Temperature and Pressure Dependence of the Diffusion Coefficients

J. Phys. Chem. B 2009, 113, 4293–4302

4293

Temperature and Pressure Dependence of the Diffusion Coefficients and NMR Relaxation Times of Mixtures of Alkanes Denise E. Freed Schlumberger Doll Research, One Hampshire Street, Cambridge, Massachusetts 02139 ReceiVed: NoVember 18, 2008; ReVised Manuscript ReceiVed: December 23, 2008

Recently, it was shown1,2 that the diffusion coefficient and nuclear magnetic resonance (NMR) relaxation times of a molecule in a mixture of alkanes follow scaling laws in the chain length of the molecule and the mean chain length of the mixture. These relations can be used to determine the chain length distribution of crude oils from diffusion and relaxation measurements. Oil reservoirs are usually at elevated temperatures and pressures, so it is important to know how these scaling relations depend on temperature and pressure. In this paper, we obtain the relation between the molecular composition of mixtures of alkanes at elevated pressures and temperatures and their diffusion coefficients Di and relaxation times T1i and T2i. Using properties of free volume theory and the behavior of the density of alkanes, we show that, for a large range of pressures, the diffusion coefficients and relaxation times depend on pressure and mean chain length of the mixture only through its density. We further show that the pressure effect can be taken into account in the power laws of refs 1 and 2 by a multiplicative prefactor that depends only on temperature and the free volumes of pure alkanes at the pressure of interest and a reference pressure. We also combine the scaling laws for Di, T1i, and T2i and the Arrhenius dependence on temperature to obtain the temperature dependence of the diffusion coefficients and relaxation times. We obtain good fits between the scaling relations and literature data. These scaling relations can be used to determine the composition of a mixture of alkanes from measurements of diffusion coefficients or relaxation times at elevated pressures and temperatures. I. Introduction Crude oils are natural fluids containing molecules with a wide range of sizes and species. Many oils are composed mainly of saturates such as n-alkanes. The composition of the oil determines its properties, including its viscosity, bubble point, and other phase changes. It is desirable to have a method to determine the composition of the oil in situ in the rock matrix in the reservoir. One set of techniques that is commonly used in the borehole, NMR relaxation and diffusion measurements, is well-suited for this purpose. In refs 1 and 2 we showed that the diffusion coefficients and NMR relaxation times for a molecule in a mixture of alkanes at a given pressure and temperature follow scaling laws in the chain length of the molecule and the mean chain length of the mixture. We also showed that for crude oils with a large percentage of saturates these relations still hold and can be used to determine the chain length distribution of the oil. Oil reservoirs are generally at elevated temperatures and pressures; in some reservoirs the temperatures can reach 175 °C and the pressures can reach 20 kpsi (138 MPa), while temperatures around 100 °C and pressures of about 50 MPa are common. Thus, to be of use in actual reservoirs, it is important to find how the scaling relations depend on pressure and temperature. Most of the common approaches for modeling diffusion and NMR relaxation in liquids are based on hard spheres or polymers. With the exception of methane and ethane, alkanes are short chains that are much smaller than polymers and have more degrees of freedom than hard spheres. Thus, neither of these approaches fully applies to alkanes or other oligomers. Several treatments have addressed the temperature and pressure dependence of the self-diffusion coefficient of alkanes,3-5 while very few have modeled the temperature and pressure dependence

of their relaxation times.6 None of these approaches easily extend to describing mixtures containing a wide range of particles, including ones with no clear distinction between solvent and solute. They also do not lend themselves to determining the composition of an oil from diffusion and relaxation measurements of the oil. In this paper, we will use ideas borrowed from polymer theory7 to model the pressure dependence of mixtures of alkanes. As we shall show, because alkanes are much shorter than actual polymers, the free volume attributed to the ends of the chain plays an important role in the pressure dependence of the diffusion coefficients and relaxation times. Somewhat surprisingly, this pressure dependence can be expressed independently of the chain length, in contrast to the temperature dependence, which is strongly affected by chain length. In ref 1, we showed that the diffusion coefficient Di of the ith component in a mixture of alkanes obeys a scaling law given by

j -β(T,P) Di ) A(T, P)Ni-νN

(1)

j is where Ni is the number of carbon atoms in the chain, N the molar mean chain length in the mixture, ν is given by 0.7, and A(T, P) and β(T, P) can depend on both temperature and pressure. The scaling with chain length Ni was based on the Rouse and Zimm models for polymer dynamics, while the scaling with the mean chain length is related to the free volume models for polymers. For pure substances, many authors have found a power law for diffusion as a function of chain length,8,4,9 and for mixtures, a dependence on mean chain length was found by Van Geet and Adamson.10

10.1021/jp810145m CCC: $40.75  2009 American Chemical Society Published on Web 03/03/2009

4294 J. Phys. Chem. B, Vol. 113, No. 13, 2009

Freed

For live oils, i.e., oils with dissolved gases such as methane j approaches one. Instead, and ethane, eq 1 is no longer valid as N the diffusion data was well fit by the modified scaling law

j + 1)-β(T,P) Di ) A(T, P)Ni-ν(N

(2)

with somewhat different values of A(T, P) and β(T, P), and where Niν is replaced by an effective radius ri for the gas components. Similarly, as discussed in ref 2, for at least some range of chain lengths, the transverse NMR relaxation time for mixtures of alkanes is given by

j )-γ(T,P) T2i ) B(T, P)Ni-κ(N

(3)

where κ ) 1.24 and again B(T,P) and γ(T,P) are temperatureand pressure-dependent parameters. For low enough frequencies the longitudinal relaxation time T1i is equal to the transverse relaxation time. The relation in eq 3 comes about because T2i depends on the rotational diffusion coefficient or, more particularly, the inverse of the rotational diffusion time. The rotational diffusion coefficient has a similar power law dependence on chain lengths as the translational diffusion coefficient, but with different exponents. The relaxation processes of dissolved gas such as methane are different than those of the longer alkane chains, so this scaling relation will not apply for the dissolved methane or when the mean chain length approaches one. There are other descriptions of diffusion and relaxation of alkanes that have been used in the literature. For example, the temperature dependence of the diffusion coefficient for pure alkanes has been found by many authors to be well fit by an Arrhenius law.4,9,11-13 The diffusion coefficient can also be described in terms of the free volume model.9,14 Thus, the diffusion coefficient can be written as

Di ) A0Ni-νe-Ea/kTe-1/f({Ni},T,P)

(4)

where now Ea is the segmental activation energy, which is j and Ni, and f({Ni},T,P) is the assumed to be independent of N free volume fraction. In ref 1 we showed that, for alkanes, the free volume fraction depended on the chain lengths of the molecules in the mixture only through the mean chain length, j , in accord with what happens for polymers.14 The coefficient N A0 in eq 4 is usually assumed to be independent of pressure and temperature.14 Thus, in order to find the pressure dependence of the diffusion coefficients and the relaxation times, it is important to know how the free volume fraction depends on pressure. In this paper, we will show how we can combine the scaling laws, Arrhenius law, and a more detailed free volume picture to explicitly obtain the temperature and pressure dependence of the diffusion coefficients and relaxation times for a wide range of temperatures and pressures. In section II, we will carefully examine the pressure, temperature, and chain length dependence of the molar volumes. We will show that, to a good approximation, for mixtures of alkanes the molar volume is a linear function of mean chain length, with the slope (or volume per segment) having practically no dependence on pressure. In section III we will show how this observation implies that the diffusion cofficients and relaxation times should depend on pressure and mean chain length only through the density, and

we show that this is the case for data from the literature. In the following section, we use the density dependence of the diffusion coefficients and relaxation times to take into account their pressure dependence by expressing Di and T1,2i as power laws in an effective chain length. Again, these results are shown to fit literature data. In section IV we use additional properties of the molar volumes to simplify these relations for Di and T1,2i. We find that at elevated pressures the diffusion coefficients and relaxation times obey the same power laws as in eqs 1 and 3, with the only modification being that the coefficients A(T,P) and B(T,P) depend on the free volume at pressure P. The free volume can be determined from density data of pure alkanes. Finally, in section V, we determine the temperature dependence of the diffusion coefficients and relaxation times by requiring them to have both an Arrhenius temperature dependence and to follow the scaling laws in chain length. With these expressions, the diffusion coefficients and relaxation times can be determined at an arbitrary temperature and pressure as long as data on alkane densities are available for that pressure and temperature. In addition, these expressions can be used to determine the composition of mixtures of alkanes and crude oils from diffusion and relaxation measurements. II. Pressure Dependence of the Molar Volumes The diffusion coefficients depend on the free volume fraction in the fluid. Thus, it is important to know how the free and occupied volumes of alkanes and mixtures of alkanes vary as a function of pressure and chain length. To address this question, in this section we will look at the pressure dependence of the molar volume. In ref 9 it was found that the density of alkanes is well described by breaking it into the volume occupied by each segment, Vs, and the extra free volume Ve, due to each end. (It is assumed that any extra occupied volume due to each end is negligible.) For long enough chains, both Vs and Ve do not depend on chain length, but they do depend on temperature and pressure. Similar observations were also made earlier in references such as ref 15. Then, as long as the volumes are additive, the molar j is volume for a mixture of alkanes with mean chain length N given by

j + 2Ve VT ) VsN

(5)

Equations for Vs and Ve at atmospheric pressure and a wide range of temperatures can be obtained from ref 9. It was found9,15 that the free volume due to the ends varies much more rapidly with temperature than the volume per segment, which makes sense if most of Vs is occupied volume. Similarly, in ref 15, it was found that Ve also depends more strongly on pressure than does Vs. However, at very high pressures this difference decreases significantly. Here, we will look more carefully at the temperature and pressure dependence of Vs and Ve. The molar volumes Ve and Vs can be determined by looking at density data at a fixed temperature and pressure and plotting the volume VT ) M/F as a function of chain length. In Figures 1 and 2 we give several examples. We focus on pressures and temperatures at which the relaxation and diffusion data in later parts of this paper were taken, but the results should apply to a wide range of pressures and temperatures. First, in Figure 1, the molar volumes calculated from the densities from ref 16 are plotted as a function of chain length at two different temperatures and three different pressures. As can be seen in the plot, for each temperature and pressure the

Diffusion Coefficients of Mixtures of Alkanes

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4295 TABLE 1: Slope Ws and Intercept 2We of the Molar Volume vs Chain Length for the Alkanes in Ref 16 temperature (°C)

pressure (MPa)

slope Vs (cm3/mol)

intercept 2Ve (cm3/mol)

25

0.1 0.14 20.7 41.4 50.0 0.1 0.14 20.7 41.4

16.27 16.32 16.18 16.01 16.07 16.91 16.84 16.74 16.54

33.20 32.61 30.00 28.45 27.06 40.49 40.84 35.56 33.01

85

j . The data are Figure 1. Molar volume VT vs mean chain length N from ref 16. The solid lines show the fits to straight lines.

TABLE 2: Slope Ws and Intercept 2We of the Molar Volume vs Chain Length for Live and Dead Alkanes temperature (°C)

pressure (MPa)

slope Vs (cm3/mol)

intercept 2Ve (cm3/mol)

30

0.1 30 40 50 0.1 23.4 50.9 0.1 30 40 50

16.33 16.08 16.09 16.09 16.54 16.12 16.20 16.65 16.22 16.25 16.27

33.81 30.31 28.79 27.53 36.24 34.08 29.19 37.45 33.72 31.74 30.12

50

60

j for dead and live Figure 2. Molar volume VT vs mean chain length N alkanes. Part a shows pure alkanes at 50 °C. The data at 23.4 MPa are shown in blue, and the data at 50.9 MPa are shown in red. In part b the data for pure alkanes are plotted with ×’s, the data for mixtures of methane and hexane are plotted with circles, and the data for mixtures of ethane with hexane are plotted with diamonds. The data at T ) 30 °C, P ) 30 MPa are plotted in blue, those at T ) 60 °C, P ) 30 MPa are in red, those at T ) 30 °C, P ) 50 °C are in green, and those at T ) 60 °C, P ) 50 °C are in magenta. For both figures, the solid lines are the linear fits to C6 and C7.

data points lie on a straight line. As the temperature and pressure are changed, clearly the intercept changes more than the slope, although the slope does change a little more as the temperature is raised. The values for the slope Vs and intercept 2Ve of the fitted lines (shown by the solid lines) are given in Table 1. For comparison, the slope and intercept calculated from the equations in ref 9 at atmospheric pressure (0.1 MPa) are also given in the table as well as values at 50 MPa, which are calculated from the NIST data17 as in the examples below.

Thus, we see clearly that for dead oils (i.e., oils with no dissolved gases) the intercept or free volume per end varies much more with pressure than the slope or volume per segment. Next, in Figure 2 we give examples that include both live alkanes and mixtures of alkanes. In Figure 2a, only pure alkanes are shown, and the temperature is T ) 50 °C and the pressure is 23.4 MPa (shown in blue) and 50.9 MPa (shown in red). In Figure 2b the molar volumes of both pure alkanes and mixtures of methane and ethane with hexane are shown at T ) 30 and 60 °C and P ) 30 and 50 MPa. The data for C1 through C7 are from the NIST webbook,17 the data for the mixtures are from the NIST mixture property database,18 and the data for C16 is from ref 19. We note that for N g 3 the fit to a straight line is pretty good, and as the pressure is raised, the fit improves for smaller chain lengths. In fact, the fit for pure alkanes is extremely good, while there is much more scatter for the plots with mixtures. This is in part due to the larger errors in the NIST mixture property database but may also reflect the fact that the volumes in the mixture are not perfectly additive. The solid lines are the fits only to the two points at C6 and C7, which extrapolate quite well to the molar volume for C16 in Figure 2a. The slopes Vs and intercepts 2Ve of these lines are given in Table 2, along with Vs and 2Ve from similar fits at 40 MPa. In addition, in Table 2, the values of Vs and 2Ve at atmospheric pressure (0.1 MPa) were found using the equations in ref 9. In both Table 1 and Table 2, it is clear that the slope Vs changes very little as a function of pressure, while the intercept 2Ve changes more rapidly. The same trend is true for the temperature dependence. However, both the slope and intercept appear to depend more strongly on temperature than on pressure. Also, Vs and 2Ve change more with pressure at higher temperatures than at lower temperatures. The fact that Vs changes so little with pressure supports the assumption, made in in the following section, that the occupied volume per segment Vso does not depend appreciably on pressure (particularly in Table 2). This is in contrast to what is found for more dense fluids such as some polymer melts.7 It is


Freed

possible that the occupied volume per segment does depend on temperature because the total volume per segment Vs does appear to vary more with temperature. This is partially justified because changes in temperature can change the intrinsic motion of the molecule, which can affect how much volume it effectively occupies.

F)

M VT

(10)

j + 2.016 is the molecular mass for a where M ) 14.016N mixture of alkanes in g/mol. With eq 5 the expression for the density becomes

III. Density Dependence of the Diffusion and Relaxation At atmospheric pressure, the diffusion coefficients and relaxation times depend only on the mean chain length or, equivalently, only on the density of the mixture of alkanes. In this section we will focus on the density dependence of the diffusion coefficients and relaxation times as both the pressure and mean chain lengths are varied. By looking carefully at the free volume picture for alkanes, we will show that the diffusion coefficients and relaxation times depend on the mean chain length and pressure only through the density of the mixture. According to section I, the diffusion coefficient depends on the free volume fraction f({Ni},T,P), given by

F)

j + 2.016 14.016N j NVs + 2Ve

(11)

Next, in both the expression for the free volume fraction and j -dependent parts the density, we can separate the pressure- and N from the rest as follows:

f)

j , P) h(N j , P) Vso + h(N

(12)

(6)

F)

j 14 + 2/N j , P) Vso + h(N

(13)

where VT is the total volume and Vf is the free volume. As seen in the previous section, for a mixture of alkanes they are given by

j ,P) is given by where h(N

and

f({Ni}, T, P) )

Vf VT

j Vs + 2Ve VT ) N

(7)

j , P) ) Vsf(P) + 2Ve(P)/N j h(N

Then the free volume fraction can be written in terms of the density as follows:

and

j Vsf + 2Ve Vf ) N

f)

(8)

j is the molar mean given by ∑ixiNi, where xi is the where N mole fraction of the ith component in the mixture. In these equations, Ve and Vs are the free volume per end and the volume per segment, respectively, as discussed in the previous section. The free volume per segment can be divided into two parts: the occupied part Vso and the free part Vsf. We will assume that the occupied volume does not depend on pressure; only the free volume does. This assumption is supported by the data in the previous section. However, once the pressure becomes too high, we can no longer expect Vso to be independent of pressure because the high pressure may affect the configuration of the molecules, which could in turn affect the occupied volume. Also, we expect the total volume and free volume to deviate from these expressions for mean chain lengths roughly less than two. The free volume fraction can be written in terms of the mean chain length and various volumes as follows:

f)

j Vsf + 2Ve N j (Vso + Vsf) + 2Ve N

(9)

We can also write the density in terms of these parameters. It is given by

(14)

j - VsoF 14 + 2/N j 14 + 2/N

(15)

To an excellent approximation, this can be written as

f)

14 - VsoF 14

(16)

j )2. Thus, where the term that is dropped is on the order of 1/(7N with the assumption that Vso is independent of pressure and chain length, all the pressure and chain length dependence of f can instead be replaced by the dependence of f on density. According to eqs 16 and 4 then, at a fixed temperature the scaled diffusion coefficient NiνDi and relaxation times NiκT1i and NiκT2i should depend only on density, regardless of the mean chain length and pressure (as long as the chain length is greater than about two and the pressure is not extremely high). In other words, the scaled diffusion coefficients and relaxation times should be functions only of density and temperature. In Figure 3, we show the T1 data for C8, C10, C12, and C16 from ref 16 as a function of density. The temperature ranges from 25 to 85 °C. For C8 through C12 the pressure ranges from about 0.1 to 41 MPa, while for C16 the pressure only goes up to about 28 MPa. In this plot, there is no clear connection between the relaxation times of each of the alkanes, except that they increase as the chain gets shorter. Next, in Figure 4, we plot the scaled relaxation time NκT1, with κ ) 1.24. As can be seen in the figure, the data now


Figure 3. Relaxation times T1 vs density for C8, C10, C12, and C16 at elevated pressures. The data are from ref 16. The data at 25 °C are plotted with circles, the data at 45 °C with plus signs, the data at 65 °C with diamonds, and the data at 85 °C with ×’s. The data for C8, C10, C12, and C16 are plotted in blue, red, green, and magenta, respectively.

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4297

Figure 5. Scaled diffusion coefficients NiνDi vs density at T ) 25 °C for pure alkanes and mixtures. The data from Harris and Dymond are at elevated pressures. All the other data are at atmospheric pressure.

Figure 4. Scaled relaxation times NκT1 vs density for C8, C10, C12, and C16. The data are the same as in Figure 3.

collapse into four lines, one for each of the four different temperatures. The main exception is for octane at the high temperature and, to a lesser extent, for hexadecane at low temperatures. Presumably these discrepancies occur because the octane is getting too close to its boiling point and the hexadecane is getting too close to its melting point, but there could be other explanations for why the range of validity of the collapse is limited. Apart from these limiting cases, the collapse is quite remarkable and demonstrates that for a range of pressures and chain lengths the scaled relaxation time depends only on temperature and density. Next, we look at the dependence of the diffusion coefficient on density. In Figure 5 we plot the scaled diffusion coefficient NiνDi for pure alkanes and some binary mixtures at 25 °C as a function of density. The data for C6 and C8 from refs 20 and 21 range from atmospheric pressure to about 350 MPa. The data for C16 from ref 19 ranges from atmospheric pressure to 27 MPa. All the other data points are at atmospheric pressure. The equation for density from ref 9 was used to obtain the density at atmospheric pressure. The data collapse reasonably well to a single line. In fact, over the entire range, the agreement between hexane and octane is quite remarkable. However, as the density nears 0.75 g/cm3, the scaled diffusion coefficients for hexane and octane at elevated pressure start to deviate noticeably from the scaled diffusion coefficient for dodecane at atmospheric pressure, and the difference between them and C16 looks significant. This is at about 250 MPa for C6 and 100 MPa for C8, so these are already rather high pressures at which our assumptions about the free volume may no longer be valid. Also, at these high

Figure 6. Scaled diffusion coefficients NiνDi vs density at T ) 25 and 30 °C for pure alkanes and mixtures. Except for the points for C1 from Harris, the data at 25 °C are the same as in the previous figure. The data from Freedman are at atmospheric pressure, and the data from Helbaek are at 30, 40, and 50 MPa.

pressures, it is not clear that the equations for the density of hexane and octane are valid either. For completeness, we will show one more example of the scaled diffusion vs density. In Figure 6 we show the previous data at 25 °C, along with data for C1 at 25 °C and additional data at 30 °C. The data for methane range from a few atmospheres to about 160 MPa. The data for C6 and C16 are at atmospheric pressure, while the data from Helbaek et al.22 for C1, C2, C6, and the mixtures C1 + C6, C2 + C6 are taken at 30, 40, and 50 MPa. The densities for the data from Helbaek are taken from the NIST webbook17 for the pure substances and the NIST mixture property database18 for the mixtures. On this scale, everything appears to lie on a single line, except for the pure methane. Plots of scaled diffusion coefficients at 60 °C show similar behavior. Finally, we note that whenever the diffusion coefficients depend only on density and not on pressure and chain length separately, eqs 4 and 16 imply that the occupied volume per segment does not change as a function of pressure. Thus, over a fairly wide pressure range, it does appear that Vso is constant, although as the pressure gets to the hundreds of MPa, this may no longer be the case.


Freed

IV. Scaling Relation at Elevated Pressures We have just shown that the free volume and, hence, the scaled diffusion coefficients and relaxation times are functions of the density. We also know that they follow power laws in the mean chain length. We can combine these two properties to find the pressure dependence of the scaling laws. A. Diffusion, Relaxation, and the Effective Chain Length. First, we will define an effective chain length. Suppose we know the equation for the density as a function of chain length at a reference pressure P0 (such as atmospheric pressure) and temperature T. If a sample is measured at another pressure P and has density F at this pressure, we can define an effective chain length Neff for the sample as the chain length that would have the same density F as the sample, but at the reference pressure P0. Thus

j , T, P) ≡ F F(Neff, T, P0) ) F(N

Figure 7. Scaled relaxation times NiκT1i vs effective chain length Neff. The data is the same as in Figure 4, with the addition of the squares for C10 at 10 °C. The black solid lines show the fits to the power law in Neff for the data at elevated pressures.

(17)

Using eq 11 for the density, we have

F)

M NeffVs + 2Ve

(18)

where M ) 14.016Neff + 2.106, and Vs and Ve are given by their values at the reference pressure. The effective chain length is then given by

Neff

2VeF - 2.016 ) 14.016 - VsF

(19)

Next, to calculate the diffusion coefficient at elevated pressure, we make use of the fact that at temperature T the scaled diffusion coefficient is a function of density only, so that

NiνDi(F(N, T, P)) ) NiνDi(F(Neff, T, P0))

(20)

Over some range of temperatures and chain lengths, the diffusion coefficient at the reference pressure has the form

Di(Neff, T, P0) ) A(T, P0)Ni-νNeff-β(T,P0)

(21)

Combining eqs 20 and 21, the diffusion coefficient at pressure P is given by

j eff-β(T,P0) Di(T, P) ) A(T, P0)Ni-νN

(22)

j eff is given by eq 19. where N A similar calculation for the relaxation times yields

j eff-γ(T,P0) T1i(T, P) ) T2i(P) ) B(T, P0)Ni-κN

(23)

where B(T, P0) and γ(T, P0) are the values of B and γ at the reference pressure. To illustrate this, in Figure 7 we have plotted the T1 data from ref 16 as a function of Neff, where P0 is at atmospheric pressure. For each value of T, we get a separate power law. The fit for the power laws are shown by the solid lines. Over the limited range of chain lengths at which the

Figure 8. Scaled diffusion coefficient NiνDi vs effective chain length Neff(P0) + 1, with P0 ) 0.1 MPa, for dead and live alkanes at 60 °C. The black solid line shows the fit to the power law from data at atmospheric pressure. The blue solid line shows the fit to the data from Helbaek and Van Geet to a power law in Neff(P) + 1, where P is atmospheric pressure. The red solid line shows the fit to data from Helbaek at 40 MPa to a power law in Neff(P) + 1, where P ) 40 MPa.

alkanes are fluids at these temperatures, the fit to the power laws are quite good. LiWe Oils and the EffectiWe Chain Length. We can obtain similar plots for diffusion coefficients. In this case the available data cover a wider range of chain lengths. However, as the chain length approaches one, things do not work as simply. In that case we found that the scaled diffusion coefficients go as a power law in Neff(P0) + 1. However, depending on the reference pressure P0, the shape of this curve will change. In addition, the expression for the molar volumes in eq 5 no longer holds for chain lengths of about two or less, and it is not clear that it makes sense to use this formula when chain lengths are this small, especially when they are less than one. For the longer chains, as found before, there is a discrepancy between the value of the diffusion coefficient for C16 at atmospheric pressure and C6 and C8 at very high pressures, so again we do not expect to find a single power law once the data include a very wide range of pressures. An example of how the diffusion coefficients depend on Neff is shown in Figure 8. The data are at 60 °C and are from the same references as the data in Figure 6, except that the value for hexadecane (shown with an x) is interpolated from the data in ref 4, and the data for C6 go up to almost 400 MPa. The scaled diffusion coefficient is plotted against the effective chain length at atmospheric pressure, Neff(P0) with P0 ) 0.1 MPa. The solid black line shows the fit to the data at atmospheric pressure, with the values of A(T,P0) and β(T,P0) found in section



V for chain lengths of six and higher. It clearly does not extrapolate properly to the lower chain lengths. However, it is possible to find much better fits. The blue solid lines show the fit to the power law in Neff(P0) + 1, with P0 equal to 1 atm, where both data at atmospheric pressure (from ref 10) and elevated pressure (from ref 22) were used. In this case, the fit is surprisingly good. We can also use another value for the reference pressure P0 in the fit. The red line shows a fit to the data from ref 22 to a power law in Neff(P) + 1, where P ) 40 MPa instead of 0.1 MPa. Again, the fit is quit good but now does not follow a straight line on this plot because the x-axis is for an effective chain length at atmopsheric pressure and not at 40 MPa. B. Scaling Law for Diffusion and Relaxation. In the previous section, we have seen that, at least over limited ranges, the scaled diffusion coefficients and relaxation times follow a scaling law in the effective chain length, which depends on the oil’s density and does not depend independently on the pressure and actual mean chain length. However, there are complications in describing what happens for live oils, and the density is not always known for a given oil. In addition, in actual crude oils, the density can be very sensitive to the amounts of non-alkanes present, such as aromatics, but the diffusion coefficients and relaxation times are not that sensitive.1,2 By looking more carefully at the implication of the free volume model and the scaling law, we will show that the pressure dependence can be simplified to a power law in the actual chain length with no explicit dependence on the density of the oil. To do this, we will revisit the effective chain length Neff. It is the chain length that has the same density at the reference pressure P0 that the sample has at its pressure P. We can write Neff in terms of the mean chain length of the sample, as follows: By the definition of Neff, we have

j , P) F(Neff, P0) ) F(N

(24)

Next, we can substitute in the expression for F from eq 13 and solve for Neff to obtain

Neff )

2Ve(P0) j N j N(Vs(P) - Vs(P0)) + 2Ve(P)

(25)

From the values of Vs and 2Ve in Tables 1 and 2, we see that the change in Vs with pressure is much smaller than the value of 2Ve, so, unless N gets very large, we can drop the first term in the denominator to obtain

Neff )

j Ve(P0)N Ve(P)

(26)

Now we can substitute this expression for Neff into eq 22 for the diffusion coefficient at pressure P to obtain

( )

Di(T, P) ) A(T, P0)

Ve(P0) Ve(P)

-β(T,P0)

j -β(T,P0) Ni-νN

(27)

In this way, we obtain the scaling law as a function of the actual j . We note that the exponent β is independent mean chain length N of pressure, so the scaling law has the form

( ) Ve(P0) Ve(P)

Di(T, P) ) A(T, P0)

-β(T)

j -β(T) Ni-νN

(28)

The pressure dependence for the relaxation times can be found in a similar way, and it has the form

( )

T1,2i(T, P) ) B(T, P0)

Ve(P0) Ve(P)

-γ(T)

j -γ(T) Ni-κN

(29)

If we have information about the density of alkanes at the desired temperature T and at both the desired pressure P and the reference pressure P0, then we can fit for Ve(P0) and Ve(P), which both depend on T. Then, as long as we know β(T) and A(P0,T), we can find the diffusion coefficient (and similarly the relaxation times) at any pressure. An example is given in Figure 9. The solid cyan line shows the scaling law at atmospheric pressure and 25 °C that is found in section V. The values of Ve(P0) and Ve(P) with P0 ) 0.1 MPa and P ) 50 MPa are from Table 2. The solid blue line is then given by eq 28, where the parameters A(P0,T) and β(T) are the values from the scaling law at atmospheric pressure. As can be seen in the figure, once we know the scaling law at atmospheric pressure and some densities at atmospheric pressure and at 50 MPa, we can find a reasonably good fit to the diffusion coefficients at 50 MPa with no fitting parameters. The main drawback to this method for calculating diffusion coefficients and relaxation times is that it relies on having good density data for the alkanes, which is not always available at high temperatures and pressures. Thus, we have found that, at any given temperature, the pressure dependence is taken into account by a multiplicative factor that depends only on the relative amounts of free volume per end for pure alkanes at that pressure and a reference pressure. As the pressure goes up and the free volume goes down, the translational diffusion coefficient decreases. In addition, the rotational correlation time increases, leading to a decrease in the NMR relaxation times. What is perhaps surprising is that the change in pressure affects all the chain lengths alike, with the same multiplicative factor. One of the consequences of this is that, on a log plot, the curves for diffusion coefficients as a function of temperature or chain length will all lie parallel to each other as the pressure is changed, The relaxation times should behave similarly. This is seen in Figure 9 and in data such as in refs 4 and 23. However, once the pressure becomes very high, the parameter β(T) will have some pressure dependence, as found by Vardag et al.4 for pressures above about 100 MPa. This presumably is an indication that the occupied volume per segment is also affected by pressure at these high pressures. As we shall show next, it is probably also an indication that the diffusion coefficient depends on pressure as well as density at these higher pressures. We conclude this section by showing that pressure independence of the exponent β(T) (over some range of pressures) is a consequence of requiring the scaled diffusion coefficient both to depend on pressure and chain length only through density and to follow scaling laws in that pressure range. This can be seen as follows: If we were to substitute into eq 22 the full expression, eq 25, for the effective chain length Neff without


Freed

Figure 9. Scaled diffusion coefficient NiνDi vs mean chain length at 25 °C. The cyan line shows the scaling law in mean chain length at atmospheric pressure. The blue line shows the scaling law in mean chain length at 50 MPa. It is obtained from the cyan line by multiplying by [Ve(P0)/Ve(P)]-β(T).

imposing any condition on the sizes of Vs and Ve, we would obtain

Di(T, P) ) A(P0, T)Ni-ν 2Ve(P0) j (Vs(P) - Vs(P0)) + 2Ve(P) N

[

]

-β(T,P0)

j -β(T,P0) (30) N

j ) to obey a strict scaling law in N j , the relation If we require D(N in eq 30 is only possible if we can replace the expression in the denominator by 2Ve(P). Thus, the scaling law is only possible j (Vs(P) - Vs(P0)) , 2Ve(P). In that case, the equation for Di if N reduces to the scaling law in eq 28. Pressure Dependence for LiWe Oils with Short Mean Chain Lengths. Finally, we make some comments about the pressure j , the dependence for live oils. As described above, for small N j , but in N j + 1. Thus, at pressure scaling law is no longer in N P the diffusion coefficient can be written in terms of the effective chain length at the reference pressure P0 as follows:

Di(T, P) ) A(T, P0)ri(Neff(P0) + 1)-β(T,P0)

(31)

where ri ) Niν for the alkanes with chain length larger than about 5, and ri is proportional to an effective hard-sphere radius for smaller chain lengths. If we substitute in the value of Neff from eq 26 into the scaling law for live oils, still assuming that j (Vs(P) - Vs(P0)) , 2Ve(P), we find N

( ) (

Di(T, P) ) A(T, P0)

Ve(P0) Ve(P)

-β(T)

j + ri N

Ve(P) Ve(P0)

)

-β(T)

(32)

In all the examples we have considered so far, the extra free energy from the edges Ve has not changed by more than about 20% as the pressure was varied. Because the intensity of the NMR signal is proportional to the number of protons, it is very j + 1 for the live oils. Thus, useful to have a scaling law in N we will replace the expression Ve(P)/Ve(P0) by 1 in eq 32 to obtain a useful approximation for the pressure dependence of live oils.

( )

Di((T, P) ≈ A(T, P0)

Ve(P0) Ve(P)

-β(T)

j + 1)-β(T) ri(N

(33)

Figure 10. Scaled diffusion coefficient (Ve(P0)/Ve(P))βriDi vs mean chain length +1 for dead and live alkanes at 30 °C. The blue line shows the scaling law at 30 MPa, the red line shows the scaling law at 40 MPa, and the green line shows the scaling law at 50 MPa. The data from Helbaek et al. include all the data plotted in Figure 6 as well as data for C8, C10, and mixtures of these alkanes with C1 or C2.

The fit to a scaling law does not seem to be that sensitive to this approximation, as we saw in Figure 8 and as in the following example. In Figure 10 we show that we can use the pressure dependence in eq 33 to collapse the data at different pressures to a single line. In these figures, we plot the full data from Helbaek et al.22 as a function of mean chain length. The data are at 30, 40, and 50 MPa. We also include data from Freedman et al.24 at atmospheric pressure (0.1 MPa). Instead of plotting the usual scaled diffusion coefficient, we include an additional scale factor of the form (Ve(P0)/Vs(P))β(T). Thus, we are plotting

( ) Ve(P0) Ve(P)

β(T)

j + 1)-β(T) riDi(T, P) ) A(P0, T)(N

(34)

versus the actual mean chain length of the alkane or mixture. We are taking the reference pressure P0 to be atmospheric pressure, but the collapse looks very similar if we use one of the elevated pressures instead. The value for β(T) is the one found in section V, and the values for Ve(P0) and Ve(P) are given in Table 2. Note that the quantity on the right-hand side of eq 34 depends only on the reference pressure P0 and on no other pressures. Thus, the data points at all four pressures should collapse to a single line. As can be seen in the figure, they collapse very well to a single line; the scatter in data at a single pressure is much larger than the scatter between data at different pressures. The solid lines show the fits done directly to the scaling law of eq 1 at the three elevated pressures. Again, they pretty much collapse to a single line. The data at 60 °C have a similar collapse. In conclusion, we have shown that the pressure dependence can be taken into account by an additional scaling of the diffusion coefficients by a factor that depends on the free volume per end of the molecules, even when the mean chain length approaches one. In the following section, we will find expressions for the temperature depence of the relaxation and diffusion coefficients. V. Temperature Dependence In this section, we will combine the power law dependence on chain length with the Arrhenius temperature dependence to obtain the temperature and chain length dependence of the diffusion coefficients and relaxation times. We begin by



reviewing the temperature dependence of the diffusion coefficients, which has already been addressed by several authors, and then we will consider the temperature dependence of the relaxation times, which has not been addressed much in the literature. According to eq 1, the diffusion coefficient follows a scaling law of the form

j -β(T) Di ) A(T, P)Ni-νN

(35)

where A(T,P) and β(T) depend on temperature. Alternatively, for pure substances the diffusion coefficient D has been found to have an Arrhenius temperature dependence of the form

D ∝ e-Ea(N)/kT

(36)

where the activation energy Ea(N) is a function of chain length.4,9,11-13 The only way for these two expressions for the diffusion coefficient to be consistent is if the activation energy is logarithmic in N, of the form

Ea(N) ) b + d log(N)

(38)

In other words, the exponent β(T) in the scaling law is given by

β(T) ) c + d/T

(39)

a ) -6.3326 b ) 143.6869 c ) -0.2442 d ) 588.4961

(40)

Because A depends on pressure, the parameters a and b can also depend on pressure. However, since β is independent of pressure, c and d should also be independent of pressure. To illustrate this, we first show data for diffusion coefficients for pure alkanes taken at a wide range of temperatures and at atmospheric pressure (or saturation vapor pressure). The diffusion coefficients as a function of reciprocal temperature are plotted in Figure 11. We have done a four parameter fit for eq 38 for the diffusion coefficient. All the data shown in the figure, apart from the data for C8 from ref 21 and the data for C78 and C154, which are blends of alkanes with mean chain length 78 and 154, respectively, were used in the fit. The fit is plotted with the solid lines, and different colors are used for different chain lengths. As can be seen, the data fits quite well to the Arrhenius plots, and the lines match up well with the appropriate chain lengths.

(41)

when the coefficient A(T,P) is given in 10-5 cm2/s. The exponent β(T) varies a fair amount with temperature. At 25 °C it is β(25 °C) ) 1.73. At 100 °C it is β(100 °C) ) 1.33, and at 200 °C it is β(200 °C) ) 1.00. This would appear to be an indication that the occupied volume/segment does depend on temperature. For these temperatures, the coefficient A(T,P) does not vary by as great a percentage, and in fact, a + b/T is not very sensitive to temperature. For T ) 25, 100, and 200 °C, we find A(T,P) ) 347.5 × 10-5, 382.8 × 10-5, and 415.3 × 10-5 cm2/s, respectively. In order to look at live oils, we also fit the data to the equation

j + 1)-(c+d/T) Di ) e-(a+b/T)Ni-ν(N

and the coefficient A(T,P) is given by

A(T, P) ) exp[a(P) + b(P)/T]

The values of the four fitted parameters were

(37)

for some temperature-independent coefficients b and d. This was in fact found in refs 12 and 9. The diffusion coefficient can then be written in terms of four temperature-independent parameters a, b, c, and d in the form

j -(c+d/T) Di ) e-(a+b/T)Ni-νN

Figure 11. Diffusion coefficient D vs reciprocal temperature for pure alkanes at atmospheric pressure. The solid lines show the result of the four-parameter fit based on the scaling law and Arrhenius law. The data from Douglass and McCall are plotted with ×’s (except the one at C16, which is from Straley). The data from Marbach et al. are plotted with diamonds. The calculated values from Ertl and Dullien are plotted with squares. The data from Harris et al. and from Dymond et al. are plotted with triangles, and the data from Vardag et al. are plotted with circles.

(42)

with the result

a ) -5.7256 b ) -212.9887 c ) -0.4636 d ) 705.1817

(43)

With these parameters, the fit to the data looks almost identical to the fit with the parameters for the first scaling law. We note, though, that for extremely large temperature ranges the diffusion coefficients no longer have an Arrhenius temperature dependence, especially for the smallest alkanes.3 Next, we will consider the temperature dependence of the NMR relaxation times. There is much less data for the relaxation of alkanes and their mixtures than there is for the self-diffusion of alkanes. The data of Zega16,25 are one of the few such sets of data and one of the most complete, so we shall use it to determine the parameters in eq 44. We begin by noting that the data at atmospheric pressure for C8, C10, C12, and C16 all fit an


Freed

Arrhenius law over the limited temperature range. Thus, the temperature dependence of the relaxation times should have the form

j -(c'+d'/T) T1i ) T2i ) e-(a'(P)+b'(P)/T)Ni-κN

(44)

where a′(P), b′(P), c′, and d′ are temperature-independent parameters. To find the values of a′, b′, c′, and d′, we can make use of the full data set in ref 16 by using the scaling as a function of Neff for the data at elevated pressures. In fact, in Figure 7, the fits for the power laws were actually a four-parameter fit to the scaling law in Neff. Again, the fit is very good and gives the following values for the parameters:

a' ) -5.75 b' ) -227 c' ) -1.43 d' ) 755

mixture. For pressure, the shift is independent of composition entirely, while for temperature the size of the shift will depend on mean chain length, with a greater temperature dependence for mixtures with larger chains. Thus, on a semilog plot, the shape of the diffusion and relaxation distributions will stay the same as the temperature and pressure are varied. The data for crude oils in refs 23 and 26 have this property. If the shape does change, it is an indication of a change of phase. In a future paper, we will show that the temperature and pressure dependence of crude oils that are high in saturates is well described by the scaling equations found in this paper and that phase changes can be detected by changes in the shapes of the distributions. Finally, in this paper we considered only mixtures of alkanes, but we expect our results should also be relevant for mixtures of other homologous oligomers. Acknowledgment. This paper is dedicated to Professor Karl Freed in honor of his 65th birthday.

(45)

with T1 in units of seconds. As with the equation for the diffusion coefficients, these parameters can be used to interpolate for relaxation times at different chain lengths and temperatures. When combined with density data, it can also be used to find the relaxation times at different pressures. VI. Conclusions In conclusion, we have shown how to obtain scaling relations for the diffusion coefficients and relaxation times of mixtures of alkanes at elevated pressures and temperatures. The pressure dependence can be taken into account by a multiplicative shift that depends only on the free volume per end of pure alkanes. Thus, the diffusion coefficients and relaxation times at elevated pressures can be determined from the scaling equations in section V, along with reliable density data for pure alkanes at the desired pressures. Once the relation between D, T1, or T2 and chain lengths is known, it can be used with the methods of ref 1 to determine chain length distributions from diffusion or relaxation measurements. The form of the pressure dependence comes about because in this pressure range (up to about 100 MPa) the occupied volume does not vary very much as pressure is changed and because the free volume per end is larger than the change in the total free volume per segment. In fact, as long as the scaled diffusion coefficients and relaxation times follow power laws as in eqs 1 and 3 and depend on pressure and mean chain length only through the density, the pressure dependence must have the form we found. However, as the pressure gets beyond about 100 or 200 MPa or the chain lengths become very long, these relations may break down, and the method may need further modification. We have also assumed that the molar volume varied linearly with chain length, which is not true for the shortest chain lengths. In addition, over a wide enough temperature range, the Arrhenius temperature dependence is known to break down3 for short chains from methane to about butane. One consequence of the scaling equations is that on a semilog plot the effect of changing both the temperature and pressure is to shift the diffusion coefficients and relaxation times by an amount that is independent of the individual components of the

References and Notes (1) Freed, D. E.; Burcaw, L.; Song, Y.-Q. Phys. ReV. Lett. 2005, 94, 067602. (2) Freed, D. E. J. Chem. Phys. 2007, 126, 174502. (3) Greiner-Schmid, A.; Wappmann, S.; Has, M.; Lu¨demann, H.-D. J. Chem. Phys. 1991, 94, 5643. (4) Vardag, T.; Karger, N.; Lu¨demann, H.-D. Ber. Bunsenges. Phys. Chem. 1991, 95, 859. (5) Assael, M. J.; Dymond, J. H.; Papadaki, M.; Patterson, P. M. Int. J. Thermophys. 1992, 13, 269281. (6) Zega, J. A.; House, W. V.; Kobayashi, R. Physica A 1989, 156, 277. (7) Flory, P. J. Statistical Mechanics of Chain Molecules; John Wiley & Sons: New York, 1969. (8) McCall, D. W.; Douglass, D. C.; Anderson, E. W. J. Chem. Phys. 1959, 30, 771. (9) von Meerwall, E.; Beckman, S.; Jang, J.; Mattice, W. L. J. Chem. Phys. 1998, 108, 4299. (10) Geet, A. L. V.; Adamson, A. W. J. Phys. Chem. 1964, 68, 238. (11) Douglass, D. C.; McCall, D. W. J. Phys. Chem. 1958, 62, 1102. (12) Ertl, H.; Dullien, F. A. L. AIChE J. 1973, 19, 19. (13) Marbach, W.; Hertz, H. G. Z. Phys. Chem. 1996, 193, 19. (14) Ferry, J. D. Viscoelastic Properties of Polymers; John Wiley & Sons: New York, 1980. (15) Kurtz, S. S., Jr. In Chemistry of Petroleum Hydrocarbons; Brooks, B. T., Boord, C. E., Kurtz, S. S., Jr., Schmerling, L., Eds.; Reinhold Publishing: New York, 1954; pp 275-331. (16) Zega, J. A. Ph.D. Thesis, Rice University, 1991. (17) Lemmon, E. W. McLinden, M. O.; Friend, D. In NIST Chemistry WebBook; NIST Standard Reference Database Number 69; Linstrom, P., Mallard, W., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, 2003 (http://webbook.nist.gov). (18) Friend, D. In NIST Standard Reference Database 14; National Institutde of Standards and Technology: Gaithersburg, MD, 1992. (19) Dymond, J. H.; Harris, K. R. Mol. Phys. 1992, 75, 461. (20) Harris, K. R. J. Chem. Soc., Faraday Trans. 1 1982, 78, 2265. (21) Harris, K. R.; Alexander, J. J.; Goscinska, T.; Malhorta, R.; Woolf, L. A.; Dymond, J. H. Mol. Phys. 1993, 78, 235. (22) Helbæk, M.; Hafskjold, B.; Dysthe, D. K.; Sørland, G. H. J. Chem. Eng. Data 1996, 41, 598. (23) Straley, C. Reassessment of correlations between viscosity and NMR measurments, SPWLA 47th Annual Logging Symposium, Paper AA, 2006. (24) Freedman, R.; Sezginer, A.; Flaum, M.; Matteson, A.; Lo, S.; Hirasaki, G. J. A new NMR method of fluid characterization in reservoir rocks: Experimental confirmation and simulation results. Paper SPE 63214 presented at the 2000 SPE Annual Technical Conference and Exhibition, Dallas, 1-4 October. (25) Zega, J. A. Master’s Thesis, Rice University, 1988. (26) Mutina, A. R.; Hu¨rlimann, M. D. J. Phys. Chem. A 2008, 112, 3291.

JP810145M

Temperature and Pressure Dependence of the Diffusion Coefficients

Recommend Documents