ) is also given diretly in Figure 4 by the right hand side part of the urve (above the ut-off value for miropore radius). Consequently the apillary pressure urve for eah system an be estimated. Figure 5 shows the apillary pressure urves of both systems as a funtion of volume and saturation. Correlation between NMR Relaxation Time T 2 and Pore Radius r At eah speed of entrifugation only drainage of the pores whose radii of aess are higher than the equivalent pore-throat radius orresponding to the applied pressure (f. Eq. 6) is allowed. Let us take as example the 950 rpm urve, shown in Figure 6. In this figure the hathed zone represents the differene of volume between the pore volume of the sample filled with water at initial ondition and the pore volume after entrifugation at 950 rpm. This volume orresponds to the pores that are emptied at this speed. It is also noted that at this rotation speed the part of the urve orresponding to the small pores remains almost unhanged, meaning that these pores are still filled with water. It is then reasonable to assume that the intersetion point between eah NMR spetrum urve (at eah speed) and the initial urve (at S w =1) orresponds to the size of the smallest pore whih was invaded by air. For eah urve the equivalent pore radius an be alulated by Eq. (6) and an be then orrelated to the relaxation time T 2 that orresponds to the point of intersetion as indiated in Figure 6. Figure 7 presents the orrelation obtained between T 2 and r alulated as desribed above.

5 A good orrelation is obtained to r 2. In this figure we also show the results obtained by Godefroy (2001) onsidering a quasi-spherial geometry of pore and water at 34 C as saturating fluid. In that work, a model was used to orrelate the relaxation time T 2 with the pore radius by T 2 =r 2 /(6D m ) where D m is the diffusion oeffiient of water (3x10-5 m 2 /s at 34 C). This model onsiders that the relaxation is dominated by the diffusion, whereas the surfae relaxation is fast. This an be the ase in large pores or when the surfae ontains paramagneti impurities. This behavior is unusual in 1µm pores, thus unexpeted in the experiments presented here. However no speial leaning was applied, and existing impurities may have enhaned the surfae relaxation. Pore Distribution of the well-onneted maropores The total maropore population onsists of well-onneted and isolated maropores. The distribution obtained by NMR at S w =1 inludes both types of maropores. It is observed that starting from 950 rpm the draining of miroporosity begins, and onsequently the distribution obtained at 950 rpm ontains the miroporosity and the isolated maropores. Thus the distribution orresponding to the well-onneted maropores an be obtained diretly by subtration V ( r ) = V ( r ) V ( r (7) M ) 0rpm 950rpm The frequeny for eah radius r is thus the proportion orresponding to eah differential volume V M (r) divided by the total volume of well-onneted maropores V M. The thik ontinuous line in Figure 8 presents these results. For omparison purposes in the same graph are also given the results of the merury injetion experiment for the maropores in terms of volume frations aessible at a given pore-throat size. It is reminded that merury injetion gives the volume distribution that is hidden behind the pore-throats, whereas magneti resonane distribution an be assoiated to the pore size distribution. It an be also notied that the minimum value from both pore size determinations is of the same order of magnitude. The mean pore size determined by this methodology (around 500 µm) is in agreement with eletron mirosopy observations and the values reported by Bousquié (1979) for the same type of rok. Nevertheless this methodology would require a broader validation on roks of various strutures and for different NMR relaxation regimes. MODELING To alulate relative permeabilities for the bimodal rok studied in the experimental part, the detailed pore size distributions of eah system (miro- maropores) have been used to onstrut an adapted dual-porosity network model (Békr i et al., 2002). First the matrix properties, P m (S w ) and K rm (S w ), are generated using a single porosity network model. Then the properties of the well-onneted maropores are reprodued to onstrut the network that is finally oupled to the matrix using the dual-porosity option.

6 Single Porosity Network Model The network model is a oneptual representation of a porous medium. The pore struture is modeled as a bi or three-dimensional network of narrow hannels interonneting pores. A real porous medium is haraterized by surfae roughness and angles, through whih the wetting phase remains ontinuous and flows simultaneously with the non-wetting phase oupying the bulk of the pores. To take into aount this displaement mehanism, pores and pore-throats are onsidered to have angular setions. The hannels (pore-throats) have triangular ross-setion and variable length. The pores are simulated as ubes. Eah pore is aessible by six idential hannels. An equivalent diameter D p and d for the pores and the hannels is assoiated to the geometry. A shemati depition of a unit ell is given in Figure 9. L represents the harateristi length of the network that orresponds to the distane between two adjaent nodes (enter of the pores). When L is defined, the marosopi properties of porosity, permeability, apillary pressure and relative permeability an be alulated. A more detailed desription of the model an be found in Larohe (1998) along with a long list of referenes on this type of models. Network Model for the Matrix Ideally, the network model is generated based on the pore and throat distributions and their orrelation. For the moment, this information annot be obtained independently. The models onstruted using measured size distrib utions have to be alibrated so that they respet the parameters measured in the laboratory: merury injetion apillary pressure, porosity and permeability. Volumes for throats and pores are related to their respetive diameters d and D p by 3 λp λ p λ V d = D AR d V p ( ) ( ) p p p ( 2 λ ) λ ( d ) ( L D ) d d p (8a) = (8b) where p and are onstants. The aspet ratio AR is the ratio of pore diameter to throat diameter. For the different ombinations of λ p, λ, p, and L it is possible to onstrut different networks with speifi properties. To determine the number based throat-size frequeny (throat-size distribution) the method presented by Larohe et al (2001) is used. The parameters that are tuned to reprodue the experimental porosity, permeability and apillary pressure of the matrix are L, p λ p and r (ut off radius). The exponent λ p is set equal to zero (the pore volume is supposed to be onstant) and by introduing r equal to 0.7 µm, it is possible to hoose two different onstant volume of the pore. All the parameters and the properties obtained for the matrix are summarized in Table 1. From merury injetion it is found that the porosity fration for the matrix is 0.48 orresponding to a porosity of 22.1% in the bimodal struture. Due to the fat that it is not possible to measure independently the matrix permeability, estimation is made onsidering an average value of the pore-throat radii from the merury injetion in the zone of miropores. This average pore throat radius is 0.05 µm. Applying the apillary tube model

7 K=r 2 Φ /8 (Dullien, 1992) the permeability of the matrix is estimated at about 0.1 md. Figure 10a presents the omparison of the experimental apillary pressure urve (for the matrix and the onneted maropores) and the alulations using a single porosity network model. The agreement is very satisfatory. Figure 10b presents drainage water/oil relative permeabilities for the matrix and the well-onneted maropores. Network Modeling for the well-onneted maropores To define the main flow system, a set of pores was randomly generated based on the pore size distribution obtained as explained in the experimental part of the paper (f. Figure 8). Effetive pore-throat sizes are already known from merury injetion, however their frequeny of appearane is unknown. In this paper the pore-throat frequeny is taken as the fitting parameter to math the experimental apillary pressure. The throats onneting the maropores are randomly generated based on an assumed frequeny distribution. They are then assigned to the pores with the only onstraint that d is smaller than D p. The mean aspet ratio AR is obtained as the ratio of the mean pore diameter <D p > to the mean porethroat diameter <d >. From merury injetion it is found that the vuggy system porosity is 14.9% (0.43 of the total pore volume). Dotted lines in Figure 8 show the pore and throat size distributions used for the network. The parameters and properties obtained with this adjustment are given in Table 2. Figure 10a presents the omparison of the experimental apillary pressure and the results using the simple network model. It an be notied that the math is very satisfatory. Figure 10b shows the relative permeability for both systems separately onsidered. These relative permeabilities are not diretly omparable beause for eah pore system they refer to its absolute permeability. Qualitatively it is observed that in the system with the higher aspet ratio the non-wetting phase flows less easily, whih is physially onsistent. Dual Porosity Network Model The dual-porosity network model approah is applied at two sales as illustrated in Figure 11. The dual-porosity model ombines transport properties data of the matrix onsidered as homogeneous (apillary pressure and relative permeabilities) with the expliit simulation of flow in the well-onneted maropores network (vugs). The bimodal porous medium is haraterized by a bimodal pore-throat radii distributio n. The NMR pore distribution is used to build a three-dimensional lattie of pores and throats representing the seondary porosity. These pores and throats are surrounded by a matrix with known properties (porosity, absolute permeability, apillary pressur e and relative permeability). Matrix permeability K m and porosity Φ m may be spatially distributed within the 3-D lattie. In this paper, it is assumed for simpliity that matrix properties are uniform.

8 The primary drainage in dual-porosity roks is simulated using invasion perolation for the single porosity model (Killins et al., 1953) with the following modifiation. The apillary pressure, P t, required for the non-wetting phase invasion into a given pore, may be determined by either the pore size r or the apillary pressure of the matrix P m. This may be expressed as P t = min[p (r), P m ] (9) where P (r) is the threshold pressure of the throat of the well-onneted maropores. When the pressure exeeds Pm, the matrix will be invaded by the non-wetting phase. The global porosity Φ of suh dual- network is alulated from the primary Φ m and seondary porosity Φ Μ : Φ = Φ M + Φ m.(1- Φ M ) (10) Similarly, ondutane g α and non-wetting phase saturation S nw are alulated for eah applied apillary pressure, P, knowing the matrix apillary pressure and relative permeability funtions (from whih the matrix ondutane g α m(s nwm (P )) is dedued from the single porosity model): Φ.S nw = Φ M.S nwm + Φ m.(1 - Φ s ).S nwm (P ) (11a) g α = g α M + g α m(s nwm (P )) (11b) where S nws, g α M are non-wetting phase saturation and ondutane of the seondary porosity, alulated using the lassial approah of pore network simulation. Knowing the ondutivity of eah phase one alulates the permeability and the relative permeabilities in the same way as for the single porosity model (Larohe, 2001). Results The total porosity and global permeability were mathed to 33.8% and 115 md respetively (the experimental values). Figure 12 shows the omparison between the apillary pressure obtained with the dual porosity model and the experimental results. The agreement is very satisfatory. Figure 13 shows the relative permeabilities for the matrix, vugs and the double porosity system as a funtion of normalized saturation. All three relative permeability sets are obtained by dividing alulated effetive permeabilities by the value of the absolute permeability of the ore (115mD). Saturation is not normalized. It is obtained by multiplying normalized saturation of figure 10b by the orresponding pore volume fration (43% for the vugs, 57% for the matrix). For the double network model saturation is the total saturation in the total pore volume. From the double porosity results, it an be seen that water relative permeability, espeially at low water saturation, is strongly affeted by the matrix, while oil relative permeability is ontrolled by the vuggy system. At the beginning of oil injetion the relative permeability to water is mainly ontrolled by the vuggy system, then when the oil invades most of the big vugs water flows essentially through the matrix, and its relative permeability is ontrolled by the matrix harateristis. Sine oil is the injeted phase and also strongly non-wetting its relative permeability is exlusively dominated by the vuggy system

9 harateristis. CONCLUSIONS Merury invasion data and NMR measurements on partially saturated samples are ombined in an original way to dedue small-sale data haraterizing pore-body and porethroat size distributions needed to build the 3-D network model. This method of interpretation of entrifugation and NMR experiments allows a realisti estimation of the orrelation between the relaxation time T 2 and the pore radius. A dual network model for dual porosity rok s was presented. It inorporates information on the primary (matrix) and the seondary (vugs, fratures) porosity. It also aounts for limited or partial onnetivity of the seondary porosity. With this model a pore network has been onstruted that satisfatorily reprodues the apillary pressure urve, the porosity and the permeability determined experimentally on double porosity rok. It was demonstrated that wetting phase relative permeability is, depending on its saturation, affeted by both the vugs and the matrix harateristis, while the non-wetting phase is mainly ontrolled by the vuggy system. REFERENCES 1. Békri, S., Larohe C. and Vizika O., Pore-Network Models to Calulate Transport Properties in Homogeneous and Heterogeneous Porous Media. XIV International Conferene on Computational Methods in Water Resoures, Delft, The Netherlands, Jun Bousquié P., Texture et Porosité de Rohes Calaires, PhD thesis, Université de Paris VI, Paris. 3. Dullien F. A. L., Porous Media. Fluids Transport and Pore Struture, Aademi Press. 4. Ehrlih R., Relative Permeability Charateristis of Vugular Cores- Their measurement and Signifiane, SPE 3553, 46 th Annual SPE meeting of the Soiety of Petroleum Engineers of AIME, New Orleans, La. 5. Godefroy S., Etude RMN de la Dynamique des Moleules aux Interfaes Solid-Liquide: des Materiaux Poreux Calibrés aux rohes Petroliferes, PhD thesis, Eole Polytehnique. 6. Ioannidis, M.A., Chatzis, I., 2000: A Dual-Network Model of Pore Struture for Vuggy Carbonates, Proeedings Soiety of Core Analysts International Symposium, Abu Dhabi, UAE, Ot. 7. Kamath J., Xu B., Lee S.H. and Yortsos Y.C., Use of Pore Network Models to Interpret Laboratory Experiments on Vugular Roks, Journal of Petroleum Siene and Engineering, 20, Killins C.R., Nielsen R.F., Calhoum J.C., Capillary Desaturation and imbibition in porous roks, Produers Monthly, De., vol. 18, no 2, Larohe C., Déplaements Triphasiques en Milieu Poreux de Mouillabilité Hétérogène, PhD Thesis, Université de Paris XI, Paris. 10. Larohe C., Vizika O., Hamon G. and Courtial R., Two-phase Flow Properties Predition from Smale-Sale Data Using Pore-Network Modeling, SCA , International Symposium of the Soiety of Core Analysts, Edinburgh, U.K., Sept. 11. Luia, F. J., Carbonate Reservoir Charaterization, Springer. 12. Tsakiroglou, C.D., Kolonis, G.B., Roumeliotis, T.C. and Payatakes A.C Merury

10 Penetration and Snap-Off in Lentiular Pores, J Colloid Interfae Si, 193, 2, Xu B., Kamath J., Yortsos Y.C. and Lee S.H., Use of Pore Network Models to Simulate Laboratory Corefloods in a Heterogeneous Carbonate Sample, SPE J., 4 (3), Sept,

11 Table 1. - Parameters and alulations of geometrial harateristis, porosity and permeability for the matrix Table 2.- Geometrial parameters and alulations of porosity and permeability for the well-onneted maropores Parameters Miropores Parameters Maropores <Dp> 1.26 µm <Dp> 647 µm <d> µm <d> 37.9 µm L 4.65 µm L 1190 µm <AR> 1.86 <AR> C Φ r 0.7 µm K md V (r<r) 9.02 µm 3 Φ K md S w % rpm Fig. 1- NMR amplitude signal (~pore volume) as a funtion of the relaxation time T 2 at the different stages of entrifugation (rpm) Fig. 2.- Evolution of the volume fration for the maropores and miropores oupied by water as a funtion of water saturation S w Fig. 3.- Volume fration orresponding to the miropores as a funtion of water saturation S w Fig. 4.- Distribution of the differential volume for the miropores and the total porosity as a funtion of the pore-throat radii

12 (a) (b) Fig Capillary pressure alulations (a) as a funtion of umulated volume and (b) as a funtion of saturation Fig. 6.- Graphial desription of the method to orrelate the pore radius and the NMR relaxation time T 2 (example for 950 rpm) Fig. 7.- Correlation between the relaxation time T 2 and the equivalent pore radius r Unit element Node V p D p Pore V d Fig. 8.- Pore size frequeny for the pores ( ) and pore-throat differential volume (D) obtained for the well-onneted maropores as a funtion of the pore radius. The dashed lines orrespond to the ones used for the network simulator. L Channels Fig. 9.- Shemati depition of a unit ell of the network model (Larohe, 1998)

13 (a) (b) Fig. 10. Curves of apillary pressure (a) and relative permeability (b) obtained for the matrix and the vugs network. Symbols are ( ) for vugs and (? ) for matrix Network of vugs/fratures Matrix: homogeneous network matrix Fig Shemati depition of a dualnetwork model Fig. 12.-Capillary pressure urves. Comparison between experimental and dual porosity model simulation Fig Water/oil relative permeability for the matrix (? ), vugs ( ) and the dualporosity system (ontinuous line).