Identification of permeable carbonates with well logs applying the Differential Effective Medium theory

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1 Ientification of permeable carbonates with well logs applying the Differential Effective Meium theory Abstract For carbonates, the porosity alone, as calculate with the combination of the Neutron an Density logs, is not enough to preict whether an interval will be permeable or not. A rock is permeable if the vois in it (matrix porosity, fractures, vugs) are connecte. The well known Differential Effective Meium theory (DEM) allows the moeling of rocks perceive to be permeable or impermeable. Spherical pores in relatively low porosity rocks are assume to be isolate an therefore resulting in very impervious rocks. Very flat ellipsoial pores (penny cracks) with a ranom orientation are postulate to form an interconnecte network of pores, resulting in a very pervious rock. These two en points can be ientifie in a Velocity (from the sonic log) vs. porosity (from the raioactive logs) iagram corresponing to the interval of interest, where the curves moele with the DEM an the Wyllie curve have also been plotte. Impervious points shoul fall high above the Wyllie curve in such a iagram, whilst the points representing permeable rocks shoul fall well below the Wyllie curve. Prouction tests have been use to check the theory. Authors - Malcolm Anerson, Hea, Department of Mathematics, University of Brunei. - Roberto Gullco, Senior Petrophysicist, CGGVeritas, Mexico. Topic Petrophysics (34) Rock Physics (36) Presentation Oral.

2 Introuction The porosity an mineralogical composition of carbonates (proportions of calcite, olomite an shale) can be calculate with ensity, neutron an gamma ray logs. However, the porosity alone is not enough to etermine if a carbonate will prouce. It is quite often to encounter carbonates with porosities of 1% or more, which prove to be impermeable when teste. On the other han, there are low porosity carbonates, which prove to be permeable an have a very goo prouction of hyrocarbons. In orer to be permeable, the pores have to be interconnecte. In carbonates with a relatively high porosity, there may be isolate cavities or vugs, which account for the porosity, but the rock is impermeable because the vugs o not form a network. In a rock of low porosity, there may be cracks or fractures which etermine an interconnection of most of the vois in the rock, resulting in a high permeability. The ientification of a zone of permeable carbonates in a well is very important from the economic point of view. The aim of this work is to ientify permeable zones in a well by means of the porosity (as calculate from the raioactive logs) an the sonic log. The basic iea is to moel the p velocity of rocks as a function of total porosity an compare the moele results with the actual ata. In orer to moel rocks perceive as permeable or impermeable, we will use the Differential Effective Meium Moel (DEM). Mavko et al (1998) provie an excellent explanation of the moel. The use of the DEM to preict certain physical properties in carbonates has been carrie out, for instance, by Kumar et al. (25), an Baechle et al.(27). The approach taken by Kumar is quite similar to the one taken in the present paper. However, in our paper we solve the system of ifferential equations analytically, either for ry or saturate rocks. The fact that we en up in possession of a set of equations, provies us an insight into the characteristics an limitations of the DEM. Furthermore, we compare the preictions of the DEM with a few prouction tests in carbonates, in an attempt to corroborate the theory. Another paper of interest is the one by Anselmetti an Eberle (1999) which, although not ealing with the DEM moel, ifferentiates the pore type of carbonates as a function of the position of the points in a velocity vs. porosity plot using the Wyllie curve as a reference. It shoul be pointe out that the answers provie by the present paper are qualitative rather than quantitative. We shoul be able to ientify in a crossplot Vp vs porosity (as erive from the neutron an ensity logs), points which represent vugs interconnecte by fractures (suppose to be permeable rocks) or isolate spherical vugs (suppose to be impermeable). However, whilst the en points are easily ientifiable, there are intermeiate points in the vicinity of the Wyllie line whose permeability cannot be iagnose in the light of the DEM theory. One of the limitations of the DEM moel consists in the fact that it is unable to moel a granular meium, that is, when the originally continuous phase is a flui with a shear moulus equal to zero, an the moel is constructe aing inclusions of a mineral. In this paper, fractures are moele as ellipsois with a very small aspect ratio (of the orer of.1 or less). It is suggeste in this work that, for very small aspect ratios, the ellipsois ten to be interconnecte an hence generate a network resulting in a permeable rock. It is postulate in the paper that there is an inverse relationship between the connectivity of the pores an their aspect ratio. However, for an aspect ratio of about.1 (the maximum value is 1, corresponing to spheres) we can reprouce the Wyllie equation quite closely. Hence, points along the Wyllie curve shoul not be very permeable accoring to the DEM an our postulates. However, it has been suggeste from empirical observations that the points falling close to the Wyllie line represent intergranular or intercrystalline porosity (Anselmetti an Eberle, 1999). These rocks may be permeable or not. Hence, there is a funamental ambiguity attache to the points falling close to the curve given by the Wyllie equation. Still, with all these limitations, it seems to be worthwhile to moel carbonates using the DEM, because in many instances it allows accurate preictions regaring the prouctivity or otherwise of certain intervals. Barcelona, Spain, June 21

3 Differential Effective Moel theory The couple system of orinary ifferential equations which characterize the DEM are: 1 1 K * * K f K * P... [1] * Q....[2] In these equations, K* an μ* are the bulk an shear mouli, respectively, K f is the flui bulk moulus an is the porosity. As written, these equations apply when the host material is the soli an the inclusions represent pores full of a flui having a bulk moulus equal to K f. The parameters P an Q epen, among other factors, on the aspect ratio of the pores (for spheres, the aspect ratio is equal to 1, which is the maximum value possible). In this work, the system of ifferential equations given by [1] an [2] has been solve analytically an the p velocity has been calculate as a function of porosity for both spherical pores an pores with low aspect ratios. Accoring to the DEM, if we epart from a soli block of a mineral (or composite of minerals) an begin to a spherical pores fille with a flui, the pores will ten not to be connecte an hence the resulting rock will not be permeable, particularly if the porosity is relatively low (no more than 2-25 %). On the other han, if the pores are very flat (have a very low aspect ratio), an they have a ranom orientation, it is quite likely that there will be intersections between the pores an hence a network of pores may exist, resulting in a permeable rock. Moelling shows that, in a velocity/porosity iagram, the theoretical curve for spherical pores lies much higher than the Wyllie curve (for a given porosity, the theoretical DEM velocity is much higher than the Wyllie velocity). If well log erive ata (the velocity from the sonic vs. the porosity from the neutron an ensity logs) is plotte in such a iagram, the points falling in the vicinity of the theoretical curve for spheres will accoring to theory represent impervious rocks. As the aspect ratio of the moel is ecrease the resulting curves ten to go ownwars an, for aspect ratios less than.1, the curves will be well below the Wyllie curve. The assumption is that, as the pores get flatter, the likelihoo that they are interconnecte will increase, resulting in permeable rocks. So the points falling well below the Wyllie curve in a velocity vs. porosity iagram will ten to be permeable. It shoul be pointe out that the DEM allows for the moelling of pores of ifferent shapes, simultaneously. That is, in the same rock we may moel spherical vugs interconnecte by very flat ellipsoial pores, resulting in a permeable rock. The stanar DEM oes not allow us to make preictions regaring those points which fall in the vicinity of the Wyllie curve, which woul represent an interparticle porosity accoring to Anselmetti an Eberle (op.cit.). These rocks may be permeable or not. The stanar DEM oes not allow us to simulate a granular meium (where we begin with an empty volume an procee to fill it with soli particles). To o so, moifications to the original theory have to be introuce, which are beyon the scope of this paper. Examples Figure 1 shows a velocity vs. porosity crossplot for an interval about 2 m thick in a carbonate sequence (the mineralogy of the interval is almost pure olomite). The relatively high porosities (a lot of points in the range ) mae the interval interesting. However, the interval prove to be absolutely impermeable. It shoul be note that the points having the maximum porosity ten to fall Barcelona, Spain, June 21

4 VELOCITY (m/sec) above the Wyllie line (in blue), relatively close to the spheres line in purple. These points are interprete to be unconnecte vugs, an hence impervious. Figure 2 shows a velocity vs. neutron porosity crossplot for a 4 metre interval of a carbonate compose mainly of calcite, free of shale. This interval was teste an prouce about 51 BOPD, with a 3/8 choke. Note that most of the points fall well below the Wyllie curve (in black), whilst the porosity is quite low (the average neutron porosity is.22). It is apparent that espite the low porosity, the permeability is quite goo, so the assumption that this interval is somewhat fracture, is quite reasonable. In this figure, the line corresponing to penny cracks with an aspect ratio of.1 has been trace in blue. As a reference, the lower Hashin-Shtriekman limit, representing the minimum possible velocity that a calcite-water system may have, has been trace in re. DISTRIBUTION OF POINTS FOR A TESTED INTERVAL ABOUT 2 M THICK. (The test prove the rock to be impermeable. The mineralogy is almost pure olomite) Vp(Obs) Vp(Sph) Vp(Wyllie) POROSITY (v/v) Figure 1: Velocity/porosity crossplot, showing the Wyllie curve in blue an the theoretical curve erive from the DEM for spheres, in purple. Barcelona, Spain, June 21

5 V EL O C ITY (m /sec) V p vs. NE U TR O N PO RO S I TY W e ll: ZZZZ P E RF O RA T ED IN TE R V AL 4 M THI CK N E UTR O N P O RO S ITY (V /V ) Figure 2: Velocity/porosity xplot, showing the Wyllie curve in black, the theoretical curve corresponing to pores with an aspect ratio of.1 in blue an the curve corresponing to the Hashin-Shtriekman lower limit in re. Conclusions The ientification of permeable zones in carbonates has been carrie out in the light of the DEM moel. The DEM theory allows for the ientification of the most impermeable points (moele as spherical cavities fille with a flui in an otherwise continuous soli phase an interprete as vuggy or molic porosity) an the most permeable points (moele as penny cracks or ellipsois with a very low aspect ratio an interprete as fractures). The theory yiels an ambiguous answer regaring the points which fall, in a velocity vs. porosity iagram, close to the Wyllie curve. References Anselmetti, F. an G.Eberle, The velocity eviation log: A tool to preict pore type an permeability trens in carbonate rill holes from sonic an porosity or ensity logs, AAPG Bulletin, V.83, No.3, Baechle, G., A. Colpaert, G. Eberli an R. Weger, 27. Moeling velocity in carbonates using a ual porosity DEM moel, SEG/San Antonio 27 Annual Meeting, Kumar, M. an D. Han, 25. Pore shape effect on elastic properties of carbonates, SEG/Houston 25 Annual Meeting, Mavko, G., T.Mukerji an J.Dvorkin, The rock physics hanbook, Cambrige University Press. Barcelona, Spain, June 21