Linking the Chemical and Physical Effects of CO2 Injection to

Geophysical Parameters

Investigators Gary Mavko, Professor (Research), Tiziana Vanorio, Asst. Professor, Geophysics; Sally Benson, Professor (Research), Energy Resources Engineering, Stanford University. Andreas Luttge, Professor, Earth Science; Rolf Arvidson, Sr. Research Scientist, Earth Science, Rice University. Abstract

This project has demonstrated techniques for quantitatively predicting the combined seismic signatures of CO2 saturation, chemical changes to the rock frame, and pore pressure. This was accomplished (i) by providing a better understanding the reaction kinetics of CO2-bearing reactive fluids with rock-forming minerals, and (ii) by quantifying how the resulting long-term, CO2-injection-induced changes to the rock pore space and frame affect seismic parameters in the reservoir. This research has involved laboratory, theoretical, and computational tasks in the fields of both Rock Physics and Geochemistry. Ultrasonic P- and S-wave velocities were measured over a range of confining pressures while injecting CO2 and brine into the samples. Pore fluid pressure was varied and monitored together with porosity during injection. The measurement of rock physics properties was integrated and complemented by those obtained via geochemical experiments to link the physical and chemical processes underlying the mechanisms triggered by CO2 injection. In this final report, we summarize results of laboratory observations and theoretical, and numerical rock physics models to quantify the P- and S-wave velocities and wave attenuation in partially saturated rocks. Introduction

Monitoring, verification, and accounting (MVA) of CO2 fate are three fundamental needs in geological sequestration. The primary objective of MVA protocols is to identify and quantify (1) the injected CO2 stream within the injection/storage horizon and (2) any leakage of sequestered gas from the injection horizon, providing public assurance. Thus, the success of MVA protocols based on seismic prospecting depends on having robust methodologies for detecting the amount of change in the elastic rock property, assessing the repeatability of measured changes, and interpreting and analyzing the detected changes to make quantitative predictions of the movement, presence, and permanence of CO2 storage, including leakage from the intended storage location. This project addresses the problem of how to interpret and analyze the detected seismic changes so that quantitative predictions of CO2 movement and saturation can be made. The main goals have been:

(a) linking the chemical and the physical changes occurring in the rock samples upon

injection; (b) assessing the type and magnitude of reductions caused by rock-fluid interactions

at the grain/pore scale;

(c) providing the basis for CO2-optimized physical-chemical models involving frame substitution schemes.

Background

Having the appropriate rock-physics model is generally a key element for time-lapse seismic monitoring of the subsurface, both to infer the significance of detectable changes (i.e. qualitative interpretation) and to convert them into actual properties of the reservoir rocks (i.e. quantitative interpretation). Nevertheless, because of the peculiar ability of CO2-rich water to promote physicochemical imbalances within the rock, we must address whether traditional rock-physics models can be used to invert the changes in geophysical measurements induced in porous reservoirs by the injection of CO2, making it possible to ascribe such changes to the presence or upward migration of CO2 plumes. Making this determination requires an understanding of the seismic response of CO2-water-rock systems. Seismic reservoir monitoring has traditionally treated the changes in the reservoir rock as a physical-mechanical problem—that is, changes in seismic signatures are mostly modeled as functions of saturation and stress variations (e.g. pore and overburden pressure) and/or intrinsic rock properties (e.g. mineralogy, clay content, cementation, diagenesis…). Specifically, modeling of fluid effects on seismic data has been based almost exclusively on Gassmann’s equations, which describe the interaction of fluid compressibility with the elastic rock frame to determine the overall elastic behavior of rock. Beginning with the bulk (Ks) and shear (µs) moduli of the mineral composing the rock, we use Gassmann’s fluid substitution scheme to compute the bulk modulus of the saturated rock (Ksat) from the bulk modulus of the fluid (Kfl) and from that of dry rock (Kdry). However, depending on the properties of the mineral composing the rock and the properties of the fluid, complex rock-fluid interactions may occur at the pore scale, leading to dissolution and formation of new mineralogical phases. All these physical modifications may compete in changing macroscopic rock properties such as permeability, porosity, and elastic velocities, which, in turn, can change the baseline properties for the Gassmann’s fluid substitution scheme. This entails two consequences: (a) a classical fluid-substitution scheme may underpredict time-lapse changes and thus mislead 4D monitoring studies; and (b) predictions of in situ velocity will compensate for the chemically softened velocities with erroneous estimates of saturations and/or pore pressure. Results What Laboratory-ˇInduced Dissolution Trends Tell Us About Natural Diagenetic Trends of Carbonate Rocks. (Tiziana Vanorio, Yael Ebert, and Denys Grombacher, Stanford University, Submitted to joint Geological Society of London-AAPG Special Publication)

The pivotal idea of this study is to unfold the processes that control the heterogeneity

in the attributes of the pore space in carbonates and, in turn, in the transport and elastic properties. Our objective is to investigate the mechanisms that may have been responsible for the pore space character. We use starting rocks of variable fabric –i.e., a depositional-dependent microstructure, induce a specific process – e.g., chemical dissolution under stress, then observe the development of the microstructure, permeability, porosity, and velocity due to the induced chemo-mechanical processes.

We find that the changes in the rock can lead to two different evolutionary trends of

permeability and velocity, depending on the effectiveness of dissolution and compaction, which in turn depend on (a) the fraction of the carbonate phase characterized by high surface area and (b) the pore stiffness of the rock. Carbonates that compact significantly upon dissolution show a reduction in contact stiffness leading to a decrease in velocity and an increase in permeability. The latter is curbed by the ongoing compaction. Carbonates experiencing minimal or negligible compaction show a slight change in porosity and velocity; however large permeability changes are observed that are related to an enhancement in connectivity or decrease in the tortuosity of the pathways.

The fundamental argument driving this study is that the spatial distribution of

carbonate reservoir properties is a product of interactions between 1) the specific diagenetic process acting upon the rock, 2) the parameters controlling fluid type and circulation (e.g., pH and flow rate) and 3) the original depositional facies defining the intrinsic mechanical response of the rock to chemical changes.

The key question is this: when we introduce a chemically reactive fluid in a carbonate

rock under stress, how does the pore space change in response to stress redistribution and microstructure adjustment as the fluid-rock system attains a new chemo-mechanical equilibrium. Because unraveling continuous sequences of diagenetic events due to dissolution and compaction is difficult and ambiguous, we use a forward modeling approach: we directly monitor the evolution of porosity, permeability, compressional and shear velocities (PPV trends) of different depositional carbonate facies. Different facies imply a different texture, and in turn, a variable pore type and pore compressibility. We induce chemical dissolution by flooding the rock samples with a carbonic acid solution (~pH=3.5) while monitoring the mechanical deformation under stress and the induced changes in transport and elastic properties. In our experiments, we monitor the evolution of both transport and elastic parameters to 1) comprehensively look at what attribute of the pore space is more likely responsible for the changes, given the original texture of the rock, and 2) learn whether the induced modifications to the microstructure lead to systematic patterns in rock properties which are characteristic of specific carbonate facies.

We conducted several laboratory experiments to investigate and monitor the

evolution of the pore network attributes responsible for variations in porosity, permeability, and P- and S-wave velocities of carbonate rocks. The experiments consist of flow-through injections of an aqueous CO2 solution into samples under confining pressure. Injections are performed under variable confining pressure, while maintaining a nearly constant downstream flow rate of 5ml/min. Effective pressure is kept constant upon injection and ranges between 2 MPa and 5 MPa depending on the trade-off between sample permeability and flow rate. Several cumulative injections are performed during each experimental run. The main objective is to expose samples to increasing pore volumes of fluids and to understand the role of the continued exposure on the trends of the measured properties. After each flooding, we characterize the samples’ helium porosity, Klinkenberg-corrected nitrogen permeability, and ultrasonic P- and S-wave

velocity. Time-lapse SEM imaging of the samples is performed after injection to relate changes in the physical properties to those in the rock microstructure. Helium gas is flowed at 150 psi through the sample after each injection for 8 hours to ensure drying within the vessel. This allows us to measure and monitor the variation of the properties of the frame alone while minimizing dispersion effects on velocity due to high frequency (Mavko and Jizba, 1991). During injection, the plugs are jacketed with rubber tubing to isolate them from the confining pressure medium. PZT-crystals mounted on steel endplates are used to generate P- and S-waves (1 MHz for P-waves and 0.7 MHz for S-waves). To monitor the rock strain driven by chemical dissolution under stress, we monitor a) the length decrease of the samples as a function of stress, which we then relate to porosity loss upon compaction and b) the concentration of calcium in the output fluid, which we relate to porosity enhancement by dissolution. The total net change in porosity is thus the resulting porosity from each component. We estimate the accuracy in velocity, porosity and permeability measurements to be approximately ±1%, ±1%, and ±2%, respectively.

Figures 1a and 1b show the changes in bulk and shear moduli of the dry frame of a

micritic grainstone/chalk (MA formation) upon injecting increasing volumes of the aqueous CO2 solution. Moduli data are colored-coded as functions of confining pressure. Several important results are apparent in these data. First, as the first pore volumes are injected (14.3 Pv), both moduli (and by proxy velocities) decrease in magnitude (Figure 1a and Figure 1b), implying either an increase in porosity, a reduction in the stiffness at the grain contacts, or both. Second, the decrease in elastic moduli levels off over time despite continued injection of the acidic solution (Figure 1a and Figure 1b). Similarly, the amount of Ca++ measured from the effluent fluid at the outlet also levels off, implying a reduced rate of calcite dissolution (Figure 1d). Third, the decrease in velocity is pressure dependent.

Figure 1c shows how the P-wave velocity of the dry frame varies as a function of

confining pressure; data are color-coded by number of injected pore volumes. For each pressure, the decrease in velocity with respect to its pre-injection value increases, while pressure decreases from 15MPa to 5MPa (Figure 1c). This translates into an increased sensitivity of velocity to pressure as injection proceeds, which implies a change in pore stiffness.

Figure 2 shows the porosity evolution of the rock sample as measured after each

injection. At first, the sample experiences mechanical compaction due solely to the effect of confining pressure (Injected pore volume=0). Porosity progressively decreases from ~25.2% (black circle) to ~23.2% (blue diamonds at 15MPa) as pressure increases (Figure 2a). As injection proceeds, the sample experiences two competing mechanisms. On the one hand, dissolution of calcite leads porosity to increase (orange circles, Figure 2b). On the other hand, the sample experiences a concurrent, pressure-dependent compaction driven by dissolution (Figure 2a). Compaction is likely due to internal creeping under constant stress resulting from the decreased rock strength. Thus, the total net evolution in porosity of the sample upon dissolution under stress (Figure 2b) is the sum of both processes and is calculated by considering the change in porosity due to each component.

Figure 2b also shows that porosity enhancement due to dissolution overwhelms the loss in porosity due to compaction, resulting in a net porosity increase after flooding.

Figure 3 shows the variations of elastic moduli, velocity, and Ca++ concentration

upon injection through a tight limestone (MSA formation). Before injection, the tight limestone shows much lower velocity-pressure sensitivity than does the micritic grainstone (Figure 1c and Figure 3c - blue diamonds), implying a stiffer pore network. The different pressure sensitivity translates to a much smaller porosity loss in the limestone due to pressure dependent compaction (Figure 2a and Figure 4a – data points at Pv equal to zero). After injection, there are three main differences between the tight limestone and the micritic grainstone. First, the magnitude of the decrease in elastic moduli (and by proxy velocity) of the tight limestone is smaller than that of the micrite-rich grainstone (Figure 1 and Figure 3). Also, the decrease does not level off as much as observed in the micrite-rich grainstones. Second, the tight limestone experiences less dissolution, as highlighted by the smaller concentration of Ca++ measured within the outlet fluid (Figure 1d and Figure 3d). This leads eventually to a smaller porosity enhancement from dissolution (Figure 2b and Figure 4b). Third, as the injected pore volumes increase, the tight limestone experiences less dissolution-induced compaction than does the micrite-rich grainstone (Figure 2a and Figure 4a). As a result, porosity enhancement due to dissolution does not strongly overwhelm the already minimal porosity loss due to compaction, resulting in a minimal porosity increase overall (Figure 4b). As do the grainstones, limestones show an increased sensitivity of velocity to pressure with flooding (Figure 3c), which is particularly visible at low pressures ranges.

In Figure 5, we compare the evolution of velocity and permeability of the treated

samples to the natural diagenetic trends of untreated (not injected) carbonate samples (gray symbols). Data are plotted against porosity, and the different colors describe different injected samples; the black-circled symbol refers to the value measured for each sample before injection. Thus, the observed trends reflect how the modifications in the pore attributes and microstructure due to dissolution under stress affect velocity and permeability. Although the number of injected pore volumes is nearly the same for each sample, the evolution trends of properties vary across facies, with some similarities among samples belonging to the same facies/formation. In chalky, micritic carbonates (MA and high-porosity FP formations), both porosity and permeability increase upon injection, while velocities strongly decrease. In limestones (tight MSA and low-porosity FP formation), porosity and velocity show negligible change upon injection, while permeability greatly increases after reaching a critical threshold. Another noteworthy and surprising result is that the evolution of both velocity and permeability of the injected samples follows the natural diagenetic trends. Dissolution seems to primarily affect the permeability of tight limestones while leaving porosity and velocity essentially unaltered. Conversely, dissolution greatly affects the decrease in velocity of micrite-rich samples with a rate that increases going from grainstones to lime mudstones. For this facies, both permeability and porosity slightly increase as dissolution proceeds.

Transport and elastic properties depend on different attributes of the pore space.

While stiffness at the grain contacts, porosity, and pore compliance control velocity

(Anselmetti and Eberli 1993, Mavko et al. 2008, Weger et al. 2009), pore connectivity, tortuosity, and pore size control permeability (Carman 1961). Any change of these parameters during diagenesis defines the evolution of permeability, porosity and velocity. This paper shows that to understand their evolution, it is vital to couple dissolution, stress, and the original pore stiffness of the rock inherited by the depositional environment, which ultimately defines the microstructure of the rock. It is clear that the combination of the initial rock microstructure and the sequence of events to which it is exposed enormously increases the number of possible deviations from first-order relationships between geophysical observables and rock properties (e.g., porosity and permeability).

Data comparison between micrite-rich grainstones and tight limestones show that the

evolution of permeability and velocity in carbonates, due to leaching and/or more advanced dissolution, is microstructure-dependent. Under constant stress, carbonates experience dissolution-induced compaction (Figure 2 and Figure 4) as a result of the decreased strength of the rock. This weakening results exclusively from the removal of carbonate phases with high surface area (i.e., micrite and cement). Dissolution and compaction compete and feed back to each other. The results depend on (1) the specific pore-stiffness of the rock, which controls the amount of compaction, (2) the amount of micrite and/or cement, which is responsible for selective dissolution, and (3) the tightness of the formation, which, by defining permeability, influences the effective fluid circulation and the balance between dissolution and precipitation. Altogether, these factors determine how velocity and permeability of carbonates evolve under stress. In micrite-rich grainstones, pores that are more compliant than in tighter limestones, as well as the presence of high-surface-area phases (i.e., microcrystalline calcite), favor dissolution and dissolution-induced compaction. Dissolution of cement and micrite causes compaction as well as grain slip and rearrangement), ultimately reducing the stiffness at the grain contacts. Therefore, injecting increasing volumes of fluid into a compacting sample does not lead to further decrease in the elastic moduli, which instead level off as injection proceeds. This is particularly important when monitoring injection of reactive fluids into carbonate reservoirs by time-lapse seismic. In addition, compaction continuously compensates for the enhancement in porosity and the enlargement of pore throats resulting from dissolution, hence curbing the increase of the permeability of these rocks. In tight limestones, the finely interlocked mosaic of micrite rhombohedra makes the deposition-inherited microstructure of these rocks very stiff (i.e., they have high pore stiffness). This, in turn, makes this rock type very insensitive to compaction. As a consequence, the enlargement of pore throats and microporosity due to dissolution is only minimally counteracted by the compaction. In addition, the relatively low permeability favors a poor dewatering of the Ca++ ion-rich water as well as supersaturation, which leads calcite to precipitate and weld micro-grains together under the effect of pressure. The direct consequence of these processes is that the enhancement of pore throats does not contribute much to the total porosity or to the decrease in velocity. However, it definitely affects pore connectivity (and likely tortuosity), leading the permeability of these tight microstructures to greatly increase.

One of the most important results of this study is that the evolution of both velocity and permeability of the injected samples follows the natural diagenetic trends of the facies. The similarities of the induced evolution of properties with those depicted by the natural diagenetic trends imply that pore attributes modify and the rock microstructure readjusts through common rules that are characteristic of each carbonate facies. Thus, the original depositional facies defines the intrinsic mechanical response of the rock to chemical changes, playing a major role in the way pore space modifies under chemo-mechanical processes. These rules seem to create systematic patterns in rock properties, whose evolution is mainly controlled by compaction, and in turn, surface area and pore stiffness. Depending on the effectiveness of compaction, changes in the rock lead either to reduction in contact stiffness (high compaction) or connectivity enhancement (low compaction). These factors are key to future model-building strategies and/or pore-scale computation of processes aiming at property predictions in carbonate reservoir characterization.

By monitoring the evolution of porosity, permeability, and velocity upon dissolution-

induced compaction, the results in this paper provide a better understanding of how chemo-mechanical processes alter the pore-space attributes. Since the experimental results mirror diagenetic trends, they also provide new insights to predict the evolution of the products of these alterations within the reservoir - i.e., porosity and permeability heterogeneity as well as acoustic response of laterally variant units. The agreement between the evolution of velocity and permeability found in the laboratory and that expressed by the natural diagenetic trends of the facies simply shows that there are basic rules controlling the way pore attributes modify upon diagenesis, rules which are dictated by the coupling between chemical and mechanical processes acting on a depositional dependent microstructure. This synergy defines how pore-space parameters change, creating the scatter in the velocity and permeability data in carbonates. Carbonate facies determine the initial microstructure, which closely controls pore stiffness and the proneness to volumetric compaction during diagenesis. Depending on the effectiveness of compaction, changes in the rock may lead either to reduction in contact stiffness (high compaction) or connectivity enhancement (low compaction); eventually leading to two evolutionary patterns of velocity and permeability of carbonates experiencing underground circulation of aggressive fluids.

Figure 1: Variation of the bulk (a) and shear (b) moduli as functions of injected pore volumes. Data refer to micrite-rich grainstone from the MA formation and are color-coded as a function of confining pressure, (c) variation of P-wave velocity as a function of confining pressure. Data are color-coded as a function of the injected pore volumes, (d) variation of the concentration of the ion calcium in the collected fluid as function of the injected pore volumes.

Figure 2: Variation of the porosity of the MA micrite-rich grainstone as a function of the injected pore volumes. The sample experiences a dissolution-driven compaction under the effect of the applied stress, eventually leading to porosity change.

Figure 3: Variation of the bulk (a) and shear (b) moduli as functions of injected pore volumes. Data refer to a tight limestone from the MSA formation and are color-coded as functions of confining pressure, (c) variations of P-wave velocity as a function of confining pressure. Data are color-coded as functions of the injected pore volumes, (d) variation of the concentration of the ion calcium in the collected fluid as a function of the injected pore volumes.

Figure 4: Variation of the porosity of the MSA tight limestone as a function of the injected pore volumes. The sample experiences a dissolution-driven compaction under the effect of the applied stress, eventualy leading to porosity change.

Figure 5: Evolution of P- and S-wave velocity (top) and permeability (bottom) of the injected sampls as functions of porosity. Data are compared to the natural diagenetic trends of untreated carbonate samples (gray circles). Data refer to dry samples measured after being injected.

Differentiating chemical effects and pressure effects on the elastic properties of the Lower Tuscaloosa sandstone in Cranfield, Mississippi by injecting carbon dioxide rich brine. (Corey Joy, The University of Texas at Austin, Tiziana Vanorio, Stanford Rock Physics Laboratory, and Mrinal K. Sen, The University of Texas at Austin. SEG Extended Abstract 2011)

Current and evolving regulatory and commercial environments governing injection

may require monitoring. Rock physics can be used to determine the effect that these injected fluids have on the reservoirs with time. Performing rock physics experiments on site-specific core plugs can give one the ability to differentiate between chemical effects and pressure effects on the elastic properties of the rock while injecting a chemically reactive substance, or reactant. Detailed Area Study (DAS) of Cranfield, Mississippi, has been selected as a candidate for carbon dioxide enhanced oil recovery. This provides an opportunity to study the geochemical and geophysical effects of injecting carbon dioxide into the subsurface. Cranfield DAS is a unique site in that a variety of methods are used to monitor carbon dioxide in the subsurface. One of such methods is time-lapse seismic monitoring. The integration of rock physics with 4D seismic technology is an effective tool to quantitatively characterize the reservoir dynamics associated with fluid injection. Nevertheless, changes in seismic velocities are mostly interpreted as variations of physical parameters— such as saturation, pore fluid pressure, temperature, and stress. Hovorka (2009) reported that in the Frio Formation, dissolved CO2 lowered the pH of the brine, dissolving calcite from the matrix and large amounts of iron (Fe) and manganese (Mn) (Kharaka et al., 2006). Furthermore, laboratory investigation of CO2-brine-rock interaction within samples from the Tuscaloosa sandstone formation at Cranfield, Mississippi reported a rapid increase of dissolved iron immediately after injection, caused by the dissolution of iron-bearing carbonate minerals and ironchlorite (Karamalidis et al., 2010). Therefore, it is essential to differentiate the effects on the elastic properties due to physical (i.e., pressure) and chemical processes occurring upon injection of carbon dioxide into the Lower Tuscaloosa formation.

Since chemical reactions depend on pressure, temperature and chemical components,

it is important to simulate in-situ conditions as well as possible. Recreating the in-situ fluid is one of the most important steps in this experimental process, because the fluid in place may promote chemical dissolution/precipitation and/or buffer the injected reactant. At the Cranfield Detailed Area Study, the in-situ brine of the Tuscaloosa formation is Na-Ca-Cl type with about 152,000 milligrams per liter of total dissolved solids (Karamalidis et al., 2010). According to La Chatelier’s principle, the presence of high calcium content in the brine may inhibit dissolution of calcite. The pH values of the synthetic Tuscaloosa brine and the CO2 rich Tuscaloosa brine were 5.8 and 3.4 respectively. The temperature and pressure of the formation also control the chemical reactions.

Confining pressure plays an important role in simulating the chemical reactions. We

use a pressure column approach to resolve the confining stress. According to well logs, the average density is roughly 2.1 grams per cubic centimeter (g/cc). The formation

under consideration is roughly 3.2 kilometers (km) deep. Therefore, the confining stress is roughly 65 MPa. The pore pressure before injection is approximately 31 MPa; the pore pressure after injection is approximately 38 MPa (Hovorka et al. 2009). However, due to the limitations of the pressure vessel used, both confining and pore pressures are scaled back by 10 MPa in order to maintain in-situ differential pressure. Therefore, the confining pressure, initial pore pressure, and final pore pressure are 55 MPa, 21 MPa, and 28 MPa respectively. Note that the experiments are performed at room temperature in order to hold one variable constant. The characterization of the samples included Helium porosity, Klinkenberg-corrected nitrogen permeability, and ultrasonic P- and S-wave velocities. The plugs were jacketed with rubber tubing to isolate them from the confining pressure medium. PZT-crystals mounted on steel endplates generated P- and S-waves (1 MHz for P-waves and 0.7 MHz for S-waves). Three linear potentiometers were used to measure length changes of the samples as a function of stress. The length changes were related to changes in porosity by assuming that pore contraction was the main cause of strain. The accuracy in velocity measurements was estimated to be about ±1%. Compressional and shear wave velocities are measured at various confining pressures after multiple injections of carbon dioxide rich brine. The velocities are measured when the sample is dry to minimize dispersion. Measurements of a dry sample also show how the frame changes with each injection. Monitoring the changes in the elastic moduli of the dry frame as a function of injected volumes is also important for fluid substitution modeling (Vanorio et al., Geophysics in press, 2011). The first step in the experimental process is to prepare the sample and record a baseline of the mechanical properties. Since the sample is sandstone with some clay, the core plug was dried at 70ºC for 72 hours. The initial geometry and mass of the core plug were recorded to note any change in bulk properties after carbonic acid injection. The initial length, diameter and mass of the sample are 27.5 millimeters (mm), 25.3 mm, and 29.1 g respectively. The initial porosity of the sample is 21%. The initial permeability of the sample is 5 millidarcies (mD). After measuring porosity and permeability, the sample is inserted into the pressure vessel where it undergoes a series of loadings, reactant injections, and unloadings. Measuring the pressure dependency of P- and S- wave velocities for the dry core plug, before any injection, establishes a baseline for the elastic properties of the rock. Both the compressional and shear velocities of the core plug are measured as a function of confining pressure, in this case ranging from 0 to 55 MPa. Smaller pressure intervals are used at low confining pressures to map the nonlinear portion of the compaction curve. The following list is the workflow used in these experiments: 1. Load sample to 55 MPa. 2. Inject sample with carbon dioxide rich synthetic Tuscaloosa brine to a pore pressure of

28 MPa. 3. Measure change in length, Vp, and Vs. 4. Flush the sample with a given number of pore volumes of carbon dioxide rich synthetic

Tuscaloosa brine while maintaining a pore pressure of 28 MPa. 5. Measure change in length, Vp, and Vs. 6. Leave saturated sample in current conditions for 12 hours. 7. Measure changes in length, Vp, and Vs.

8. Drain fluid and dry with gaseous helium at 150 psi for five hours. Now the pore pressure is zero.

9. Measure changes in length, Vp, and Vs while decreasing confining pressures to 20 MPa.

10. Repeat steps 1-9 to monitor the effect on Vp and Vs while increasing pore fluid volumes Measuring velocity at various differential pressures after injection may simulate

changing the pore pressure. Knowing the impact of pore pressure on the elastic properties of the core plug can enable one to resolve the change in velocity due to the chemical reaction and the change in velocity due to the change in pore pressure. However, loading and unloading a core plug in this manner may permanently deform the microstructure of the sample. In order to understand if the sample was experiencing any damage revealed by hysteresis, we performed an experimental test in which we loaded and unloaded a “twin” sample with the same stress path. The “twin” core plug comes from the depth as the injected sample and has the same porosity and permeability. SEM images of the top of the samples (i.e., injection side) are taken before and after the injection experiments to monitor changes in the rock’s microstructure. The fluid is regularly sampled at the outlet to measure pH and iron content using the titration method. We use a digital titrator (HACH LANGE 16900) and a commercial TitraVer 0.0716 M solution as titrant.

The compaction curves show that velocities increase with increasing confining

pressure (Figure 6). As typically observed in sandstones, the compressional and shear wave velocities increase quickly with confining pressure up to 20 MPa. At a confining pressure of 20 MPa the curve flattens and is approximately linearly proportional to confining pressure. After the injection of the first 25 pore volumes, the P and S-wave velocities of the dry frame decreased by roughly 10%. Small changes in velocities are observed after subsequent carbonic acid injections. The compaction curves following injection behave similarly to the compaction curves pre-injection; from 20 to 55 MPa the velocity is linearly proportional to differential pressure (Figure 6).

The elastic moduli decrease after each injection. The largest decrease in elastic

moduli is observed after the first injection. The bulk and shear moduli remain fairly constant after the second injection (Figures 7 and 8). Using the linearity of the compaction curves, we found the P and S-wave velocities at a differential pressure of 26 MPa, which is the current in-situ differential pressure of the injection site. After adjusting density for change in volume, the bulk and shear moduli were extracted from the velocities (Figures 9 and 10). The equations for the shear and bulk moduli as a function of injected pore volumes, Pv, are

µ Pv( ) =11.722 + 0.7404*exp −1.1316*Pv( )K Pv( ) =16.713+ 2.2827*exp −0.041093*Pv( )

After the final injection and drying in the oven for three days at 60ºC, the final length,

diameter, and mass of the core plug are 27.3 mm, 25.3 mm, and 29.1 grams respectively. The final porosity and permeability are 20.2% and 10 mD respectively. The sample has

hardly changed in the context of size, weight, and petrophysical properties. However, the elastic properties have greatly changed, which are a manifestation of change in microstructure. Microstructural changes are observed in the core plug after the injection of CO2 rich brine (Figures 11 and 12). The sand grains are not affected by the reactant and do not compact. On the other hand, the pore filling material experiences significant deformation. Dissolution of the material under pressure occurs at grain contacts and induces cracks. The same stress path was applied to the “twin” as the injected sample. The compaction curves of the “twin” do not show change in P or S-wave velocities while loading and unloading within error.

The purpose of this paper is to study the effects of chemical reactions and pressure

effects on the elastic properties of the Lower Tuscaloosa sandstone to better quantify the amount of carbon dioxide in the subsurface using seismic methods. The effects of changing the differential pressure did not have a large effect on the compressional and shear wave velocities for this rock. Decreasing the confining pressure from 33 to 26 MPa decreased P and S-wave velocities by roughly 60 and 20 m/s respectively.

Figure 6: Compaction curves of the dried sample pre and post-injections. Pore pressure is zero.

Figure 7: Shear modulus of the dry frame versus injected pore volumes colored by confining pressure.

Figure 8: Bulk modulus of the dry frame versus injected pore volumes colored by confining pressure.

Figure 9: Shear modulus of the dry frame versus injected pore volumes for a differential pressure of 26 MPa.

Figure 10: Bulk modulus of the ddry frame versus injected pore volumes for a differential pressure of 26 MPa.

Figure 11: SEM image of the core plug before CO2 rich brine.

Figure 12: SEM image of the core plug after CO2 rich brine.

How predictable are dissolution rates in natural and industrial systems? (Cornelius Fischer, Rolf S. Arvidson, and Andreas Lüttge, Rice University)

The large discrepancy between field and laboratory measurements of mineral reaction rates is a long-standing problem in earth sciences, often attributed to factors extrinsic to the mineral itself. Nevertheless, differences in reaction rate are also observed within laboratory measurements, raising the possibility of intrinsic variation as well. Critical insight is available from analysis of the relationship between the reaction rate and its distribution over the mineral surface. This analysis recognizes the fundamental variance of the rate. The resulting anisotropic rate distributions are completely obscured by the common practice of surface area normalization. In a simple experiment using a single crystal and its polycrystalline counterpart, we demonstrate the sensitivity of dissolution rate to grain size, results that undermine the use of “classical” rate constants. Comparison of selected published crystal surface step retreat velocities (JORDAN & RAMMENSEE, 1998) as well as large single crystal dissolution data (BUSENBERG & PLUMMER, 1986) provide further evidence of fundamental variability. Our key finding is the loss of the classical assumption regarding a single-valued “mean” rate or rate constant as a function of environmental conditions. Reactivity predictions and reservoir long-term stability calculations based on laboratory measurements are thus not directly applicable to natural settings without a probabilistic approach. Such a probabilistic approach must incorporate both the variation of surface energy as a general range (intrinsic variation) as well as constraints to this variation owing to heterogeneity of complex material (density of domain borders etc.). [submitted to Geochimica et Cosmochimica Acta]

Does the stepwave model predict mica and clay mineral dissolution kinetics? (Inna Kurganskaya, Rolf S. Arvidson, and Andreas Luttge, Rice University)

Micas and clay minerals are a unique class of minerals because of their layered structure. A frequent question arising in mica dissolution studies is whether this layered structure fundamentally changes the dissolution mechanism. We addressed this critical question in this work, using VSI and AFM data from experiments using muscovite to evaluate crystallographic controls on mica dissolution. These data provide insight into the dissolution process, and reveal important links to patterns of dissolution observed in framework minerals. Under our experimental conditions (pH 9.4, 155°C), the minimal global rate of surface-normal retreat observed in VSI data was 1.42 × 10−10 mols ⁄ m2 ⁄ s (σ = 27%) while the local rate observed at deep etch pits reached 416 × 10−10 mols ⁄ m2 ⁄ s (σ = 49%). Complementary AFM data clearly show crystallographic control of mica dissolution, both in terms of step advance and the geometric influence of interlayer rotation (stacking periodicity). These observations indicate that basal/edge surface area ratios are highly variable and change continuously over the course of reaction, thus obviating their utility as characteristic parameters defining mica reactivity. Instead, these observations of overall dissolution rate and the influence of screw dislocations illustrate the link between atomic step movement and overall dissolution rate defined by surface retreat normal to the mica surface (Figure 13). Considered in light of similar observations available elsewhere in the literature, these relationships provide support for application of the stepwave model to mica dissolution kinetics. This approach provides a basic mechanistic link between the dissolution kinetics of phyllosilicates, framework silicates,

and related minerals, and suggests a resolution to the general problem of mica and eventually clay mineral reactivity. [submitted to Geochimica et Cosmochimica Acta]

A Comprehensive Stochastic Model of Phyllosilicate Dissolution Kinetics (Inna Kurganskaya and Andreas Luttge, Rice University)

Phyllosilicate dissolution is a complex process that consists of a large number of basic and/or elementary reactions operating on the mineral surface. Despite this fact, the common kinetic models reduce the dissolution process to only one reaction that involves the breaking of a precursor complex. Here, we have developed a comprehensive model of phyllosilicate dissolution that is based on our original work on feldspars (e.g., Lasaga and Luttge, 2004a;b; Zhang and Luttge, 2007, 2008, 2009a;b). We applied kinetic Monte Carlo methods that treat dissolution as a many body problem and a sequence of randomly chosen reaction events (Figure 14A). We used muscovite as an example illustrating the capabilities of the model. The topographical data of dissolving muscovite surfaces were obtained from experiments and used as a comparison. In this way, we were able to test the model (Figure 14D). The results of the simulations with four different model versions are then used to explain specific features of the etch pits developing on the (001) face. The different versions of the model can be distinguished by a number of parameters and increasing complexity, i.e., chemistry and the number of first nearest neighbors, lattice resistance to hydrolysis (influence of the cluster size or site connectivity on bond hydrolysis), the role of second nearest neighbors and bond orientation (steric factors). The simplest version of the model is able to reproduce basic etch pit features, e.g. the overall shape, orientation and effects of the silicate stacking sequence on step superposition. The more complex versions of the model consider second coordination and other steric effects (Figure 14B). They can reproduce small scale features, such as detailed step morphology and anisotropy in dissolution rates as a function of crystallographic direction (Figures 14C,14D). The effects of the second coordination sphere on the stability of experimentally observed steps are explained by the structure and relative reactivity of the “ledge” and “kink” sites specific for micas. The new model in combination with experimental observations showed its unique effectiveness in providing a mechanistic insight into the dissolution kinetics.

Kinetic Monte Carlo Model of Sheet Silicate dissolution: Parameterization of ab initio Calculations (Inna Kurganskaya and Andreas Luttge, Rice University)

Despite the intensive development of ab initio and Molecular Dynamics approaches to study reactivity of surface sites and species, most of the KMC models of mineral dissolution either remain at the oversimplified level of coarse-graining (e.g., Kossel crystal) or designed specifically to the given crystal structure or effect. In this paper we are discussing the role of the factors influencing reactivity of the surface sites inferred from the ab initio/MD studies and their implementation into the KMC models of sheet silicate dissolution. These factors include chemistry, structural position, coordination and charge of a surface site, size of the cluster etc. The lack of the ab initio data for a substantial number of surface reactions happening on mica surface creates a problem for a correct model parameterization. We made an attempt to solve this problem by making a series of assumptions about energetic effects of the factors described above and assigning them some energetic parameters. Systematic variation of values of these parameters in

the KMC simulations produces a spectrum of different etch pit structures and surface morphologies (Figures 15A,15B, 15C) observed in a wide range of experiments conducted at different environmental conditions. The quantitative relationships between the rates or probabilities of the key surface reactions lead to the formation of the given surface structure. Thus, the structure of the steps and their relative velocities can be explained by the different probabilities to nucleate and dissolve a kink site (Figures 15D, 15E). The results of the simulations provided here illustrate the ultimate importance of the correct values of bond hydrolysis activation energies for modeling dissolution of complex silicate structures, such as phyllosilicates.

Figure 13. Schematic illustration of stepwave formation on muscovite {001} surface. A. Consecutive steps moving outwards from a dislocation center. B. Steps emanating from two major etch pits (linear defects) and surrounding monolayer pits (point defects) form a single stepwave of complex morphology. C. Multilayer etch pit formation, showing layer-by-layer dissolution in two principal directions.

A B

C D

Figure 14. An illustration to Kinetic Monte Carlo simulations of muscovite dissolution. A. Polyhedral model of a small cluster representing part of the muscovite surface. Dissolution probabilities are assigned to an each surface site according to its chemistry, structural position and coordination number. B. Size of a coordination sphere used to differentiate reactivity of the tetrahedral sites. C. An example of an etch pit on muscovite (001) face produced in the simulations. D. An etch pit formed on muscovite (001) face after reaction in water pH=9.4, T=155˚C.

A B C

D E

k1k2

k1

k2

k1

k2

L

L

L

Figure 15. An illustration to the role of activation energy parameters on etch pit morphologies. A,B,C. Examples of the etch pits produced in the KMC simulations using different values of activation energies. D. A model of the reactive octahedral “ledge” and “kink” sites at the 3 primary {hk0} faces on muscovite surface. “Ledge” sites are labeled by light gray semitransparent squares, “kink” sites are labeled by yellow semitransparent squares. E. Structure of a (110) step reproduced in KMC simulations (only Si and Al cations are shown). Light gray spheres represent tetrahedral cations in the upper TOT layer, dark gray spheres represent octahedral cations, black spheres represent tetrahedral cations in the lower TOT layer.

Evaluation of the role of grain boundaries in mineral dissolution, with application to sequestration reservoirs (Rolf S. Arvidson and Andreas Luttge, Rice University)

Despite the significant increase in the understanding of mineral dissolution and growth enabled by real-time, microscopic observations of reacting mineral surfaces, these observations typically involve relatively large grains or particles. The kinetics of such surfaces are dominated by the distribution of defects, e.g., screw dislocations, that provide a critical means of sustaining step movement, kink creation, and thus dissolution and growth processes. However, in natural settings such as sequestration reservoirs, the discontinuities created by physical grain boundaries and contacts may constitute a critical contribution to “whole rock” reactivity, a point that has been recognized in the past but in which there is no formalized approach. Because of the complex relationship between mineral surface characteristics such as area, roughness, and reactivity, this problem requires a modeling approach that permits examination of key aspects that distinguish the reactivity of grains of arbitrary size. Here, we use kinetic Monte Carlo techniques to explore the dynamics of dissolution involving a discrete, three dimensional crystal (Figure 16). This approach has several important motivations: for example, we may question if there is a critical grain size, governed by surfaces kinetics (and not simply thermodynamic properties), below which the reactivity of grains is dominated by contributions from boundary sites. A key result of this approach is the demonstration of the decoupling of surface area and its related density of reactive sites such as steps and kinks, from the overall rate: e.g., as diameter decreases, the diffusion length of mobile surface components will eventually exceed the spacing of step and kink sites, leading to diffusion limitations that are not predicted in rate models parameterized simply by surface area (Figure 5). This success of this approach also suggests a powerful strategy for the simulation of complex reservoir frameworks, in which diverse grains reside in complex contact with one another.

Figure 16. Detail of corner and monolayer pits of 3D Kossel-Stranski cubic crystal, showing terrace atoms in red-brown, step edge sites in tan, kink sites in yellow, and 2-bonded chains in red.

Figure 17. Dissolution time series. Left and right panels compare system stapshots at constant mass remaining (0.99 to 0.10) as a function of bond breaking activation energy: left panel,

�

Φ /kT = 4 ; right panel ,

�

Φ /kT = 11. Surface diffusion is prevented by setting ϵD ⁄ kT = 100.

Variation of P-wave modulus with frequency and water saturation: extention of dynamic-equivalent-medium-approach for high fluctuation media. (Yuki Kobayashi and Gary Mavko)

Recent improvement of the quality of seismic data allows us to quantitatively

evaluate injected CO2 in sequestration projects using time-lapse seismic surveys (e.g., Chadwick et al. 2010, Daley et al. 2008, and Ivanova et al. 2012). Accurate knowledge of the relationship between P-wave velocity (VP) and water saturation (SW) is one of the key requirements for these methods. Formerly, most quantitative interpretation of seismic data has relied on amplitude information (Foster et al., 2010); however, interpretation techniques that actively use frequency information have been introduced and are increasingly used for oil and gas field exploration and reservoir characterization (e.g., Castagna et al., 2003, Burnett et al., 2003).

The effects of pore-filling fluid on the elastic properties of a saturated rock have been

studied extensively since the pioneering work of Biot (1956) and Gassmann (1951).

Gassmann’s (1951) theory allows us to perform “fluid substitution,” in which the bulk modulus of a rock saturated with one fluid can be predicted from the measured modulus the same rock saturated with another fluid. Under the assumption that the increment of pore fluid pressure induced by external stress is identical throughout the pore space, the equations exactly predict the modulus of the gas-filled rock starting from the information of a brine-filled rock.

The Gassmann equations assume a condition that the pore fluid consists of only one

phase. For multiple phase pore-fluid systems, where gas and water coexist in a rock, prediction of the saturated bulk modulus by the Gassmann equations cannot be achieved directly. A widely accepted approximation to relax the requirement is to use the bulk modulus of an “effective” fluid that is assumed to represent the fluid mixture. One way to estimate the bulk modulus of the effective fluid is the Reuss average (Reuss, 1929) of the bulk moduli of each fluid. The effective modulus is then substituted into the Gassmann equation to obtain the bulk modulus of a saturated rock. This approach is referred as “the Gassmann-Wood (GW)” model.

Another approach is based on the idea that a rock sample is divided into multiple

pieces (Mavko et al., 2009), each containing only one fluid phase. The Gassmann equations can be applied to each single-phase portion of rock. The upscaled “average” over the many small regions is achieved using Hill’s average (Hill, 1963). This approach is referred as “the Gassmann-Hill (GH)” model.

Studies of the effect of wave-induced fluid flow on seismic velocity and attenuation

have explained theoretically why the bulk modulus of a partially saturated rock sometimes falls between the GW and the GH models. Pore-fluid pressure gradients between the stiff (e.g. water-filled) and the soft (e.g. gas-filled) portions of a rock will generate pore-pressure diffusion during the passing of seismic wave, leading to attenuation and modulus dispersion. When the seismic frequency is very high, the induced pore-pressure differences between the various portions do not have time to diffuse and equilibrate, resulting in a composite rock being in the so-called “un-relaxed” state. At the low-frequency limit, where there is enough time for pore pressure to equilibrate, induced pore pressure becomes uniform throughout the composite. Since the bulk modulus of fluid mixture in isostress condition is given by the harmonic average of those of the constituents, the resulting bulk modulus at this limit is given by the GW bound, and the rock at this limit is in the “relaxed” state. In general the bulk modulus of a rock saturated by two fluid phase lies between the two bounds.

Modulus dispersion due to the pressure diffusion mechanism is generally referred as

the wave-induced fluid flow (WIFF) effect and is specifically called mesoscopic-scale WIFF when the scale of heterogeneity is much larger than pore or grain scale but still much smaller than seismic wavelength (Müller et al., 2007). For random heterogeneous media, Gurevich and Lapatnikov (1995) introduced a statistical approach to address the mesoscopic WIFF mechanism and developed expressions for velocity dispersion and attenuation. Müller and Gurevich (2005a) extended the statistical approach to three-dimensional heterogeneous random media. One strong limitation of these statistical

approaches is that they solve Biot’s poroelastic equations with spatially varying coefficients under the approximation that fluctuations of poroelastic parameters are small. One of the major advantages of the statistical approach is the use of spatial correlation function and the correlation length, which can be measured with core sample.

According to Müller and Gurevich (2005a), the complex-valued wavenumber, k1 , of

Biot’s fast wave (P-wave) passing through a three-dimensional, statistically isotropic, heterogeneous medium, is given by

�

˜ k 1

= k1* 1 + Δ

2+ Δ

1˜ k 2*2 rB(r)exp(i

0

∞

∫ ˜ k 2*r)dr

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ , (1)

where k1* and k2

* are the wavenumbers of Biot’s fast and slow waves for the background (reference) homogeneous medium, and B(r) is spatial correlation function of mesoscale heterogeneity. Other mechanisms causing intrinsic attenuation are not considered here, so that the background P-wave wavenumber is a real-valued quantity. Asterisks refer to the reference system, and

�

˜ X  denotes a complex-valued. Equation 1 assumes that all auto- and cross-correlation functions have the same functional form, B(r), and only differ by their variances. In equation 1, Δ1 and Δ2 are fluctuation parameters, which can be expressed as the combination of variance and cross-variance of poroelastic parameters:

�

Δ1 =α *2M*

2L*σHH2 − 2σ

HC2 + σ

CC2 +

3215

G*2

H*2σGG2 −

83G*

H*σHG2 +

83G*

H*σGC2⎛

⎝ ⎜ ⎞

⎠ ⎟ , (2)

�

Δ2 = Δ1 +12σHH2 −

43G*

H*σHC2 + 4 G

*

H*+1

⎛

⎝ ⎜ ⎞

⎠ ⎟ 415

G*

H*σGG2 , (3)

where σXY represents cross-variance (or variance if X=Y) between poroelastic parameters X and Y. Necessary parameters are listed in Table 1. Spatial heterogeneity is characterized by the correlation function and the variance and the cross-variance of poroelastic parameters. Equation 1, together with equations 2 and 3, is a general result, and heterogeneity can be attributed to rock frame, pore-filling fluid, or both.

Equation 1 can be simply rewritten by

�

˜ k 1 = k1* 1 + Δ2 − Δ1

˜ F { } , (4)

where

�

˜ F is the “interpolation function,” which contains information on frequency dependence. If one adopts an exponential correlation function, i.e., B(r) = exp(-r/a) where a is the spatial correlation length,

�

˜ F is given by

�

˜ F =4s3(1 + s)

(1 + 2s + 2s2)2 + i 2s2(1 + 2s)(1 + 2s + 2s2)2 . (5)

In equation 5, s represents the relative scale between the heterogeneity and the wavelength of Biot’s slow wave.

Table 1: List of required parameters

Variables Meaning

�

L Dry P-wave modulus

�

α Biot coefficient

�

M

�

M = [(α −φ) /Ks + φ /K f ]−1 , where

�

K f is the fluid bulk modulus

�

C

�

C = αM

�

H P-wave modulus of saturated rock

�

k1* Wave number of Biot’s fast wave

�

˜ k 2* Wave number of Biot’s slow wave

When heterogeneity is attributed to the fluid distribution only, the variance and the

covariance relating to the shear modulus, G, vanish from equations 2 and 3, and since

σHH2 − 2σHC

2 +σCC2 = εH − εC( )2 = α 2M *

H * εM − εM⎛

⎝⎜⎞

⎠⎟

2

= L2

H *2 σMM2 , (6)

and

�

σHH2 =

α 2M*

H*εM

⎛

⎝ ⎜ ⎞

⎠ ⎟

2

=α 4M*2

H*2σMM2 . (7)

Δ1 nd Δ2 can be written as

�

Δ1 =Lα 2M*

2H*2σMM2 , (8)

�

Δ2 = Δ1 +12σHH2 . (9)

Since  

�

˜ F

�

˜ F tends from 0 as 1 as frequency tends from 0 to ∞, the low- and the high-frequency limits of

�

˜ k 1 are expressed with the variance of M and H as

�

˜ k 1_ LF= k1

* (1 + Δ2) = k1* 1 +

α *2M*

2H* σMM2⎛

⎝ ⎜ ⎞

⎠ ⎟ , (10)

�

˜ k 1_ LF= k1

* (1 + Δ2 − Δ1) = k1* 1 +

12σ

HH2⎛

⎝ ⎞ ⎠ . (11)

At the low-frequency limit, where wave-induced pore pressure has enough time to equilibrate between the two phases, the effective P-wave modulus of a rock with multiphase fluid is given by the Reuss average of the bulk moduli of pore-filling fluids, followed by the Gassmann equation:

�

HGW

= L + α 2MW

, (Gassmann’s equation) (12)

MW is given by the harmonic average of M:

�

MW =1M

−1 (Reuss average) (13)

and can be rewritten with the variance of porespace stiffness,

�

σMM2 as

�

MW =1

M* (1+ εM )

−1

≈ M* 1− εM + εM2 −1

≈ M* (1− σMM2 ) (14)

for small fluctuation,

�

εM << 1. Therefore, HGW and the corresponding wavenumber, kGW, are obtained by

�

HGW

= L + α 2MW

= H* 1− α 2M*

H*σMM2⎛

⎝ ⎜ ⎞

⎠ ⎟ , (15)

and

�

kGW

= ωρ*

HGW

= ωρ*

H*1+

α 2M*

H*σMM2⎛

⎝ ⎜ ⎞

⎠ ⎟ . (16)

At the high-frequency limit, where there is no time for pressure diffusion to occur, the effective P-wave modulus can be calculated using the Hill’s average,   which can also be written in terms of the fluctuation of P-wave modulus as

�

HGH

= H* 1− σHH2( ) . (17)

The corresponding wavenumber is given by (for small fluctuations)

�

kGH

= ωρ*

HGH

= ωρ*

H*1+

12σHH2⎛

⎝ ⎞ ⎠ (18)

Equations 4 with 5 give the frequency-dependent complex-valued wavenumber for heterogeneously saturated rock, and has low- and high-frequency limits that agree with the known rigorous bounds, as long as the fluctuation

�

εM is small. However, higher than second-order

�

εM cannot be neglected if

�

εM is large, and the accuracy of equation 4 deteriorates. This could occur, for example, when one of the pore-filling fluids is gas or supercritical CO2, whose bulk modulus is quite different from that of water. To overcome this limitation and keep the modeled P-wave modulus within the rigorous bounds, we modify equation 4 as follows.

From equations 10 and 11, the real parts of the low- and high-frequency limits of P-

wave modulus are given by

HLF =VP2ρ = ω 2ρ*

R !k1*{ }2 1+ Δ2( )2

≈ H * 1− 2Δ2( ) (19)

and

�

HHF

=ω 2ρ*

R ˜ k 1*{ }2

1 + Δ2 − Δ1( )2 ≈ H* 1− 2Δ2 + 2Δ1( ) . (20)

By scaling the original DEMA with the GH and the GW bounds, BA-DEMA ensures the predicted P-wave modulus lies between the two bounds. However, there is still the question of the validity of the use of correlation length “a” for large fluctuation media. To examine this point, the theoretical predictions are compared with the laboratory measured P-wave velocities. As far as we are aware, Cadoret’s (1993) measurements are only available data set that is suitable for the study of mesoscopic scale WIFF effect: low and high frequency measurements of both velocity and attenuation and computer tomography (CT) scanned images of core samples. Cadoret (1993) first performed velocity and attenuation measurements of extensional and shear waves by the resonant bar technique. The lowest resonant frequency is about 1kHz. He also conducted ultrasonic measurements of compressional and shear waves on the same sample at four different frequencies from 50 kHz to 1MHz, among which we only use the results at 50kHz because dispersion mechanisms other than mesoscopic-scale WIFF may become significant at higher frequencies. Among ten samples in Cadoret (1993), we selected the data set of the drying experiments on the “Estaillades (S7)” for the comparison with the theory since Cadoret (1993) also reports the CT-scanned images of the sample at different water saturations.

To apply BA-DEMA, the correlation length of the medium heterogeneity is required.

We extracted empirical correlation lengths from the two CT-scanned images on partially saturated core samples (Cadoret, 1993). Figure 18 shows the extracted correlation functions at two different water saturations. We fitted an exponential correlation function to estimate the scale of heterogeneity. We observe that the estimated correlation length slightly increases with increasing water saturation. We calculated P-wave modulus at 50kHz by BA-DEMA. Model predictions are plotted in Figure 19 together with the laboratory measurements. The large difference between the model (blue curve) and the laboratory measurement (square symbol) is clear. The modeled dispersion curves in Figure 19 (a) and (c) are shifted to higher frequency compared to the laboratory measurements. From this observation, it is concluded that the shape of dispersion curve is not significantly affected by medium fluctuation, but that the frequency scale of the curve has to be adjusted for high fluctuation medium.

Figure 18: (a) and (b) CT-scanned images on the partially saturated core samples at water saturation SW = 0.92 and 0.61, respectively. (c) and (d) The empirical correlation length in x and y directions (black curves) and fitted exponential curve, exp(-r/a), by least-square method (blue curve) for the two images shown in (a) and (b), respectively. (e) Measured correlation length, a, at the two water saturations. Dashed line is a linear line connecting them. (f) Measured correlation length over water saturation for three limestone samples. CT-scanned images (a and b) are from Cadoret (1993) and (f) is a replication of Figure 7 of Toms-Stewart et al. (2009). Note that both Cadoret’s samples and those of Toms et al. (2009) show similar increase of correlation length as water saturation increases.

Figure 19: Impact of the introduced variable LC model on P-wave modulus and inverse quality factor. Red curves are the theoretical prediction by BA-DEMA with the variable LC model, and blue curves are by BA-DEMA with measured correlation length. Notice that the improvement of theoretical prediction by the variable LC model. Squares represents laboratory-measured P-wave modulus and inverse quality factor at two different frequencies (Red: 50kHz, Yellow: 1kHz) of the sample S7 from Cadoret (1993). The location of the modeled dispersion curve is totally determined by the

interpolation function, !F . Figure 20 shows the shape of the imaginary part of  

�

˜ F (=Fi) for an exponential correlation function. The horizontal axis is given by the relative scale, s. As Figure 20 shows, -Fi becomes its maximum when

�

s ≈ 2 . Since s=ak2, the characteristic frequency, at which the inverse quality factor becomes its maximum, is given by

�

FC =κN*

πη*1

α / 2( )2=

D

α / 2( )2 , (21)

where D is the effective diffusivity of the rock.

The disagreement between the model prediction and the laboratory measurements suggests that LC must be be modified for high fluctuation media. One clue comes from White’s concentric sphere model (White, 1975). The characteristic frequency of the White’s model is approximately given by (Vogelaar et al., 2010)

FC = D(R − r)2

, (22)

where R and r are the radii of outer shell and inner sphere. Therefore, R-r corresponds to the thickness of water-filled outer shell. This means that, for high fluctuation media, LC is more or less equivalent to the size of water-filled region. This is because energy dissipation due to pressure diffusion phenomenon mainly occurs in a portion with stiffer fluid.

Figure 20: Shape of the imaginary part of the weight function for exponential correlation function. Suppose that the spatial correlation length of the heterogeneous permeability field of

the host rock is given by a0. The representative size of water-filled region can be regarded as infinity before non-wetting phase invades the rock, i.e. at SW = 1, since the rock is fully occupied by water. On the other hand, LC becomes zero when SW = 0, as there exists no water-filled region. Furthermore, it is expected that LC is somehow controlled by the correlation length of the host rock a0 when SW = 0.5, that is, a half portion of the host rock is filled by water, and is expressed by Ca0, where C is a positive constant. Any expression satisfying the three criteria, such as

LC = Ca0SW

1− SW

⎛⎝⎜

⎞⎠⎟

2

, (23)

would become a candidate for LC, with positive number n. LC then defines the relative scale of heterogeneity, s, as

s = 2LCk2 , (24)

instead of s = ak2.The form of the interpolation function is unchanged.

It is of course possible to assume other functional forms that satisfy the above conditions; however, we found that this simple expression is sufficiently accurate to explain the laboratory measured VP-SW relationship. Figure 2 shows the measured P-wave modulus and the theoretical prediction with n = C = 1 (the variable LC model, red curves). They agree with each other very well.

We examined the variable LC model by comparing with other samples from Cadoret

(1993). Figure 21 shows the measured data and the modeled curves for the two different samples in the drying experiments at 50kHz and 1kHz. They are also in good agreement.

In Cadoret (1993), there is no clear CT-scanned image on the sample S3 and V3;

therefore we estimated the dry rock correlation scale, a0, for the two samples by using VP data at 50kHz. Though we determined a0 so that the modeled and the measured VP at 50kHz match, accurate predictions of VP and  QP

−1at 1kHz are very good.

Figure 21: Modeled P-wave modulus and inverse quality factor for sample S3 and V3 of Cadoret’s data set (Cadoret, 1993) of the drying experiments. Red and green circles are measured data at 50kHz and 1kHz, respectively. Solid curves are the theoretical predictions by BA-DEMA with the variable LC model (red at 50kHz, green at 1kHz).

In summary we have developed a new model of the relationship between complex-valued P-wave modulus and water saturation by considering the scale of fluid distribution, from the viewpoint of the effect of mesoscopic WIFF on a rock saturated with multiple-phase fluids. We extended the Dynamic-Equivalent-Medium-Approach for high fluctuation media. The two key modifications on the original DEMA are scaling by the rigorous bounds and replacing correlation length with characteristic length. The characteristics of this approach are summarized as follows: 1) it explicitly contains the effect of frequency which cannot be evaluated by the previous models, such as the GW, the GH models, or Brie’s empirical fluid mixture relationship, 2) both VP and QP

-1 can be modeled at the same time in a consistent manner, 3) there are clear physical and geological meanings on the parameter, a0: the correlation length of host rock. Each of the characteristics is important in practical application of the theory of WIFF.

We consider LC as the representative size of water-filled regions. In fact, LC is not a

pure geometrical property but an imaginary length that is also affected by rock frame and pore-fluid properties.

In this study, all analysis and model calibration were done with Cadoret’s (1993)

measurements, which, as far as we know, are only available data set that is suitable for the study of mesoscopic scale WIFF effect. Experimental confirmation of the variable LC model with different samples is desirable for further possible improvement. Laboratory, numerical and theoretical studies on the characteristic length variation will further improve our quantitative interpretation of seismic velocity and attenuation.

[We would like to thank GCEP and INPEX CORPORATION for their financial support of this study.]

Conclusions

This project has demonstrated techniques for quantitatively predicting the combined seismic signatures of CO2 saturation, chemical changes to the rock frame, and pore pressure. This research has involved laboratory, theoretical, and computational tasks in the fields of both Rock Physics and Geochemistry. Extensive laboratory measurements revealed that the chemical disequilibrium associated with the injection of brine can have substantial changes in the porosity, permeability, and elastic properties of porous rocks, especially carbonates. Work on chemical kinetics of CO2-bearing reactive fluids with rock-forming minerals provided observation evidence of mechanisms of dissolution and allowed development of stochastic models of the reaction. Ultrasonic P- and S-wave velocities were measured over a range of confining pressures while injecting CO2 and brine into the samples. The observed permanent changes to the rock frame were reflected in changes to the elastic rock properties, which must be incorporated when using seismic data to monitor the injection of CO2. These changes to the rock frame result from mineral dissolution and precipitation, as well as mechanical compaction of the weakended frame.

Rock physics models were developed for improved elastic modeling of the CO2-related changes. Observed permanent changes to the rock frame mimicked natural diagenetic trends, thus leading to strategies to calibrate these trends with well logs.

Robust models for seismic attenuation and dispersion resulting from mixed CO2-brine phases were also developed and validated.

Publications and Patents

1. Dralus, D. E., T. Vanorio, and G. Mavko, 2012, Anomalous strain behavior in CO2-saturated zeolithic tuffs, AGU Fall Meeting 2012.

2. Grombacher, D., T. Vanorio, and Y. Ebert, 2012, Time-lapse acoustic, transport, an NMR measurements to characterize microstructural changes of carbonate rocks during injection of CO2-rich water, Geophysics, 77, WA169-WA179.

3. Kobayashi, Y., and G. Mavko, 2014, Characteristic length of a rock heterogeneously saturated with two immiscible fluids, SEG Annual Meeting Extended Abstract, 2014.

4. Fischer, C., R. S. Arvidson, C. Fischer, and A. Luttge, 2012, Does the stepwave model predict mica dissolution kinetics? Geochimica et Cosmochimica Acta, 98, 177-185.

5. Kurganskaya, I., R. S. Arvidson, and A. Luttge, 2012, How predictable are dissolution roates of crystalline material? Geochimica et Cosmochimica Acta, 97, 120-130.

6. Kurganskaya, I., R. S. and A. Luttge, 2013, A comprehensive stochastic model of phyllosilicate dissolution: structure and kinematics of etch pits formed on muscovite basal face, 120, 545-560-130.

7. Kurganskaya, I., R. S. and A. Luttge, 2013, Kinetic Monte Carlo simulations of silicate dissolution: model complexity and parameterization, The Journal of Physical Chemistry C, 117, 24894-24906

8. Vanorio, T., A. Nur, and Y. Ebert, 2011, Rock physics analysis and time-lapse rock imaging of geochemical effects due to the injection of CO2 into reservoir rocks, Geophysics, 76, O23-O33.

9. Stéphanie Vialle, Jack Dvorkin, and Gary Mavko (2013). ”Implications of pore microgeometry heterogeneity for the movement and chemical reactivity of CO2 in carbonates.” GEOPHYSICS, 78(5), L69-L86. doi: 10.1190/geo2012-0458.1

10. Vialle, S., T. Vanorio, and G. Mavko, 2010, Monitoring the changes of rock properties in a micritic limestone upon injection of a CO2-rich fluid, SEG Annual Meeting Extended Abstract, 2010.

11. Vialle, S. and T Vanorio, 2011, Laboratory measurements of elastic properties of carbonate rocks during injection of reactive CO2-saturated water, Geophysical Research Letters, 38, 10.1029/2010GL045606.

12. Vialle, S., J. Dvorkin, and G. Mavko, 2012, Flow paths and chemical reactivity of CO2 in carbonates using mercury-intrusion capillary pressure data and dimensionless numbers, AGU Fall Meeting 2012.

13. Vialle, S., T. Vanorio, and G. Mavko, 2010, Linking the acoustic properties of carbonate rocks to induced chemical changes in the pore system upon injection of CO2-rich water, AGU Fall Meeting 2010.

14. Vialle, S. and G. Mavko, 2012, Geophysical signatures of carbonate rocks flooded with CO2-rich water: effect of the presence of oil in the pore space, EGU General Assembly 2012, held 22-27 April, 2012 in Vienna, Austria., p.6248

15. Vialle, S., J. Dvorkin, and G. Mavko, 2013, Relating rock microstructure to flow paths and chemical reactions: application to CO2 injection in carbonates, Proceedings, Second International Workshop on Rock Physics, Southampton, 4-9 August 2013.

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Contacts

Gary Mavko: mavko@stanford.edu Tiziana Vanorio: tvanorio@stanford.edu Denys Grombacher: denysg@stanford.edu Sally Benson: smbenson@stanford.edu Stephanie Vialle: svialle@stanford.edu Andreas Luttge: aluttge@rice.edu Rolf Arvidson: rolf.s.arvidson@rice.edu