Implications of pore microgeometry heterogeneity for the movementand chemical reactivity of CO2 in carbonates

Stéphanie Vialle1, Jack Dvorkin2, and Gary Mavko2

ABSTRACT

We studied the heterogeneity of natural rocks with respect totheir pore-size distribution, obtained from mercury-intrusioncapillary pressure (MICP) tests, at a scale about one-fifth ofthe standard plug size (2.5 cm). We investigated two Fontaine-bleau sandstone and two limestone samples. We found that atthe scale of the MICP tests, heterogeneities are practically non-existent. Still, there are large differences in the capillary curvesfrom one rock type to another. Also, carbonate rocks, unlikeFontainebleau sandstone, show heterogeneities at a scalesmaller than the scale used in MICP tests, as seen by the com-plexity in the mercury saturation versus pressure curves. Weused this diversity between the capillary curves and this com-plexity within a single capillary curve to obtain informationabout the movement and chemical reactivity of CO2 in carbon-ates. The method consists of three steps: first, subdividing the

carbonate pore system into microstructural facies, each of themhaving a specific range of pore throat size (e.g., tight micrite,microporous rounded micrite, small vugs, : : : ); second, gettinga characteristic value of their petrophysical properties (namelyporosity, effective surface area, and permeability) from the col-lected MICP data; and third, computing, for experimentalconditions corresponding to a transport-controlled system, thedimensionless Péclet and Damköhler numbers, expressed as afunction of the aforementioned permeability and effective sur-face area. These numbers allowed us to infer the dominantprocess (i.e., diffusion, advection, or kinetics) controlling thedissolution/precipitation reaction induced by the carbonic acid.Because of heterogeneities in the pore microstructure, we foundthat either diffusion or advection is locally the dominant mecha-nism, which renders some zones (e.g., vugs or, to a lesser extent,microporous rounded micrite) chemically more reactive thanothers (e.g., tight micrite or spar cement).

INTRODUCTION

Natural rocks are heterogeneous from the pore scale upward. Inthe laboratory, controlled physical experiments are conducted onsamples generally a few centimeters in size, and computational ex-periments deal with even smaller samples, about 1 mm in size. Thequestion is how to use such results to quantify rock properties(including the transport properties) at a larger scale that is relevantto a reservoir. This problem is essential to many fundamental andapplied issues, from the dynamic processes of fluid flow in theearth’s crust, to the exploration and production of hydrocarbons,to groundwater management.Permeability is an important reservoir parameter, because it con-

trols the movement of fluids. Although it is common practice to

estimate permeability from permeability-porosity relationships ob-tained from laboratory core analysis, these correlations may break atlarger scales, especially in the presence of strong spatial hetero-geneity (e.g., Brace, 1980; Lucia, 1995). One reason is that thehydraulic permeability strongly depends on pore connectivity(e.g., Guéguen and Dienes, 1989), and this connectivity may haveextremely strong variation in space. Parameters describing rockmicrostructure, such as grain size, pore-body or pore-throat size,specific surface area, and tortuosity, are required to better estimateflow paths and permeability.This dependence of the pore-fluid flow velocities on the connec-

tivity within the pore space, or in other words, on the flow redis-tribution within the different pores and pore throats as the overallpressure gradient varies, has been observed and discussed in

Manuscript received by the Editor 29 October 2012; revised manuscript received 15 March 2013; published online 19 August 2013.1Formerly Stanford University, Department of Geophysics, Stanford, California, USA; presently Stanford University, Department of Geology and

Environmental Sciences, Stanford, California, USA. E-mail: svialle@stanford.edu.2Stanford University, Department of Geophysics, Stanford, California, USA. E-mail: dvorkin@stanford.edu; mavko@stanford.edu.© 2013 Society of Exploration Geophysicists. All rights reserved.

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GEOPHYSICS, VOL. 78, NO. 5 (SEPTEMBER-OCTOBER 2013); P. L69–L86, 11 FIGS., 5 TABLES.10.1190/GEO2012-0458.1

network representations of porous media (Simon and Kelsey, 1971;Sorbie et al., 1989; Damion et al., 2000; Bijeljic et al., 2004) as wellas in packed beds experiments using magnetic resonance imaging(Sederman et al., 1998). A consequence of this effect is that, to pre-dict flow properties at the pore scale, we must assess the connec-tivity of the different pores, which is only possible by couplinghigh-resolution 3D images (i.e., micro or even nano CT scans) withdigital simulation using pore-network models or pore-scale models.Conversely, having an average value of flow velocity (or of the per-meability) at the core scale (the scale at which the laboratory experi-ment is conducted) will not provide enough information about theheterogeneities of the sample to explain the different observedbehaviors (e.g., different fluid saturation or different types of chemi-cal reactions).In reactive systems (e.g., carbonates subject to CO2 injection),

there is another, time-dependent process that further complicatesthis question, which is the possibility of chemical alteration ofthe pore space. Various reactions of dissolution and precipitationmay occur, depending on the mineralogical composition of the res-ervoir rock (e.g., Bachu et al., 1994; Baines and Worden, 2004;O’Connor et al., 2004; Rochelle et al., 2004; Xu et al., 2004; Gauset al., 2005; Giammar et al., 2005; Palandri et al., 2005; André et al.,2007; Matter et al., 2007; Andreani et al., 2008; Wigand et al., 2008;Daval et al., 2009; Goldberg and Slagle, 2009; Hangx and Spiers,2009; Matter and Kelemen, 2009; Gratier et al., 2012; Luquot et al.,2012). In carbonate rocks, dissolution and eventual reprecipitationof carbonate minerals are likely to result when CO2 dissolves in theformation water, rendering it acidic. Experimental and theoreticalworks have shown that, for a given chemical composition of thefluid and of the rock, the dissolution pattern (i.e., whether the dis-solution is homogeneous or heterogeneous, the latter causingwormhole generation) is influenced by the injection rate (e.g., Gol-fier et al., 2002; Singurindy and Berkowitz, 2003a, 2003b; Luquotand Gouze, 2009). Indeed, in many cases, the transport and chemi-cal (as well as mechanical and biological) processes are stronglycoupled. A typical example is the oxidation reduction zone thatmay develop in an aquifer downstream from an organic-rich landfill(Steefel et al., 2005); there, the metabolic activity of the biofilmconsortia at the pore scale may depend on some combination ofadvective and diffusive transport combined with local biogeochem-ical reactions providing electron donors and acceptors. These proc-esses may then result in changes in the physical properties of themedium through biological growth and/or mineral precipitation ordissolution, providing feedback between transport and reaction.In reactive systems, an important parameter is the reactive surface

area, the fraction of the pore surface area that is in contact with thereactive fluid, because the mineral-fluid interface partially controlsthe kinetic behavior in many geologic systems (e.g., Lasaga andKirkpatrick, 1981). Usually, the reactive surface area is obtainedfrom adsorption isotherms (Brunauer et al., 1938) or geometricalconstructions (Canals and Meunier, 1995; Le Gallo et al., 1998;Colon et al., 2004), but this approach does not account for thehydrologic accessibility of the reactive species within the porestructure (Maher et al., 2006; Peters, 2009). Emerging techniquesto tackle this problem use 3D X-ray microtomography at the corescale, coupled with 2D scanning electron microscopy (SEM), en-ergy-dispersive X-ray spectroscopy, and focused ion-beam SEM, atvarious scales from nanometer to millimeter (Landrot et al., 2012).In areas of the pore space where reactions rates are fast in compari-

son with the transport of the solutes into or out of the consideredarea, localized microenvironments are created (Steefel et al., 2005).These microenvironments are likely to develop in physically hetero-geneous systems, where the local flow rates can be very slow, andthe chemical reactions are controlled by diffusion. This can have asignificant influence on the overall behavior of the system at largescale, as shown by Molins et al. (2012). Consequently, upscalingfrom the pore scale to the field scale, while taking into accountthe nanoscale phenomena that are likely to have an impact at thelarge scale, is not an easy task. It is thus necessary to consider differ-ent scales simultaneously, as suggested, for example, by the studiesof Kechagia et al. (2002) and Battiato and Tartakovsky (2011).In this study, to explore natural heterogeneity of rock, we use four

standard-size samples (about 2.5 cm in length and diameter): twocarbonates and two sandstones. We subdivide each sample intofour subsamples and conduct mercury-intrusion capillary pressure(MICP) tests on each of the subsamples. The goal of these measure-ments is to assess pore-space heterogeneity at several scales. Theimplications of these tests go beyond quantifying the heterogeneityof the test results within a sample. The pore-size and pore-throat-size spatial distribution affects the effective flow propertiesand, moreover, the reactive flow properties, essential to understand-ing injection of CO2 into reactive rocks, such as carbonates.We use MICP tests because each MICP volume-versus-pressure

curve itself carries information about the heterogeneity of the porespace. Hence, by taking separate portions of the same curve, we cansubdivide the rock into several microstructural facies linked to thepore-throat sizes, such as tight micrite, high-porosity micrite, smallvugs, and grains. Then, always by using MICP data, we can infer aneffective porosity and an effective pore surface area for each of themicrostructural facies and estimate an effective permeability usingthe Kozeny-Carman equation for each facies. To get informationabout the movement and chemical reactivity of CO2; i.e., to relatethe aforementioned petrophysical microstructure parameters to thereactive transport of CO2, we use the dimensionless Péclet andDamköhler numbers, expressed as a function of the permeabilityand specific surface area, and thus infer the magnitude of thedifferent dissolution regimes — those controlled by diffusion,by advection, or by kinetics (e.g., de Marsily, 1986; Golfier et al.,2002) — in each microstructural facies.We applied this aforementioned method to the laboratory CO2

injection experiment of Vialle and Vanorio (2011), which uses flowvelocities that typically can be found within a few feet of the in-jection well. These experiments show that, for the same flow rateand chemical composition of the injected CO2-bearing brine, andfor the same mineral composition of the rock sample (pure calcite),the mineral dissolution in some samples was more pronounced thanin the others. Furthermore, within the same rock sample, dissolutionaffects the microporous micritic matrix of the carbonate samplemore than the spar cement, as previously observed (Noiriel,2005; Noiriel et al., 2009). One explanation is that this observeddifference reflects the differences in microstructure, namely poresizes and shapes, controlling the overall permeability and pore sur-face area.In this study, we investigate how heterogeneities in the rock

microstructure locally affect permeability and pore surface areas,and how the spatial variation in these parameters affect the geo-chemical evolution of the fluid/rock system, locally and at a largerscale.

L70 Vialle et al.

STUDY OF PORE MICROGEOMETRYAND HETEROGENEITIES

Because information about the pore microgeometry (such as porethroat size distribution derived from MICP tests) are usually ob-tained on samples much smaller than the plug that will be used laterin the laboratory (in a reactive percolation experiment, for example),it is always questionable whether the rock properties measured onthe smaller subsample are representative of the whole core plug. Forthis reason, we first carefully measure helium-grain densities andporosities as well as mercury capillary curves on several (four toeight) subsamples of the same rock plug. Results are presentedand discussed in the first part of the paper, which allows us toappreciate the homogeneity/heterogeneity of each core plug(∼12 cm3) at the subsample scale (∼0.5 cm3). This step is essentialto evaluate the validity of our conclusions about the effects of thepore microgeometry on the fluid-solid chemical reactions that arethe object of the second part of the paper and to correctly upscalethose results.

Sample description and preparation

Geologic settings and rock microstructure

Four rock samples were selected for this study: two sandstonesamples, sand-A and sand-B, and two carbonate samples,carb-A and carb-B. The two sandstone samples are Fontainebleausandstones, from the Île-de-France region, near Paris (France). TheFontainebleau sandstone is an early Oligocene (36–27 Ma) unit,50–80-m thick, of quartz arenite. Quartz-cemented horizons haveformed from silicification controlled by water-table variations ina Pliocene-Quaternary hydrogeologic event, related to spring lines(Thiry et al., 1988). The Fontainebleau sandstone is characterizedby a well-sorted, uniform grain size (∼250 μm indiameter) as shown in the SEM images inFigure 1, and a monomineralitic composition(∼100% quartz).The two carbonate samples come from the

Upper Cretaceous carbonate system of theGargano-Murge region, which belongs to the sta-ble foreland of southern Italy. Carb-A is a lime-stone from the Paleocene-Eocene Peschiciformation, and carb-B is a micritic mudstonefrom the Late Cretaceous Monte Acuto forma-tion (Martinis and Pavan, 1967; Cremonini et al.,1971). Both carbonates are composed of nearly100% calcite. SEM images showing the micro-structure of these two samples are given inFigure 2. Both carbonate samples display a ma-trix of micrite (microcrystalline calcite) whosegrain size is typically 1–4 μm (Moshier, 1989),but whose textures are different. Following theclassification of micrite microtexture proposedby Lambert et al. (2006) and Deville de Periereet al. (2011) based on SEM images observations,the micrite in sample carb-A is mainly a “tightmicrite,” anhedral compact to fused, with grainstypically 1–2 μm in diameter, whereas micrite insample carb-B is a “porous micrite,” rounded tosubrounded, with anhedral to subhedral, rounded

grains typically 2–4 μm in diameter. Beside a micrite matrix, samplecarb-A exhibits vuggy pores either rounded, up to about 60 μm, ormore elongated, up to 300 μm in length. Sample carb-B exhibits aspar calcite cement of grains typically 10 to several hundredmicrometers in diameter, with narrow, tubular contacts betweenthem, as well as rounded vugs up to about 100–200 μm.Prior to studying heterogeneities at the subcore scale, we con-

ducted experiments on the four intact core plug samples (2.5 cmin diameter and length). He-grain density, bulk density, and the re-sulting porosity, as well as air permeability, were measured at roompressure and temperature. The error bars in the measured density,porosity, and permeability do not exceed �0.5%, �1%, and �2%,respectively. The P- and S-wave velocities, at 1 and 0.7 MHz fre-quency, respectively, were acquired on dry samples under increas-ing (up to 30 MPa) and decreasing hydrostatic stress. Velocitieswere measured by using a pulse-transmission technique. The errorbars for VP and VS are about�1%. The results are listed in Tables 1and 2.In Figure 3, porosity and permeability measurements are com-

pared with previously reported data for Fontainebleau sandstone(Bourbié and Zinszner, 1985; Gomez et al., 2010) and for the car-bonate data set to which the two carbonate plugs belong (Scotalleroet al., 2008). In the case of carbonates, the permeability-porosityrelation does not show a good correlation, especially for low poros-ity values (less than 0.1); we attribute the poor correlation to thecomplexity of the carbonate microstructure. Indeed, because of theirpropensity for chemical, physical, and biological alterations, car-bonate rocks experience continuous changes during sedimentationand postdepositional diagenesis, leading to various and complextextures and fabrics that complicate any first-order relations thatmight have existed between geophysical observables and rockproperties.

Figure 1. SEM images of Fontainebleau sandstone samples sand-A (right) and sand-B(left) at different magnifications. The variable-pressure backscattered electrons (VP-BSE) mode was used with a vacuum pressure of 15 Pa and a beam accelerating voltageof 15 kV. The VP-BSE mode allows imaging samples at high resolution without coatingthem.

Microstructure heterogeneity L71

Measured grain densities (2.63–2.65 [�0.01]g∕cm3 for sand-A and sand-B, and 2.69–2.70[�0.01] g∕cm3 for carb-A and carb-B) are con-sistent with data reported in the literature forquartz and calcite crystals, respectively (e.g.,Defoe and Compton, 1925; Katz et al., 1970).Measured P- and S-wave velocities, color

coded by the pressure, are plotted against poros-ity in Figures 4 and 5, for the sandstone and car-bonate samples, respectively. For comparison,Figure 4 also shows sandstone data from Han(1996) and Gomez et al. (2010), and Figure 5shows carbonate data from Scotellaro et al.(2008). Both sandstone samples show strongpressure dependence: An increase of confiningpressure from 0 to 30 MPa causes the P-wavevelocity to increase by 52% (sand-A) and 19%(sand-B), and the S-wave velocity by 52%(sand-A) and 17% (sand-B).Velocities for sample carb-A show small pres-

sure dependence (the P-wave velocity increasesby only 4%, whereas the S-wave velocity showsno change), whereas the P- and S-wave velocitiesfor sample carb-B increase by 28% and 31%,respectively. Qualitatively, this difference inbehavior can be explained by the different

Figure 2. SEM images of carbonate samples carb-A (a, b) and carb-B (c, d) at differentmagnifications. (a) Microstructure of sample carb-A is composed of a tight micritic ma-trix (light gray) and rounded to subrounded vugs, up to 100–200 μm. (b) Micrite ismainly anhedral compact (arrow A) to fused (arrow B), sometimes porous microrhombic(arrow C). Micritic grains are 1–2 μm in diameter. Carb-B shows some vugs, a sparcement (in light gray on image c) and a matrix of microporous rounded micrite (imaged), formed by rounded grains of calcite, up to 4 μm in diameter.

Table 1. Measurements at benchtop conditions, for all four studied samples: porosity, bulk, and grain densities (in g∕cm3) andpermeability (in mD). Permeability of plug carb-A is below the sensitivity level of the apparatus used (∼0.1 mD).

Sample Porosity Bulk density (g∕cm3) Grain density (g∕cm3) Permeability (mD)

Sand-A 0.097� 0.001 2.39� 0.01 2.65� 0.01 35� 3

Sand-B 0.153� 0.002 2.23� 0.01 2.63� 0.01 540� 20

Carb-A 0.167� 0.002 2.24� 0.01 2.69� 0.01 <0.1

Carb-B 0.294� 0.003 1.90� 0.01 2.70� 0.01 60� 5

Table 2. Pressure dependence of the elastic-wave velocities for all four studied samples. Pressure is in megapascal, and P- andS-wave velocity are in kilometers per second.

Pressure

Sand-A Sand-B Carb-A Carb-B

VP VS VP VS VP VS VP VS

0 3.280 2.131 3.943 2.709 5.007 2.835 2.769 1.754

2.5 3.934 2.607 4.302 2.937 5.045 2.838 3.295 2.032

7.5 4.353 2.841 4.516 2.990 5.091 2.835 3.477 2.191

10 4.399 2.932 4.558 3.066 5.100 2.834 3.504 2.226

20 4.804 3.085 4.637 3.132 5.116 2.822 3.551 2.295

30 4.958 3.237 4.696 3.166 5.113 2.821 3.553 2.290

25 4.942 3.211 4.682 3.187 5.114 2.821 3.548 2.284

15 4.780 3.065 4.630 3.133 5.085 2.822 3.527 2.268

5 4.353 2.819 4.480 3.002 5.048 2.840 3.384 2.195

0 3.355 2.280 4.051 2.797 5.033 2.842 2.792 1.785

L72 Vialle et al.

microstructures observed on the SEM images: Sample carb-A mi-crostructure is composed of a tight micritic matrix and rounded orhigh-aspect-ratio vugs, whereas sample carb-B microstructure re-veals tubular, microcrack-like pore connections.

Preparation of subsamples

For each of the four rock plugs, eight subsamples were extracted.The subsamples are also cylindrical plugs, about 0.9 cm in diameterand 0.5 cm in length. After coring and sawing, the subsamples werepolished with sandpaper or diamond smoothing disks to ensuresmooth, flat surfaces and make their faces parallel to within�0.1 mm. Samples were dried for 24 hours in a vacuum oven(1 bar) at 70°C prior to further analyses. Their diameters and heightswere measured with a high-precision caliper to compute their vol-umes, and their masses were measured on a high-precision balance.This careful preparation ensured (1) precise volume calculations ofthe subsamples and (2) minimization of subsample surface effectsduring MICP tests.

Experimental procedure

Helium porosity

Porosity is computed from the measured grain volumes of eachsubsample. Grain volumes were determined using a Boyle’s lawdouble-cell helium pycnometer (AccuPyc II 1340 from Microme-ritics). After calibration of the apparatus, which yields the referencechamber volume (Vr) and the sample chamber volume (Vc), thecore-plug sample is placed in the sample chamber. Helium gas isadmitted into the reference chamber at a predetermined pressure(about 20 psi) and allowed to equilibrate. The pressure P1 inthe reference chamber is recorded. The gas is then allowed toexpand into the sample chamber. The resulting lower pressure(P2) is measured in the sample chamber after the system hasreached equilibrium. The procedure is repeated 10 times for eachsample.Using the gas law (Boyle’s equation) and a mass balance of gas

within the reference and sample chamber and assuming isothermalconditions, the grain volume of the sample is given by

Vg;He ¼ Vc − VrðP1∕P2 − 1Þ; (1)

where Vc is the sample chamber volume, Vr isthe reference chamber volume (both determinedduring the initial calibration), P1 is the fillingpressure in the reference chamber, and P2 isthe pressure in the sample chamber after expan-sion of the gas and equilibration. Knowing themass of the sample ms, the grain density is thencalculated as

ρg;He ¼ ms∕Vg;He: (2)

Helium porosity was then derived by using thegeometric volume of the sample, obtained bymeasuring length and diameter of the cylindri-cal-shape sample with a high-precision digitalcaliper:

ϕHe ¼VS;geom −ms∕ρg;He

VS;geom: (3)

The maximal standard deviation was 0.0003 cm3 for the mea-sured grain volumes, 0.002 g for the measured mass, and between2 and 36 mm3 for the geometrical volumes, depending on the sam-ple. Density was thus obtained with a maximal relative uncertaintyof �0.2% and porosity with a relative uncertainty between �0.4%

and �0.8%, depending on the sample.Note that the geometric volume determined here may be different

from the “true” bulk volume. Particular care was given to subsam-ples preparation (sawing, drilling and smoothing), to have sampleswhose volume is as close as possible to a circular cylinder. Still, the

Figure 3. Permeability versus porosity for the two Fontainebleauplugs under examination, sand-A (large black dot) and sand-B(large gray dot), and for carbonate plug carb-B (large black square).Permeability of plug carb-A is below the sensitivity level of the ap-paratus used and, hence, is not plotted. These data are comparedwith the Gomez et al. (2010) data (black small dots) and the Bourbiéand Zinszner (1985) data (light gray small dots), for the Fontaine-bleau samples, and to the carbonate data set (Scotallero et al., 2008)to which the two carbonate plugs belong (gray diamonds).

Figure 4. Pressure dependence of dry (a) P-wave and (b) S-wave velocity for the twoFontainebleau plugs under examination, sand-A and sand-B. Velocity data are colorcoded as a function of applied confining pressure (in MPa) under hydrostatic stressconditions. Also plotted for reference are Fontainebleau data from Gomez et al.(2010) (triangle symbols) and Han (1986) (square symbols) both at 30 MPa.

Microstructure heterogeneity L73

edges might have been slightly eroded. Hence, the porosity deter-mined from equation 3 might have been slightly overestimated.

Mercury intrusion capillary tests

MICP measurement is commonly used to measure the pore-throat (or pore-access) size distribution of a sample. Measurementswere carried out with an automated mercury porosimeter (AutoPoreIV 9500 from Micromeritics) having low- and high-pressure analy-sis ports, allowing the development of pressures between 0.22 and33,000 psi. As a result, pore diameters between about 980 μm andabout 0.0065 μm can be measured. Measurements were conductedon four subsamples from each plug, with the subsample volumesvarying between 0.3 and 0.5 cm3.An equilibrating time of 10 s for the low pressures and 20 s for

the high pressures was used before the volume of mercury intrudedinto the samples was recorded. A total of 68–76 pressure stationsregularly spaced in the log scale were generated. The resulting cu-mulative intruded volumes of mercury versus pressure are known asthe mercury capillary curves.A closure correction was applied to the generated data. This cor-

rection consists in subtracting all mercury intrusion volumes re-corded up to the entry pressure; these data correspond to theeasy invasion of mercury into the supercapillary pores (megapores)at the outer surface of the permeable rock sample under very lowpressures. These megapores are considered to be surface featuresthat are not representative of the internal microstructure.The pressure was converted into the pore-throat diameter via the

Washburn equation (Washburn, 1921), which assumes that poresare cylindrical and mercury is a nonwetting fluid:

P ¼ −4γcos θ

d; (4)

where P is the applied pressure in psi, γ is the surface tensionof mercury in dynes∕cm, θ is the contact angle, and d is thepore-throat diameter in micrometers. We used a surface tensionof 485 dynes∕cm and an advancing contact angle of 140° for theair/mercury system, which are commonly used values (e.g., Goodand Mikhail, 1981).

Relative uncertainty in pore-throat diameter is due to the resolu-tion of the pressure transducer, which depends on the value of gen-erated pressure. For the low-pressure analysis port (pressure lessthan 30 psi or pore-throat diameter greater than 7 μm), the resolu-tion is 0.01 psi, and thus the maximum relative uncertainty in thediameter values is 4.5%; for the high-pressure analysis port, the res-olution is 0.1 psi for pressures ranging from 30 to 3000 psi and0.2 psi for a pressure value greater than 3000 psi. This leads toa maximum relative uncertainty of 0.33% for diameters between0.07 and 7 μm and 0.007% for the diameters less than 0.07 μm.The volume of mercury intruded in the sample at the end of the

experiment (for which the applied pressure is 33,000 psi) is as-sumed to be the (connected) pore volume of the sample, VS;Hg,as suggested by the fact that no more intruded mercury was detectedfor pressures greater than 15,000 psi.In addition to the caliper-measured sample volumes, we also used

the MICP data, which, it may be argued, produce more accuratevalues. The MICP-computed volume of the sample VS;Hg is

VS;Hg ¼ Vpene −mpene;s;Hg −mpene;s

ρHg− Vext; (5)

where Vpene is the volume of the penetrometer (calibrated),mpene;s isthe mass of the penetrometer that contains the sample during theMICP analysis taken before starting the experiment, mpene;s;Hg isthe mass of the penetrometer filled with the sample and with mer-cury, ρHg is the density of mercury at the experiment temperature,and Vext is the volume of mercury that has to be subtracted due toclosure correction, which is supposed to be related to surfacefeatures.The grain density of the sample is then

ρg;Hg ¼ms

Vg;Hg; (6)

and the porosity is

ϕHg ¼VS;Hg − Vg;Hg

VS: (7)

The error bars of these estimates are less than�0.5% for the sample volume, less than �0.5%

for the grain density, and less than�0.6% for thesample porosity.Let us emphasize again that the sample vol-

ume fromMICP is very accurate because the vol-ume of air that might have been trapped at therock/mercury interface is minimized by applyinghigh vacuum before filling the penetrometer withmercury.Errors in density and porosity obtained from

MICP result from the errors in the pore-volumeestimation. Because of the sample’s compactiondue to applied pressure, some pore throats maybecome unconnected, preventing the mercuryfrom filling the entire pore volume. This wouldlead to an underestimation of porosity, as op-posed to its overestimation from geometricalmeasurements.

Figure 5. Pressure dependence of dry (a) P-wave and (b) S-wave velocities for the twostudied carbonate plugs, carb-A and carb-B. Velocity data are color coded as a functionof applied confining pressure (in MPa) under hydrostatic stress conditions. Also plottedare benchtop velocity data of the carbonate data set (Scotellaro et al., 2008) to which thetwo carbonate plugs belong (gray dots).

L74 Vialle et al.

Corrections to the porosity

To get a porosity value closer to the true value, we combined thedensity obtained by He pycnometry with the volume of the samplederived from MICP tests. The “corrected” values of porosity arethen

ϕcorr ¼VS;Hg −ms∕ρg;He

VS;Hg: (8)

The relative error for the corrected porosity does not ex-ceed �1%.

Results

Capillary curves

The measured capillary curves are shown in Figure 6a for all foursubsamples of each sample, sand-A, sand-B, carb-A, and carb-B.The volume of intruded mercury has been normalized by its finalvalue (which is the pore volume of the subsample measured by mer-cury tests) to give the mercury saturation and is plotted as a func-tion of increasing applied pressure, and decreasing correspondingpore-throat radius on the secondary axis. In Figure 7a, the mercuryvolume (obtained by equation 4) has been normalized to the sub-sample volume (obtained from equation 5), which gives the cumu-lative porosity of the subsample, as a function of intrusion pressure(and decreasing corresponding pore-throat radius [obtained byequation 4] on the secondary axis).For each of the four rock types, the capillary curves show little

variation among the subsample. This means that for the four sam-ples under examination, and arguably for any rock of the same type,the MICP properties are fairly homogeneous throughout each sam-ple, at least at the scale of 0.5 cm, which is one-fifth of the scale ofthe intact sample.

Still, as expected, we observe large variations in the MICP prop-erties from one rock type to the other. The sandstone capillarycurves are characterized by a broad plateau, followed by a sharpincrease, then a rapid leveling to a constant value, as is typicallyobserved in well-sorted rocks with a narrow, monomodal pore sizedistribution. The sharp increase occurs for higher capillary pres-sures (lower pore-throat values) in sand-A. This is consistent withthe lower porosity of sample sand-A, which is the result of quartz-grain overgrowth during diagenetic cementation, resulting in a de-crease of the size of the pores (Thiry et al., 1988). It is also worthnoticing that sample sand-A levels off less sharply, which could in-dicate the presence of very small pore accesses and microcracks.Capillary curves of sample carb-B display a broad plateau followedby a sharp increase, but then they continue to steadily increase be-fore leveling off. Compared with the sandstones, carb-B has smallerpore throats, as well as a broader range of pore-throat values. Capil-lary curves of sample carb-A are characterized by a steady increaseover a pressure range spanning three orders of magnitude. Thisbehavior indicates an even broader range of pore-throat sizesand thus a more complex pore microgeometry than in Fontainebleausandstone.To gauge the homogeneity or heterogeneity among the different

capillary curves obtained for each rock type, we plot the averagedcapillary curves (black line) and their spread, in terms of standarddeviation (shaded area, one side), in Figure 6b for the mercury sat-uration and Figure 7b for the porosity. It can been seen in Figure 6bthat samples sand-A, sand-B, and carb-B show constant and smallstandard deviations (typically less than 1%), except for samplessand-B and carb-A in the portion of the curve before the steepincrease, where standard deviation can go up to 3%. For samplecarb-A, however, the standard deviation is small (about 0.1%) inthe plateau region, then it increases in the 10–800 psi range, upto 7.5% at about 160 psi, and then is very small again (less than0.1%) at higher pressures.

Figure 6. (a) Capillary curves for all studied subsamples. The saturation of mercury (volume of mercury intruded normalized to its final value)is plotted as a function of increasing applied pressure (lower x-axis), and corresponding decreasing pore throat radius (upper x-axis). (b) Foreach rock type sand-A, sand-B, carb-A, and carb-B, the black line represents the geometric average of the four capillary curves obtained foreach subsample, and the shaded area represents the standard deviations (one side).

Microstructure heterogeneity L75

Density and porosity

Table 3 reports the subsample volume, grain density, and effec-tive porosity obtained by the different methods presented in an ear-lier section.We first observe that there is a noticeable difference between the

subsample geometric volume and those obtained by MICP tests,and that the former is always greater than the latter. As discussedearlier, this is because the geometric volume includes an “external”volume between the surface of the grains and the “geometrical”lines forming the circular cylinder measured by the caliper. Poros-ities using the geometric volume (i.e., ϕHe) are thus higher thanporosities using the volume determined by MICP (i.e., ϕMICP

and ϕHecorr).To better appreciate and compare the different measurements,

grain densities and porosities are shown in Figures 8 and 9, respec-tively, for each subsample of the four studied rock types, as well asfor the entire plug.For sand-A, sand-B, and carb-B, the He-porosity values of each

subsample are, within the experimental error, identical. For samplecarb-A, the He-porosity values of each subsample show differencesthat are slightly above the experimental error. Densities derived fromMICP tests are also, within the experimental error of this method,similar among the studied subsamples. We notice, however, thatthe density values for sand-A and carb-B are systematically signifi-cantly lower than the ones obtained by He pycnometry. This may bethe result of subsample deformation and closure of some flow pathswith increasing mercury applied pressure, which isolates parts of therock subsamples. Figures 4 and 5 indeed showed that samples sand-Aand carb-B have strong pressure dependence.The porosities deduced from He pycnometry and the subsample

geometric volumes (in blue, Figure 9) display large variations, but,as discussed in the method section, this is a consequence of thedifficulty of measuring the subsample geometrical volumes. Thesevariations are reduced with the use of the other methods (MICP

tests, black triangles in Figure 9, and combination of MICP testand He pycnometry, gray dots in Figure 9), especially for thetwo Fontainebleau samples, sand-A and sand-B.To summarize, the main conclusions of this study are thus that

between the subsamples of each rock type, there are very smalldifferences in the capillary curves, grain densities, or porosities,usually within the estimated experimental error, or, for samplecarb-A, only slightly above. One of the outcomes of this resultis that each subsample can be seen as representative of the wholeplug. There are, however, big differences from one rock type to an-other (sand, carb-A, and carb-B), with each formation displayingsignificantly different capillary curves. Another main observationis that the capillary curves of the Fontainebleau sandstones are typ-ical of a simple, monomodal pore-size distribution, whereas thecapillary curves of both carbonate rocks are far more complex,as a result of a plurimodal pore-size distribution.In the next section, we use the experimental conditions of the

experiments of Vialle and Vanorio (2011), which consisted in in-jecting CO2-rich water in carbonate samples, some of them beingsimilar to carb-A and carb-B, to seek answers to the following twoquestions:

1) What is the effect of the complexity of the pore microgeometryobserved in carbonates (compared with that of Fontainebleausandstone) for the fluid-solid chemical interactions?

2) How do different initial pore microgeometries (resulting in dif-ferent capillary curves between carb-A and carb-B) affect fluid-solid chemical interactions?

IMPLICATIONS OF PORE MICROGEOMETRYFOR FLUID-SOLID REACTIVITY

Many experimental and theoretical studies have found thatthe flow velocity of the injected fluid has a large effect on

Figure 7. (a) Capillary curves for all studied subsamples. The cumulative volume of mercury intruded into the sample has been normalized tothe subsample volume (obtained from equation 5), giving the cumulative porosity of the subsample, as a function of applied pressure (lowerx-axis), and corresponding decreasing pore throat radius (upper x-axis). (b) For each rock type sand-A, sand-B, carb-A, and carb-B, the blackline represents the geometric average of the four capillary curves obtained for each subsample, and the shaded area represents the standarddeviations (one side).

L76 Vialle et al.

Table 3. Sample bulk volume, skeletal density, and effective porosity obtained by the different techniques used: Vgeom is the bulkvolume obtained from calipering, and VMICP is the one deduced from mercury immersion during MICP analysis; ρHe is theskeletal density deduced from helium pycnometry, and ρMIP is the one deduced from MICP analysis; ϕHe is the effectiveporosity computed from ρHe and Vgeom, ϕMIP is the one deduced from MICP analysis, and ϕHe;corr is the one computed from ρHeand VMICP. Refer to the text for the details of the calculation. Volumes are in cm3 and densities are in g · cm−3.

Sample

Sample volume Skeletal density Effective porosity

Vgeom VMICP ρHe ρMIP ϕHe ϕMICP ϕHe;corr

Sand-A1 0.508� 0.007 0.490� 0.003 2.646� 0.005 2.616� 0.013 0.111� 0.002 0.068� 0.001 0.078� 0.001

Sand-A2 0.462� 0.010 — 2.644� 0.005 — 0.101� 0.002 — —Sand-A3 0.461� 0.004 — 2.648� 0.005 — 0.128� 0.001 — —Sand-A4 0.465� 0.019 — 2.644� 0.005 — 0.141� 0.006 — —Sand-A5 0.433� 0.035 0.408� 0.002 2.648� 0.005 2.643� 0.013 0.127� 0.010 0.072� 0.001 0.073� 0.001

Sand-A6 0.486� 0.005 — 2.643� 0.005 — 0.113� 0.001 — —Sand-A7 0.478� 0.003 0.456� 0.002 2.646� 0.005 2.628� 0.013 0.120� 0.001 0.073� 0.001 0.078� 0.001

Sand-A8 0.346� 0.018 0.327� 0.002 2.641� 0.005 2.638� 0.013 0.123� 0.006 0.070� 0.001 0.071� 0.001

Average N/A N/A 2.645� 0.003 2.631� 0.013 0.121� 0.012 0.071� 0.002 0.075� 0.004

Plug 12.641� 0.632 N/A 2.650� 0.013 N/A 0.097� 0.001 N/A N/A

Sand-B1 0.472� 0.004 — 2.632� 0.005 — 0.183� 0.002 — —Sand-B2 0.444� 0.003 — 2.634� 0.005 — 0.179� 0.001 — —Sand-B3 0.427� 0.003 — 2.626� 0.005 — 0.184� 0.001 — —Sand-B4 0.400� 0.017 0.381� 0.002 2.621� 0.005 2.639� 0.013 0.182� 0.008 0.138� 0.001 0.132� 0.001

Sand-B5 0.444� 0.029 0.430� 0.002 2.625� 0.005 2.615� 0.013 0.173� 0.011 0.136� 0.001 0.141� 0.001

Sand-B6 0.414� 0.036 0.391� 0.002 2.626� 0.005 2.645� 0.013 0.191� 0.017 0.149� 0.001 0.143� 0.001

Sand-B7 0.426� 0.011 — 2.625� 0.005 — 0.167� 0.004 — —Sand-B8 0.426� 0.008 0.407� 0.002 2.632� 0.005 2.612� 0.013 0.192� 0.004 0.145� 0.001 0.151� 0.001

Average N/A N/A 2.628� 0.005 2.628� 0.017 0.181� 0.009 0.142� 0.008 0.142� 0.006

Plug 14.192� 0.710 N/A 2.629� 0.013 N/A 0.153� 0.002 N/A N/A

Carb-A1 0.451� 0.003 0.445� 0.001 2.699� 0.005 2.689� 0.013 0.134� 0.001 0.121� 0.001 0.123� 0.001

Carb-A2 0.449� 0.020 — 2.689� 0.005 — 0.137� 0.006 — —Carb-A3 0.386� 0.010 — 2.681� 0.005 — 0.171� 0.005 — —Carb-A4 0.453� 0.004 — 2.689� 0.005 — 0.166� 0.002 — —Carb-A5 0.416� 0.005 0.387� 0.001 2.687� 0.005 2.685� 0.013 0.186� 0.002 0.125� 0.001 0.126� 0.001

Carb-A6 0.363� 0.005 0.350� 0.001 2.677� 0.005 2.691� 0.013 0.146� 0.002 0.119� 0.001 0.114� 0.001

Carb-A7 0.447� 0.005 — 2.697� 0.005 — 0.139� 0.002 — —Carb-A8 0.454� 0.003 0.434� 0.001 2.688� 0.005 2.668� 0.013 0.188� 0.001 0.122� 0.001 0.129� 0.001

Average N/A N/A 2.688� 0.007 2.683� 0.013 0.158� 0.022 0.122� 0.003 0.123� 0.006

Plug 12.314� 0.616 N/A 2.685� 0.013 N/A 0.167� 0.002 N/A N/A

Carb-B1 0.412� 0.002 — 2.697� 0.005 — 0.280� 0.002 — —Carb-B2 0.399� 0.010 0.376� 0.002 2.694� 0.005 2.642� 0.013 0.303� 0.008 0.246� 0.002 0.261� 0

Carb-B3 0.340� 0.002 — 2.691� 0.005 — 0.327� 0.002 — —Carb-B4 0.378� 0.001 0.345� 0.002 2.694� 0.005 2.659� 0.013 0.334� 0.002 0.266� 0.002 0.271� 0

Carb-B5 0.373� 0.003 — 2.701� 0.005 — 0.333� 0.003 — —Carb-B6 0.427� 0.005 0.399� 0.002 2.699� 0.005 2.644� 0.013 0.317� 0.004 0.265� 0.002 0.270� 0

Carb-B7 0.400� 0.019 0.372� 0.002 2.691� 0.005 2.666� 0.013 0.312� 0.015 0.254� 0.002 0.261� 0

Carb-B8 0.408� 0.005 — 2.703� 0.005 — 0.278� 0.004 — —Average N/A N/A 2.696� 0.005 2.653� 0.013 0.311� 0.022 0.258� 0.010 0.266� 0.006

Plug 13.438� 0.672 N/A 2.696� 0.013 N/A 0.294� 0.003 N/A N/A

Microstructure heterogeneity L77

dissolution/precipitation mechanisms and the subsequent changesin microstructure and petrophysical parameters. For example, theexperimental work of Singurindy and Berkowitz (2003a, 2003b)shows that, depending on the flow rate of injection, the same car-bonate rock sample can be dominated by either dissolution proc-esses or precipitation processes or can be subject to one andthen the other alternately. More precisely, dissolution depends onthe local flow conditions (i.e., the pore-velocity distributions),which depend themselves on the pore (throat) size distribution(i.e., microstructure) (Bijeljic et al., 2004). However, accurate pre-diction of pore-fluid velocities at the pore scale is a complex taskand requires, for example, knowing the connectivity of the differentpores, which is generally achieved using high-resolution 3D im-ages. On the other hand, knowing only the average fluid velocity(or flow rate of the injected fluid) at the core scale will not allow usto take into account heterogeneities at the subcore scale and explain,for example, why cement and matrix do not undergo the same dis-solution/precipitation processes, unless the injection rate is so fast(compared to the calcite rate dissolution rate) that the system is con-trolled not by transport processes but by the surface kinetics. In thislatter case, pore surface area would be the main factor controllingthe dissolution and smaller grains would dissolve preferentiallycompared with bigger grains. We will justify in the following para-graphs that our experimental conditions (Vialle and Vanorio, 2011),which use flow conditions typical of those within a few feet of aninjection well, lead to a transport-controlled system.

Rock micromodel

We propose here to work at a scale that is intermediate betweenthe pore scale and the core scale, and we can use “effective param-eters” at that scale.To do so, we subdivide the rock samples into microstructural fa-

cies, based on the qualitative analysis of the SEM images and on thepore throat sizes deduced from MICP tests. Traditionally, the twomost commonly and equally used petrographic classifications forcarbonate rocks are those of Folk (1959) and Dunham (1962). Theyboth divide carbonate rocks in three main components: (1) a frame-work of skeletal grains, (2) a matrix of microcrystalline calcite (i.e.,micrite), whose grains are typically 1–4 μm in diameter (Moshier,1989) and (3) a spar calcite cement, with grains usually more than10 μm in diameter (Moshier, 1989). However, micrite can displayvarious textures: from porous micrite with rounded-to-microrhom-bic crystals, to tight micrite with anhedral compact-to-fused crystals(Lambert et al., 2006; Deville de Periere et al., 2011). The differenttextures of micrite, because of the differences in the crystal shapesand type of intercrystalline contacts, will thus display differentpetrophysical properties, from poor to excellent porosity and per-meability (Deville de Periere et al., 2011). Because we are interestedin relating microstructure to flow paths (and reactivity), instead ofsubdividing carbonate rocks in term of grain types, we divide themin terms of pore (or pore-throat) types: (1) “vugs” with pore-throatdiameters >10 μm, (2) intermediate pore-throat diameters, between1 and 10 μm, and (3) small pore throats, less than 1 μm in diameter.

Figure 8. Grain densities obtained by the different methods used, for the subsamples (∼0.5 cm3) of the four types of studied rocks. Eightsubsamples were analyzed by He pycnometry (in gray diamonds), and four subsamples by MICP technique (in black triangle). Also reported isthe grain density measured by He pycnometry on the whole plug (∼12 cm3) (in light gray).

L78 Vialle et al.

The values of the pore throats delineating the three rock compo-nents were chosen by analyzing the SEM images so that the com-ponents we defined (2 and 3) can also be related to the componentsin the Folk and Dunham classifications. Component 2 is a porousmicrite with rounded to subrounded crystals, and component 3 iseither a tight micrite, for sample carb-A, or a spar cement, for sam-ple carb-B. A schematic representation of the microgeometry of thetwo types of carbonate, carb-A and carb-B, is given in Figure 10.Note that here component (3) encompasses also the grains.As classically done for interpreting mercury-intrusion capillary

tests, we represent pores/pore throats as circular tubes. The valuesLi, ri, Ai, and Vi, which stand for the length, the radius, the wallsurface area, and the volume of each pipe i, respectively, arerelated by

Vi ¼ 2πriAi; (9)

Ai ¼ 2πriLi; (10)

and

Ai ¼ SiVS; (11)

where Si is the specific surface area of the capillary tube i in m−1

and VS is the volume of the rock sample under investigation.

Using this aforementioned pipe model, we then characterizedeach microstructural facies j in terms of porosity and effective poresurface area, plotted in Figure 11, inferred from the MICP data, asfollows:

ϕj ¼X

i

VHg;i

VS(12)

and

Sj ¼X

i

Ai

VS¼ 2

X

i

VHg;i

VS · ri¼ 2

X

i

ϕi

ri; (13)

where ϕj is the contribution of the microstructural facies j to thetotal porosity of the carbonate subsample in PU units, VHg;i is theincremental volume of intruded mercury in m3 over the pressurerange corresponding to the microstructural facies j, VS is the vol-ume of the carbonate subsample in m3, Ai is the incremental poresurface area in m2, and ri is the pore-throat radius in meters (in-ferred from equation 4). The subscript i runs over the pore-throatsize range of the component j.An effective permeability can then be obtained, for each micro-

structural facies j, with the help of the Kozeny-Carman equation(Carman, 1961):

Figure 9. Porosities obtained by the different methods used, for the subsamples (∼0.5 cm3) of the four types of studied rocks and the wholeplug (∼12 cm3). The gray diamonds represent the porosities obtained from the eight He-pycnometry-measured grain densities and geometricvolumes, and the average value is given by the dashed gray line; the black triangle represent the porosities obtained from the four MICP-measured grain densities and volumes; the gray dots represent the porosities obtained from the He-pycnometry-measured grain densities andthe four MICP-measured volumes. The shaded area is the He-porosity value and its error bar, measured on the whole plug.

Microstructure heterogeneity L79

kj ¼ϕ3j

2 · τ2 · s2j; (14)

where τ is the tortuosity (defined as the ratio of the total flow pathlength to the length of the sample). Hydraulic tortuosity typicallyranges up to about five in consolidated rocks (e.g., Dullien, 1992),but it is beyond the scope of this paper to give a precise value, es-pecially because, as developed later, we aim at getting a first-orderestimate of the presented parameters.The Kozeny-Carman relation is highly idealized and assumes that

pores are cylindrical pipes. Nevertheless, in practice it often givesreasonable results, especially for clastic rocks (Mavko et al., 2009).Indeed, the carbonates we are dealing with are mainly made ofgrains, and they are not typical, vuggy, heterogeneous carbonates;it has been shown that the Kozeny-Carman relation is appropriatefor these rocks (Dvorkin et al., 2011).At that stage, we have a rock micromodel made of three micro-

structural facies, and its microstructure parameters: porosity,specific surface area, and permeability.

Dimensionless analysis

To relate these parameters to flow paths and fluid-rock chemicalinteractions, we then estimate two dimensionless numbers, the Péc-let and the Damköhler numbers, which have been recognized asuseful criteria for determining the different dissolution regimes(e.g., de Marsily, 1986; Golfier et al., 2002), that is, whether dif-fusion, advection, or kinetics controls the chemical reactions andsubsequently the generated dissolution pattern. The dissolutionof a solid into solution consists of different physical chemical proc-esses, which can be broadly summarized as three steps: (1) transportof the reactants toward the surface, (2) surface reactions (adsorp-tion, chemical reaction at the surface, and desorption) and (3) trans-port of the products away from the surface. An important principlethat governs all multistep reactions is that the overall rate of thereaction cannot exceed the rate of the slowest step, the so-calledrate-limiting step. If the rate of the chemical reaction is slower thanthe mass transfer rate, the reaction of dissolution is limited by thereactions at the surface (surface-controlled system). On the contrary,if the rate of the chemical reaction is faster than the mass transferrate, the reaction of dissolution is limited by the mass transfer

(transport-controlled system). In this latter case,depending on how fast (or whether) the fluid isflowing, diffusion or advection processes can bethe limiting processes. The two dimensionlessnumbers that allow one to determine the dissolu-tion regime are the Péclet and Damköhler num-bers. The Péclet number compares the advectionrate to the diffusion rate, and the Damköhlernumber compares the reaction rate to the advec-tion rate. In a homogeneous porous medium,flow rates much higher than the reaction rates(which corresponds to a system in which disso-lution is controlled by the kinetics) will generatea homogeneous dissolution pattern, because thefluid will be forced into all pores and will staychemically undersaturated over long distancesbecause of its low residence time. Intermediateflow rates, for which the dissolution rate is con-trolled by advection, will create heterogeneousdissolution patterns (e.g., wormholes), due tothe initial and growing heterogeneity of thevelocity field: important dissolution will occurin the high-velocity flow paths, whereas no dis-solution and/or precipitation may occur in themore “stagnant” regions. At very low flow rates,for which the dissolution rate is controlled by dif-fusion, the fluid (except in the very near region ofthe injection) is close to thermodynamic equilib-rium with the minerals forming the rock, and thedissolution is the so-called face dissolution be-cause the acid is completely consumed in the firstmillimeter of the core. These dissolution struc-tures have been classified by Fredd and Miller(2000) based on experimental observations.Expressions of the Péclet and Damköhler

numbers are, respectively,

Pej ¼vdf;jLc;j

Df(15)

Figure 10. Schematic representation of the pore system of the two studied types ofcarbonate, carb-A and carb-B; pore throats are divided into three subsets based onthe size of their radius, obtained fromMICP tests: big pores, more than 10 μm (in white),intermediate pores (in light gray), and small pores (in black), less than 1 μm. Thesesubdivisions delineate three microstructural facies: vugs, microporous rounded micrite,and tight micrite (for carb-A) or spar cement (for carb-B), respectively.

L80 Vialle et al.

and

Daj ¼αSr;jkc;jð1 −ΩjÞLc;j

Ceq;j · vdf;j; (16)

where vdf;j is the pore fluid velocity inm · s−1, Lc;j is the character-istic length of the system under investigation in m, Df is themolecular diffusion coefficient in m2 · s−1 (which is taken equalto 10−9 m2 · s−1 as a characteristic value for the major ions in aque-ous solution), α is a stoichiometric coefficient (1 for the dissolutionof calcite), Sr;j is the reactive surface area (inm−1), kc;j is the kineticconstant of the reaction in mol · m−2 · s−1 (which depends on thelocal pH in microstructural facies j and temperature), Ωj is thedegree of saturation, and Ceq;j is the calcium concentration inmol · m−3 at the thermodynamic equilibrium between the fluid satu-rating the microstructural facies j and the calcite.To compute these two dimensionless numbers for the laboratory

experiments performed by Vialle and Vanorio (2011), we upscalethem for core plug samples 2.5 cm in length and diameter, the typ-ical size used in these experiments.The characteristic length Lc;j is thus taken as

Lc;j ¼ Ls · f1∕3j ; (17)

with Ls being the core sample length (2.5 cm here) and fj is thevolume fraction of the microstructural facies j.We then need to express the pore fluid velocity in such a way that

it will be explicitly function of the microstructure parameters ofeach of the three microstructural facies. The pore velocity is relatedto the injection flow rate Qj by

vdf;j ¼Qj

ϕjAsj; (18)

where Asj is the cross-sectional area.Using Darcy’s law

Qj ¼kjΔPjAsj

ηLj; (19)

the pore fluid velocity can be rewritten as

vdf;j ¼ΔPη

kjLjϕj

; (20)

or, by using equations 10, 11, and 13, as

vdf;j ¼4π · ΔPηVS

kjfj S2j

; (21)

where ΔP is the pressure drop across the sample in Pa, η is the fluidviscosity in Pa · s, and VS is the core plug sample volume in m3.The value of ΔP is obtained from the flow rate of injection Qexp,monitored during the experiment:

ΔP ¼ QexpηLs

Askplug; (22)

where As is the cross-sectional surface area of the core plug sampleand kplug is the core plug permeability, which has been measuredbefore the injection experiment. The pressure drop is indeed hetero-geneous throughout the sample and should differ for each of thethree microstructural facies, but we keep the value measured acrossthe sample as the characteristic value (i.e., ΔPj ¼ ΔP for all threemicrostructural facies). Doing so also implicitly assumes that thetree microstructural facies are in parallel.By substituting equation 21 in equations 15 and 16, we can thus

provide the following expression for Pe and Da, for each micro-structural facies j:

Pej ¼4π · ΔPηDf

·Ls

VS· f−2∕3 ·

kjS2j

(23)

and

Daj ¼η

4π · ΔP· VSLs ·

kc;jð1 −ΩÞCeq;j

· f4∕3j ·S3jkj

: (24)

As written above, these expressions show that the Péclet andDamköhler numbers depend on the one hand on the rock micromo-del parameters, namely, its permeability and its pore surface areaand on the other hand on some parameters related to the experimen-tal conditions (fluid viscosity, flow rate through the pressure dropacross the sample), the size of the sample, and, for the Damköhlernumber, some geochemical parameters.

Application to laboratory experiments(Vialle and Vanorio, 2011)

To evaluate the Damköhler number, one first needs to evaluatesome chemical parameters that may vary from one microstructural

Figure 11. Cumulative specific surface area for each subsample ofcore plugs carb-A and carb-B, as a function of increasing appliedpressure of mercury (lower x-axis), and corresponding decreasingpore-throat radius (upper x-axis). The cumulative specific surfacearea is deduced from the MICP curves, by modeling the rock poresystem as a bundle of parallel capillary tubes of variable size. Notethat the scale is 10 times larger for subsamples carb-A. The pore sizerange of each of the three microstructural facies is also indicated.

Microstructure heterogeneity L81

facies to another, because the completeness of the chemical disso-lution of calcite at thermodynamic equilibrium is affected by therate at which the reactive species arrive to and depart form thecalcite surface, which itself is affected by the (heterogeneous)pore-size distribution. Indeed, even at the core scale, the sole meas-urement of the calcium concentration at the output, as performed inVialle and Vanorio (2011), does not make it possible to infer thedegree of saturation Ω because the carbonate system requires thattwo parameters be fully determined. However, with reasonable as-sumptions, it is easy to provide an estimate of the Damköhler num-ber for each of the three microstructural facies. For the vugs(component 1), we can assume that the circulating fluid has apH very close to the initial pH of 3.2 because the fluid flowstoo fast for the calcite dissolution to appreciably raise the pH;the degree of saturation will be thus very small. For the “tightmicrite/spar cement” (component 3), we can assume the opposite:that the composition of the fluid will be very close to thermody-namic equilibrium with respect to calcite because pore velocitieswill be very small. The degree of saturation will be close to onein this type of microstructural facies. The “microporous micrite”(component 2) will be in an intermediate thermodynamic state.We use a degree of calcite saturation of 10−5 for component 1,0.5 for component 2, and 0.99 for component 3. Using the thermo-dynamic constants associated with the carbonate system (Plummerand Busenberg, 1982) and an initial pH of 3.4 (a consequence ofdissolution of CO2 in the water), a first-order calculation gives acalcium concentration ranging from ∼2.10−4 mol · L−1 (for thevugs component) to ∼2.10−2 mol · L−1 (for the tight micrite com-ponent) and a pH ranging from 3.5 to 5.5, for the same components,respectively. The kinetic constants for each of the three components,which depend on the pH of the saturating fluid, were deduced fromthe database of Arvidson et al. (2003) and range from 10−4 mol ·m−2 · s−1 (pH of 3.5) to 10−5.5 mol · m−2 · s−1 (pH of 5.5).For the calculation of the Péclet and Damköhler numbers, a pres-

sure drop of 0.4 MPa and a fluid viscosity of 1 cP were used. Themass diffusion coefficient was taken at 10−9 m2 · s−1. The volumefraction of each of the three microstructural facies j was determinedby a qualitative analysis of SEM images of the subsamples. Valuesof the Péclet number and of the Damköhler number are given inTable 4 for each of the four subsamples of the two carbonate sam-ples, carb-A and carb-B (upscaled to core plugs 2.5 cm in lengthand diameter), and for each of the three microstructural facies. Alsogiven in Table 4, for each microstructural facies, are its volume frac-tion, its contribution to the total porosity, its specific surface area,and its permeability. Note that in the calculation of permeability,tortuosity has been taken equal to the unit because we are interestedin getting an order of magnitude of the Péclet and Damköhlernumbers.In Table 4, we can see that there are only small differences among

the subsamples of each type of carbonate, carb-A and carb-B, interm of porosity, specific surface area, permeability, and Pécletnumbers. This is another argument for considering each subsampleto be representative of the entire plug to which it belongs. Thus, inTable 5, we give an order of magnitude of the petrophysical proper-ties and dimensionless numbers for each microstructural facies andeach type of rock.The Damköhler numbers are all much greater than 1. This means

that the reaction rate is much faster than the convective masstransport rate and that the dissolution is controlled by transport

mechanisms, for all three types of microstructural facies and bothtypes of rocks.In the vugs component, values of the Péclet number greater than

one mean that the advection rate is faster than the diffusion rate. Thefluid circulating in these zones will remain in thermodynamic dis-equilibrium with the rock-forming minerals because of the highpore-fluid velocities; the pH of the fluid will remain close to thepH of the injected fluid. In these zones, subject to efficient masstransfer of the reactants and products, calcite dissolution will thusbe important. In the tight micrite (for carb-A) and the “spar cement”(for carb-B), the advection rate is much smaller than the diffusionrate. This implies that calcite dissolution is slow in these types ofmicrostructure, and it is controlled by the species diffusion; the satu-rating fluid, being not efficiently renewed, will become close to cal-cite saturation, and the pH will be close to the pH of equilibriumbetween the saturating fluid and the calcite, which will stronglyretard calcite dissolution.Eventually, the dissolved calcium and bicarbonate ions may ac-

cumulate and recombine, giving rise to the reprecipitation of newlyformed calcite. The “microporous micrite” component is subject toa mixed regime, in which diffusion and advection rates are of thesame order of magnitude. Values of the Péclet number are aroundfive for carb-B and around one for carb-A. We need to keep in mindthat the Péclet values are an average over a certain range of pore-throat sizes, which means that some parts of the microporous mi-crite (the ones with the highest pore-throat size and/or the most con-nected) will be more dominated by advection and some parts will bemore dominated by diffusion. These values might also suggest theexistence of main flow paths within the microporous micrite, espe-cially for carb-A. The comparison of the Péclet number values be-tween the three rock components thus shows that the differentmicrostructural facies of the rock sample are exposed to distinctlydifferent chemical forcings: efficient dissolution in the biggest andintermediate pore-size components (vugs and part of the “micropo-rous rounded micrite”) and little or no dissolution and/or repreci-pitation in the smallest pore-size component (tight micrite andspar cement), which can be seen as stagnant zones. These resultsare consistent with the experimental observations of Noiriel et al.(2009) and Vanorio et al. (2011).It is worth pointing out again that the microstructural facies with

the highest specific surface area are not always the components thatare the most affected by the dissolution, depending on the type ofrock, because the pore surface area, to be reactive, needs to behydrologically accessible by the reactive species (Maher et al.,2006), which is not the case for all pore (throats) size ranges ina transport-controlled system. The tight micrite of the carb-A rockhas the highest specific surface area — higher than the micropo-rous rounded micrite — but experiences little dissolution or rep-recipitation of calcite.This quantitative analysis, based on dimensionless numbers, also

allows comparison of the two types of rock, in term of relationshipsbetween microgeometry of their pore space and implications for thefluid-rock interactions.As a consequence of the discussion above, we can assume that

dissolution happens only in the vugs and microporous rounded mi-crite components. The reactive surface area of the whole core plug(i.e., the fraction of the pore surface area in contact with the chemi-cally reactive fluid) is thus the pore surface area of the vugs plus thatof the microporous rounded micrite components. The reactive

L82 Vialle et al.

surface area Sr is thus ∼105 m−1 for carb-A and ∼2 105 m−1 forcarb-B:

Sr;B ≅ 2 · Sr;A: (25)

Using the following equation, based on transition-state theory, todescribe the rate of the calcite dissolution R (Lasaga and Kirkpat-rick, 1981):

R ¼ Q · ½Ca2þ� ¼ Ar · kc · ð1 −ΩÞ ¼ SrVS

· kc · ð1 −ΩÞ;(26)

we have

½Ca2þ�B ≅ 2 · ½Ca2þ�A; (27)

by assuming that, with the same flow rate Q, the values of thekinetic constant and of the degree of saturation are the same forboth carbonates (i.e., the average pH of the fluid in the vugs andthe microporous rounded micrite components is the same for bothrocks).The measured calcium concentration in the fluid sampled at

the outlet, after percolation through the whole core plug, was∼2.5 103 mol · L−1 for carbonate rocks whose mineralogy andmicrostructure were similar to carb-A, and ∼3.75 103 to5.70 10−3 mol · L−1 for carbonate rocks whose mineralogy and mi-crostructure were similar to carb-B (Vialle and Vanorio, 2011). Thisis consistent with equation 27, within the experimental error and thevarious approximations and hypotheses made in the method pre-sented above.Last, when a rock initially similar to carb-B was partially filled

with crude oil and rendered oil wet (Vialle and Mavko, 2011),

Table 4. Results of the quantitative analysis relative to CO2 reactive transport in subsamples carb-A and carb-B. For each ofthe three microstructural facies (vugs, microporous rounded micrite, and tight cement), f is the corresponding volume fraction,ϕ is the contribution to the subsample porosity (in PU), S is the contribution to the subsample specific surface area (in m−1), kis the derived Kozeny-Carman permeability (in mD), Da is the Damköhler number, and Pe is the Péclet number (see text for thedetails of the calculations).

VugsMicroporous

rounded micrite Tight Micriteþ grains VugsMicroporous

rounded micrite Spar cementþ grains

Carb-A1 Carb-B2

f 0.15 0.15 0.70 f 0.10 0.40 0.50

ϕ 0.01 0.05 0.07 ϕ 0.09 0.15 0.01

S 6 102 1 105 2 106 S 2 104 2 104 8 104

k 200 2 0.03 k 600 35 0.02

Pe 4 106 0.6 2 10−5 Pe 5 103 7 9 10−3

Da ≫ 1 ≫ 1 ≫ 1 Da ≫ 1 ≫ 1 ≫ 1

Carb-A5 Carb-B4

f 0.15 0.15 0.70 f 0.10 0.40 0.50

ϕ 0.01 0.05 0.07 ϕ 0.11 0.15 0.01

S 11 102 1 105 3 106 S 3 104 2 105 8 104

k 200 2 0.03 k 700 35 0.01

Pe 1 106 1 1 10−5 Pe 4 103 7 6 10−3

Da ≫ 1 ≫ 1 ≫ 1 Da ≫ 1 ≫ 1 ≫ 1

Carb-A6 Carb-B6

f 0.15 0.15 0.70 f 0.10 0.40 0.50

ϕ 0.01 0.05 0.07 ϕ 0.11 0.15 0.01

S 3 102 1 105 3 106 S 3 104 2 105 11 104

k 200 2 0.03 k 700 32 0.01

Pe 2 107 0.8 1 10−5 Pe 5 103 7 2 10−3

Da ≫ 1 ≫ 1 ≫ 1 Da ≫ 1 ≫ 1 ≫ 1

Carb-A8 Carb-B7

f 0.15 0.15 0.70 f 0.10 0.40 0.50

ϕ 0.01 0.04 0.07 ϕ 0.09 0.15 0.01

S 5 102 1 105 2 106 S 3 104 2 105 10 104

k 460 2 0.03 k 500 35 0.03

Pe 1 107 7 10−1 2 10−5 Pe 4 103 8 8 10−3

Da ≫ 1 ≫ 1 ≫ 1 Da ≫ 1 ≫ 1 ≫ 1

Microstructure heterogeneity L83

the measured calcium concentration in the output fluid was3.2 103 mol · L−1. This value is lower than the range of values ob-tained on clean, similar rocks (3.75 10−3–5.70 103 mol · L−1),which means that the presence of oil in the pore space made therock less reactive. It was observed, through SEM imaging combinedwith confocal microscopy (Vialle and Mavko, 2011), that some oilwas trapped in the microporous rounded micrite, thus preventing thereactive fluid from flowing through it and decreasing the reactivesurface area.

CONCLUSION

The question that initiated this experimental work was how dif-ferent the MICP curves could be within a single rock plug. In otherwords, how heterogeneous can natural rock be with respect to pore-size distribution at a linear scale about one-fifth of the standard plugsize. We have established that within each of the four different plugsunder examination, this heterogeneity is practically nonexistent.Still, the capillary curves strongly differ between the four plugs.We used this diversity to assess the spatial intensity of chemical

transformation in the two carbonate samples due to CO2-rich waterinjection. To do so, we used the Péclet and Damköhler dimension-less numbers Pe and Da, which help characterize the different dis-solution regimes, i.e., whether diffusion, advection, or kineticscontrols the chemical reactions. The key point was to subdividethe pore structure of each carbonate sample into microstructural fa-cies (e.g., tight micrite, microporous rounded micrite, small vugs,and grains), and express Pe and Da for each microstructural facies,as a function of its intrinsic (effective) petrophysical properties(namely, permeability and effective surface area), that were inferredfrom the MICP data and classical rock-physics relationships. Bydoing so, we quantitatively linked parameters characterizing the

pore space of each microstructural facies to the magnitude of thedifferent transport and kinetic processes. This shows that the mainfactors controlling the movement and chemical reactivity of CO2

are permeability, which controls the effective mass transfer ofthe reactants and dissolved products, and reactive surface area,the fraction of the pore space in contact with the reactive fluid.For laboratory experiments that uses conditions typically foundwithin a few feet of the injection well (high flow velocities andlow pH) and low effective stress, we found that the studied systemwas controlled by the transport processes (Da ≫ 1). The values ofthe Péclet number were different for each of the microstructuralfacies by several orders of magnitudes, ranging from values ≪1

(diffusion-controlled system) to ≫1 (advection-controlled system).Hence, because of the existence of structural heterogeneities (i.e.,the existence of different microstructural facies with the same rocksample), the rock is locally exposed to different chemical forcings(e.g., diffusion-controlled, advection-controlled, : : : regimes),which makes some zones chemically more reactive than others.More generally, at the reservoir scale, the rock formation will be

exposed to advection-dominated regimes close to the injection well,where the pore fluid velocities are high and the fluid is far from thethermodynamic equilibrium, and diffusion-dominated (or eventu-ally kinetics-dominated) regimes far from the well, where the porefluid velocities are low and the fluid is close to the thermodynamicequilibrium. These distinct dissolution regimes will cause differentmodifications of the pore microgeometry, and consequently differ-ent modifications of the porosity, permeability, and mechanicalproperties of the rock.

ACKNOWLEDGMENTS

This work was sponsored by the Stanford Rock Physics andBorehole Geophysics Project, the Stanford Global Climate and En-ergy Project Award 55, and DOE contract DE-FE0001159. Wewishto thank T. Vanorio for providing the samples, S. Benson for the useof her laboratory equipment, L. Joubert (CSIF Stanford MedicalSchool) for advice in SEM image acquisitions, and R. Pini for tech-nical assistance in sample preparation and fruitful discussions dur-ing MICP tests. We thank Y. Guéguen for his invaluable commentsas well as the associate editor C. Morency and an anonymousreviewer.

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