Characterization and scaling of mesoscale heterogeneities in sandstones


  • Ronny Pini,

    Corresponding author
    1. Department of Energy Resources Engineering, Stanford University, Stanford, California, USA
    • Corresponding author: R. Pini, Department of Energy Resources Engineering, Stanford University, 367 Panama Street, 065 Green Earth Sciences, Stanford, CA 94305, USA. (

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  • Sally M. Benson

    1. Department of Energy Resources Engineering, Stanford University, Stanford, California, USA
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[1] A core-flooding technique to measure core- and subcore-scale capillary pressure curves is applied to the characterization of a naturally heterogeneous sandstone core. N2 and water are used as the nonwetting and wetting fluid phases, and X-ray imaging is applied to precisely visualize multiphase flow at the subcentimeter scale. The spatial variability in the Pc(S) relationship is represented in terms of scaling factors, which are further translated into a corresponding distribution of permeabilities, thus creating a unique and consistent framework where multiphase and rock properties are integrated. The analysis introduced here provides a practical and efficient method for characterizing subcore-scale heterogeneities, whose effects can be substantial on both single and multiphase flows.

1 Introduction

[2] The heterogeneity of geological formations varies over a wide range of length scales and represents a major challenge for predicting the movement of fluids in the subsurface. For instance, the millimeter- to centimeter-scale features that are commonly observed in sedimentary rocks [Murphy et al., 1984; Klise et al., 2008] have been shown to greatly influence fluid transport over much larger observational scales (tens to hundreds of meters) [Ronayne et al., 2010]. These features are often represented by a spatial association of so-called lamina and beds, and give rise to capillary phenomena, which in turn affect the behavior of multiphase flows, as reported in various experimental [Wardlaw and Cassan, 1979; Kueper et al., 1989; Chaouche et al., 1994; Perrin and Benson, 2010; Krevor et al., 2011; Shi et al., 2011] and modeling studies [Ataie-Ashtiani et al., 2002; Saadatpoor et al., 2010; Krause et al., 2011; Kuo et al., 2010]. From a practical perspective, they appear in the form of “finger-like” displacements [Illangasekare et al., 1995] and control process-relevant parameters, such as sweep [Ringrose et al., 1993] or trapping efficiencies [van Lingen et al., 1996].

[3] To account for this complexity, field-scale flow models adopt upscaled (“effective”) flow functions (e.g., relative permeability and capillary pressure) with the aim of capturing the effects of heterogeneity at the subgrid scale. As compared to the size of a grid block used in a coarse reservoir model (∼100 m), so-called “fine-grid” simulations typically use 0.1–1 m grid blocks, i.e., a size that is still significantly larger than the characteristic length scale required to capture capillary effects [Ringrose et al., 1993]. As a consequence, multistage upscaling approaches have been recently developed, which link the core scale to the size of major reservoir units through progressively larger models [Lohne et al., 2006]. We argue that the lack of access to information about rock property heterogeneity at the subcore scale has restricted the ability to fully take advantage of these methods, whose inputs are typically derived from well-logs and core samples. In fact, properties derived from the latter are inherently “effective,” their spatial resolution being limited to a minimum of several centimeters by the measurement or sampling technique. However, making accurate predictions of multiphase flows requires making measurements at the full range of relevant spatial scales, thus referring to the internal structure of the sample and including the effects of the laminations described above [Ringrose et al., 1993; Krause et al., 2011]. Essential components in this description include continuum properties that are related to the rock (porosity and permeability), to the fluids (saturation), and to both of them (capillary pressure-saturation relationship); the ability to create a link among all these properties is key to a physically sound description of these naturally complex systems.

[4] One way to accomplish this is by adopting an integrated approach that combines displacement experiments in naturally heterogeneous core samples with the simultaneous imaging of flow as well as with the support of detailed numerical simulations [Krause et al., 2013]. Methods have been developed that correlate the observed variability in the saturation distribution at the subcore scale to spatial variations in the capillary pressure-saturation relationship at the same scale [Pini et al., 2012]. Corroborated by detailed numerical simulations [Krause et al., 2011, 2013], these findings prove that subcore-scale capillary heterogeneity is the key in controlling saturation distributions during multiphase flow. Built on this knowledge, a methodology is presented here that allows for the characterization of mesoscale heterogeneities within a laboratory sample based solely on experimental observations. Spatially distributed N2/Water capillary pressure curves are measured at a scale of ∼4–10 mm3 in a 8.5 cm long, 5 cm diameter Berea Sandstone sample; it is shown that capillary heterogeneity is significant and that it can be represented in terms of scaling factors. Additionally, upon precise visualization of fluid saturations and porosity distributions, the latter are translated into a corresponding distribution of permeabilities, thus creating a unique and consistent framework where multiphase and rock properties are integrated.

2 Methods and Materials

2.1 Capillary Pressure Measurements

[5] As explained in a previous publication [Pini et al., 2012], the pressure drop across a core, ΔP, that is measured at steady state during the constant rate (qinj) injection of a nonwetting phase (in our case N2) to displace the resident wetting phase (water) corresponds to the capillary pressure at the core's inlet face (x=0), i.e.,

display math(1)

where inline image and P2=Pwx=0=Pwx=L are measured in the end caps mounted just outside of the inlet and outlet faces of the core. By performing experiments at increasing injection rates, increasingly higher capillary pressures can be attained. A capillary pressure curve, Pc(S), can be constructed by relating the latter to saturations measured at the inlet face of the core. In fact, one of the benefits of the method described in Pini et al. [2012] resides in the use of X-ray computerized tomography (CT) scanning to observe and quantify fluid saturations in a rock core, thus allowing for the measurement of Pc(S)-curves at various positions and spatial scales within the sample. In practice, the functional relationship between capillary pressure and saturation is obtained for the inlet slice of the rock using the average saturation and capillary pressure measured there, and subcore-scale capillary pressure curves are constructed by linking the measured capillary pressure value to the saturation observed in each subset (voxel) of same slice. This is possible because fluids are in capillary equilibrium at the experimental conditions applied in this study (see section 2.3). We show here that the method can be readily extended to the subcore-scale characterization of the whole core sample upon deriving a capillary pressure profile along its length from the known core-scale Pc(S) functional relationship and measured slice-averaged saturations.

2.2 Scaling of the Capillary Pressure Curve

[6] With the aim of simplifying the description of the statistical variability of a spatially distributed property such as the capillary pressure, hydrologists have adopted scaling approaches [Warrick et al., 1977; Hopmans, 1987]. The latter make use of so-called scaling factors, αj, to relate the capillary pressure at a given location, Pc,j, to a representative mean, inline image, i.e.,

display math(2)

For the core-flooding experiment considered here, the data set consists of j=1,...,Nvox locations (voxels) and k=1,...,Npc capillary pressure values (Npc=Nq×Nsl, with Nq and Nsl being the number of flow rates and slices, respectively) and Nvox×Npc saturation values. The scaling factors for all voxels can be found by minimizing the following objective function,

display math(3)

The characteristic (reference) capillary pressure function, inline image, is obtained by applying the method described in section 2.1 to the inlet slice of the core, and the assumption is made that this function is representative of the entire core (see sections 3 and 4 for a further discussion). Although various functional forms are available, a spline interpolation through the experimental points has been chosen here, this providing a perfect match to the measured data.

[7] Spatial variations in a multiphase flow property, such as the capillary pressure, can be linked to the corresponding variability in properties of the rock (such as porosity and permeability) through the well-known J-Leverett function, J(S), a scaling law of the same form as equation (2) [Leverett, 1941],

display math(4)

where γ12 is the interfacial tension of the given fluid pair, while ϵ and k are the rock's porosity and permeability, respectively. While in its original form, this function was intended to scale data among lithologically similar core samples [Brown, 1951], a number of investigators have subsequently applied it to capture capillary heterogeneity at the subcore scale [Chaouche et al., 1994; Krause et al., 2011]. This last approach is followed here and allows obtaining an expression for the scaling factor αj as a function of porosity and permeability. In fact, equating equation (4) applied to the core and to a given voxel j, and combining with equation (2) gives

display math(5)

In other words, permeability heterogeneity at the subcore scale can be readily quantified, once the core's average porosity (inline image) and permeability (inline image), as well as the distribution of porosities and scaling factors are known.

2.3 Rock, Fluids, and Experimental Procedure

[8] A 8.5 cm long, 5 cm diameter Berea Sandstone core, fired at 700°C, was used in this study that has an average permeability to water of 325 mD and a porosity of 18.8%. A mercury intrusion capillary pressure curve (MICP, Micromeritics Autopore IV) was measured on a 10 mm long, 6 mm diameter plug that was drilled from a section adjacent to the inlet face of the core; the curve was corrected upon application of the “true” (Helium) skeletal density (inline image g/cm3 measured with a Micromeritics AccuPyc II 1340) and of an entry pressure of 53.5 kPa. The MICP curve is fitted to the core-flooding data obtained for the inlet slice of the rock, and a cubic spline interpolation (function “spline” in MATLAB) through the converted mercury intrusion data is used as a representative capillary pressure curve, inline image, in equation (2). The equation proposed by Purcell [1949] is used to compare Pc-curves measured with the two fluid pairs. Hereby, the interfacial tension for the N2/Water and Mercury/Air systems is assumed to be 65 m N/m (at 50°C and 2.4 MPa) [Yan et al., 2001] and 485 m N/m, respectively; the contact angle θ12 for the N2/Water fluid pair is used as a fitting parameter, while for Mercury/Air, it takes a value of 40°.

[9] A detailed description of the apparatus used for the core-flooding experiments is reported in a previous publication [Krevor et al., 2012], whereas details regarding the experiment preparation are summarized in the supporting information. For the actual capillary pressure experiments, a total of 10 Pore Volumes are allowed to circulate through the core at each selected flow rate (qinj= 2, 4, 8, 14, and 25 mL/min) in order to ensure that steady state conditions are achieved, followed by the acquisition of three complete X-ray scans. The image that is obtained upon averaging among them is further processed by applying a (2×2×3) mm3 coarsening scheme, thus reducing the uncertainty associated to the computed porosity and saturation values at the voxel scale down to σϵ≈ 0.35% absolute and σS≈ 1.8% absolute, respectively [Pini et al., 2012].

3 Results

[10] Figure 1 shows core- (circles) and voxel-scale (squares) capillary pressure curves measured at the inlet face of the core. As examples of general validity, four curves are plotted that represent four distinct positions in the slice; these are indicated by the 2-D saturation map shown in the inset of the main plot and are characterized by an early or late invasion of N2 as compared to the average slice saturation. Results are also shown from an independent mercury intrusion experiment (empty circles) that have been converted to the N2/Water system upon using the contact angle as a fitting parameter (θ12= 38.8°). The agreement between the two techniques is very good, and the obtained contact angle suggests that strong wetting conditions prevail, its value being almost identical to the one imposed for the mercury/air system. Additionally, at a given capillary pressure level, the four millimeter-scale capillary pressure curves are shifted toward higher or lower saturation values but retain a similar shape as the mean curve; this is evidenced by the colored lines in the figure that are drawn primarily to guide the eye but that result merely from a shift of the mercury data toward higher or lower capillary pressure levels.

Figure 1.

N2/water capillary pressure curves as a function of the water saturation: converted mercury intrusion data (empty black circles, measured on a small plug of ∼1.1 cm3) are compared to observations at the slice (∼1.9 cm3, black-filled circles) and voxel scale (∼4 mm3, colored squares) as obtained from the core-flooding technique described in this study. The black line represents a spline interpolation through the MICP data and is used to describe the behavior at the voxel scale (colored lines). The saturation map in the inset is constructed by averaging among 15 repeated X-ray scans taken at the inlet slice of the core and by applying a (2×2×1) mm3 coarsening scheme (σS≈ 1.6% absolute). Vertical and horizontal error bars indicate two standard deviations of uncertainty in the measured pressure drop and saturation, respectively.

[11] Figure 2 shows snapshots of the fluids' saturation distribution in the core upon attainment of steady state conditions at four distinct injection flow rates, and the corresponding saturation histograms are shown below each map. Note that the large density contrast between N2 and water allows for a precise visualization of saturations at the resolution considered here, i.e., σS<2% absolute for a voxel size of 12 mm3. Although not visible in the dry or fully saturated state, features in the form of stratifications that are oriented subparallel to the axis of the core appear as soon as N2 is injected to displace the resident water. This again confirms that, unlike single-phase flow properties, the effect of microstructural heterogeneity on multiphase flow is large [Arns et al., 2003]. The latter are very likely associated to finer-textured strata that are often observed in Berea Sandstone cores [Krause et al., 2011; Brooks and Corey, 1964] and are particularly evident at the lowest to moderate flow rates, i.e., when that portion of the capillary pressure curve is covered where saturation is very sensitive to the value of capillary pressure. Accordingly, the histograms plots show a bimodal saturation distribution that turns into a skewed distribution upon increasing the flow rate. In fact, we can use the mean (μS) and the mode (mS) of the distribution to quantify the effects related to the presence of heterogeneity: the former gives the actual average saturation in the core, while the latter represents its most prominent value, thus providing an indication for the saturation of a perfectly homogeneous core. As reported in the figure, the difference between these two measures is significant (3–10% absolute); for horizontal flows, such as those examined here, the effect of neglecting the presence of fine-scale heterogeneities would be to significantly overestimate displacement efficiency, particularly at low flow rates. Moreover, as explained below, these saturation distributions can be understood by considering spatial variations in the capillary pressure characteristic curve, and they can be directly related to a corresponding distribution of permeabilities.

Figure 2.

Steady state 3-D saturation maps (cross section along the core length) obtained upon injection of N2 at four different flow rates in the core initially saturated with water. These are constructed by averaging among three repeated X-ray scans of the whole core and by applying a (2×2×3) mm3 coarsening scheme. Below each map, histograms are shown of the corresponding saturation distribution; the width of the bins w has been chosen, so that w=2σS (voxel size ∼12 mm3 and σS≈1.8% absolute). For comparison, a normal distribution curve (black line) is plotted on top of each histogram that is centered on the mode of the distribution.

[12] Unscaled capillary pressure values Pc,j at Nvox=11,637 locations within the rock sample (27 slices and 431 voxels/slice) are given in Figure 3a (left); saturation values are plotted for all capillary pressure levels (Npc=135) thus giving a total of more than 1.5 million data points. In the figure, the solid line represents the characteristic capillary pressure curve. For the unscaled data, a large spread of saturation values is observed: at Pc=10 kPa, Sw ranges from about 0.4 down to 0.05. Figure 3a (right) shows the same data scaled by minimizing equation (3); it can be seen that the originally scattered data now coalesce nicely into a relatively narrow band around the characteristic capillary pressure curve. The effectiveness of the scaling is reflected in the value of the sum of squares Φ, which is reduced to only 15% of the value for the unscaled data (Φunsc≈266 versus Φsc≈40). In other words, an approach based on constant (location dependent) scaling factors is able to capture the variability of the Pc curve at the subcore scale, further implying that the description of capillary heterogeneity can be simplified considerably.

Figure 3.

(a) Unscaled (left) versus scaled (right) N2/water capillary pressure curves; circles are results from the mercury intrusion (empty black symbols) and core-flooding (filled black symbols) experiments, and represent the mean curve against which observations at the voxel scale (plus gray symbols, voxel size ∼12 mm3) are scaled. To this aim, the converted mercury data have been interpolated by a spline function (black line). (b) Distributions of the scaling factors αj in histogram form; two normal distributions are plotted that correspond to two voxel families, which are characterized by a mean permeability that is lower (brown curve) or higher (blue curve) than the core average permeability. (c) The 3-D maps of the scaling factors and corresponding permeabilities, as obtained from equation (5). The 3-D maps are reconstructed with a resolution of (2×2×3) mm3.

[13] The spatial variability of the capillary pressure curve within the rock sample can be appreciated by looking at the 3-D distribution map of the scaling factors, as shown in Figure 3c (top). The similarity with the saturation maps shown in Figure 2 is as anticipated: the same strata that remain filled with water during the drainage process are now characterized by small scaling factors, i.e., at these particular locations, a larger capillary pressure is required to reach the same saturation as the core-averaged value. The graphical representation of the same distribution in a histogram (Figure 3b) evidences that the majority of the scaling factors αj lie within a ±15% range from the mean and that they follow a bimodal distribution, as observed in the saturation maps. Note that the values of the slice-averaged scaling factors are always very close to unity (deviations less than 2%), thus supporting the assumption that the same capillary pressure curve can be applied to each slice (in our case, the one measured at the inlet of the core). As given by equation (5), permeabilities at the voxel scale can now be computed based solely on the experimental observations, i.e., by combining the derived scaling factors with the porosity distribution at the voxel scale (from the X-ray images), and with measured core-average porosity and permeability. The resulting 3-D map is given in Figure 3c (bottom); it is not surprising that the exact same features appear as for the distribution of the scaling factors and that deviations from the mean are now larger (±25%). Interestingly, two main families of voxels can be identified within the rock sample (given by the brown and blue curves plotted on top of the histogram), whose statistics are normally distributed around two mean scaling factor (or permeability) values, which are, respectively, smaller and larger than the corresponding mean properties of the core.

4 Discussion

[14] Ultimately, a thorough understanding of multiphase flow in natural porous media is reflected by the ability to successfully predict the distribution of fluids within their porous structure. Ideally, such an effort should rest on observations at a scale where flow is represented by continuum-scale physics, these corresponding to the same framework that is used to describe the dynamics of large floods. Of particular interest are small-scale (mm to cm) features whose combined effects can be substantial on the flow behavior over much larger scales. This study incorporates N2/Water core-scale displacement experiments with simultaneous precise quantification of saturation distributions at the subcentimeter scale (∼4–10 mm3). A recently developed technique to measure core- and subcore-scale capillary pressure curves is applied and extended to the characterization of the whole rock sample, and the variability in the capillary properties of the rock has been further related to variations in permeability and porosity through a scaling approach. Hereby, we applied the capillary pressure curve measured at the inlet slice of the core to the remaining slices; although reasonable for the relatively homogeneous Berea Sandstone core considered here, this assumption can be easily relaxed. For instance, a parametrized form of the Pc(S) relationship could be used, whose parameters are fitted together with the scaling factors and let vary from slice to slice. Beside its application to capture heterogeneities at a larger (field) scale [Warrick et al., 1977] this approach might be necessary for those rock samples that contain major structural heterogeneities.

[15] The main outcomes from this study are summarized as follows:

  1. [16] The spatial variability of the capillary pressure curve at the subcore scale is captured by means of a single stochastic parameter αj (the scaling factor), thus simplifying considerably the description of heterogeneity. It is remarkable that αj shows variations up to 20% even a homogenous rock, such as a Berea Sandstone.

  2. [17] Although not required by the proposed technique, the use of the J-Leverett relationship is validated to scale capillary pressure curves at the subcore scale, as suggested from previous numerical simulation studies [Krause et al., 2013]. This is in contrast to other commonly applied scaling relationships (such as the Kozeny-Carman), which break down at the subcore scale [Krause et al., 2011].

  3. [18] Multiphase and rock properties are evaluated within a unique and consistent framework, thus providing a new level of diagnostic information into the effects of small-scale heterogeneities on fluid displacement. These effects can be quantified through the analysis of statistical measures of the saturation distribution, while representative populations of voxels within the core sample can be identified through the distribution of the scaling factors. Besides their exploitation within an upscaling process, the latter define terms for comparison among different rock types.

  4. [19] Core-scale X-ray CT combined with core-flooding experiments has the potential of creating the necessary link between pore- and reservoir-scale transport properties: precise visualization of multiphase flow is achieved at a voxel scale equivalent to volume elements typically analyzed in microscale X-ray CT studies [Andrä et al., 2013], and continuum-scale parameterization of subcore-scale properties can be determined; these can be related to core-averaged properties, which correlate to the spatial scale measured by wireline logs and provide a starting point for multistage upscaling methods.


[20] This research has been supported by The Global Climate and Energy Project (GCEP) at Stanford.

[21] The Editor thanks two anonymous reviewers for assistance evaluating this manuscript.