Pore geometric modeling for petrophysical interpretation of downhole formation evaluation data Authors M. Gladkikh, o 1. D. Jacobi, 1. F. Mendez 1. First published: 30 November 2007Full publication history DOI: 10.1029/2006WR005688View/save citation Cited by: 2 articlesRefresh citation count Citing literature Abstract [1] An accurate description of water- or oil- bearing reservoirs strongly depends on a robust determination of their petrophysical parameters, e.g., porosity, permeability and fluid distribution. Downhole logging measurements are the primary means to formation evaluation; however, they do not directly provide the petrophysical properties of interest. To interpret well logging data, a range of empirical models are usually employed. These empirical relationships, however, lack scientific basis and usually represent generalizations of the observed trends.
Transcript
Pore geometric modeling for petrophysical interpretation of downhole formation evaluation dataAuthors
M. Gladkikh,o
1.
D. Jacobi,1.
F. Mendez1.
First published: 30 November 2007Full publication history
DOI: 10.1029/2006WR005688View/save citation
Cited by: 2 articlesRefresh citation count Citing literature
Abstract[1] An accurate description of water- or oil-bearing reservoirs strongly depends on a robust determination of their petrophysical parameters, e.g., porosity, permeability and fluid distribution. Downhole logging measurements are the primary means to formation evaluation; however, they do not directly provide the petrophysical properties of interest. To interpret well logging data, a range of empirical models are usually employed. These empirical relationships, however, lack scientific basis and usually represent generalizations of the observed trends. Since macroscopic rock properties vary depending on their microstructure, we suggest using a pore-scale approach to establish links between various petrophysical properties of sedimentary rocks. We outline a method for computing formation permeability using the proposed rock models. The method utilizes NMR (Nuclear Magnetic Resonance) logging data for the information about porosity and grain size. We also present an approach for prediction of acoustic velocities of model rocks. The proposed methodology is applied to the field data, and the corresponding interpretation results are included in this paper.Close the feedbackYou are viewing our new enhanced HTML article.
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1. Introduction[2] Macroscopic formation properties (such as porosity, absolute and relative permeabilities, capillary pressure curves, and fluid saturations) are fundamental to formation evaluation, assessment of reserves, quality of aquifers and water resources, and production forecast. Some formation properties are easily obtained from logging data, for example, porosity and bulk density. Other properties are very difficult if not impossible to measure directly. These properties include grain size distribution, absolute and relative permeabilities, capillary pressure curves, and reservoir fluid types and saturations. They are usually derived from empirical correlations with the logging data and lack strong scientific basis. In this paper we present the methodology that is based on modeling of physical processes directly at the pore scale. We create model rocks with known pore geometric structure, thus relating petrophysical properties of interest with measured data and providing a link between microscopic rock structure and its macroscopic parameters.[3] The idea is to consider pore-scale physical processes in simple but physically representative model rocks [Mason , 1971; Bryant and Blunt , 1992 ; Bryant et al. , 1993 ; Mason and
Mellor , 1995 ; Bryant and Pallatt , 1996 ; Bryant and Johnson , 2003; Gladkikh and Bryant , 2005 ; Gladkikh , 2005 ]. Our approach to log interpretation is based on this previous research. The model is a numerically generated random
dense packing of equal spheres. Results of authigenic processes (isopachous [Bryant et al. , 1993 ] and pore-filling cement [Gladkikh , 2005 ], presence of clay minerals) are further considered in this model sediment. The grain space and the void space in the packing are completely defined, thus providing all the necessary geometric information of rock microstructure. The goal of this work is to construct an approach for the successful routine petrophysical interpretation of downhole logging measurements utilizing pore-scale techniques developed previously. This particular problem, to our knowledge, has never been considered before.
2. Physically Representative Model Rocks[4] We consider a random dense packing of equal spheres as model sediment [Mason , 1971 ; Bryant et al. , 1993 ]. It represents an “ideal soil” model, suggested in the literature previously [Slichter , 1899 ; Hackett and Strettan , 1928 ]. We create such a packing numerically. The collective rearrangement algorithm with periodic boundary conditions, based on the work by Clarke and Wiley [1987] , was implemented to pack spheres of equal size into the unit cube. Cylindrical core with length equal to diameter was then cut out of the unit cube. This model sediment core contains about 5,000 grains and has a porosity of 36.8%, which corresponds to the porosity of unconsolidated sands. The number of grains and geometrical dimensions (i.e., length and diameter) of the model core were chosen as a result of a series of numerical simulations. These numbers should be large enough in order for the model sediment to
be physically representative and provide fair comparison with experimental core data, and at the same time small enough to ensure fast numerical computations. The cylindrical core used in numerical simulations is shown in Figure 1.Figure 1.
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Numerically generated, random dense packing of equal spheres. Cylindrical core, used in simulations. Number of grains is approximately 5,000; porosity is 36.8%.
[5] The spheres in the packing represent detrital grains of model sediment. In this work we assume that material of these detrital grains can be either quartz or feldspar. The material properties are therefore assigned to grains on the basis of the specified mineral composition (namely, quartz and feldspar mass percentages of the total mass of solid material) of the model rock. The assumption of two types of detrital grain material is not a limitation of the approach: it can easily be extended to multimineral composition.[6] The results of diagenetic processes can also be modeled in the packing, which allows creating simple models of sedimentary rocks [Bryant et al. , 1993 ]. Isopachous cement occurs in the form of uniform overgrowths at the surfaces of detrital grains. The example of this type of cementation is quartz overgrowths. We model isopachous cementation by uniform increase of radii of all spheres without changing any of the spatial coordinates of grain centers [Bryant et al. , 1993 ].[7] Pore-filling cement occurs in the form of large poikilotopic crystals (e.g., calcite cement), which completely occupy clusters of neighboring pores. We
model pore-filling cementation by allowing such clusters of pores to be completely filled with cement [Gladkikh , 2005].[8] The occurrence of authigenic clay minerals is common for sedimentary rocks, most often due to the alteration of plagioclase or potassic feldspar mineral grains. We consider two types of clay occurrence: pore-lining (e.g., chlorite) and pore-filling (e.g., illite). We consider that pore-lining cement exists in the form of thin coatings on the surfaces of detrital grains. These coatings may be porous, but their thickness is assumed to be much smaller than the radius of detrital grains. Knowing mass percentage (in total mass of solid material), specific surface area, thickness, density, and porosity of pore-lining clay, makes computing part of the total grain surface area, covered by clay coatings, uncomplicated. Then, by assuming that individual grains are completely covered by pore-lining clay, we compute the number of these coated grains.[9] We assume that pore-filling clay completely occupies individual pores. It may be porous; in this case we need to specify its porosity. By knowing mass percentage (in total mass of solid material), density, and porosity of pore-filling clay, it is easy to compute part of pore volume (and therefore number fraction of pores), filled by this type of clay.[10] If a model rock has a complex composition (i.e., it has overgrowth cement, pore-filling cement, and clay minerals), we assume that overgrowth cementation precedes pore-filling cementation, and that clay minerals are introduced in the pore space last. This sequence of authigenic events is common for sedimentary rocks [Land et
al. , 1987 ]. In this case, we first compute porosity loss
caused by pore-filling clay and pore-filling cement, and remaining porosity loss is assumed to be due to isopachous cement. The surface area of the resulting model rock is further computed and pore-lining clay is distributed accordingly.
3. Model of Fluid Flow[11] In order to model fluid flow in the packing, we subdivide pore-space into a network of pore bodies and throats using Delaunay tessellation of sphere centers [Mason , 1971 ]. We implement an algorithm for the Delaunay tessellation following the work of Thompson [2002] . The resulting network has “sites” (pore bodies), which correspond to the internal regions of tetrahedral Delaunay cells and “bonds” (pore throats), which correspond to the faces of those cells [Bryant et al. , 1993]. All geometric features of these pore bodies and throats (such as their volume, surface area, etc.) follow directly from the known coordinates of sphere centers [Gladkikh , 2005 ]. In this work, a total number of 5,000 grains yields a network with about 25,000 pores and 52,000 pore throats. It is known that sometimes Delaunay tessellation results in “flat” tetrahedral cells (tetrahedron having a very large obtuse angle) [Al-Raoush et al. , 2003 ]. Following the work by Al-Raoush et al. [2003] , we implement an algorithm for merging such tetrahedra into larger pores. The number of such “flat” cells, however, is small (about 1% of all the pores in the packing) and their presence does not influence macroscopic properties of model rock.[12] The model of fluid flow through porous media is based on the methodology suggested in [Bryant and Blunt , 1992; Bryant et al. , 1993 ]. The idea is to compute fluid flow
numerically in the network extracted from the packing using modified Delaunay tessellation. We compute hydraulic conductance of each throat in the packing, write a mass balance equation for each pore, and solve the resulting system of linear equations using a modified method of conjugate gradients. The detailed description of the method, discussion of its accuracy, and comparison with experimental data can be found, for example, in the works by Bryant and Blunt [1992] , Bryant et al. [1993] , and Gladkikh [2005] .
4. Model of Acoustic Velocities[13] Acoustic methods are used widely to investigate the properties of rocks. Seismic surveys are very important to hydrocarbon reservoir and aquifer exploration. Acoustic measurements are routinely performed in well logging, and measured elastic velocities are usually correlated with formation porosity. This method is one of the most widely used porosity estimates. The ability to predict acoustic (namely, compressional and shear) velocities in model rocks cannot only improve interpretation of logging data, but also provide results for the independent assessment of constructed model rocks and developed methodology in general.[14] In order to predict acoustic velocities in the constructed model rocks, we use the following algorithm. When constructing a model rock, we specify mineral composition of the rock, namely, mass percentages of different minerals in total mass of solid rock material. Using elastic moduli of these mineral constituents, we compute bulk and shear moduli of composite multimineral rock matrix using
Hashin-Shtrikman bounds for multiphase systems [Berryman , 1995 ; Knackstedt et al. , 2005 ]:
where
Here Nmin is the total number of minerals in model rock; Ki and μi are their bulk and shear moduli, respectively; mi are mineral mass fractions in total mass of solid material.[15] Knowing the elastic moduli of the composite matrix, we apply the grain contact theory of Digby [1981] to compute elastic moduli of model rock skeleton, as was suggested by Bryant and Raikes [1995] in application to sphere packings:
where R is the average grain radius; Z is the average coordination number of grain contacts; a is the average cross-sectional area of grain contacts at zero confining pressure; ν and μ are the Poisson's ratio and bulk modulus of grain material; ϕ is the porosity of model rock. We compute average coordination number and average cross-sectional area of grain contacts directly from the pore geometric model of rock, assuming that all porosity loss is due to overgrowth cementation.[16] Further, we use Gassmann's equations [Gassmann , 1951 ] to compute acoustic velocities in the model rock (dry and saturated with water):
Here Vp and Vs are compressional and shear velocities, respectively; KB and μB are bulk and shear moduli of rock skeleton; K and μ are bulk and shear moduli of grain material (composite matrix); ϕ is the porosity of model rock; KW is the bulk modulus of water; ρdry and ρsat are the densities of dry and water saturated rock, respectively.[17] Figure 2 presents the comparison between the measured acoustic velocities for the water saturated Fontainebleau sandstone [Bourbie and Zinszner , 1985 ] and predictions using equations (1)–(8). The required material properties for equations (1)–(8) for this case (Fontainbleau sandstone is composed of quartz grains) are the densities and bulk moduli of quartz and water, and the shear modulus of quartz. These properties were taken from the work by Bryant and Raikes [1995] : quartz density is 2.65 g/cm3, water density is 1.03 g/cm3, bulk modulus of quartz is 41 GPa, bulk modulus of water is 2.2 GPa, shear modulus of quartz is 39 GPa. Figure 2 illustrates a good agreement between the predictions and the measured data.Figure 2.
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Prediction of acoustic velocities in water saturated model rock (porosity loss is due to overgrowth cement) and comparison with experimental data for Fontainebleau sandstone [Bourbie and Zinszner ,
1985].[18] Figure 3 presents a comparison between measured acoustic velocities for clay-rich sandstones by Han et
al. [1986] (small dots), finite element numerical simulations in corresponding clay-rich model rocks by Knackstedt et
al. [2005] (black lines), and predictions by the methodology described in this work (large dots). Following Knackstedt et
al. [2005] , we assume that the detrital grain material is quartz, having the following material properties: density of 2.65 g/cm3, bulk modulus of 37 GPa, and shear modulus of 44 GPa. Material properties of clay (assumed to occur as overgrowth cement) are also the same as those that Knackstedt et al. [2005] use: density is 2.60 g/cm3, bulk modulus is 20.8 GPa and shear modulus is 6.9 GPa. Water has a density of 1.0 g/cm3 and bulk modulus of 2.2 GPa. Figure 3illustrates good agreement not only between predictions and measurements (which can only have a qualitative character, since we do not have any information about type of clay and rock structure of samples used in the measurements by Han et al. [1986] ), but also between our predictions and accurate finite element numerical simulations of Knackstedt et al. [2005] . The latter suggests that the approximate acoustic theory that we use (equations (1)–(8)) adequately describes elastic properties of model rocks under assumed conditions.Figure 3.
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Prediction of acoustic velocities in water saturated cemented packing (overgrowth cement and clay) and comparison with experimental data for clay-rich sandstones [Han et al. , 1986 ] and finite element numerical simulations [Knackstedt et al. , 2005 ].
5. Model of NMR Response[19] Downhole measurement of Nuclear Magnetic Resonance (NMR) signal is probably the only logging data that contain information related to grain size. It is impossible to make an accurate prediction of the absolute permeability of the formation without such information. Numerical simulations of fluid flow, described above,
provide means to compute dimensionless permeability, and therefore their results must be scaled appropriately to obtain desired dimensional properties. NMR logging data therefore provide an opportunity to infer grain size for the successful scaling of dimensionless predictions. NMR techniques are usually employed in the petroleum industry to predict permeability, for fluid typing, and to obtain formation porosity, which is independent of mineralogy. The former application uses a surface relaxation mechanism to relate measured relaxation spectra with surface-to-volume ratios of pores and then the latter is employed to estimate permeability. The common approach is based on the model by Brownstein and Tarr [1979] . They have found an exact solution of the diffusion equation for total magnetization with surface-like sinks in an isolated spherical pore. This solution is a sum of decreasing exponential functions. Moreover, they have shown, that in the fast diffusion limit, given by the expression
where ρ is surface relaxivity of pore wall material, r is the radius of spherical pore, and D is water diffusivity, the lowest relaxation mode has much larger amplitude than other modes, and in this case total magnetization decays in time as a single exponential function:
where M0 is the initial magnetization and the transverse relaxation time T2 is given by
where S/V is the surface-to-volume ratio of the pore. The model, described by equations (9)–(11), is applied widely in the petroleum industry, often disregarding assumptions that were used in its derivation. The real pore space, for
example, does not consist of isolated spherical pores, but is rather a complicated network-like structure of converging-diverging sections. Nonsphericity of pores questions the derivation ofequations (9)–(11). Higher relaxation modes of a single pore may give significant contribution in the case of real pore space, when the fast diffusion limit (10) is not satisfied. The connections between pores make pore-to-pore coupling possible. Finally, the possibility of the rock having multimineral composition questions the use of a uniform value for the surface relaxivity even within a single pore.[20] There exist numerical models that use random-walk Monte Carlo algorithms to simulate relaxation process [Straley and Schwartz , 1996 ; Toumelin et al. , 2003 ]. This approach automatically resolves all complications in NMR modeling, mentioned above. Unfortunately, high computation costs preclude its application to log interpretations. The problem is that grain size is usually unknown a priori, and one needs to derive it from the measured relaxation spectra. In this case much faster algorithms are needed.[21] Here we suggest the following approach. It is based on predicting NMR transverse relaxation spectrum in a random dense packing of spherical particles of uniform size (Figure 1). This uniform size is an approximation of various grain size distributions that occur in real sediments; in the case of such distributions, the mean value of grain size can be used in the model. Modified Delaunay tessellation subdivides pore space of the packing into tetrahedral pores (see section 3 above). Surface-to-volume ratio is easily computed for each pore [Gladkikh , 2005 ]. If the values of grain size and surface relaxivity are specified,
the application of equation (11) immediately yields T2 spectrum for the packing fully saturated with water. Figures 4 and 5depict a comparison between experimental T2 spectrum (unconsolidated and fused glass beads, grain diameter is 100 μm [Straley and Schwartz , 1996], Figure 4) and predictions for the numerical packing with isopachous cement (Figure 5). Predictions were made using values of surface relaxivity, reported by Straley and
Schwartz [1996] . The agreement between measured and simulated spectra is very good.Figure 4.
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Experimental NMR T2 distribution for fused glass bead packs [Straley and Schwartz , 1996 ].Figure 5.
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Predicted NMR T2 distribution for numerical packing with isopachous cement.[22] In the case of multimineral composition of grains (i.e., when the pore is bounded by grains composed of quartz and feldspar), equation (11) is modified as
where the sum is taken over all grains that form the pore, and ρi and Si are the corresponding values of surface relaxivity and surface area for each grain.[23] If pore-filling cement is present in the neighboring pore, equation (12) is modified to include the term, corresponding to the surface area of cement at the pore throat that subdivides these two pores, multiplied by surface relaxivity of the pore-filling cement (e.g., calcite).
[24] Presence of authigenic clay minerals is accounted for in similar fashion. Pore-lining clay exists as a thin coating layer on the surface of grain i and thus contributes to equation (12). The surface area of such coating that contributes to the NMR relaxation is computed from the specific surface area of clay mineral (usually available as an independent measurement for pure clay specimen in m2/g by BET adsorption method, see, for instance, Brunauer
et al. [1938] , specified porosity, density, and thickness of the coating. Pore-filling clay is assumed to occupy individual pores completely; therefore we apply equation (11) to compute NMR transverse relaxation time of the pore, filled with clay. In this case we use a single value for the surface relaxivity of clay. The surface area S of the pore filled with clay in equation (11) is computed from the specific surface area of the clay mineral (knowing the volume of the pore, clay porosity and density).
6. Prediction of Permeability From NMR T2 Distribution[25] One of the most important applications of NMR logging is permeability estimation. The most widespread model used to compute permeability from measured NMR transverse relaxation time spectrum is the Timur-Coates equation [see, e.g., Coates et al. , 1991 ], which employs the subdivision of the relaxation spectrum into the contributions from “movable” (Bulk Volume Movable, BVM) and “capillary bound” (Bound Volume Irreducible, BVI) fluids:
Here C is an empirical constant and BVM + BVI = ϕ (porosity of the rock). For a given T2 distribution, the values
of BVM and BVI are defined by some threshold value of transverse relaxation time T2cutoff. This threshold is also purely empirical; the value of 33 ms is commonly used for sandstones. The basis ofequation (13) is, in fact, a Kozeny-Carman type [see, e.g., Carman , 1956 ] relationship for permeability derived from the bundle of capillary tubes model. In order to use equation (13), one must first determine two empirical parameters (T2cutoff and C), which question practical applicability of the model.[26] In this work, we suggest a different approach: instead of attempting to derive an equation for computing permeability from NMR data, we invert relaxation spectrum to infer grain size. First, we create the model rock on the basis of the information about porosity (which can either be inferred from acoustic measurements or taken as a sum of amplitudes for full T2 distribution) and mineralogy. Then we compute dimensionless absolute permeability of the model rock, as described above (section 3). Further, we compute NMR T2 distribution numerically for a range of grain sizes, as described in the previous section. As an estimate for mean grain size, we choose the value that provides the closest fit to the measured data. As a criterion for fitting we use the minimization of net quadratic error for all bins in the relaxation spectrum. This value of the grain size is used to scale computed dimensionless permeability to its dimensional value.[27] It should be pointed out here, however, that this technique is based on the Brownstein and Tarr [1979] model (equations (9)–(11)) and therefore inhibits all the limitations described above. It is a subject of future work to develop the technique further to account for nonspherical pores, pore-to-pore coupling, and different relaxation
regimes (when fast diffusion limit, equation (9), is not satisfied).[28] We first test our methodology for cores with known microstructure, mineral composition, and macroscopic properties. For this purpose, we present here a comparison between predictions and measurements for two Berea sandstone cores, labeled here BSS1-1 and BSS8-8. The Berea sandstone is a good candidate for such a comparison, because it does not contain abundant clay minerals or paramagnetic minerals which could significantly complicate NMR relaxation process. Thus we expect that our pore-scale model would adequately describe the pore geometry of the Berea sandstone cores. Figures 6 and 7present microphotographs of thin sections of these samples in plain nonpolarized light. The obvious difference between these two cores is that sample BSS8-8 (fine sand) has much smaller grain sizes than BSS1-1 (coarse sand). This is confirmed by the laser grain size analysis for these cores (Figure 8 shows grain size distribution for BSS1-1, and Figure 9 shows that for BSS8), as well as the analysis of thin sections. Average grain radius that we infer from these data to use for the simulations is approximately 200 μm for BSS1-1 and 80 μm for BSS8-8. BSS1-1 has porosity about 31% and permeability to air 13,263 md. BSS8-8 has porosity about 20% and permeability to air 43 md. Thin section analyses indicate that sample BSS1-1 contain abundant feldspar minerals including K-feldspar and plagioclase, while BSS8-8 is quartz-rich. The abundance of feldspar in BSS1-1 is evident from Figure 6 (grey turbid grains). This is confirmed by the X-ray diffraction mineralogical data, which provides the composition of sample BSS1-1 as 47% quartz, 29%
feldspar, 13% calcite, 9% illite, and 2% anhydrite. XRD data for sample BSS8 is the following: 78% quartz, 2% feldspar, 14% calcite, and 6% illite.Figure 6.
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Nonpolarized light microphotograph of Berea sandstone sample BSS1-1. The scale on the photograph is 1.6 mm.
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Nonpolarized light microphotograph of Berea sandstone sample BSS8-8. The scale on the photograph is 1.6 mm.
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Laser grain size analysis for Berea sandstone sample BSS1-1.
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Laser grain size analysis for Berea sandstone sample BSS8-8.
[29] We computed absolute permeability and NMR relaxation spectrum for both samples using two simulations for each core. In the first simulation we did not use the mineral composition data and assumed that the model rock consists of pure quartz and all porosity loss is due to isopachous cement. Furthermore, using measured average grain size (200 μm for BSS1-1 and 80 μm for BSS8-8), we computed the absolute permeability of the model rocks and corresponding NMR relaxation spectra. For the latter
simulation we have chosen the value of quartz surface relaxivity so that computed and measured spectra fit. These values are 100 μm/s for BSS1-1 and 50 μm/s for BSS8-8. The value for BSS1-1 is higher than usually reported for sandstones [Roberts et al. , 1995 ]. Corresponding computed permeabilities in the model rocks are 47,976 md for BSS1-1 and 1,757 md for BSS8-8, which is substantially higher than the measured values for both cores.[30] Figures 10 (BSS1-1) and 11 (BSS8-8) present the comparison between the measured T2 relaxation spectrum for the cores and results of two types of simulations. The first type (pure quartz) was described above. For the second type we used mineral composition taken from the X-ray diffraction data. For BSS1-1, we used the following parameters. Mineral composition: 57% quartz, 30% feldspar, 8% calcite, 5% illite; surface relaxivity: quartz, 50 μm/s; feldspar, 200 μm/s; calcite, 10 μm/s; illite, 0.3 μm/s; specific surface area of illite, 93 m2/g; its porosity, 36%. Computed permeability is 17,490 md and corresponding NMR relaxation spectrum is shown on Figure 10. There is a good agreement with core measurements for both NMR relaxation spectrum and absolute permeability.Figure 10.
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Comparison between measured NMR T2 relaxation spectrum, prediction with the assumption of monomineral quartzose sandstone, and prediction using mineral composition data. Berea sandstone sample BSS1-1.[31] The amount of calcite that we use to create a model rock is smaller than that indicated by XRD data. This is because (1) the porosity of the sample BSS1-1 is too high
(31%) to accommodate 13% (of total solid mass material) of calcite cement and (2) the high degree of calcite cementation is not confirmed by thin section analysis, which does indicate localized sparse calcite cement, but no whole-scale cementation. On the other hand, thin section analysis of sample BSS8-8 confirms presence of calcite cement extensive enough to partially occlude intergranular porosity.[32] It should also be pointed out here that much larger values of surface relaxivities for quartz, feldspar, and calcite as compared to that of illite are to be expected. We use equation (12) with geometric surface areas (assuming smooth spherical grains) for detrital grains. At the same time we use specific surface area of illite as measured by the BET adsorption method [Brunauer et al. , 1938 ], which usually gives surface areas of several orders of magnitude higher than corresponding geometric value.[33] When computing absolute permeability in a model rock containing porous pore-filling clay, we account for the fluid flow through the pores filled with such clay. We assume that pore-filling clay can be represented as a random close sphere packing with particle size of 1 μm, further cemented to accommodate for the specified porosity of clay. Fluid flow through the pore filled with such clay is modified on the basis of precomputed permeability of the sphere pack, using algorithms, described in section 3. The values of hydraulic conductances of pore throats, neighboring such a pore, are then modified accordingly. Such an approach for hydraulic conductivity of pore-filling clay is by no means accurate, but a more realistic model would demand a construction of detailed structure of clay minerals, which is outside of the
scope of this paper. It should be pointed out here, however, that if size of detrital grains is much larger as compared to clay particles, the contribution to permeability from the pores and throats filled with clay will be negligible.[34] For sample BSS8-8, we used the following parameters. Mineral composition: 78% quartz, 2% feldspar, 14% calcite, 6% illite; surface relaxivity: quartz, 50 μm/s; feldspar, 200 μm/s; calcite, 10 μm/s; illite, 1μm/s; specific surface area of illite, 93 m2/g; its porosity, 36%. Computed permeability is 30 md and corresponding NMR relaxation spectrum is shown on Figure 11. There is a good agreement with core measurements for both NMR relaxation spectrum and absolute permeability. Measured T2 spectrum has wide distribution, whereas simulation results show two distinct peaks: one corresponding to pore-filling clay (at about 10 ms), and the other to clay-free pores, bounded by detrital grains. This difference can be attributed to diffusive coupling between clay-free and clay-filled pores, which is neglected in our model. Considering this effect is a subject of future research.Figure 11.
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Comparison between measured NMR T2 relaxation spectrum, prediction with the assumption of monomineral quartzose sandstone, and prediction using mineral composition data. Berea sandstone sample BSS8-8.[35] Computed values for absolute permeabilities are in very good agreement with core measurements for both cores. The comparison between measurements and predictions for both NMR spectra and absolute permeabilities illustrates the applicability of the proposed
approach and indicates that knowledge of mineral rock composition can significantly improve petrophysical predictions.[36] It should be pointed out here that the choice of the value of surface relaxivity is very important to estimate grain size and therefore permeability. Surface relaxivity, however, is not an adjustable parameter, and has a clear physical meaning. Surface relaxivities of different minerals can be measured in the laboratory and data are available in the literature [see, e.g., Roberts et al. , 1995 ]. When simulating T2distribution for Berea sandstone samples, we are choosing values of surface relaxivity so that T2distribution matches the experimental data and then compare computed and measured permeabilities in order to estimate the accuracy of the pore-scale model of the particular rock. At the same time, we are trying to be consistent with the values of surface relaxivity that we choose so that they remain the same for all simulations. The two differences are (1) higher surface relaxivity of quartz in a pure quartz model for the sample BSS1-1 (which we have to change in order to fit experimental T2 distribution) and (2) surface relaxivity of illite in the simulations for two Berea samples. The second, however, influence neither the position of the main peak in T2 distribution nor the computed value of permeability. It is possible that illite in these two samples might have slightly different physical properties (which results in different position of the peaks).
[37] Figure 12 presents the log of a well from Johnson City and its petrophysical interpretation. Johnson City Stribling #3 well was drilled using a 6 11/16'' bit size, through well-consolidated formations. The borehole traverses various clean to very shaly sands comprising fresh water aquifer zones. The portion of the well shown on Figure 12 consists primarily of clean quartzose and feldspathic sandstones. The well has been extensively cored during the drilling. Data obtained from the cores include porosity, grain density, and permeability. The core data were depth-matched, interpolated to the level spacing of the logs (4 samples/foot), and filtered (using an 11 points weighted filter, the standard one used for 4samples/foot logs) in order to provide a fair comparison with the log data (for the information on NMR logging tool principles [see, e.g., Edwards , 1997 ]). Since only water is present in the formation, the single phase model described above is applicable here. Track 1 shows depth in feet. Track 4 shows NMR T2 relaxation spectrum in milliseconds. Track 2 presents porosity: total NMR porosity (MPHS, the integration of the complete T2 distribution curve is assumed to represent the total porosity of the rock); NMR Clay Bound Water (MCBW, signal below 3 ms); NMR Bound Volume Irreducible (MBVI, signal below 33 ms), and NMR Effective Porosity (MPHE, total minus MCBW). Track 5 shows basic mineralogy of the formation (as mass percentages in total solid material), derived independently by other means. It comprises quartz, K-feldspar, and calcite. Track 3 depicts permeability: core data (brown dots are actual core permeabilities and red curve is filtered core data); prediction using Timur-Coates model, equation (13) (blue curve), and predictions of the model described in
this paper (magenta curve, with the use of mineralogy on Track 5; green curve, without it). When making petrophysical interpretation employing mineral composition data, we compute the ratio of quartz/feldspar grains from the quartz/feldspar ratio on Track 5. Calcite is assumed to be present only in the form of pore-filling cement. If total NMR porosity is smaller than can be accounted for by the presence of calcite cement, remaining porosity loss is assumed to be due to quartz overgrowths. In the prediction without use of mineralogy (green permeability curve), we assume that the formation consists only of quartz grains and the only type of cement is quartz overgrowths.Figure 12.
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Logging data from the Johnson City well and prediction of permeability.
[38] All permeability models were calibrated to one point (depth 1236 ft). Core measurements at this depth provide porosity of 9.3% and permeability of 31 md (MPHS reads porosity of 8.5%). Parameter C inequation (13) was chosen so that permeability prediction matches core data. This value is C = 6.5 (the default value of C to use in equation (13) is usually equal to 10). MPHE was used for ϕ in equation (13), MBVI for BVI, and MBVM = MPHE-MBVI for BVM. For pore-scale modeling, we used the value of 30 μm/s for surface relaxivity of quartz (which provides the good agreement with the measurement at 1236 ft for the monomineral model and is within the range of values, reported in the literature for sandstones [Roberts et
al. , 1995 ]; the value of 30 μm/s for surface relaxivity of
calcite; and the value of 100 μm/s for surface relaxivity of feldspar (which should be higher than that of quartz because of weathered surfaces of K-feldspar grains).[39] The comparison between the three predictions and core permeabilities shown in Figure 12 allows several important observations. All models are able to provide successful predictions in a highly permeable quartzose zone, where the NMR signal exhibits sharp single peak (1084–1094 ft, indicated by the red box on Figure 12). Equation (13), however, being extremely sensitive to the value of BVI, fails when NMR log shows a substantial signal at shorter times. See, for example, depths around 1164 ft (indicated by the blue box on Figure 12), where the core data indicate a permeable formation and the Timur-Coates model yields basically zero permeability. This happens because of the very large value of BVI (almost all the signal). This shift of T2 relaxation spectrum toward shorter times is most likely due to the presence of weathered feldspars and iron oxides. These minerals form and are absorbed to the grain surfaces, and have nothing to do with permeability decrease. Another problem of equation (13) is evident when BVI appears to be unreasonably small (see the region around the depth 1202 ft, indicated by the green box on Figure 12; BVI here is approximately 1% of total rock volume and permeability computed by equation (13) is about 400 md). In this case (which is in all probability simply an effect of large grain size) equation (13) overestimates permeability by more than an order of magnitude. Interpretation based on pore geometric models, on the other hand, even without using mineralogy (green curve), is more consistent with core data. If mineralogy is used (magenta curve), predictions are significantly
improved just on the basis of the amounts of quartz and K-feldspar (compare predictions in region 1068–1080 ft (magenta box) when significant amount of feldspar is present).[40] It has to be pointed out here that one has to upscale pore-scale predictions (mm) to a log scale (ft). We assume that the homogeneous pore-scale model that we use (Figure 1) is representative of the formation at each given depth, and their macroscopic properties therefore correspond to each other. This upscaling is only possible in homogeneous formations that do not exhibit significant heterogeneity or layering below vertical resolution of logging tools. To test our methodology, we have chosen the portion of the well that is very homogeneous and formation properties change smoothly from depth to depth.[41] Pore-scale approach, described in this work, is based on simple, but reasonable geological and physical assumptions. Having in mind these assumptions, it is possible to identify the reasons for the failure of the model in the zones where it does not work. It should be pointed out here, that the presented pore geometric model is only applicable for reservoir sandstone rocks (for example, we do not expect this model to be useful in shales). In such cases more geologic data are necessary in order to build a more representative pore-scale model. On the other hand, empirical equations, like equation (13), do not provide any possibility for different physical interpretation, and one usually adjusts empirical coefficients (C and T2cutoff for the case of equation (13)) in the zones where this empirical equation fails.[42] The application of the presented technique is not limited to the case when only one phase is present in the
formation, as was considered throughout this paper. Fluid typing, based, for example, on diffusivity contrast [Kleinberg
and Vinegar , 1996 ], allows subdividing relaxation spectrum into contributions from different fluids (i.e., water and hydrocarbon). Having thus a signal only from the wetting phase (which is usually water) and knowing therefore its saturation, we simulate drainage (or imbibition, depending on the reservoir conditions), until this saturation is reached. For this purpose we use previously developed algorithms [Bryant and Johnson , 2003 ; Gladkikh and Bryant , 2005 ; Gladkikh , 2005]. Surface areas and volumes of different wetting phase configurations are computed directly from pore geometry [Gladkikh , 2005 ], thus contributing to NMR relaxation spectrum by means of equation (11).[43] Moreover, presented methodology can be combined with the previous research in pore-scale modeling to predict capillary pressure curves [Mason and Mellor , 1995; Bryant and Johnson , 2003 ; Gladkikh and Bryant , 2005; Gladkikh , 2005 ], relative permeabilities [Bryant and
Blunt , 1992 ; Gladkikh , 2005 ], and electrical properties of the formation [Bryant and Pallatt , 1996 ; Gladkikh , 2005 ; Gladkikh et
al. , 2006 ].
8. Conclusions[44] This work presents the approach for petrophysical interpretation (prediction of absolute permeability) of downhole logging measurements using detailed geometric pore-scale description of physically representative model porous medium (numerical dense random packing of equal spheres). Porosity and mean value of grain size of the model rock are derived from logging data (Nuclear Magnetic Resonance transverse relaxation time spectrum).
The methodology was applied to the selected set of well logging data. The predictions of absolute permeability of the formation were found consistent with existing core data. It was shown also that predictions can be significantly improved if the mineral composition of the formation is known.Notation
BVIBound Volume Irreducible, fraction of porosity.
BVMBulk Volume Movable, fraction of porosity.
Cempirical constant of Timur-Coates equation (13).
Dwater diffusivity, μm2/s.
Kelastic bulk modulus of composite matrix, GPa.
KBelastic bulk modulus of rock skeleton, GPa.
KCabsolute permeability computed by Timur-Coates equation
(13), md.Ki
elastic bulk modulus of the mineral i, GPa.KW
elastic bulk modulus of water, GPa.mi
mass fractions of mineral i in total mass of solid material.M0
initial total magnetization, A/m.M(t)
total magnetization as a function of time, A/m.Nmin
