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Kerogen maturation data in the Uinta Basin, Utah, USA, constrain predictions of natural hydrocarbon seepage into the atmosphere

2014; Wiley; Volume: 119; Issue: 6 Linguagem: Inglês

10.1002/2013jd020148

2169-8996

Marc L. Mansfield,

Methane Hydrates and Related Phenomena

Journal of Geophysical Research: AtmospheresVolume 119, Issue 6 p. 3460-3475 Research ArticleFree Access Kerogen maturation data in the Uinta Basin, Utah, USA, constrain predictions of natural hydrocarbon seepage into the atmosphere Marc L. Mansfield, Corresponding Author Marc L. Mansfield Bingham Research Center, Utah State University, Vernal, Utah, USA Correspondence to: M. L. Mansfield, [email protected]Search for more papers by this author Marc L. Mansfield, Corresponding Author Marc L. Mansfield Bingham Research Center, Utah State University, Vernal, Utah, USA Correspondence to: M. L. Mansfield, [email protected]Search for more papers by this author First published: 12 February 2014 https://doi.org/10.1002/2013JD020148Citations: 4AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract Natural seepage of methane from the lithosphere to the atmosphere occurs in regions with large natural gas deposits. According to some authors, it accounts for roughly 5% of the global methane budget. I explore a new approach to estimate methane fluxes based on the maturation of kerogen, which is the hydrocarbon polymer present in petroleum source rocks and whose decomposition leads to the formation of oil and natural gas. The temporal change in the atomic H/C ratio of kerogen lets us estimate the total carbon mass released by it in the form of oil and natural gas. Then the time interval of active kerogen decomposition lets us estimate the average annual formation rate of oil and natural gas in any given petroleum system, which I demonstrate here using the Uinta Basin of eastern Utah as an example. Obviously, this is an upper bound to the average annual rate at which natural gas seeps into the atmosphere. After adjusting for biooxidation of natural gas, I conclude that the average annual seepage rate in the Uinta Basin is not greater than (3100 ± 900) tonne yr−1. This is (0.5 ± 0.15)% of the total flux of methane into the atmosphere over the Basin, as measured during aircraft flights. I speculate about the difference between the regional 0.5% and the global 5% estimates. Key Points H/C ratios of petroleum source rocks fix the total mass of generated petroleum Duration of active petroleum generation constrains rate of natural gas venting Natural seepage rate for Uinta Basin, Utah, less than 3000 tonne methane/year 1 Introduction Studies indicate that a substantial amount of fossil methane is reaching the atmosphere without being burned as fuel [Lowe et al., 1988; Wahlen et al., 1989; Lassey et al., 2007; Neef et al., 2010; Simpson et al., 2012; Peischl et al., 2013]. Furthermore, emissions measurements in two petroliferous basins in the western USA indicate that methane flux to the atmosphere is 4% and 9%, respectively, of total natural gas production [Pétron et al., 2012; Karion et al., 2013]. Because methane is a potent greenhouse gas, these studies have implications for global warming. The obvious explanation for all this fossil methane in the air is leakage from natural gas production and distribution. However, many authors have argued that spontaneous, nonanthropogenic venting of natural gas from the lithosphere to the atmosphere is also a contributing factor. (In the jargon of the petroleum industry, “production” refers to the extraction of fossil fuels from the crust. I follow that usage here. To avoid confusion, I will use terms such as “formation” and “generation,” but never “production,” to speak of the creation of hydrocarbon fluids in source rocks in the crust.) For example, Johnson and Roberts [2003] suggest that geologic formations in Colorado and Utah, USA, have been leaking natural gas since the early Tertiary. Furthermore, Etiope and Klusman [2010] have catalogued ground-to-air methane flux measurements from around the world. Oil production from one of the fields that they catalogued, the Rangely Anticline in western Colorado, began in 1902 in the vicinity of natural seeps [Spencer, 1995], and more recently, Klusman [2003a, 2003b] reported significant methane fluxes in the same region. Global extrapolations of regional measurements such as those catalogued by Etiope and Klusman [2010] lead to predictions for natural seepage in the neighborhood of tens of teragrams of methane per year (Tg yr−1) [Kvenvolden et al., 2001; Etiope and Klusman, 2002, 2010; Etiope, 2004, 2005, 2009, 2012; Kvenvolden and Rogers, 2005]. Here, I will take 30 Tg yr−1 as the consensus value representing these studies. However, these measurements represent only a few petroliferous regions around the globe, and global extrapolations may therefore be questionable. Furthermore, using statistical arguments to be published later, I will show that the regional estimates are very poorly constrained because natural methane seepage is temporally intermittent and spatially variable. For these reasons, better measurements and estimates of natural seepage are called for. The total annual methane budget is estimated to be in the hundreds of teragrams annually, with 600 Tg yr−1 being a consensus value [Cicerone and Oremland, 1988; Fung et al., 1991; Khalil and Rasmussen, 1994; Crutzen, 1995; Stern and Kaufmann, 1996; Hein et al., 1997; Lelieveld et al., 1998; White et al., 2007; Adushkin and Kudryavtsev, 2010; Kai et al., 2011]. If both sets of estimates are reasonable, then flux due to natural methane seepage is roughly 5% of the total budget. In this paper, I formulate an approach to regional estimates of natural seepage based on the chemical composition of oil and gas source rocks. Modern hydrogen-to-carbon atomic ratios in the rocks permit an estimate of the total amount of hydrocarbon fluids that have been generated over time. The geologic moment at which the rocks first began generating these fluids can also be estimated. The ratio of these two estimates gives an estimate of the average annual generation of hydrocarbon fluids by the source rocks, which is, presumably, at least an order-of-magnitude estimate of the current generation rate. (The mass-to-age ratio provides a natural scale for the generation rate at any given time, and the probability is high that the modern rate will be within an order of magnitude of this. A similar argument is used by Kvenvolden et al. [2001].) If we assume that the entire petroleum generating system has achieved steady state, with a balance between generation of fluids in the source rocks and loss of fluids to the atmosphere, then the current generation rate equals the rate of flux to the atmosphere. On the other hand, if steady state has not yet been achieved, and hydrocarbon fluids continue to accumulate in the crust, then the current generation rate provides an upper bound to the atmospheric flux rate. Either way, this line of argument helps us determine how much of the leakage could possibly be natural. As reported below, I predict that natural seepage from one petroliferous basin is probably less than (0.5 ± 0.15)% of the total methane flux from that same basin. If a comparable ratio applies to all petroliferous basins, then the 5% global estimates of natural seepage mentioned above are perhaps too high. However, more than anything, the discrepancy points out the difficulties associated with extrapolations from the regional to the global level, and the need for additional study. Below I comment on this discrepancy. Kerogen is the solid hydrocarbon polymer that forms from organic matter in sediments as these latter are transformed into sedimentary rocks. For example, it is the primary organic constituent of oil shale. During the process of basin formation, such rocks are eventually buried deeply enough that they reach temperatures at which the kerogen begins to decompose thermally, generating both oil and natural gas [Tissot and Welte, 1984; Allen and Allen, 2005]. Another important degradation pathway is methanogenic biodegradation, which occurs in many petroleum accumulations [Milkov, 2011]. As kerogen decomposes, its chemical composition changes. Indeed, the H/C atomic ratio is one of several practical indicators of kerogen maturity. Over time, kerogen becomes richer in carbon because of the correlation between molecular size and H/C ratio in hydrocarbons: The decomposition products, being lighter, carry away excess hydrogen [Vandenbroucke and Largeau, 2007]. Therefore, within the kerogen itself is a record of the amount of carbon that has been released over time through decomposition. I present an analysis of the kerogen found beneath the Uinta Basin of Duchesne and Uintah Counties, Utah, USA; see Figure 1. (“Uinta” and “Uintah” are common alternative spellings.) This is a petroliferous basin covering approximately 2.3 × 104 km2, bounded to the north and west by the Uinta and Wasatch Mountain Ranges, to the south by the San Rafael Swell and the Uncompahgre Uplift, and to the east by the Douglas Creek Arch. Natural gas and oil source rocks beneath the basin are the Green River and related formations, which formed in lower to middle Tertiary; the Mesaverde Group, which formed during the Upper Cretaceous; and the Mancos Shale, also Upper Cretaceous [Tissot et al., 1978; Anders et al., 1992; Fouch et al., 1992; Nuccio and Roberts, 2003]. Figure 1Open in figure viewerPowerPoint The Uinta Basin of eastern Utah. (a) Regional map, showing the 0.1° × 0.1° integration grid (red), which coincides with the footprint of the basin itself. (b) Location of the basin in the western USA. There is abundant empirical evidence that anthropogenic activities have modified the natural flow of hydrocarbons to the surface. In the early years of the petroleum era, wells were frequently drilled in the vicinity of obvious surface seeps, and production from such wells almost always led to the cessation of seepage. In general, it appears that fossil fuel production relieves formation pressures, which slows or suspends seepage. On the other hand, repressurization of reservoirs by fluid injection is a very common practice, for at least four different purposes: (1) enhanced oil recovery, (2) waste disposal, (3) natural gas storage, and (4) CO2 sequestration. There are examples of repressurization stimulating leakage [Horvitz, 1985; Araktingi et al., 1984; Arp, 1992; Jones and Burtell, 1996; Tedesco, 1999; Bailey and Grubb, 2006]. In this paper, my aim is to constrain estimates of surface seepage of natural gas in the preindustrial era, before the perturbing effects of anthropogenic activities. Because of discrepancies between different literature data sets, or simply because of a lack of adequate information, it is occasionally necessary to choose between different options as I develop the estimates below. In the interest of arriving at a firm upper bound to the true emission flux, I intentionally resolve such dilemmas so as to maximize the flux estimate. However, I will always point out when such judgments are being made. The Uinta Basin is one of only two regions worldwide that are known to produce significant atmospheric concentrations of winter ozone [Schnell et al., 2009; Stoeckenius and Ma, 2010; Martin et al., 2011; Lyman and Shorthill, 2013]. Another purpose of this paper is to determine if natural seepage contributes significantly to ozone precursor emissions in the region. 2 Mechanisms for Natural Gas Seepage Pore pressures in the lithosphere (i.e., hydrostatic pressure of the fluids found in pores and fractures of the rock) generally obey (1)where Pp, g, ρW, zs, and z represent pore pressure, acceleration of gravity, density of water, elevation at the surface, and elevation at depth, respectively. (See Table 1 for a full list of symbols used in this work.) Equation 1 is the pressure-depth relationship expected for any body of water and indicates that any volume element in the water column supports the weight of all the water above. This is significant because it implies that the water in the lithosphere generally forms a continuous phase, in spite of the tortuosity of the pore-fracture-fault network in the crust. Table 1. List of Symbols Used in This Work Symbol Definition D Diffusivity dV = dx dy dz Volume element within the source rock f0 Initial H/C atomic ratio of immature kerogen fR H/C atomic ratio of the hydrocarbons released from kerogen during its maturation f Modern H/C atomic ratio of kerogen F Initial mass fraction of carbon in source rock; TOC when the rock was immature g Acceleration of gravity G Geothermal gradient (slope of temperature-depth relationship) MR Total mass of carbon released by kerogen by decomposition MT Total mass of the formation Pl Lithostatic pressure Pp Pore pressure r = (x, y, z) Displacement vector of a position in the formation R0 Vitrinite reflectance (an empirical measure of kerogen maturation) TOC Total organic carbon of a source rock; mass fraction of the organic carbon in the rock VT Total volume of a formation x, y, z Cartesian coordinates: east, north, up, respectively; z measured relative to sea level z1, z2 Elevation of the base and top of the formation, respectively zs Surface elevation α Fraction of carbon lost by kerogen during its maturation 〈α〉 Mean of α over an entire formation θ Celcius temperature ρW, ρR Densities of water and of the source rock, respectively τ Time before present at which kerogen decomposition began in a formation ΦF Rate of surface seepage of the carbon mass of hydrocarbons ΦR Rate of release of carbon mass from kerogen ΦS Rate of surface seepage of carbon mass (hydrocarbons plus CO2) Lithostatic pressure refers to the stress in the rocks themselves generated by the weight of the overburden. It obeys (2)where ρR represents the density of the rock. There are conditions under which pore pressure exceeds equation 1. Such formations are said to be overpressured [Neuzil, 1995]. However, pore pressure never exceeds lithostatic, because then the fluids are able to relieve stress by fracturing the rock. Natural gas in the lithosphere can exist in two different physical states, namely, either as a separate gas phase or dissolved in the dominant liquid phase, which is either oil or water. Two mechanisms have been identified that provide for the transport of natural gas in the lithosphere, related to these two physical states. The fact that the carrier phase of water is continuous provides one of the mechanisms. Natural gas trapped in a reservoir, at partial pressures of hundreds of bars, can dissolve in the water phase and outgas to the atmosphere, where its partial pressure is much less. If the water column is stationary, then transport of dissolved hydrocarbon down the ensuing concentration gradient is governed by Fick's Laws. If the water column circulates in the crust, then transport is even faster, with circulation promoting mixing. The time scale for Fickean transport over a distance Δx is order-of-magnitude equal to (Δx)2/D, where D is the diffusivity. D has strong temperature dependence, but in the relevant temperature range, D for methane in bulk water is around 0.1 km2 Myr−1, or (Δx)2/D ≈ 10 My if Δx = 1 km. D for methane in water-saturated solid rock can be from 1 to 3 orders of magnitude smaller, depending on the porosity and permeability of the rock [Witherspoon and Saraf, 1965; Schlömer and Krooss, 1997; Sachs, 1998]. We expect that the effective diffusivity for methane in a water-saturated fracture network lies between the value for bulk water and solid rock. These considerations imply a mega-year to mega-decade time frame for the process. However, we expect the total flux of dissolved gases to depend on the fracture state of the rock column and on water flow patterns, neither of which are well known and both of which vary widely from one petroleum system to another, making it difficult to better constrain the time interval. Nevertheless, it has been asserted that natural gas reservoirs will empty out by dissolution over mega-year time scales, unless they are actively replenished by generation of new gas [Leythaeuser et al., 1982; Krooss and Leythaeuser, 1996; Schlömer and Krooss, 1997]. The other mechanism for natural gas transport might be called “phase invasion,” in which natural gas invades the water column as an intact gas phase. Details about the form of the invading phase (“microbubbles” versus plug flow) remain controversial. One major difference between the dissolution and phase invasion mechanisms is the response to pressure. The pressure dependence of the dissolution mechanism is expected to follow Henry's law, i.e., to be linear in the partial pressure of methane in the reservoir. Phase invasion, on the other hand, is expected to be highly nonlinear: An intact second phase cannot leave a reservoir until its pressure is high enough to surmount a capillary barrier. The driving force for transport is gas buoyancy, and ascent times are believed to be on the order of 100 to 1000 m yr−1, much faster than the dissolution mechanism [Price, 1986; Klusman and Saaed, 1996; Klusman, 1997; Saunders et al., 1999; Tedesco, 1999; Brown, 2000]. However, if dissolved natural gas is diffusing away from underground reservoirs, we should ask where, and under what form, it reaches the atmosphere. I pose this question because in many regions, deep crustal water is saline, and if dissolved methane can diffuse away, then certainly electrolytes can also. In fact, because of higher solubility and diffusivity, their transport should be faster. If dissolved methane can reach the surface by this mechanism, then electrolytes should also. However, we all know that groundwater is fresh, not saline. We can resolve this issue by remembering that groundwater is meteoric, and it flows downslope in aquifers, which are nothing more than rock formations with sufficient porosity to permit such flow. When solutes from deeper formations, diffusing slowly upward, encounter the aquifers, we can expect that they become diluted in the faster aquifer flow and that they are carried away by that flow. Because transport of hydrocarbons in solution is inherently slow, and because meteoric water probably contains oxygen and methanotrophic bacteria, it is highly likely that dissolved hydrocarbons are oxidized to CO2 before they can reach the atmosphere. According to this argument, natural gas bubbling up as an intact gas phase stands a chance of getting through the surface layer of meteoric water, but dissolved natural gas may not. 3 Estimates of Total Carbon Released During Kerogen Decomposition Mass Balance Considerations Assume that a volume of kerogen, which originally contained NC and NH carbon and hydrogen atoms, respectively, is later found to contain NC1 and NH1 atoms, because NC2 and NH2 atoms of carbon and hydrogen, respectively, have been released through decomposition. Let f0 = NH/NC represent the initial H/C atomic ratio of the immature kerogen, fR = NH2/NC2 that of the hydrocarbons that have been released by decomposition, and f = NH1/NC1 that of the mature, remaining kerogen. (The instantaneous fR ratio depends on the age of the kerogen, since oil and gas are primarily released earlier and later, respectively. However, in the absence of additional data, we are forced to assume that a single fR value applies throughout the course of maturation.) The two conservation relations NC = NC1 + NC2 and NH = NH1 + NH2 lead to this relationship (3)for the fraction of carbon atoms that were released while the kerogen matured to its present state. Kerogen samples are classified as Types I, II, or III, depending on the ancient depositional environment from which they evolved, and the value of f0 depends on the kerogen type [Tissot et al., 1978; Vandenbroucke and Largeau, 2007]. Table 2 summarizes the type classification of Uinta Basin kerogen [Nuccio and Roberts, 2003], and the f0 values that were used in this study [Anders et al., 1992; Nuccio and Roberts, 2003]. During kerogen maturation, the value of f changes from the f0 values cited in Table 2 to 0.5 or lower [Vandenbroucke and Largeau, 2007]. Precise values of fR are not known, but we do know that for saturated, acyclic hydrocarbons, it is exactly 2 + 2/n, where n is the total number of carbon atoms in the molecule. Therefore, it is 4 for methane, 2.67 for propane, 2.25 for octane, and it approaches 2 for large, saturated acyclic hydrocarbons. We also know that both unsaturation and cyclization lead to smaller fR: For example, it is 1 and 0, respectively, for benzene and graphite. We also know that it must be greater than f0, since α in equation 3 cannot be greater than 1. Uinta Basin crude oil has a high paraffin content [Tissot and Welte, 1984, p. 420], which argues for a value nearer to 2, but the basin also produces considerable natural gas, which argues for a value nearer to 4. In this paper, I assume that the appropriate value lies between the extremes of 2 and 4. Table 2. Results of the Integrations, Equations 4, 6, and 7, for Each of the Formations Formation Kerogen Type f0 MT, 1018 g MR, 1018 g 〈α〉 τ, Myr ΦR, 104 tonne/yr Green River I 1.6 82 0.60 ± 0.20 10% to 20% 23 2.6 ± 0.9 Mesaverde III 0.9 33 0.08 ± 0.03 8% to 14% 40 0.20 ± 0.08 Mancos Mixed II and III 1.1 67 0.20 ± 0.07 10% to 20% (?) 60 0.33 ± 0.11 Total 182 0.88 ± 0.21 3.1 ± 0.9 Immature kerogen also contains some oxygen, most of which leaves during decomposition as CO2 [Tissot and Welte, 1984]. Because my analysis assigns all missing carbon to oil and natural gas, it probably slightly overestimates the amount of carbon leaving in the form of hydrocarbon. Consider a volume element within the source rock having volume dV = dx dy dz at the position r = (x,y,z), as shown in Figure 2. The x and y coordinates are horizontal, east and north, respectively, while z measures the altitude relative to sea level. The volume element has mass ρR dV, where ρR ≈ 2.4 g cm−3 = 2.4 Pg km−3 is the density of the rock. The mass fraction of carbon within such rocks is a ratio known as the total organic carbon, or TOC. TOC refers to carbon found not only in the insoluble kerogen but also in bitumen, the soluble organic component of the rock. Bitumen consists of lower hydrocarbons which have been separated from the kerogen by decomposition, but which have not yet been expelled from the source rock. Here, we let F(r) represent the total mass fraction of carbon in the rock when it still contained only immature kerogen, i.e., its initial TOC value. With these definitions, the total mass of carbon found initially in the kerogen in the volume element dV is ρRF(r) dV, while the net mass of carbon to have been released from the kerogen over time is ρRF(r)α(r) dV. We let VT represent the total volume of the formation, and we integrate ρRF(r)α(r) dV over the entire volume to determine the net mass of released carbon, MR. The function α appears with r dependence since, as already pointed out, deeper rocks have spent more time in the temperature window for thermal decomposition and therefore have released more carbon. F also appears as a function of r because it can vary considerably from one point to another in a formation [Law, 1984; Creany and Passey, 1993]. Technically speaking, ρR is also a function of r, but its variability is low, and so is treated as a constant. We obtain (4) Figure 2Open in figure viewerPowerPoint Each grid cell consists of a column extending from the base of the formation, z = z1, to the top, z = z2. An element of volume, dx dy dz, is also shown. Its total mass, the initial mass of carbon, and the mass of carbon lost from its kerogen as a result of decomposition are also displayed. In the above, (5)is our notation for integrating an arbitrary function g(r) over the volume of the formation. In this notation, the total mass of the formation may be written as (6) As a measure of the overall maturity of the source-rock formation, we also calculate the mean of α over the entire volume: (7) The integrals shown in equations 4, 6, and 7 were performed numerically with a resolution of 0.1° longitude by 0.1° latitude in the x and y variables, respectively, which defines the grid system shown in Figure 1. Each individual grid cell consists of a rectangular column extending from z = z1 to z = z2, as shown in Figure 2, representing respectively the altitudes at the base and the top of the formation. For each grid cell, we also define z = zs as the altitude of the surface. The values of z1, z2, and zs are assumed to be constant throughout an individual cell and are assigned as explained below. Obviously, in each cell we have zs ≥ z2 ≥ z1. The integrals along z within each grid cell extend from z1 to z2 and were performed by trapezoidal rule integration. Within any one grid cell, α(r) is assumed to be a function of z and zs, as explained below. At the latitude of the Uinta Basin, the cross-sectional area, A, of each grid cell is 94.7 km2, and 232 grid cells are required to cover the basin. The literature contains other calculations of the total hydrocarbon mass released by kerogen in other petroleum systems [Cooles et al., 1985; Pepper and Corvi, 1995a, 1995b; Pepper and Dodd, 1995]. As far as I know, the current contribution is novel in its use of carbon and hydrogen mass balances to arrive at estimates of MR. Contribution From the Green River and Related Tertiary Formations Because of differences in the way that maturation data have been reported in the literature, it is convenient to consider each formation independently. We begin with the Green River and related Tertiary formations [Tissot et al., 1978; Anders et al., 1992; Fouch et al., 1992; Nuccio and Roberts, 2003], and results for all three formations are summarized in Table 2. Values of z1 and z2 in each grid cell were assigned based on data from Fouch et al. [1992, Figure 14] and Anders et al. [1992, Figure 2]. Values of zs were assigned using elevation data from Google Earth. Kerogen maturation data for the Green River Formation beneath the Uinta Basin have been obtained from core samples and are reported by Anders et al. [1992]. However, much of these are actually in the form of another maturation indicator, the so-called vitrinite reflectance index, or R0 [Vandenbroucke and Largeau, 2007]. Fortunately, Anders et al. [1992, Figure 6] also provide the following correlation between R0 and f: (8)Equation 8 appears plotted in Figure 3. Figure 3Open in figure viewerPowerPoint Empirical correlations between H/C atomic ratio and vitrinite reflectance employed in this work for kerogen Types I and III. The curves correspond to equations 8 and 13, respectively. Kerogen maturation varies with the depth, zs − z, of the sample. I have been unable to find a direct relationship between R0 and depth, but the paper of Anders et al. [1992] provides enough information to formulate one. They identify four stages in the kerogen aging process: onset of oil production, peak in oil production, end of intense catagenesis, and end of liquid hydrocarbon preservation; they associate both temperature (95°C, 110°C, 135°C, and 150°C) and vitrinite reflectance (0.7, 0.9, 1.2 and 1.4%) with each stage. These four data points all lie essentially on the line (9)where θ represents the Celcius temperature. We will use equation 9 to relate R0 to θ. Furthermore, over relevant depths within the upper crust, it is appropriate to assume linearity between depth and temperature: (10)where zs − z represents depth below the surface. G is the so-called geothermal gradient, which varies from about 11 to 38 °C/km between the north and the south of the basin. G was taken to be constant within each grid cell, and the values for each grid cell are obtained from Anders et al. [1992, Figure 9]. Using equations 8–10, we are able to estimate f(z), i.e., the H/C atomic ratio as a function of altitude in each grid cell, which then gives α(z) according to equation 3. However, a slight modification is still required. The above formulas have been developed for the window of active decomposition and are inappropriate at shallower depths. The symptoms of the problem are either a negative R0 or a negative α, both of which are physically meaningless. To avoid this problem, we assume that α is exactly zero at any of the shallow depths for which the above formulas produce either the condition R0 < 0 or α < 0. A significant source of uncertainty in these calculations is the value of F(r). As mentioned above, F is equal to the initial value of total organic carbon (TOC), expressed as a mass fraction. TOC values fluctuate widely with depth in a single core sample and from core to core in a given formation [Creany and Passey, 1993]. Therefore, an appropriate F(r) is difficult to ascertain, and a common practice in basin modeling is to apply a constant, coarse-grained average value throughout a formation, but because of high variability in the measurements, even that is difficult to determine accurately [McPherson, 1996]. Furthermore, we are interested in an initial, ancient value, before the sample lost any carbon to decomposition. Also, modern TOC numbers include contributions from both kerogen and bitumen in the rock, whereas H/C or vitrinite reflectance data apply only to kerogen. This fact would complicate any effort to project an initial TOC based on the differen