Environ. Sci. Technol. 1999, 33, 2879-2884
Measurement and Prediction of Effective Diffusivities through Spreading and Nonspreading Oils in Unsaturated Porous Media C H A R L E S E . S C H A E F E R , * ,† PAUL V. ROBERTS,‡ AND MARTIN J. BLUNT† Department of Petroleum Engineering, Stanford University, Stanford, California 94305-2220, and Environmental Engineering and Science Program, Department of Civil and Environmental Engineering, Stanford University, Stanford, California 94305-4020
The difference in phase continuity between spreading and nonspreading oils at low saturations was examined by measuring the oil phase effective diffusivity. Results indicated that for the sand and water saturations studied, the spreading oil (1-octanol) remained continuous at saturations as low as approximately 0.035 (the lowest oil saturation measured), while the nonspreading oil (n-dodecane) became discontinuous at a saturation of approximately 0.055. 1-Octanol also showed higher effective diffusivities than the n-dodecane throughout the saturation range examined, indicating that the spreading oil was more highly connected. Two separate models, one for spreading and another for nonspreading oils, were used to describe the behavior of the oils in the unsaturated porous medium. Each model gave a reasonable prediction of the experimental data and was able to describe the difference in continuity between nonspreading and spreading oils.
Introduction Environmental and human health hazards from petroleum hydrocarbon releases are a continuing public and private concern. Soil-water systems may become contaminated by, for instance, accidental surface spills of crude or refined oils, broken pipelines, or leaking underground storage tanks. Upon entering the soil, the bulk oil phase may migrate downward through the vadose zone toward the water table due to gravitational and capillary forces. The presence of oil at or below the water table will contaminate the water and may compromise drinking water. The oil often becomes entrapped in pores as small blobs or ganglia during its downward migration (1). Some oils may form thin wetting layers (2). The oil may fill anywhere from 5 to 40% of the soil pore volume (3). The oil may persist in the soil for several years due to its low volatility and water solubility. The bulk oil phase is often composed of several constituent hydrocarbon contaminants. Many of these constituent contaminants are more volatile and water soluble than the bulk oil. In these cases, the oil will serve as a source for the release of these contaminants into the vapor or aqueous phases (4). As a result, concentration gradients may develop * Corresponding author e-mail:
[email protected]. † Department of Petroleum Engineering. ‡ Department of Civil and Environmental Engineering. 10.1021/es981087f CCC: $18.00 Published on Web 07/14/1999
1999 American Chemical Society
in the bulk oil phase. This can lead to diffusion of constituent hydrocarbon components through the bulk oil phase. For low permeability porous media where bulk oil movement is negligible, contaminant transport of hydrophobic and nonvolatile contaminants may be dominated by diffusion through the oil phase. Also, diffusion through the bulk oil phase can significantly contribute to the overall contaminant mass transport since the bulk oil phase will often contain most of the contaminant mass. Models have incorporated solute diffusion through the oil phase, but the parameter Deff (the effective diffusivity through the bulk oil phase) was not investigated (5, 6). While many studies have been carried out to determine the effective diffusivities of solutes through the water and vapor phases in unsaturated soils (7), only recently have studies been carried out to investigate effective diffusivities through oils (8). Results of that study indicated that diffusion through the oil phase is a function of the oil saturation as well as the pore size distribution. The transport of contaminants is fundamentally controlled by the configuration of the fluids in the pore space. Recent experiments in micromodels, which are two-dimensional representations of a porous medium, have studied the generic arrangements of oil, water, and air in water-wet and oil-wet media for different displacements (9-11). These results have shown to be consistent with theoretical considerations based on capillary equilibrium and experiments on single capillary tubes (2, 12, 13). Using this work, several three phase network models have been developed which predict the fluid movement for different sequences of water, air, and oil invasion (13, 14). The key features at the pore scale are the different mechanisms by which one fluid displaces another. These mechanisms are governed by the local capillary pressure and the presence of oil layers wedged between air and water in the corners of the pore space. For some oils, these oil layers can provide continuity of the oil phase down to low oil saturations, i.e., 1% and lower (2). The saturation at which the oil becomes discontinuous (and thus residually trapped in the porous media) has been shown to be inversely related to the spreading coefficient (15). The spreading coefficient is defined as
C is ) γgw - γow - γgo
(1)
where C is is the initial spreading coefficient [N/m] and γxy is interfacial tension between phases x and y (where x and y are either g (gas), w (water), or o (oil)) [N/m]. C is is the initial spreading coefficient measured on fluids before the three phases are brought in contact. If C is > 0, oil will spread on a flat water surface as a molecular film, lowering γgw. If C is < 0, no film forms. In thermodynamic equilibrium, the spreading coefficient is C eqs e 0 (16). A spreading oil is one with C is > 0 and C eqs ≈ 0. A nonspreading oil has C is, C eqs < 0. In porous media, molecular films of nanometer thickness have a negligible effect on advective transport (17). However, oil can form wetting layers between water and gas in corners or crevices in the pore space. These layers may be several micrometers across, and transport through them can be significant. It is these layers that provide continuity of the oil phase down to low saturation. Nonspreading systems may form oil layers but these layers become unstable, allowing oil to be trapped and leaving a residual oil saturation that is typically in the range 3-10% (2, 18). A simplified illustration of oil layers in square pore corners is shown in Figure 1. The real angle of the pore corner would determine the thickness of the oil layer (13). VOL. 33, NO. 17, 1999 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 1. Thin oil layers may form in the crevices and corners of pore spaces. The consequences of this pore-level phenomena in terms of the macroscopic behaviorsrelative permeability and capillary pressuresare reasonably well understood for waterwet media (13). However, this work has been largely, although not exclusively, pursued for petroleum engineering applications. In this work, column experiments will be conducted to measure the effective diffusivity through the bulk oil phase as a function of both oil saturation and oil type. The effect of oil spreading coefficient on both the continuity and effective diffusivity through the bulk oil phase will be investigated. A model will also be presented which predicts the effective diffusion coefficient through the oil phase as a function of oil saturation and spreading coefficient. This study will lead to a better understanding of diffusive transport of contaminants though bulk oil phases, and it will ultimately improve the predictions of multiphase transport models.
Model Development
to the oil-filled macropores which remain oil-filled at residual oil saturation, and “inter” refers to interparticle pores which connect the macropores at oil saturations above residual saturation. The resistances (R1, R2, and R3) are diffusional resistances through the interparticle, macropore, and interparticle-to-macropore regimes, respectively. These resistances are inversely related to the “connectedness” of the pore regime(s) of interest. With macro,oil determined from laboratory diffusion experiments (the saturation at which diffusion across the bed is no longer observed), eqs 3-6 may be used to predict the oil phase diffusional resistance as a function of oil content (7, 19). When the oil saturation is reduced to residual saturation (inter,oil ) 0), Roil becomes infinite (or, from eq 2, the effective diffusivity is 0). Diffusion through spreading oils in porous media may not be accurately described by these equations. Spreading oils will not become discontinuous at low saturations. Trapped blobs or ganglia may not occur. This could cause lower diffusional resistances (Roil) at low oil saturations for spreading oils than for nonspreading oils. These differences have been noted by Sahni et al. (18) in regard to measurement of relative permeabilites and oil saturations. Thus, a different approach for describing the diffusional resistance through spreading oils may be required. Unlike the model presented in eq 3, in which pores are divided into continuous and noncontinuous regimes, spreading oils will be continuous throughout the entire porous medium down to virtually 0% saturation. As the saturation of a spreading oil approaches zero in a porous medium, diffusion through the bulk oil phase will be controlled by the oil layers. Since no oil becomes entrapped, the path length across the porous medium through the oil phase should be independent of the oil saturation (i.e., tortuosity is independent of oil saturation). However, the cross-sectional area for diffusion will vary linearly as the oil saturation decreases. This may be expressed, in terms of diffusional resistance through the oil phase, as (20, 21):
Diffusional resistance of a contaminant through the bulk oil phase may be defined as (7, 8, 19)
Roil ) D0/Deff
(2)
where D0 is the contaminant diffusivity through the oil [m2/ s], and Deff is the effective contaminant diffusivity through the oil phase within the porous medium [m2/s]. A parallel resistance model to predict the diffusional resistance through the gas phase, Rgas, has been previously developed (7). This gas phase model has been shown to give a reasonable prediction of the gas phase effective diffusivity through unsaturated soils, as well as predicting residual saturation of the gas phase (i.e., the gas saturation at which the gas phase is no longer continuous across the soil bed). Prediction of the diffusional resistance allows the effective diffusivity to be calculated (eq 2). Applying the parallel gas phase resistance model to the oil phase, the diffusional resistance may be expressed as (7, 8)
Roil )
(
)
1 1 + R1 (R2 + R3)
-1
(3)
2/3 2/3 R1 ) (inter,oil inter,oil )-1
(4)
R2 ) 1
(5)
2/3 2/3 R3 ) (macro,oil inter,oil )-1
(6)
where inter,oil equals the fraction of total bed volume occupied by oil minus the volume fraction of oil at residual saturation (macro,oil), and macro,oil equals the volume of oil left at residual saturation divided by the total bed volume. “Macro” refers 2880
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Roil ) (S)-1τ
(7)
where is porosity [m3 void/m3 bed], S ) oil saturation [m3 oil/m3 void], and τ ) tortuosity factor. Assuming pores are randomly arranged, a value of 2 has been shown to be a reasonable estimate of the tortuosity factor (22).
Experimental Design Experiments were carried out to validate the models presented in eqs 2-7 for diffusion through oil at low saturations. Sand with a particle range from 100 to 450 µm, and a mean particle size of approximately 300 µm, was used in all experiments. Two organic liquids, n-dodecane and 1-octanol, were used as the model bulk oil phases. These oils were chosen to show a range of spreading coefficient and were relatively nonvolatile and insoluble in water. Some of their properties are listed in Table 1. Separate experiments were performed for each oil. Approximately 32 sand column experiments were carried out in total. A water saturation of approximately 25% was maintained, and the saturation of the oil (either n-dodecane or 1-octanol) was varied from 3.5 to 20%. Values of the diffusional resistance measured in the experiments using n-dodecane were then compared to values of the diffusional resistance measured in experiments containing 1-octanol. Thus, the effective diffusivity was measured as a function of oil saturation and oil type (spreading or nonspreading) in these experiments.
Experimental Method The method used for measurement of the diffusional resistance was a modified version of the diffusion tube
TABLE 1. Properties of Selected Oilsa
oil
γow (N/m)
γog (N/m)
γgw (N/m)
C iS (N/m)
vapor pressure (kPa) @ 25 °C
water solubility (kg/m3) @ 25 °C
viscosity (kg/m/s) @ 25 °C
density (kg/m3) @ 20 °C
n-dodecane 1-octanol
52.8 8.52
24.9 27.5
72.7 38.1
-5.0 2.1
0.016 0.010
3.7 × 10-6 5.4 × 10-1
1.4 7.4
750 830
a γ , γ , and γ ow og gw are the oil-water, oil-gas, and gas-water interfacial tensions, respectively. The interfacial tension and density values listed in Table 1 have been previously tabulated (23). Vapor pressure, water solubility, and viscosity data are taken from Riddick et al. (24).
FIGURE 2. Only the section of the soil bed at constant oil and water saturation is used for the diffusion column experiments. method, adapted for the situation with three fluid phases in a porous medium (8, 19). 14C-Labeled oil served as the diffusing tracer through the oil phase. Dry sand was packed into vertical cylindrical columns by slowly pouring the sand through a large funnel. This allowed for uniform packing and gave bed porosities between 39% and 41%. The columns were 7-9 cm long, with a cross-sectional area of 5.3 cm2. The columns were then saturated with water by slowly pipetting deionized water from the top of the column. Experimentally, water saturation was measured to be at least 95%. Once saturated, air was pumped into the column at 3 psi. This caused water to be displaced in the column, leaving the column at a uniform saturation. Next, oil (either n-dodecane or 1-octanol) was added to the top of the sand column and allowed to drain due to gravity. As the oil drained through the column, a saturation profile developed. On the basis of the sand grain diameter and column diameter, it has been shown that fingering would not occur in this system (25). To analyze diffusional resistance through the oil phase, a constant oil saturation as a function of column length was needed. To accomplish this, the water and oil saturation profiles were analyzed in the column. Water and oil saturation profiles were analyzed by extruding the sand column and slicing the soil into thin disks. These disks were placed in 5 mL of 2-propanol. Both the water and oil were completely soluble in the 2-propanol. The propanol solution was then analyzed on a gas chromatograph with a thermal conductivity detector. More details of this method will be given in the Analytical Methods section. At small length scales within the column (usually about 1.5 cm), both the water and oil saturations were found to be constant. Once the location of this section of column length was determined, the soil was extruded from the column on either end, leaving only the small section of column at constant liquid saturations (Figure 2). To obtain different oil saturations, both the amount of oil added to the column and the amount of drainage time were varied. Approximately 0.5 g of sand was mixed with 0.025 mL of 14C-labeled oil (either n-dodecane or 1-octanol). This served as the spiking sand. The oil saturation of the spiking sand was approximately 0.08. About 0.3 g of this spiked sand mixture was placed and packed (lightly) on top of the sand column (the section at constant oil and water saturation). The column was then placed in a horizontal position. Separate experiments were carried out to ensure that placing the
FIGURE 3. Obtaining a 14C diffusion profile in the sand column. column horizontally did not cause the fluids to move in either the axial or radial direction. Both ends of the soil column were sealed with Parafilm and stoppers to minimize moisture loss (19). The 14C-labeled oil then diffused through the sand column. Because of the low volatility and water solubility of the oils (Table 1), diffusion was assumed to occur only through the bulk oil phase. After a diffusion period of 2 to 5 days, the diffusion tube was extruded and sliced into thin disks. Each slice was analyzed for 14C in disintegrations per minute (DPM) via scintillation counting. The diffusion profile obtained from the analysis of this column was used to calculate the effective diffusivity of the tracer through the oil. This method is shown schematically in Figure 3. To ensure that no sorption of 14C to the sand occurred, batch adsorption experiments also were carried out to measure any mass of 14C that might sorb to the sand. Batch adsorption isotherms have been carried out by many investigators (26-28). All experiments (diffusion and adsorption) were carried out at a room temperature of approximately 21 °C. Calculation of Oil Phase Diffusional Resistance (Roil). The diffusion column experiments were described by considering unsteady-state diffusion through one (oil) phase and assuming the only transport mechanism is diffusion through the oil phase.
∂Co ∂2Co ) -Deff 2 ∂t ∂z VOL. 33, NO. 17, 1999 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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Deff is the effective diffusivity through the oil, and Co is the solute concentration in the oil. The following boundary and initial conditions were then applied to the system
I.C.
Co ) 0 at z > 0 at t ) 0
B.C.1
Co ) 0 at z ) ∞ (semi-infinite column)
B.C.2
dCo/dZ ) 0 at z ) 0 (spiked end of column)
where Co is the 14C concentration in the oil phase (DPM per bed volume). In order for B.C. 1 to be valid, the column had to be sliced soon enough so as not to allow the concentration profile to reach the end of the soil bed (19). B.C. 2 is obtained using the method of reflection and superposition (i.e., B.C. 2 is valid because the solute diffuses only in the positive z direction through the column) (29). An analytical solution has been derived for this differential equation (29)
Co )
( )
-z2 M exp 0.5 4Defft (πDefft)
(9)
where Co is the concentration of 14C [DPM/m3 bed], M is the total amount of 14C [DPM], t is the diffusion time [s], and z is the length from the spiked end of the column [m]. All parameters in eq 9 were either known or measured except for Deff. This was determined from a nonlinear regression of eq 9 to the experimental diffusion profiles. Equation 9 was then combined with eq 2 to obtain the diffusional resistance. Calculation of D0. Experiments to measure D0 through both n-dodecane and 1-octanol were also carried out. The capillary method, described in detail in Cussler (30) and Dunlop (31), was used to obtain values of the diffusion coefficient. A small capillary tube of 14C-labeled oil (length ) 3.8 cm, inside diameter = 0.1 cm) in a stirred bath (41 mL) of unlabeled oil was used in this study. The experiments were carried out for diffusion times of 24-48 h. Analytical Methods. A Hewlett-Packard 5880 gas chromatograph with a thermal conductivity detector was used to analyze for water and oil concentrations in the sand slices. Helium was used as the carrier gas. An Alltech stainless steel packed column (Porapak Q, 60/80, 6 ft × 1/8 in.) was used. Injector and detector temperatures were 210 °C. The oven temperature was initially set at 180 °C for 2 min, then increased at a rate of 30 °C/minute until a final temperature of 280 °C was reached. Water and oil standards were prepared in a 2-propanol solvent. Standards were made such that the concentrations bracketed the samples. Scintillation counting was performed on a Packard 2500 TR/AB liquid scintillation counter. All samples, both in the diffusion column and adsorption experiments, were extracted with 15 mL of Ultima Gold scintillation cocktail. Controls consisting of only cocktail fluid and sand were carried out to determine the background DPM of the samples. A Packard 14C reference standard of 137 000 DPM was used to normalize the scintillation counting. Other experiments were also carried out to ensure that no measurable quenching of the 14 C occurred during the analysis. Such methods in analyzing diffusion tubes have been previously employed (8).
Results and Discussion Results of the adsorption experiments indicated that there was no measurable sorption of 14C to the sand for equilibration times of up to 6 days. A D0 value of 1.10 × 10-9 m2/s was obtained for n-dodecane. Duplicate experiments were carried out for 1-octanol to ensure the repeatability of the experimental procedure. The values obtained were 2.15 × 10-10 and 2.25 × 10-10 m2/s. An average D0 value of 2.20 × 10-10 m2/s was used for the 1-octanol. Previous experiments have 2882
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FIGURE 4. Saturation profile of 1-octanol in a sand-packed column. The water saturation is constant across the bed at approximately 0.25 saturation (mL water/mL void). These profiles were shown to be repeatable.
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FIGURE 5. Diffusion profile and regression for n-dodecane with a saturation of 0.075. shown that diffusion coefficients of nonelectrolytes are inversely proportional to the viscosity of the liquid solvent (32). Thus, the large difference in the value of D0 between n-dodecane and 1-octanol (whose molecules are roughly the same size) can be primarily explained due to their large difference in viscosity (Table 1). A typical oil saturation profile for 1-octanol is shown in Figure 4. The indicated section was used for the diffusion column experiments (carried out separately). These saturation profiles were found to be repeatable for a given water and oil saturation, as for well as for a given drainage time. Saturations (S) as low as 0.055 were obtained for n-dodecane, and saturations as low as 0.035 were obtained for 1-octanol. Water saturations for all experiments ranged between 0.2 and 0.3. A typical diffusion profile is shown in Figure 5. Also shown is the regression of the data to eq 9. Typical diffusion times ranged from 1 to 5 days, depending on the oil saturation and oil type. Only a 5% difference in the regressed value of the effective diffusivity occurs if the data point at 0.15 cm is not used in the regression fit. Thus, lack of additional data points near the spiked end of the tube did not have a significant effect on the measured values of the effective diffusivity. Figure 6 shows the diffusional resistance (eq 2) versus oil saturation for n-dodecane and 1-octanol. Both oils show an increase in diffusional resistance with decreasing oil saturation. This resistance versus saturation relationship has been observed in other liquid phase diffusion studies (19, 33). As the oil saturation decreases below about 0.10, the n-dodecane begins to exhibit much higher diffusional resistance than the 1-octanol. The vertical line at 0.055 represents the
FIGURE 6. Diffusional resistance (Roil) versus oil saturation (S) for n-dodecane (0) and 1-octanol (b). Error bars indicate the standard deviation obtained upon regression to eq 9. The solid line is the model prediction for a nonspreading oil (eqs 3-6), and the dashed line is the model prediction for a spreading oil (eq 7). The solid vertical line represents the saturation at which diffusion through the n-dodecane phase was no longer observed (infinite diffusional resistance). saturation at which no diffusion was observed for two n-dodecane experiments (infinite resistance). The gas phase, which is still continuous across the sand bed at this oil saturation, is shown not to significantly contribute to the overall diffusion. This validates the assumption that gas phase diffusion is not significant for these experiments (1-octanol is also assumed not to have significant diffusion through the vapor since it has an even lower vapor pressure than the n-dodecane). Diffusion was measured at saturations as low as 0.035 for 1-octanol. Lower saturations were not investigated. To ensure that this was caused by diffusion through the 1-octanol and not diffusion through the water, one experiment with 14Clabeled octanol was carried out on a water-saturated column (the 1-octanol saturation through the column was 0). No diffusion was measured in this column experiment, thus indicating that no measurable diffusion occurred through the water phase. The difference in the behavior of the two oils is believed to be caused by the fact that 1-octanol (with a positive spreading coefficient) forms connected oil layers in the pore space. The n-dodecane, which does not spread on water, is no longer continuous across the length of the sand column at a saturation of 0.055. Even before S ) 0.055 is reached, some of the n-dodecane is believed to become discontinuous (or, disconnected), thus causing a reduction in the effective diffusivity. It should also be noted that the saturation value at which this discontinuity occurs may vary for different porous media and different water saturations (7, 18, 19). Error bars on the data points represent the standard deviation in the value of the diffusional resistance. The standard deviation of the diffusional resistance was obtained by first regressing eq 9 to each experimentally obtained diffusion profile (such as shown in Figure 5) and calculating the standard deviation of the effective diffusivity. The standard deviation of the diffusional resistance was then related to the standard deviation of the effective diffusivity by eq 2. A considerable amount of scatter in the 1-octanol data is observed below the saturation at which the n-dodecane becomes discontinuous. It is believed this scatter is due to the fact that the effective diffusivity is controlled by the thin 1-octanol oil layers. Another possible cause for this error is oil saturation heterogeneity on small (µm) length scales. This could be significant at low oil saturations. However, we cannot offer a more complete and quantitative explanation for this
scatter in the data at present. Similar scatter in measuring the relative permeabilities of spreading oils at low saturations has been observed (18). Also shown in Figure 6 are the model predictions for both the nonspreading (solid line) and spreading (dashed line) oils. The nonspreading oil was modeled using eqs 3-6, with macro,oil determined from the experimental data (insular saturation at S ) 0.055). The value of macro,oil used was 0.021 (obtained by multiplying 0.055 by the porosity). The spreading oil (1-octanol) was modeled using eq 7. Both the model predictions for the spreading and nonspreading oils provide a reasonable prediction of the experimental data. While the scatter in the data poses a barrier to unambiguous interpretation, the models appear to explain the difference in behavior between the two oils on the basis of their spreading properties. The difference in phase connectivity of two oils (nonspreading and spreading) at low saturations was demonstrated by measuring the effective diffusivities. Results indicated that the spreading oil was still continuous at low saturations, whereas the nonspreading oil became discontinuous across the sand column at a saturation of 0.055. This difference was believed to be caused by the presence of oil layers. Two separate models were used to describe the behavior of the oils, based on their spreading properties, in the unsaturated porous medium. Each model gave a reasonable prediction of the experimental data. In porous media contaminated with a bulk oil phase at low (near residual) saturations, diffusive mobility of constituent oil contaminants might be significantly higher for spreading oils than for nonspreading oils. This may impact fate and transport modeling, as well as the selection and design of remediation technologies.
Acknowledgments The authors thank Brad Daniels, Jeff Davis, and Jamie Hardt for their assistance in carrying out the diffusion experiments. Also, David DiCarlo furnished guidance in developing the experimental methods. Funding for this project was provided by the Stanford University Gas Injection Affiliates Program (SUPRI-C) and from the Department of Energy Grant DEFG22-96BC14851.
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Received for review October 21, 1998. Revised manuscript received March 16, 1999. Accepted May 27, 1999. ES981087F
