a rock physics model for tight gas sand

1484 The Leading Edge December 2010

SPECIAL SECTION: T i g h t g a s s a n d sT i g h t g a s s a n d s


SPECIAL SECTION: T i g h t g a s s a n d sSPECIAL SECTION: R e v e r s e t i m e m i g r at i o nSPECIAL SECTION: T i g h t g a s s a n d s

Tight gas reservoirs are often defined as gas-bearing sandstones or carbonates

having in-situ permeabilities to gas less than 0.1 mD (Holditch, 2006; Smith et al., 2009). Tight gas reservoir rocks can be at different in-situ physical conditions: deep or shallow; over- or underpressured; high temperature or low temperature; and under different stress states. The reservoir-forming rock can have different textures such as shaley and silty unconsolidated sandstones or clean-cemented sandstones. These different rocks produce gas at low rates. Tight reservoir rocks can be blanket or lenticular, homogeneous or heterogeneous, and can contain a single layer or multiple layers, be fractured or unfractured, and mainly produce dry natural gas.

Tight gas reservoirs pose textural fea-tures below the resolution of conventional seismic data and rock microstructures and mineralogy com-plexity that may obscure the presence of gas in the pore space. If the elastic properties and behavior of these low-porosity rocks are understood, feasibility studies can be conducted and the effect of fluids, heavy and soft minerals, and rock micro-structural parameters can be analyzed at seismic scales. If rock physics modeling is conducted using theoretical approxima-tions consistent with the specific microstructure, then it may be possible to detect prospective areas.

One distinguishing feature of low-porosity and low-per-meability sandstones is the departure of the behavior of their elastic properties from that of more conventional porous and permeable sands. Smith et al. showed examples of such be-havior. In particular, in a typical velocity versus porosity plot, the zero-porosity intercept crosses the velocity axis at a value lower than that consistent with the pure mineral composi-tion of the rock. Smith et al. postulated that this is due to the presence of cracks in these rocks, and suggested using either single- or multiple-aspect-ratio cracks to model this behavior.

In this study, we deal with low-porosity and low-perme-ability consolidated and well-cemented rocks that may con-tain randomly distributed microcracks, or crack size below well-log resolution, by expanding on the idea first suggested by Smith et al. Several different effective medium approxi-mations (e.g., O’Connell and Budiansky, 1974; Kuster and Toksöz, 1974; Berryman, 1980; Norris, 1985) can be used to estimate the elastic moduli of rocks with low porosity and poor pore connectivity. In this paper, we use the self-consis-tent model (SC) of Berryman in our analysis, instead of the Kuster-Toksöz model used by Smith et al. We are using the isotropic assumption because the well logs we have available do not include cross-dipole shear-wave logs and thus we have no information about the anisotropy of the formation. Smith

FRANKLIN RUIZ and ARTHUR CHENG, OHM Rock Solid Images

et al. presented cases where such models can be applied to anisotropic rocks if cross-dipole shear-wave logs are available. Both pores and grains are modeled as randomly oriented ob-late spheroid inclusions, with aspect ratios ( ) between 0 and 1.

To determine the elastic properties of tight gas sands, we propose an approach based on the ideas of Smith et al. This approach consists of dividing the rock pore space (total po-rosity) into two pore spaces: the soft pore space (soft porosity) and the stiff pore space (stiff porosity). The soft porosity ( soft) is defined here as the volumetric fraction of soft inclusions with a fixed aspect ratio = 0.01; the stiff porosity ( stiff) is defined as the volumetric fraction of spherical inclusions with a fixed = 1. The elastic properties of the rock matrix are estimated using the average of the multimineral version of the Hashin-Strikman (1963) upper and lower bounds (HSA). Once a rock microstructure has been assumed, the methodol-ogy consists of matching well-log sonic data (at each depth location) with the SC velocities and finding the soft porosity required to accomplish this match. After calculating soft, we can conduct fluid substitution by filling the pores with dif-ferent fluids, spatially distributed in diverse ways. Finally, we conduct fluid substitutions using effective medium approxi-mations and compare them with the results from Gassmann’s equations. The mathematical flexibility of inclusion theories enables such exercises because by varying soft (concentration of pores with = 0.01) and stiff (concentration of pores with

= 1) it is possible to match almost any observed well-log data point.

We compare our approach with the one proposed by Ruiz and Dvorkin (2010) where the measured velocities are matched with the effective medium velocities (DEM or SC) by looking for the single aspect ratio ( ) of the inclusions

Figure 1. (a) Soft-porosity model (SPM). (b) Soft and stiff pores are filled with a mixture of brine and gas. The effective bulk modulus (kf) of this fluid mixture is given by the iso-stress average of kw and kg of the in-situ brine and gaseous phases, respectively. (c) Brine filling the soft pores and the remaining water filling some of the stiff pores. Gas fills the rest of the spherical pores.

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December 2010 The Leading Edge 1485

T i g h t g a s s a n d s

and the only variable is soft(zi).soft is determined by minimizing the difference between

Vpoband Vp

SC, as a function of soft, generally to less than 1%. The process of finding soft(z) also allows us to determine VS

SC, and thus predict the S-wave velocity if one is not available.

Once we find soft(z), fluid substitution can be conducted, treating fluid inclusions as another constituent of the rock. Different fluids, such as brine and gas, can be placed in differ-ent parts of the pore space—for instance water in soft pores and gas in stiff pores, or water in soft pores and part of it in stiff pores (Figure 1). The way we distribute the fluids in the

required at each well-log depth location. The difference between these two method-ologies is that, when looking for soft, the aspect ratio of the soft and stiff pores is constant at all depths, whereas in Ruiz and Dvorkin’s methodology different values of

are used at different depths. These two approaches produce comparable but slight-ly different results.

We proceed by applying the soft-poros-ity model (SPM) and demonstrate its util-ity on logs from a Venezuelan well available in the literature (Graterol at al., 2004; Ruiz and Dvorkin, 2010).

Soft-porosity model (SPM) and single-aspect-ratio model (SAR)To estimate the effective elastic properties of a low-porosity tight gas sand, we pro-pose the soft-porosity model (SPM). This model treats the tight multimineral rock as a composite with a continuous rock matrix (host) with embedded spherical pores (stiff pores) and crack-like pores (soft pores). Both pore types are randomly distributed and oriented in space (Figure 1). SPM di-vides the total pore space (total porosity) into two spaces: the stiff-pore space (stiff porosity) and the soft-pore space (soft po-rosity). Soft porosity ( soft) and stiff porosi-ty ( stiff ) are defined here as the volumetric fraction of oblate spheroid inclusions of as-pect ratio 0.01 and spheres of aspect ratio 1, respectively. We picked the aspect ratio of 0.01 for the soft porosity based on an empirical fit to the data. For small-aspect-ratio cracks, the important parameter is the crack density (O’Connell and Budiansky), which is pro-portional to the crack porosity divided by the aspect ratio. The exact aspect ratio chosen is of secondary importance. The choice of aspect ratio 1 for stiff porosity is based on the fact that spherical pores are the stiffest of any pore shape (Mavko et al., 2009).

The relation between total, stiff, and soft is:

total = soft ( = 0.01) + stiff ( = 1) (1)

The process of finding soft begins by estimating the effec-tive elastic moduli of the solid multimineral matrix using the HSA. Then, the effective elastic moduli of the hypothetical rock (Figure 1), at a certain concentration of stiff pores ( stiff) and soft pores ( soft), are calculated using the SC model. The methodology consists of matching well-log sonic P-wave ve-locity data (Vp

obs) with theoretical SC-velocities (VpSC) and de-

termining the amount of soft porosity ( soft), at each well-log depth zi, required to accomplish this match. Here the aspect ratios of the soft and stiff pores are kept constant at all depths

Figure 2. VP and VS prediction for a Venezuelan well. (left) Mineral and fluid (brine) volume fractions. (middle) VP predicted (red) from measured VS and VS predicted (blue) from measured VP using the single-aspect-ratio model (SAR); the measured velocity logs are black. (right) Same as the middle panel for the soft-porosity model (SPM).

Mineral k (GPa) μ (GPa) ρ (g/cm3)

Quartz 36.6 45 2.65Clay 21 7 2.58Calcite 76.8 32 2.71Pyrite 147.4 132.5 4.93

Table 1. Bulk (k) and shear (μ) moduli of the minerals used in the modeling (Mavko et al., 2009).




pore space (Figure 1b-c) depends on the in-situ hydrocarbon and brine saturations, Sh(z) and Sw(z), respectively, the rock’s wetability, and other in-situ physical conditions at each well-log depth. In the minimization process, Sh(z) and Sw(z) are considered observable input parameters, so we need to specify how the fluids are placed in the pore space. The simplest case is using an effective fluid modulus (kf) for the pore fluid that is an iso-stress average of the bulk moduli kw and kh of the in-situ brine and hydrocarbon phases, respectively (Figure 1b).

If the P-wave velocity log VP instead of the S-wave log VS is unavailable, this approach can be used to predict VP based on the measured VS as well.

This approach is similar to the single-aspect-ratio model (SAR) proposed by Ruiz and Dvorkin. SAR assumes that the total pore space, at a specific well-log depth location, con-sists of oblate spheroidal pores of the same aspect ratio. SAR methodology matches measured data and mineralogy with theoretical values derived using an effective medium theory (e.g., SC or the 1985 DEM model of Norris) and finding the single aspect ratio of the inclusions required to achieve this match.

Although the actual rock may not be like a physical real-ization of our model (SPM), we find the elastic equivalency

between the theoretical model and data. The hypothesis is that if such elastic equivalency can be established for many log data points, we can find an idealized physical analog to real rock and then use this analog to show a relationship be-tween different well-log properties.

An advantage of the SPM approach compared with SAR is that instead of an average aspect ratio for the entire pore space, the SPM model separates it into an equal-dimensional part (stiff porosity) and a crack-like part (soft porosity). The stiff porosity can be thought of as intergranular pores and the soft porosity as grain contacts (see Figure 6b of Smith et al.). Thus it can be argued that SPM represents the pore space more realistically.

The way we test these models (SPM and SAR) using well data is by predicting the elastic properties in the logged inter-val and comparing them with measured data. In the logged interval, we assume we know the type and volumetric frac-tions of minerals, porosity, fluid type and saturations Sh(z) and Sw(z). These properties are commonly available from a standard suite of logs.

ResultsIn this section, we show the results of using SPM to make

Figure 3. Predicted versus measured VP and VS. (top) Predicted versus measured VP (left) and VS (right) from SAR. (bottom) Predicted versus measured VP (left) and VS (right) from SPM.




velocity predictions and conduct fluid substitution. The results are compared with those from SAR. We show a modeling example using well-log data from a Venezuelan oil field that in-cludes measured VP and VS (Graterol at al., 2004).

Comparison of velocity prediction us-ing SPM and SAR. In the entire well in-terval, the rock’s mineral constituents are quartz, calcite, clay (mixed clays), and small amounts of pyrite (< 0.01), and the pore fluid is brine, (i.e., Sw(z) = 1). The elastic moduli and density of the mineral constituents used are shown in Table 1. The total poros-ity ( total) is calculated from the mea-sured bulk density ( ) and the min-eral constituents densities. In this well, the soft porosity and stiff porosity are filled with fluid with salinity, density, and bulk modulus, which are constant with depth. The salinity of the brine is 40,000 ppm and is assumed constant in the entire well interval. The bulk modulus and density of the brine are calculated according to Batzle and Wang (1992) using site-specific mea-sured temperature and pressure.

In Figure 2, we compare the mea-sured VP and VS with estimated curves using SPM and SAR. In both cases we fit one of the velocities, either VP or VS, and predict the other. Both approaches produce predictions that are in overall agreement with the measured data, but the SPM results appear slightly better.

To further examine the fit of the two methodologies, we crossplotted the predicted velocities versus observed velocities for both P- and S-wave velocities for both models (Figure 3). Although both the SAR and SPM model predictions agree with measured data quite well (the crossplotted points cluster around the 1:1 line), the SPM model does a slightly better job. The SAR results appear to overpredict the P-wave velocities and underpredict the S-wave velocities. We thus believe that the SPM model is more appropriate for these low-porosity, low-permeability sands.

Another way to look at how the model behaves is to look at the soft-porosity log, soft(z), generated from the fit to the data. Figure 4 shows the two soft(z) logs, one generated from the fit to the P-wave log and the other from the fit to the S-wave log. There are minor differences in the two soft-porosity logs, and these are attributable to the differences in the origi-nal P- and S-wave logs. Both logs show a qualitative relation-ship between increasing crack or soft porosity with increasing quartz content. In general, the two logs are quite consistent. This shows that this methodology is capable of generating

stable and reproducible results.Comparison of fluid substitution approaches using SPM,

SAR, and Gassmann. Gassmann’s relations are used to esti-mate the variation of the low-frequency elastic moduli of porous media caused by a change of pore fluids (Mavko et al., 2009). At low seismic frequencies, pore pressures are equilibrated throughout the pore space. Gassmann’s equa-tion assumes that: the solid matrix moduli are homogeneous; the pore space is isotropic; the rock is fully saturated; and all pores are connected. However, it is free of assumptions about the pore geometry. Inclusion models (e.g., SC and DEM) can be used to estimate the variation of the high-frequency elastic moduli of porous media caused by a change of pore fluids. Inclusion models are applicable only to statistically homo-geneous media, are based on idealized representations of the rock microstructure and pore-space morphology, and are not free of assumptions about pore geometry. The inclusions are isolated with respect to flow, so they simulate the behavior of higher-frequency saturated rock. Fluids are treated as another

Figure 4. (left) Mineralogy and porosity (filled with brine). (middle) Soft porosity derived from matching measured VP with predicted VP from SPM. (right) Soft porosity derived from matching measured VS with predicted VS from SPM.




constituent of the rock, so different fluids can be randomly placed in the pore space: for instance, water in low-aspect-ratio pores and gas in high-aspect-ratio pores. Well-log sonic frequencies may or may not be within the range of validity of Gassmann’s equation, depending on the connectivity of the pores and the fluid viscosity (Mavko et al.). Thus, to account for the effect of fluids in rocks with low or poor pore con-nectivity, effective medium models are more appropriate than Gassmann’s relations.

In this work, we conduct fluid substitution using: (a) Gassmann’s equations, (b) single-aspect-ratio SAR, and (c) the soft-porosity model (SPM). In the latter two cases, the soft porosity is calculated using the self-consistent SC model. We consider two saturation conditions: (1) rocks fully satu-rated with brine (the in-situ fluid) and (2) rocks saturated with a mixture of 80% gas and 20% brine (Figure 5).

The bulk moduli of the brine and gas vary with the in-situ physical conditions at different depths, and are calculated according to Batzle and Wang. The brine salinity is 40,000 ppm, the gas-water ratio is 0, and the gas gravity is 0.65. In the example shown in Figure 5, we assume that the brine and gas phases are mixed at the finest scale and use the effective fluid modulus (kf) for the fluid mixture that is an iso-stress av-erage of the moduli kw and kg of the in-situ brine and gaseous

phases, respectively. We can observe in Figure 5 that the SPM and SAR mod-

els predict a different behavior of the rock in the Poisson’s ratio (PR)-acoustic impedance (AI) crossplot. There is not much difference in the behavior between the fully saturated (open circles) and partially saturated rock (solid circles) un-der Gassmann fluid substitution. This is expected since these rocks have low porosity. However, because of the addition of nonspherical crack-pore spaces in both the SAR and SPM models, substantial differences are predicted in the behavior of the fully and partially saturated rocks. The same can be said about the predictions in the AI-density crossplots.

The SAR and SPM predictions have subtle but significant differences. Some lower PR/AI values for partial saturation are predicted by SPM and, in general, there is slightly more separation between the fully and partially saturation rocks in the AI-density crossplot for the SPM model.

Discussion and conclusionThere have been quite a number of rock models using cracks or low-aspect-ratio pores. We have based our soft-porosity model on a suggestion in Smith et al. Other analogous ap-proaches were proposed by Xu and White (1995) and Mar-kov et al. (2005). Xu and White estimate the velocities of dry

Figure 5. Fluid substitution results from log data shown in Figure 2. Open circles represent brine saturation, and closed circles partial gas (80% gas and 20% water) saturation. (top) Poisson’s ratio (PR) versus acoustic impedance (AI) using Gassmann (left), the single-aspect-ratio model (middle), and the soft-porosity model (right). (bottom) Acoustic impedance versus density for the same three cases.




rocks in shaley sandstones using a DEM approach based on the Kuster and Toksöz approximation and Gassmann’s equa-tion to estimate the low-frequency effective velocities of wet rock. In their model, the effective rock solid matrix velocities are estimated by the Wyllie time average. The approach of Markov et al. assumes that the rock is composed of a ho-mogeneous matrix with randomly distributed small-scale-disconnected primary pores and large-scale-connected sec-ondary pores. In this model, fluid flow is allowed only in the secondary pore space. The effective elastic moduli of the rock are estimated by calculating the effective elastic properties of the matrix with the primary small-scale pores and then treat-ed as a homogeneous isotropic host in which the large-scale secondary pores are embedded using DEM. By varying the ellipsoid aspect ratios, they model different secondary poros-ity types. Thus their approach is more analogous to the SAR model. To account for the effect of fluid in the secondary pore system on the elastic moduli, the elastic moduli of the dry frame were calculated first, and Gassmann’s equations then applied, for the estimation of the low-frequency effec-tive velocities of wet rock.

We have shown that inclusion models can mimic the elas-tic properties of real rocks, if appropriate shapes and volumet-ric fractions of inclusions are assigned to the model. Both the soft-porosity model (SPM) and the single-aspect-ratio mod-els (SAR) can predict accurately the VP and VS trends when the rock constituent’s type and volume fraction, and physical conditions are available as a function of depth. The amount of soft porosity or the inclusion aspect ratio is used as a fitting parameter in order to capture the rock elastic behavior. Thus, the interpretation of absolute values of the derived amount of soft porosity or pore aspect ratios may not be unique, because we adapt them in order to match precisely available observed data and predict the unavailable data, without considering uncertainties in the observed data. The advantage of our model is that by matching the great complexity of multimin-eral rock behavior with a single parameter, soft, with ideal-ized elastic models, SPM can be predictive for various fluid substitution scenarios in low-porosity and low-permeability rocks. We have shown that addition of soft porosity can help us understand the elastic behavior of these rocks.

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Acknowledgments: This work is partly supported by the members of OHM Rock Solid Images Lithology and Fluid Prediction Con-sortium. Arthur Cheng would like to express his gratitude to Tad Smith for many illuminating discussions on this and related rock physics topics over multiple cups of coffee.

Corresponding author: [email protected]


a rock physics model for tight gas sand

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