Article pubs.acs.org/EF
Heuristics for Equilibrium Distributions of Asphaltenes in the Presence of GOR Gradients Denise E. Freed,*,† Oliver C. Mullins,† and Julian Y. Zuo‡ †
Schlumberger-Doll Research, One Hampshire Street, Cambridge, Massachusetts 02139, United States HPS Schlumberger, 150 Gillingham Lane, MD-3, Sugar Land, Texas 77478, United States
‡
ABSTRACT: Production of crude oil from subsurface reservoirs is greatly impacted by many complexities such as reservoir fluid flow, connectivity, viscosity gradients, and tar mat formation. In situ fluid analysis in oil wells has enabled facile measurement of fluid gradients of dissolved gases and dissolved solids in reservoir crude oils; these gradients have proven very useful for analysis of reservoir complexities. The analysis of solution gas generally uses the cubic equation of state. However, until recently, there had been no predictive equation of state to model asphaltene gradients. Recently, the different nanostructures of asphaltenes in crude oil have largely been resolved and codified in the Yen−Mullins model. In turn, this has enabled equation of state development for asphaltene gradients in crude oils. For example, the Flory−Huggins−Zuo EoS is now ubiquitously utilized in modeling asphaltene gradients. Here, the magnitude and dependencies of the three terms of this EoS, gravity, solubility, and entropy, are considered in detail. Simple expressions for ratios of these terms are obtained as a function of the gas/oil ratio of the crude oils. In particular, the transition from gravity dominance to solubility dominance is examined. A variety of heuristics are developed to guide interpretation of asphaltene gradients that are so routinely measured. Expressions are given that could be used for real-time interpretation upon measurement of these gradients. The utility of EoS modeling of asphaltene gradients is significantly enhanced when incorporating these heuristics.
1. INTRODUCTION Oil reservoirs exhibit a wide variety of variations and complexities that directly impact the efficiency and cost of oil production.1 For example, one of the most important questions that remains difficult to answer is the extent of flow connectivity across an oil field. Less connectivity generally requires drilling more oil wells, often significantly reducing the value of the oil field. The distribution of asphaltenes in reservoirs is also very important as oil viscosity depends exponentially on asphaltene content. In many reservoirs, the contained fluids are not in thermodynamic equilibrium, potentially yielding a variety of reservoir concerns such as substantial gradients in viscosity, gas/oil ratio (GOR), and phase transition pressures. Large spatial variations of crude oil properties in a reservoir generally lead to corresponding large temporal variations in production. All of these issues have an immediate and substantial effect on production. For example, pressure reduction in oil production is often limited by phase transitions, thereby affecting maximum oil flow rates. Oilfield facilities must be designed with knowledge of gas and liquid handling requirements. Consequently, the evaluation of these reservoir and fluid complexities is key. Measurement of the fluid gradients vertically within wells and laterally across the oil field has proven very useful for deciphering reservoir and fluid complexities. In particular, new technology under the banner of downhole fluid analysis has enabled the measurement of many fluid properties within oil wells.1 Realtime determination of fluid complexities allows matching the number of measurement locations in the reservoir to the complexity of the oil column, thereby increasing efficiency of this difficult sample analysis task. Reservoir crude oils are composed of dissolved (hydrocarbon) gases, liquids, and dissolved (or stably suspended) solids, the asphaltenes. In this paper, we will © XXXX American Chemical Society
refer to the stably colloidally suspended asphaltenes as dissolved solids. Generally, the dissolved gas and dissolved asphaltene content of the oil show the greatest variation of fluid properties, and in situ fluid analysis has focused on these measurements. Measurements of these fluid gradients mandate corresponding analyses based on first principles. The cubic equation of state (EoS), for example, the Peng−Robinson EoS,2 has long been used with great success to analyze dissolved gas gradients in crude oil. However, until recently, there had been no equation that successfully modeled asphaltene gradients. This significant limitation was due to the unknown size of asphaltenes in crude oils (and in laboratory solvents). Even asphaltene molecular weight had been the subject of uncertainty over several orders of magnitude.3 Fortunately, significant advances in asphaltene science have taken place recently and have been codified in the Yen−Mullins model (Figure 1).4 Confirmation of various aspects of this model have been obtained recently. The average aggregation number of the asphaltene nanoaggregates has been shown to be about seven by surface-assisted laser desorption/ionization mass spectrometry (SALDI),5 in close agreement with six shown in Figure 1. These studies prove that laser-based mass spectrometry probes asphaltene molecules within nanoaggregates, thus reinforcing previous laser desorption/ionization mass spectrometry (L2MS) confirmation of the dominance of an island molecule architecture as shown in Figure 1.6 Nuclear magnetic resonance (NMR) studies of asphaltene diffusion and spin relaxation find consistency with the model in Figure 1, especially giving small Received: March 26, 2014 Revised: June 30, 2014
A
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using a cluster size of 5.1 nm as the only adjustable variable, in close agreement with 5.0 nm shown in Figure 1.18 Other oil fields with mobile heavy oil exhibit the same gradient with a cluster size of 5.0 nm.19 Moreover, laser mass spectrometry studies of these different crude oils showed an invariant nanoaggregate size with an aggregation number of approximately six in agreement with Figure 1.20 In these field studies, the particle size of asphaltenes (nanoaggregates or clusters) remained invariant even though there was often a large variation of asphaltene concentration. In addition, in the FHZ EoS, it is presumed that the asphaltene chemistry remains invariant in these oil columns. This has been explicitly established for the biggest reservoir with the largest asphaltene gradient that has been examined by these methods, in Saudi Arabia.21 Indeed, the molecular weight and sulfur chemistry of the asphaltenes throughout this giant reservoir are the same. Consequently, the application of the FHZ EoS is reinforced. Application of a cubic EoS with several types of asphaltenes with differing interaction parameters, estimated critical constants, etc., is shown to be inappropriate to account for this large gradient in asphaltene concentration.21 It might appear surprising that a formalism based on the Flory−Huggins model, which depends on a simple solubility parameter, works so well in describing gradients in asphaltene concentration. The Flory−Huggins theory has successfully been used to account for phase transitions of asphaltenes.22 With regard to asphaltene gradients in oil columns, for crude oils of low GOR, the gravity effect has been found to dominate so that the detailed nature of the solubility and entropy effects is of secondary importance.14,15,18,19 Moreover, in most crude oils, the asphaltenes are in nanoaggregate or cluster form, so high energy sites on the asphaltene molecules would be consumed in the aggregate structures, enabling simple expressions for asphaltene solubility parameters to apply. Even for high GOR crude oils where the asphaltenes are molecularly dispersed, the dominant effect is the variation of the liquid phase solubility parameter due to large variations in dissolved gas content. Again, the liquid phase variation is well treated within the solubility parameter formalism. Indeed, the success of the FHZ EoS in treatment of asphaltene gradients in diverse crude oils in many field studies is supportive of the simple concepts underpinning the Yen− Mullins model for the dominant role of the polarizability interaction (with some dipole interaction) in asphaltene intermolecular chemistry. In this paper, we explore in more detail the properties of the Flory−Huggins−Zuo equation for asphaltene gradients. As mentioned above, this equation describes the effect of gravity, solubility and entropy on the asphaltene gradient. We show that the ratio of the contributions from solubility and entropy to that from gravity can be well approximated by simple expressions that depend on the methane content, and we give methods for estimating the methane content from the GOR. We find that at low GORs, the gravity term will dominate, whereas at higher GORs, the solubility term becomes significant. In this paper, we give heuristics for determining when the solubility term becomes important, and we give methods for estimating this term when the GOR and gravity term are known. In Section 2, we review the model for asphaltene gradients and its dependence on energies arising from gravity, solubility, and entropy. In the next section, we compare the magnitude of these three effects for two examples, a near-critical oil and a black oil with varying amounts of dissolved methane. In Section 4, we find simple approximations for the ratios of the energies from the solubility and entropy to the gravitational energy. We show that
Figure 1. Yen−Mullins model representing the dominant asphaltene molecular and nanocolloidal structures found in crude oil.4 At low concentrations, asphaltene molecules (left) form a true molecular solution. At higher concentrations, such as found in typical black oils, asphaltenes are present as nanoaggregates (middle). At even higher concentrations, such as in mobile heavy oils, asphaltenes are dispersed as clusters of nanoaggregates (right).
aggregation numbers for the clusters.7−9 In addition, a recent breakthrough in understanding oil/water interfacial properties is explicitly consistent with the asphaltene molecular and nanoaggregate structures shown in Figure 1.10,11 In particular, the interfacial tension for oil−water systems was found to be given simply in terms of the relative asphaltene coverage, independent of aging time, asphaltene concentration, or viscosity. A universal curve was obtained for the surface tension vs asphaltene relative coverage; this curve can be fit with the Langmuir equation, and one of the resulting parameters, the molecular contact area, matches the Yen−Mullins model.10,11 Most importantly, as will be discussed shortly, the species of asphaltenes shown in Figure 1 have been observed in field studies by measurement of asphaltene gradients in reservoir crude oils. With the resolution of molecular and colloidal properties of asphaltenes, an EoS for asphaltene gradients could then be developed specifically for reservoir crude oils with dissolved gas.12,13 The preferred EoS, which is now called the Flory− Huggins−Zuo (FHZ) EoS, takes into account the effect of gravity and uses the Flory−Huggins model to describe the solubility of the asphaltene in the oil. The EoS depends on the size of the asphaltene molecules, nanoaggregates, and clusters, which are given by the Yen−Mullins model. Field studies have repeatedly shown the applicability of the FHZ EoS combined with the Yen−Mullins model for delineating gradients in reservoir fluids from condensates to mobile heavy oils.13 For crude oils of low GOR, the gravity term dominates, thereby giving a close look at the specific asphaltene species in the oil. Insitu measurements of asphaltene gradients in many wells in a billion-barrel field operated by Chevron showed that asphaltenes with average diameters of slightly less than 2 nm were dominant.14 Similar results were obtained in a different field operated by Pioneer.15 Centrifugation experiments with a live black oil with moderate GOR also obtained a 2 nm nanoaggregate size.16 Recently, technologists at Shell employed the FHZ EoS in a complex reservoir with a large asphaltene gradient. Detailed analysis of in situ measurements of asphaltene gradients in the complex reservoir showed excellent agreement with a single nanoaggregate size of 2 nm, even though the asphaltene concentration varied substantially.17 This important paper illustrated how fault-block migration could be monitored via measurement of asphaltene gradients.17 In a definitive study, asphaltene clusters were shown to dominate in a mobile heavy oil rim in a huge field in Saudi Arabia. The heavy oil rim in a four-way sealing anticline has a height of 50 m, length of 100 km, and an asphaltene gradient of a factor of 10. The extensive data from seven wells penetrating this heavy oil rim showed an excellent fit B
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expressions were derived using a constant pressure condition. The two expressions are the same in the limit as the asphaltene concentration becomes small. It is also possible to show that the full expression in reference 12 can be made consistent with the above equations and those in reference 23, even away from this limit. In addition, if h − h0 is small enough, the right-hand side in eq 2 above can be approximated by −vag(h − h0)Δρ(h), as in reference 23. In the above equations, Grav(h, h0) is the effect due to gravity. It is often referred to as the Boltzmann effect when describing gravitational segregation. It depends on the difference in densities, Δρ, of the asphaltene and the bulk oil. In eq 2, we have taken h to be equal to the depth. As expected for the Boltzmann effect, the Grav(h, h0) term causes the concentration of the asphaltenes to increase with increasing depth in the oil column. The second term, Sol(h, h0) is due to the solubility effects and depends on the difference between the solubility parameters of the asphaltene and bulk oil. Because oils with significant GORs generally have more of the light components higher in the oil column, and these light components do not dissolve the asphaltene as well, this solubility term also tends to increase the amount of asphaltene lower in the column. Lastly, the third term, Ent(h, h0), comes from the entropy of mixing. Most of the entropy of mixing is already accounted for by the left-hand side of eq 1. The expression in eq 4 adds in the effect of the change with depth of the volume of the bulk oil. Because the bulk oil will have a larger molar volume higher in the column, this latter contribution from the entropy tends to decrease the gradient in the asphaltene concentration. It is important to note that all three terms also depend on the molar volume of the asphaltene, va. Thus, to determine the magnitude of the asphaltene gradient, it is essential to know the size of the asphaltene. However, the ratios of the magnitudes of the Boltzmann effect, the solubility effect, and the entropy effect do not depend on the size of the asphaltene, and one can determine which is the dominant effect, even without knowing va. In addition, to calculate the gradients in the asphaltene concentration, the density and solubility parameters of the asphaltene and bulk oil, as well as the molar volume of the bulk oil, are also required. The density of the oil is commonly measured as a function of depth, but the solubility parameters and molar volume are not. These values can be calculated from composition measurements using an EoS tuned to additional fluid properties. Alternatively, in reference 29, we showed how the solubility parameter of the oil can be related to the density. In this paper, instead, we will show how we can estimate the relative magnitudes of the gravity, solubility, and entropy terms and the gradients in asphaltene concentration from the methane fraction or GOR, which can be measured by formation evaluation logging tools.
they depend mainly on the gradient in the methane fraction. In Section 5, we show how the asphaltene gradients can be calculated from the three energy terms. Then, in Section 6, we show how the gradient in the methane fraction can be related to the GOR and its gradient. This makes it possible to estimate the relative magnitudes of the solubility, entropy, and gravity effects given measurements of the GOR. We find that when both the GOR and its gradient are low, the solubility and entropy terms tend to cancel each other, leaving only the gravity term. However, even for a GOR as low as 500 scf/stb, the gradient in the GOR can be large enough to cause a significant contribution from the solubility term.
2. FLORY−HUGGINS−ZUO MODEL FOR ASPHALTENE EQUILIBRIUM DISTRIBUTIONS To model the asphaltene distribution in the oil column, we treat the bulk oil as a two-component mixture of the maltene and the asphaltene. The maltene itself is a mixture that can be treated by an EoS with methods given in reference 23 and references therein. The effect of the maltene on the asphaltene can be described by a Flory−Huggins type of model, combined with gravity, as explained in references 12, 23, and 24. Although the Flory−Huggins model was originally derived for large chain molecules in a solvent, several authors have shown that it still applies for larger rigid molecules mixed with smaller ones.25−27 As the size difference between the types of molecules increases, the entropy of mixing decreases. Here, as in references 12 and 23, we consider the case in which the maltene composition can vary with height. Thus, the maltene properties and the bulk oil properties, such as the density ρ(h), molar volume v(h), and solubility parameter δ(h) depend on the height. In this paper, we will assume that the asphaltene properties, including its density, ρa, molar volume, va, and solubility parameter, δa, do not depend on height. It is possible to also treat the asphaltene component as a distribution that does vary with height.23,28 In the model, va refers to the molar volume of the relevant asphaltene species, e.g., of the molecules, nanoaggregates, or clusters. In the two-component model, the volume fraction ϕa(h) of asphaltenes at a height h can be expressed in terms of the volume fraction ϕa(h0) at a reference height h0 as ϕa(h) ϕa(h0)
= exp{−[Grav(h , h0) + Sol(h , h0) + Ent(h , h0)]/kT }
(1)
where Grav(h , h0) = −vag
∫h
h
Δρ(h)dh 0
Sol(h , h0) = va[(δa − δ(h))2 − (δa − δ(h0))2 ]
(2) (3)
3. COMPARISON OF THE GRAVITY, SOLUBILITY, AND ENTROPY TERMS In this section, we will compare the ratios of the magnitudes of the gravity, solubility, and entropy terms, Grav, Sol, and Ent, and look in more detail at their effect on the asphaltene gradient. For this comparison, we will consider two examples. The first example is a near-critical oil from a well in the North Sea, which is described in detail in reference 23. This well has a large variation in the GOR, so the solubility effects are the main cause of the gradient in the asphaltene or heavy-resin concentration. Heavy resins are those resins with the largest aromatic ring systems. As such, they are somewhat colored. The second example is a
and ⎡ 1 1 ⎤ Ent(h , h0) = kTva⎢ − ⎥ v(h) ⎦ ⎣ v (h 0 )
(4)
In the above equations, Δρ = ρa − ρ(h) is the difference between the asphaltene density and the density of the bulk oil, k is the Boltzmann constant, and T is the temperature. We note that in reference 12, which derived a form of eq 1 by calculating the Helmoltz free energy and using a constant volume condition, the maltene density, molar volume, and solubility were used in place of the bulk oil values in eqs 2−4. These latter C
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Table 1. Values of the Gravity, Solubility and Entropy Terms for the Near-Critical Oil12 depth (m)
Grav/(vakT) (nm−3)
Sol/(vakT) (nm−3)
Ent/(vakT) (nm−3)
Sol/Grav
Ent/Grav
x680 x685 x690 x695 x700 x705 x710 x715
−0.005 −0.011 −0.016 −0.022 −0.027 −0.032 −0.037 −0.042
−0.214 −0.426 −0.637 −0.846 −1.054 −1.261 −1.466 −1.670
0.050 0.098 0.146 0.193 0.239 0.284 0.328 0.372
39.4 39.4 39.4 39.4 39.5 39.5 39.5 39.5
−9.1 −9.1 −9.0 −9.0 −8.9 −8.9 −8.8 −8.8
assume that the GOR gradient is measured, thereby taking into account the main temperature effect.) The reference point has a pressure of 6000 psia and a depth of 3000 m. The compositional gradients with depth were estimated using the Peng−Robinson EoS with the Peneloux volume shift, as in the case for the condensate. Additional properties, including the GOR, density, solubility parameters, molar volume, and partial molar volume of methane, were calculated. The resulting GOR, density, molar volume, and solubility parameters are shown in Figure 2 as a
simulated black oil with various gas/oil ratios. In this example, when the GOR is small, there is not much variation in the GOR with depth, and, instead, gravity is the main cause of the gradient of the asphaltene distribution. As the GOR increases, its variation with depth increases, and the size of the solubility term increases. For the largest values of the GOR considered, it is again the dominant cause of the asphaltene gradient. For the two examples, we can compare the relative magnitudes of the gravity term, entropy term, and solubility term in eqs 2−4. Because the volume of the asphaltene particle va only occurs in these terms as an overall multiplicative factor, the ratios of their magnitudes do not depend on the molar volume of the asphaltene. Accordingly, we will compare the values of Grav/ vakT, Sol/vakT, and Ent/vakT. These expressions depend on the temperature and the asphaltene’s density and solubility parameter, which we have taken to be 135 °C, 1.2 g/cm3, and 21.85 MPa1/2, as in references 12 and 30. They also depend on the density, solubility parameter, and molar volume of the bulk oil. These values were obtained from the Peng−Robinson EoS2,31 with volume translation32 using the characterization procedure of references 33 and 34, based on the measured fluid composition and tuned to pressure/volume/temperature (PVT) properties. The results are described in detail in reference 23. In Table 1, we give the sizes of Grav/vakT, Sol/vakT, and Ent/ vakT and the ratios Sol/Grav and Ent/Grav as a function of depth for the condensate. A depth of x670 m is taken as the reference depth. As shown in the table, both the gravity and solubility terms are negative, while the entropy term is positive. Thus, the gravity and solubility terms increase the asphaltene concentration deeper in the well, while the entropy term decreases this compositional gradient, as expected. One salient feature of Table 1 is that the ratio between the gravity and solubility terms and the ratio between the gravity and entropy terms are all nearly constant over the entire range of depths. The reasons for this will be explored below in more detail. For the condensate, the gravity term is about 40 times smaller than the solubility term, whereas the entropy term is only about 4 times smaller than the solubility term, as noted in reference 12. Thus, for this example, the solubility effects are the major cause of the asphaltene gradient, and the entropy has a larger effect than the Boltzmann term. For the second example, the compositional gradient in a black oil was simulated as a function of depth for several different gas/ oil ratios. The composition at the reference height of the column was based on the measured composition of a surface oil and gas from a Gulf of Mexico black oil with a GOR of about 600 scf/stb. Six different recombined fluids were modeled with GORs of 0, 500, 1000, 1500, 2000, and 3000 scf/stb. The reservoirs were assumed to be isothermal with a temperature of 100 °C. (Temperature gradients slightly affect the GOR gradient, which will affect the value of the solubility parameters. However, we
Figure 2. Oil properties vs depth for various GORs. In plot a, the GOR vs depth is shown. In plot b, the density ρ(h) vs depth is shown. In plot c, the molar volume v(h) vs depth is shown, and in plot d, the solubility parameter δ(h) vs depth is shown. In all four plots, the oils with a GOR of 0, 500, 1000, 1500, 2000, and 3000 scf/stb at the reference depth of 3000 m are shown in blue, red, green, magenta, yellow, and black, respectively.
function of depth. The plots in this figure make it clear that as the GOR increases, the density, molar volume, and solubility parameter of the oil decrease. In addition, as the GOR increases, the gradients in all these parameters also increase. From these parameters, the gravity, solubility, and entropy terms can be calculated. Again, we take the asphaltene solubility parameter to be 21.85 MPa1/2. As in the first example, the ratio of the solubility term to the gravity term and the ratio of the entropy term to the gravity term are nearly constant as a function of height, but these ratios do depend on the GOR. In Figure 3, these ratios, along with the ratio (Sol + Ent)/Grav, are plotted as a function of GOR. In this example, when the GOR is small enough, the gravity term dominates, but once the GOR is above about 500 scf/stb, the solubility term becomes larger, and once the GOR is above 1000 scf/stb, the combined solubility and entropy terms are larger than the Boltzmann term. This is to be expected. D
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and replace the remaining h’s by their maximum value, hmax. Then the solubility term becomes Sol(h)/va = −Δhδ12(δa − δ ̅ )
(6)
where δ̅ = [δ(h0) + δ(hmax)]/2. Similarly, the gravity term becomes Grav(h)/va = −g[ρa − ρ ̅ ]Δh
(7)
where ρ̅ =[ρ(h0) + ρ(hmax)]/2. The entropy term becomes v Ent(h)/va = ΔhkT 12 [1 − (v1/v0)hmax /2] v0 (8) where, for this equation, we have replaced the second factor of Δh by its average value and taken h0 = 0. We note that in these expressions, the only dependence on the height h is in the Δh appearing in each term. Thus, if we take the ratio of the solubility and gravity terms, Sol/Grav, and the ratio of the entropy and gravity terms, Ent/Grav, these should be independent of height, as was found in the previous section. With the approximations given above, the ratio of the solubility term to the gravity term is given by
Figure 3. Ratio of energies vs GOR for the black oil example. At lower GORs, the entropy term (Ent/Grav) and the solubility term (Sol/Grav) will almost cancel each other because they have opposite signs and similar magnitudes.
As will be shown in the following section, the ratio Sol/Grav depends on the change in the solubility δ with depth. In this example, for low GORs, there is very little gradation of the methane and other components of the maltene. Thus, there should be negligible changes in the solubility of the asphaltene in the oil as a function of depth. This is confirmed in Figure 2a,d. As the GOR increases, the variation with depth of the methane content starts to increase, which should cause the solubility parameter to vary with depth. Again, this is what is seen in Figure 2a,d. This, in turn, should increase the magnitude of the solubility term, relative to the gravity term. Although the ratio Sol/Grav is above 4 at 3000 scf/stb, it is still significantly lower than the value of 40 in the previous example. This is to be expected, because the condensate in the previous example is near-critical. Thus, the gradient in its GOR should be much higher than for the black oil, even though the GORs in the two examples are similar.
Sol/Grav =
2δ1(δa − δ ̅ ) g (ρa − ρ ̅ )
(9)
Thus, this ratio depends only on the difference between the bulk oil and asphaltene solubility parameters, the difference between the oil and asphaltene densities, and the gradient of the oil solubility parameter. We note that the solubility parameters of stock tank oils at standard conditions have been found to depend linearly on the oil’s density:35,36 δ = aρ + b
(10)
where a and b are approximately constant for a wide range of oils. Even for live oils, this relation has been found to hold.29 For example, for the enlivened black oil considered here (at 100 °C and 12 000 psia), the solubility parameter again has a dependence on the density, which is close to linear, as shown in Figure 4. The fit to this line is given by eq 10, with a = 12.2
4. LINEARIZED THEORY AND APPROXIMATIONS FOR SOL/GRAV AND ENT In this section, we will look in more detail into the behavior of these ratios and derive some heuristics for when the different effects will play a dominant role. To begin, we note that in many cases, including the two examples considered above, the molar volume, solubility parameter, and density of the oil are well fit by a linear function in height over the range of heights under consideration. (This approximation is somewhat better for the black oil parameters shown in Figure 2 than for those of the nearcritical fluid.) To first order in Δh, they are given by v(h) = v0 + v1Δh Figure 4. Solubility parameter of the oil vs density of the oil for the black oil example. The linear fit is shown with the solid line.
δ(h) = δ0 + δ1Δh ρ(h) = ρ0 + ρ1Δh
(5)
MPa1/2 cm3/g and b = 6.37 MPa1/2. The values for the asphaltene solubility parameter δa = 21.85 MPa1/2 and density ρa = 1.2 g/cm3 that we are using here also lie close to this line. This is not too surprising, because as the oil becomes more dense, it typically becomes more aromatic and, perhaps, more asphaltene-like. If both the asphaltene parameters and the oil parameters lie close to the same line, then the ratio (δa − δ̅)/(ρa − ρ)̅ is very
where v0, δ0, and ρ0 are the values of the bulk oil volume, solubility parameter, and density at the initial height, h0, and Δh = h − h0. When the solubility, entropy, and gravity terms in eqs 2, 3, and 4 are expanded in Δh, the lowest-order term is linear in Δh. Because the next-order correction is too large to ignore altogether, we instead expand to second order, factor out the Δh, E
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nearly a constant. For the black oil, it is close to 14 MPa1/2 cm3/g and varies from 13.6 to 15.4 MPa1/2 cm3/g (less than 15%) over the entire range of depths and GORs considered here. More generally, this ratio is unlikely to vary by much more than 50% for a typical range of oils, pressures, and GORs, whereas δ1, the gradient in the solubility parameter of the oil, can vary by over an order of magnitude. Even in Figure 4, we can see that although the data points all lie close to the straight line, the slope of the segments of adjacent data points, which is related to δ1, varies significantly. Thus, the main cause for the change in the ratio Sol/Grav as the GOR is varied is the change in the slope of the oil’s solubility parameter. In the following sections, we will investigate in more detail the effect of the gradient in the oil’s solubility on the different ratios of energy terms. Heuristics for the Solubility Term. In this section, we will relate the slope in the oil’s solubility parameter, δ, to more readily measured parameters. To highlight the main effects, we will use a simple model for the oil. To begin with, we will treat the maltene as consisting of two components, the methane and the rest of the oil. We will refer to this second part of the maltene, plus the asphaltenes, as the dead oil, even though it can contain ethane, etc. We single out methane as one of the two components because the variation in its concentration can have a large effect on asphaltene solubility. In addition, it is readily measured by formation evaluation tools. We can then approximate the solubility parameter of the oil as δ = ϕC δC1 + (1 − ϕC )δdo 1
1
Figure 5. Ratio Sol/Grav vs the gradient of the methane volume fraction (in percent) for the black oil example. The data are plotted with circles, and the various approximations are plotted with solid lines. From right to left, the data points are at a GOR of 0, 500, 1000, 1500, 2000, and 3000 scf/stb. The ratio Sol/Grav is equal to the ratio of the average gradients ϕ̅ ′S /ϕ̅ ′G , as described later in the text.
In the plot, we have taken (δa − δ̅)/(ρa − ρ̅) and δC1 − δdo to be constants, equal to their average values of 14 MPa1/2 cm3/g and 8.5 MPa1/2, respectively. For each GOR, we used a single, average value of ϕC1. For all but the highest GOR, this is a reasonable approximation, but at a GOR of 3000 scf/stb, the methane volume fraction ϕC1 varies from 43.5% to 38.9% over the range of heights considered. Thus, at this GOR, taking one constant value is not such a good approximation, which is a large reason why the blue line does not agree as well with the data point at the largest GOR. Lastly, at each GOR, the parameters ϕC1 and δC1 are all quite linear in h, so, at each GOR, a single value could be used for each of their slopes, while, for all GORs, the gradient of δdo with zero dissolved gas was used. As can be seen in Figure 5, the fit to the data points is excellent, apart from the point at the highest GOR, as mentioned above. The red line is the plot of the approximation to eq 13 where only the term with the gradient in the methane volume fraction is included:
(11)
where ϕC1, δC1, and δdo are the volume fraction of methane, the solubility parameter of methane, and the solubility parameter of the dead oil, respectively. (We note somewhat more complicated expressions have been used to more accurately calculate the solubility parameter of oils mixed with methane, but for determining the main causes of the gradient in the asphaltene concentration, this simple expression suffices.) Then, the slope in δ is given by δ1 =
∂ϕC ∂δC1 ∂δ ∂δ 1 = (δC1 − δdo) + ϕC + (1 − ϕC ) do 1 ∂h 1 ∂h ∂h ∂h
(12)
In general, we expect the gradient in the methane volume fraction with respect to height to be much larger than the gradients in the solubility parameters. In addition, the difference between the solubility parameter of the methane and dead oil should be greater than one, so we expect the first term in eq 12 to be the dominant effect. The difference in solubilities should not depend much on height or GOR, so the main parameter governing the gradient in the oil solubility parameter and, hence, the ratio Sol/Grav is the gradient in the methane volume fraction. The one exception to this is for gas/oil ratios near zero, where only the last term, proportional to the gradient in the dead oil solubility, is non-negligible. In Figure 5, we plot the ratio Sol/Grav for the black oil example as a function of ∂ϕC1/∂h. As can be seen in the figure, this ratio is very close to linear in the gradient of the methane volume fraction, except at the highest GOR. In the figure, the solid lines show the various approximations for the ratio. The blue line is a plot of the full equation Sol/Grav =
Sol/Grav =
2(δa − δ ̅ ) ∂ϕC1 (δC1 − δdo) g (ρa − ρ ̅ ) ∂h
(14)
Even though it is much simpler than the equation for the blue line, it still has the correct trend and just systematically underestimates the ratio Sol/Grav. It shows that the gradient in the methane volume fraction is the dominant contribution to the ratio Sol/Grav. The black line comes from also taking into account the gradient in the methane solubility parameter: Sol/Grav =
∂δC1 ⎞ 2(δa − δ ̅ ) ⎛ ∂ϕC1 ⎟ ⎜⎜ (δC1 − δdo) + ϕC 1 ∂h ⎟ g (ρa − ρ ̅ ) ⎝ ∂h ⎠
(15)
Finally, the green line comes from also taking into account the gradient in the dead oil solubility parameter: Sol/Grav =
∂δC1 ∂δ ⎞ 2(δa − δ ̅ ) ⎛ ∂ϕC1 ⎜⎜ + (1 − ϕC ) do ⎟⎟ (δC1 − δdo) + ϕC 1 ∂h 1 ∂h g (ρa − ρ ̅ ) ⎝ ∂h ⎠
∂δ ⎞ 2(δa − δ ̅ ) ⎛ ∂ϕC1 ⎜⎜ (δC1 − δdo) + (1 − ϕC ) do ⎟⎟ 1 ∂h g (ρa − ρ ̅ ) ⎝ ∂h ⎠
(16)
As can be seen in the figure, this last approximation works well for a low GOR, where the gradient in the dead oil solubility
(13) F
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becomes a significant factor, whereas the previous approximation works better for a high GOR, where the gradient in the methane solubility becomes more important. In summary, we have found that the ratio Sol/Grav can be approximated by Sol/Grav ≈ −A
∂ϕC
1
(17)
∂h
where, for the black oils considered here, eq 14 gives A ≈ 240 m, when ϕC1 is the volume fraction in percent and h is the depth in meters. A fit to the data in Figure 5 (or to the approximation given by eq 15) gives A ≈ 280 m, instead. Finally, the fit to the full expression, eq 13, gives Sol/Grav = A∂ϕC1/∂h + C, with A ≈ 270 and C ≈ 0.21. We note that the parameters A (and C) will vary much less with the type of oil, GOR, pressure, etc. than does the gradient in the volume fraction of methane, ∂ϕC1/∂h. Thus, the above equation can be used to get a rough estimate of Sol/Grav for a wide range of oils and conditions. Because the density of the oil is often measured and the asphaltene density only falls within a small range, it is usually straightforward to obtain the value of the gravity term, divided by the asphaltene molar volume. Using the methods given above, we can estimate the value of Sol/Grav. By combining these two values, we can then obtain an estimate for the solublity term, again divided by va. Entropy Term. Next, we consider the entropy term. In the spirit of looking only for the parameters with the most significant contribution to this term, we note that, to leading order Ent/(vaΔh) = kT
v1 v02
= −kT
∂ ⎛⎜ 1 ⎞⎟ |h = h ∂h ⎝ v ⎠ 0
Figure 6. Entropy term Ent/(vaΔhkT) vs the gradient of the methane volume fraction (in percent) for the black oil example. The data are plotted with circles, and the theoretical approximation is plotted with the solid line.
values for the different GORs are shown by the circles, and the approximation given by eq 21 is shown with the solid line. As can be seen in the figure, the approximation is reasonably good, but always underestimates the actual gradient. As in the case for the solubility term, the main contribution to the entropy term depends on the gradient of the methane volume fraction, except near zero GOR. For the entropy term, though, the corrections to this approximation involve gradients of one over the molar volumes of methane and the dead oil, which vary considerably with the GOR. When the GOR is close to zero, the gradient in the dead oil properties (the last term in eq 20) becomes important. Because of this term, the scaled entropy energy does not go to zero as the gradient of the methane fraction goes to zero, as shown in Figure 6. This contribution to the entropy term has the opposite sign to the part of the solubility term that also comes from the gradient of the dead oil properties. In many cases, these two terms tend to cancel each other. For example, in Figure 3, at zero GOR, the combined solubility and entropy terms sum very nearly to zero. This further ensures that at very low GORs, the asphaltene gradient is usually determined only by the gravity term. In summary, we find that the entropy term can be approximated by
(18)
We can express the molar volume of the oil in terms of the volume fraction of methane as follows: ϕC (1 − ϕC ) 1 1 1 = + v vC1 vdo
(19)
where vC1 and vdo are the molar volumes of the dissolved methane and the dead oil, respectively. In that case, the derivative of the reciprocal of the molar volume of the oil is given by ∂ϕC ⎡ 1 1 ⎤⎥ ∂ 1 ∂ 1 1 ⎢ − − ϕC = 1 ∂h ⎢⎣ vC1(h) vdo(h) ⎥⎦ ∂h vC1(h) ∂h v(h) − ϕdo
∂ 1 ∂h vdo(h)
∂ϕC Ent 1 ≈ −B vaΔhkT ∂h
(20)
Again, the gradient of the methane fraction will be much larger than the gradient of the inverse molar volumes of methane and the dead oil, so the first term will dominate. Thus, we can approximate the entropy term by Ent/(vaΔh) ≈ kT
∂ϕC
1
∂h
1 1 − vC1 vdo
(22)
where, for the black oils considered here, B ≈ 0.08 nm−3 when the volume fraction of methane is in percent and the molar volume of the asphaltene is in cubic nanometers. Again, although the molar volume, particularly of methane, can vary with the depth and type of oil, etc., these variations will generally be much less than the variation in the volume fraction of methane, so again eq 22 can be used to get a rough estimate of the entropy term for a range of oils and conditions. If we take the ratio of the entropy term and the gravity term, we then find that
h (21)
where vC1 and vdo are the molar volumes of the dissolved methane and the dead oil, respectively, and ⟨⟩h denotes an average over the heights under consideration. The right-hand side of the above equation is essentially the gradient of the entropy term. As we will show in the next section, Ent/(vaΔhkT) is very similar to the part of the gradient of log ϕa that is due only to the entropy term. In Figure 6, we plot this quantity versus the gradient in ϕC1. The
∂ϕC Ent kT 1 ≈B Grav g (ρa − ρ ̅ ) ∂h G
(23)
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for Grav/Δh, Sol/Δh, and Ent/Δh in the preceding sections, except that now these gradients depend on the depth h instead of just being an average over a range of depths. Thus, from the preceding sections, we know that the contribution from solubility effects will become important compared to the gravity term once the gradient of the methane volume fraction becomes large enough. We also know that the dominant contribution to (∂Sol/∂h)/(∂Grav/∂h) and (1/kT)∂Ent/∂h is proportional to the gradient of the methane volume fraction. If we average the gradients over the heights, then Sol/Grav is the same as ϕ ̅ ′E /ϕ ̅ ′G , and Ent/Grav is the same as ϕ ̅ ′S /ϕ ̅ ′G . We note that ϕ′S/ϕ′G is also equal to the ratio of the part of the gradient of the relative amout of asphaltene due to the solubility and gravity, and similarly for ϕ′E/ϕG′ . Then, Figure 3 shows the ratios of the average gradients, ϕ ̅ ′S /ϕ ̅ ′G and ϕ ̅ ′E /ϕ ̅ ′G as a function of GOR. Similarly, Figure 5 shows the ratio ϕ ̅ ′S /ϕ ̅ ′G as a function of the gradient of the methane volume fraction. This ratio is very well approximated by a linear function of ∂ϕC1/∂h, as shown in Figure 5. In Figure 7, we plot 1/(vaϕa(h)) times the gradient of ϕa as a function of height for the different gas/oil ratios. We also plot the
As in the case of the ratio of the solubility term to the gravity term, the asphaltene volume drops out of this expression. For the ratio of the entropy term to the gravity term, we need to know the density of the asphaltene and the oil (with the asphaltene included), in addition to the gradient of the methane content.
5. ASPHALTENE GRADIENT In this section, we will turn our attention to the asphaltene gradient. The asphaltene volume fraction at a height h is related to the asphaltene volume fraction at height h0 as follows: ϕa(h) = ϕa(h0)e−(Sol + Grav + Ent)/ kT
(24)
The gradient in the asphaltene volume fraction is then given by ∂ ϕ (h) = ϕ′G + ϕ′S + ϕ′E ∂h a
(25)
where
ϕ′G = − ϕ′S = −
ϕa(h) ∂Grav kT
∂h
(26)
ϕa(h) ∂Sol kT
∂h
(27)
and ϕ′E = −
ϕa(h) ∂Ent kT
∂h
(28)
We note that the gradient of the asphaltene volume fraction at height h, ∂ϕa(h)/∂h, is proportional to the volume fraction of asphaltene at this height, ϕa(h). Thus, the more asphaltene there is, the larger the gradient will be. Often, we are interested only in the relative amounts of asphaltene. For example, when asphaltene content is measured by optical density, it gives the relative amount of asphaltene. In that case, we are interested in the quantity ϕa(h)/ϕa(h0). If we take a derivative of this and then evaluate it at h0, we find ∂ ϕa(h) ∂h ϕa(h0)
=− h = h0
1 ⎛⎜ ∂Grav ∂Sol ∂Ent ⎞⎟ + + kT ⎝ ∂h ∂h ∂h ⎠
(29)
Figure 7. Scaled gradient of the asphaltene volume fraction vs depth at various GORs for the black oil example. The values of the full scaled gradient of the asphaltene volume fraction, 1/(vaϕa)dϕa/dh, are shown with the solid lines. The values of the scaled gradient that come only from the solubility term, 1/(vaϕa)ϕ′S, are shown with the dashed lines.
Thus, the derivative of the relative amount of asphaltene, evaluated at the reference depth, is exactly equal to (1/ϕa(h0))dϕa(h0)/dh. We note that this also equals the gradient of the log of the asphaltene volume fraction. The gradient of ϕa(h) is also proportional to the molar volume of the asphaltene, because Sol, Grav, and Ent are all proportional to va. Thus, the larger the molar volume of the asphaltene, the larger the gradients. Lastly, the gradient of ϕa(h) depends on the sum of the gradients of Grav, Sol, and Ent, which are given by ∂Grav = −gva(ρa − ρ(h)) ∂h
(30)
∂Sol = −2vaδ1(δa − δ(h)) ∂h
(31)
portion of (1/vaϕa(h)) ∂ϕa/∂h that comes from only the solubility term. In other words, we plot (−1/kTva) (∂Sol/∂h). As can be seen in the plot, these gradients are almost constant as a function of height, except at the highest GOR, where the height dependence is more noticeable. At low GORs, the contribution from the solubility term is much less than the entire gradient, which is due mainly to the Boltzmann term. As the GOR increases, the contribution from the solubility term increases relative to the total gradient, and, at the highest GORs in the figure, it actually exceeds the total gradient by a small amount. At these higher GORs, the contribution from the entropy term is no longer smaller than the gravity term. Because it has the opposite sign compared to the solubility and gravity terms, the total gradient is less than the gradient due only to the solubility term. In conclusion, when the gradient in the methane term becomes large, the solubility term will have a large effect on the asphaltene gradient, and the effect of the entropy term will also be enhanced.
and ∂Ent ∂ 1 = −kTva ∂h ∂h v(h)
(32)
Thus, the gradient in the relative amount of asphaltenes, (1/ϕa(h1))dϕa/dh, is the sum of −1/kT times these gradients. These gradients are very similar to the approximations we used H
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6. GRADIENT OF THE METHANE VOLUME FRACTION AND GOR As we have seen in the preceding sections, the parameter that is most important in determining the magnitude of the solubility and entropy terms, compared to the gravity term, is the gradient of the methane volume fraction. Once this is above about 0.005/m, the solubility term is larger than the gravity term, so it has a significant effect as long as the molar volume of the asphaltene is large enough. The entropy term is considerably smaller, but it, too, can be larger than the gravity term when ∂ϕC1/∂h is greater than about 0.015/m. The question then becomes, how does this gradient of the methane volume fraction depend on parameters of the oil that are typically used? To guide estimates of when the asphaltene will show significant gradients and when the solubility effect will be important, we make use of correlations between the GOR and methane mass fraction that are given in the literature. One that is derived in reference 37 is GOR =
AmC1 mdo − BmC1
with a and b defined in eq 37. For this equation, we have ignored any depth dependence of ρdo and ρC1. According to eq 38, the square root of (∂GOR/∂h)/(∂ϕC1/∂h) should be linear in the GOR, with a slope of b/√a and an intercept of a−1/2. In Figure 8, we plot this quantity versus the
scf/bbl (33) Figure 8. Square root of the ratio of the gradient of the GOR and the gradient of the asphaltene volume fraction vs GOR for the black oil example. The values calculated from the data are shown with circles and the linear fit is shown with a solid line.
where mC1 is the mass fraction of methane and mdo is the mass fraction of the dead oil. A and B are constants that are fairly independent of pressure, temperature, and type of oil. However, because this is a simple two-component model of the oil, they do depend on assumptions such as how much of the dead oil is in the gas phase and what the density of the dead oil is. In reference 37, they are found to be
A = 8930 B = 0.193
average value of the GOR, averaged over height. The fit to a straight line, shown with the solid line, is excellent. The slope and intercept are 2.25 × 10−3 and 5.31, which give b = 4.23 × 10−4(scf/stb)−1
(34)
AϕC ρC 1
1
ρdo − (ρdo + BρC )ϕC 1
1
(35)
To obtain this equation, we have used the fact that mC1M = ρC1ϕC1v, and similarly for mdo, where M is the molar mass of the bulk oil. We also used the fact that ϕC1 + ϕdo = 1. Equation 35 can be inverted to obtain the expression for the methane volume fraction as a function of GOR, with the result
ϕC = 1
aGOR 1 + bGOR
(36)
with a=A
ρC
1
ρdo
⎛ ⎞ 1 ⎜ ρdo ⎟ + B ⎟ A ⎜⎝ ρC ⎠ 1
b=
(37)
The derivative of ϕC1 with respect to h is then given by ∂ϕC
1
∂h
=
a ∂GOR 2 (1 + bGOR) ∂h
a = 0.0355(scf/stb)−1
(39)
If, instead, we use eq 37 for a and b, we obtain b = 3.65 × 10−4 (scf/stb)−1 and a = 0.0344 (scf/stb)−1, where we have used ρC1 = 0.29 g/cm3 and ρdo = 0.89 g/cm3, and where ϕC1 is measured in percent. The values for b found by fitting to the data in Figure 8 and calculated from eq 37 agree within 15%, and the values for a agree within 4%. This agreement is quite good, given the simplicity of the model. With these values of a and b, we can get an estimate for the gradient of the methane volume fraction as a function of the GOR and the gradient in the GOR. When the GOR is small compared to 1/b, the gradient in the methane fraction is just proportional to the gradient in the GOR. However, once the product of the GOR and b is not much less than one, the gradient in the methane volume fraction is reduced and also depends on the GOR. Once we have an expression for the gradient of the methane volume fraction as a function of the GOR and its gradient, we can estimate the quantities Sol/Grav, Ent/Grav, Ent/ΔhkTva, ϕ′S/ϕ′G, and ϕ′E/ϕ′G directly from the GOR. These quantities depend on the GOR and its derivative through the combination given in eq 38, where a and b are given by eq 39. In Figure 9, we plot the ratio ϕ ̅ ′S /ϕ ̅ ′G as a function of a/(1 + bGOR)2dGOR/ dh. The circles show the data for the black oil and the x shows the data for the condensate. The solid lines show the various approximations found in Section 4. The black line shows the simplest approximation, which depends only on the term containing the gradient of the methane volume fraction as in eq 14. The red line shows the approximation that contains both the gradient of the methane volume fraction and the gradient of the methane solubility parameter. The green line shows the full approximation, given by the linear fit to eq 13, which contains all
More involved approximations, involving the mass fractions of C2−C5 and CO2 also exist to improve on this correlation. If the GOR is given by eq 33, then it is related to the methane volume fraction and the densities of methane and the dead oil as follows: GOR =
and
(38) I
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based on only a few oil samples. Using more oil samples and more sophisticated relations could help improve the estimate of the asphaltene gradients based on the knowledge of the GOR and its gradient.
7. CONCLUSIONS In this paper, we looked at the relative magnitudes of the contribution to the asphaltene gradient due to gravity, solubility, and entropy, and we derived heuristic equations for estimating the magnitude of the asphaltene gradient. We found that, to a good approximation, the ratio of the solubility term to the gravity term depends linearly on the gradient in methane content, and the ratio of the entropy term to gravity term roughly depends linearly on the methane content and also on the density of the oil. The constants of proportionality in these relations depend only very weakly on the oil properties, whereas the variation in methane content and gradient can be quite large. Thus, the values for these constants that we found for the oils considered in this paper should work for a wide range of oils. We also showed how the gradient in methane content can be related to the GOR and its gradient. Thus, we have found a method for estimating asphaltene gradients and the magnitude of the different contributions from gravity, solubility, and entropy from the GOR and its variation with height. These heursitics for estimating the magnitude of the asphaltene gradients should help in identifying equilibrium distributions in oils with a variety of gas/oil ratios. Currently, in the field, the Flory−Huggins−Zuo equation with only the gravity term is used to determine whether oil columns are in equilibrium when the GOR is quite low. As we have seen in the paper, at low GORs, the solubility and entropy terms are small compared to the gravity term and partially cancel each other. This makes it possible to obtain a good approximation for the asphaltene gradients with just the gravity term. As the GOR becomes larger, the solubility term increases, and the entropy term no longer cancels it. For example, for the black oil considered in this paper, when the GOR is 600 scf/stb, the sum of the solubility and entropy terms is 65% of the gravity term. If only the gravity term is used to fit the data, this will give a 65% error in the volume of the asphaltene particle, which corresponds to an 18% error in the diameter. The precise values of the error will depend not only on the GOR, but also on the variation of GOR with height. In general, though, once the GOR becomes higher than about 600 scf/stb, the effects of solubility should be taken into account to properly model the gradients in asphaltene concentrations.
Figure 9. Ratio of the average gradients ϕ̅ ′S /ϕ̅ ′G vs the function of GOR and its gradient, a/(1 + bGOR)2dGOR/dh, with a and b given by eq 39.
three terms. The parameters for the approximations were all found using the values for the black oil. However, our assumptions that most of the parameters in eqs 13 and 14 and did not vary much as temperature, pressure, and type of oil was varied are borne out by the figure: the data point for the condensate is still well approximated by the theoretical lines, particularly the one that takes into account all three terms in the theoretical expression for ϕ′S/ϕ′G. This is remarkable because the ratio of gradients is about 10 times higher for the condensate than for the black oil with a similar GOR. Finally, in Figure 10, we show the value of the ratio of gradients ϕ ̅ ′S /ϕ ̅ ′G as a function of both GOR and ∂GOR/∂h. In the plots,
■
AUTHOR INFORMATION
Corresponding Author
*D. E. Freed. E-mail:
[email protected]. Figure 10. Ratio of the average gradients ϕ̅ ′S /ϕ̅ ′G vs GOR and its gradient.
Notes
The authors declare no competing financial interest.
■
we have used the approximation in eq 15 for Sol/Grav. These plots can be used to estimate the value of this ratio if the GOR and its gradient are known. If the molar volume of the asphaltene and the densities of the asphaltenes and the maltene are known, then the part of the gradient due to the Boltzmann term and the solubility term can be estimated. A similar plot can be made for the gradient of the entropy term, but its effect should always be considerably less than the solubility effect. In this section, we gave only a very simple model for the relation between ϕC1 and the GOR, and the numbers in eq 39 are
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