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Steam fingering at the edge of a steam chamber in a heavy oil reservoir.

INTRODUCTION

An increasing number of heavy oil and bitumen reservoirs, not amenable to surface mining, are being recovered by the steam-assisted gravity drainage (SAGD) process; displayed schematically in vertical-section in Figure 1. In this process, steam injected into the top wellbore enters the reservoir and collects in a steam depletion chamber in the region above and surrounding the wellbores. The injected steam flows convectively to the outer edges of the chamber and releases its latent heat to the cool oil sand at the edges warming the oil, rock, and connate water. The bitumen, now heated, becomes more mobile because its viscosity drops by several orders of magnitude. It then flows under gravity to the lower wellbore and is produced to the surface.

The performance of SAGD process is often measured by the volumetric cumulative steam, expressed in cold water equivalent (CWE), to oil ratio (cSOR). Typically, in field operations the cSOR ranges from 2 to 5 [m.sup.3] CWE injected steam per [m.sup.3] oil produced (Aherne, 2006; Gates and Chakrabarty, 2006; Gupta and Gittins, 2006). The larger the cSOR, the greater the amount of injected steam, in other words energy, required per unit volume bitumen produced. In the first stage of SAGD before the chamber has reached the cap rock, all of the latent heat released by steam ends up heating oil sand at the edges of the chamber. In this stage, the process is thermally very efficient. After the chamber reaches the top of the reservoir, a significant fraction of the latent heat is delivered to the cap rock and therefore the thermal efficiency of the process drops. To better control the operation of SAGD, it is important to monitor the performance of SAGD as it evolves in the reservoir. In some SAGD projects, the steam chamber growth is estimated from temperatures obtained from thermocouples located at several depths within observation wells. The steam chamber vertical growth or rise rate primarily depends on porosity, permeability, oil saturation, steam properties, bitumen viscosity behaviour with respect to temperature, and relative permeability (Butler, 1997). Table 1 summarizes available rise rate data for several SAGD projects and oil sand reservoirs. The data show that the rise rate depends on the oil viscosity which in turns depends on the steam temperature. In Athabasca reservoirs, the data reveals that the rise rate can be as high as 20 cm/day.

[FIGURE 1 OMITTED]

The steam chamber grows by continuously stripping off layers of bitumen from the edges of the steam chamber. As the steam-bitumen interface moves into the oil sand, heat is conducted from the hot steam chamber to the cooler oil sand beyond. At the edges of the chamber, it has been suggested that there are steam fingers, shown in exaggerated view in Figure 1, which penetrate the oil sand ahead of the chamber (Butler, 1987, 1994).

Ito (1984) postulated that change of the formation due to geomechanical effects provide a significant contribution to the generation of steam fingers. In the underground test facility (UTF) Phase B, the analysis of thermocouple data by Ito et al. (2001a) revealed that the three separate steam chambers that evolved from the three wellpairs were joined at the top of the reservoir but were prevented from further vertical growth by the Inclined Heterolithic Stratification unit that sits above the reservoir. This indicates that if steam fingers control the vertical growth of the steam chambers, the existence of fingers are prevented by impermeable layers associated with IHS. From analysis of results from Ito and Hirata (1999), Ito and Suzuki (1999), and Ito et al. (2001b) on the Hangingstone SAGD project, the length of steam fingers that extend into the oil sand are predicted to be about 2 m. This estimate was determined from temperature measurements recorded at many locations in several observation wells placed among the SAGD wellpairs. Recently, Ito and Ipek (2005) examined SAGD steam chamber growth and steam fingers by analyzing field data from four SAGD projects: UTF Phases A and B, Hangingstone, and Surmount. On average, the Surmount project was operated at 3500 kPa whereas the UTF A and B phases and Hangingstone SAGD projects were operated at on average 2450, 2600, and 5300 kPa, respectively. Ito and Ipek simulated and analyzed the field observations from the UTF Phase-A SAGD wellpairs by using a thermal reservoir simulator. They found that a sand deformation model (Ito, 1984), a model to represent dilation of oil sand at the edges of the steam chamber, was required to history-match field data.

Ito suggested that the sand deformation model represented the behaviour of steam fingers in the oil sand. However, the sand deformation model, as implemented by Ito (1984), simply adjusts the value of the porosity as a function of pressure, that is, within a particular pressure range, the higher the pressure, the larger the porosity. That a pressure-induced porosity enhancement was required to history-match the process suggests that dilation, not necessarily steam fingers, is a key component of chamber growth. From their calculations, Ito and Ipek (2005) determined that the steam chamber growth rate is proportional to the permeability of the formation and that solution gas has a relatively small impact on the growth of the chamber.

Ito and Ipek (2005) found that steam chamber growth depends on the historical evolution of the operating pressure and that the higher the operating pressure, the faster the chamber growth. This is expected for two reasons: First, higher operating pressure implies higher steam saturation temperature and thus greater bitumen mobility (both due to lower viscosity and geomechanically enhanced permeability). Second, higher pressure means larger porosity which in turn implies enhanced permeability. Thus, the overall mobility of the oil phase was raised with higher operating pressure.

Ito and Ipek (2005) also remarked that as the operating pressure dynamically dropped during injection, the top of the steam chamber, estimated from thermocouple data, descended within the formation. Given that temperature is the actual data measured and not oil saturation, the data does not suggest that the top of the steam chamber descended but rather that the temperature contour dropped as the operating pressure was lowered. In earlier analysis of the Surmount project, Ito and Singhal (1999) concluded that the steam chamber growth slowed after the operating pressure dropped. The reduction of the chamber growth rate is associated with the higher oil viscosity associated with the reduction of the saturation temperature and lowered oil mobility due to geomechanically reduced permeability that accompanies lowering of the chamber operating pressure.

Ito and Ipek (2005) claimed that thermocouple data from an observation well in the Hangingstone pilot suggests that the length of the fingers is roughly 2 m. Given that the thermal diffusivity of Athabasca oil sand is equal to about 0.06 [m.sup.2] /day, it is expected that the conductive heating length ahead of the expanding steam chamber is of the order of 1-3 m. This suggests that it remains unclear whether the thermocouple data reflect that steam fingers are 2 m long or rather that this is simply conductive heating.

Butler (1987) developed an analytical model to predict steam finger rise rate and finger length and reported rise rates that range from 0.4 cm/day for Athabasca bitumen at 100[degrees]C to 19 cm/day for Lloydminster crude at 300[degrees]C. In his study, Butler (1987) derived the following equations to predict the maximum rise rate and dimensions of steam fingers (symbols defined in nomenclature):

V = [kk.sub.ro]g/(m+1)[v.sub.os] [phi][DELTA][S.sub.o] {(1 - [X.sub.i])[X.sub.i]/(1 - A) [X.sup.2.sub.i] + (A - B) [X.sub.i] + B} (1a)

where A and B are defined by:

A = [k.sub.ro][v.sub.g])[T.sub.s] - [T.sub.r])/ [k.sub.rg][v.sub.os][lambda][phi][DELTA][S.sub.o][[rho].sub.o] {[[rho].sub.c[C.sub.c] + [[rho].sub.o[C.sub.o][phi][DELTA] [S.sub.o] (m + 1/m + 2) - [[rho].sub.r[C.sub.r]/2}

and

B = [k.sub.ro][v.sub.g])[T.sub.s]/ [k.sub.rg][v.sub.os] [lambda][phi][DELTA][S.sub.o][[rho].sub.o] {[[rho].sub.r[C.sub.r]/2}.

The maximum rise rate is calculated from Equation (la) with [X.sub.i] given by:

[X.sup.max.sub..sub.i] = [square root of B(1 +2B) - B/1 + B] (1b)

which is not at the centreline of the finger. In other words, the maximum rise rate of the finger interface does not occur at the centreline of the finger. Indeed, Butler's theory calculates the rise rate at the centre of the finger to be equal to zero. This particular result of Butler's theory appears to be counter-intuitive and is inconsistent with experimental data which shows that this is not the case (Sasaki et al., 2002).

Butler approximated the shape of the rising finger by a parabolic function and calculated finger dimensions of 3.2 to 6.3 m length for different relative permeability sets, [k.sub.ro] = 0.5, [k.sub.rg] = 0.4 and [k.sub.ro] = 0.25, [k.sub.rg] = 0.2 respectively, in Athabasca bitumen reservoirs. He estimated the characteristic dimension of the steam finger, [f.sub.D], by:

[f.sub.D] = [pi][alpha]/V {[kk.sub.ro]g/[THETA][DELTA][S.sub.o] V(m + 1)[v.sub.os] - [R.sup.']/m + 1 [k.sub.ro][v.sub.g]/ [k.sub.rg][v.sub.os] (2)

where [R.sup.'] is the mass-based steam-to-oil ratio where the shape of the finger is given by y = [x.sup.2]/[f.sub.D]. He reported his results in rise rate curves for heavy oil and bitumen reservoirs of permeability equal to 1D and oils having viscosities of Athabasca (300 cSt. at 100[degrees]C) and Cold Lake (100 cSt. at 100[degrees]C) bitumen and Lloydminster (30 cSt. at 100[degrees]C) heavy oil. The characteristic dimensions of the parabolic-shaped steam fingers, predicted by Equation (2), at this temperature are between 2.1 and 3.2 m. Estimates for the rise rate calculated from Butler's (1987) theory are much smaller than that obtained from field thermocouple data. In this study, a corrected analytical model has been derived to estimate the rise rate of steam fingers and steam chamber in terms of steam properties, bitumen thermal and flow properties, and formation thermal properties. The rise rates calculated from the new theory is compared to those obtained from field SAGD thermocouple measurements and an SAGD physical model experiment. An analysis of conductive heating versus convective steam fingers will also be examined.

THEORY

Figure 2 displays a cross-section view of a steam finger. Below the finger is the steam chamber which is rising through the reservoir as oil drains from its edges and is removed from the reservoir by the production wellbore. As described by Butler (1987) and Ito and Ipek (2005), steam fingers grow ahead of the overall bulk of the expanding chamber. Steam flows upward through the lower boundary of the finger. The latent heat of the steam provides heat to the interface zone between the steam finger and cool oil sand. Heated bitumen and condensate (condensed steam) flow countercurrent to steam along the inner surface of the fingers towards the lower boundary of the finger. The rate of oil drainage depends on the oil viscosity which in turn depends on steam temperature. Consequently, the higher the steam temperature, the greater the steam finger rise rate. The rise rate thus depends on heat delivered to the bitumen which is measured by the local finger volumetric steam-to-oil ratio (fSOR). The fSOR is the volume of steam required in the finger per unit volume oil drained from the finger. The fSOR depends on steam temperature, oil viscosity dependence on temperature, initial and residual oil saturations, and oil sands thermal properties (e.g., thermal conductivity and heat capacity). At the edge of the finger, the heat delivered to the oil sand is conductively transferred to cooler oil sand. Due to buoyancy effects, the steam tends to rise whereas the condensate and mobilized bitumen falls under gravity. Figure 3 displays a steam finger with the notation used here. The steam finger is inclined at angle [THETA] to the horizontal and there is a plane of symmetry located at x = [x.sub.1] .

[FIGURE 2 OMITTED]

Steam Finger Rise Rate

In Figure 3, at Point P, in the mobile layer of oil adjacent to the steam finger, the volumetric flow rate of oil is given by Darcy's law:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where k is the absolute permeability, [k.sub.ro] is the relative permeability with respect to oil, g is the acceleration due to gravity, [[rho].sub.o] is the oil density, and [v.sub.o] is the oil kinematic viscosity. The volumetric flow rate of steam within the finger upward across the horizontal plane through Point P is given by:

[Q.sub.g] = ([kk.sub.rg]x/[[micro].sub.g]) {[partial derivative]P/[partial derivative]y - [[rho].sub.g]g} (4)

where [k.sub.rg] is the gas phase relative permeability, [[rho].sub.g] is the gas phase density, and [[micro].sub.g] is the gas (steam) phase viscosity. After re-arranging Equation (4) to obtain an expression for the pressure gradient and then substituting the result into Equation (3), the oil flow rate is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

By defining the volumetric finger steam-to-oil ratio, fSOR = [Q.sub.g]/[Q.sub.o], Equation (5) becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[FIGURE 3 OMITTED]

In Equation (6), if the oil viscosity dependence on temperature is known and because the temperature distribution through the mobile layer is known versus distance, then the integral term in Equation (6) can be evaluated. Within a limited range, the dependence of oil phase kinematic viscosity on temperature can be represented by (Butler, 1985):

1/[v.sub.o] = 1/[v.sub.os] [[T - [T.sub.r]/ [T.sub.s] - [T.sub.r]].sup.m]. (7)

where in is a dimensionless parameter that depends on the nature of the bitumen, [T.sub.r] is the initial reservoir temperature, and [v.sub.os] is the oil kinematic viscosity at steam temperature [T.sub.s]. The temperature profile into the oil sand can be approximated by the first term of the solution to the conduction heat equation into a semi-infinite slab (Carslaw and Jaeger, 1959; Closmann and Smith, 1983):

T - [T.sub.r]/[T.sub.s] - [T.sub.r] = 1/2erfc [xi]/ 2[square root of [alpha]t] for [xi] [greater than or equal to [square root of [pi][alpha]t] (8)

where [alpha] is the thermal diffusivity of the oil sand and t is the time. The error function can be approximated by a Maclaurin series expansion (Eric, 2006). Butler (1987) only used the first term of the error function expansion which leads to a linear profile of temperature with position which for conductive heat transfer into an infinite slab is not possible. In this study, seven terms were used to ensure convergence of the series. Thus, the integral term of the Equation (6) becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 9)

which yields on integration:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a simple function of the viscosity parameter, m. The relationship between [A.sub.m1] and in has been plotted in Figure 4a revealing that m has strong effect on [A.sub.m1]. After inserting Equation (10) into Equation (6), the oil flow rate at point P at distance [x.sub.i] on the interface is given by:

[Q.sub.io] = [kk.sub.ro] sin [THETA] {(1 - [[rho].sub.g]/[[rho].sub.o]) g - [Q.sub.io][[micro].sub.g]/[kk.sub.rg][x.sub.i][[rho].sub.o]fSOR} [square root of [[pi][alpha]t]/[v.sub.os] [A.sub.m1]. (11)

From geometry, as shown in Figure 3, we have:

[[xi].sub.1] = [square root of [pi][alpha]t] = [x.sub.1 - [x.sub.i]/ sin[THETA] (12)

which when substituted into Equation (11) yields:

[Q.sub.io] = [kk.sub.ro] {(1 - [[rho].sub.g]/ [[rho].sub.o]) g -[Q.sub.io][[micro].sub.g]/ [kk.sub.rg][x.sub.i][[rho].sub.o] fSOR} [x.sub.1] - [x].sub.]/[v.sub.i] [A.sub.m1] (13)

[FIGURE 4 OMITTED]

From mass conservation, this oil flow rate must be equal to overall oil rate which is being displaced by steam finger. The oil flow rate at distance [x.sub.i] on the interface is given by:

[Q.sub.io] = [phi][DELTA][S.sub.o][x.sub.i]u (14)

where [phi] is the porosity, [DELTA][S.sub.o], is the difference between the initial and residual oil saturations, and n is the rise rate of the steam finger. By substituting Equation (14) into Equation (13) and rearranging the result, the rise rate is given by:

u = [A.sub.m1] k [k.sub.ro]/[[mu].sub.o][kg.sub.r]/ [[mu].sub.g] ([[rho].sub.o] - [[rho].sub.g]) g 1/[DELTA][S.sub.o][phi] 1 - [X.sub.i]/[X.sub.i] + [A.sub.m1]fSOR([k.sub.ro]/[mu.sub.o])(1 - [X.sub.i]) (15)

where [X.sub.i] = [x.sub.i]/[x.sub.1] is a dimensionless interface location and [k.sub.ro]/[[mu].sub.o] and [k.sub.rg]/[[mu].sub.g] are the oil and steam phase mobilities. The derivation presented here is similar to that done by Butler (1987) yet Equation (15) presents a result that has a fundamental and important difference with Butler's theory, Equation (1a). In the derivation done here the maximum rise rate occurs at the centreline of the finger at [X.sub.i] = 0 and drops to zero at the edge of the finger. In Butler's theory, the maximum rise rate is not at the centreline, rather it is equal to zero there.

Equation (15) shows the effect of gravity potential difference between the two phases and the integrated viscosity profile represented by [A.sub.m1] in the mobilized oil layer. Additionally, the rise rate is proportional to the oil and gas phase mobility not simply the reservoir permeability. At the symmetry plane at [X.sub.i] = 0, the rise rate is given by:

u|[X.sub.i]=o = k[k.sub.rg]/[[mu.sub.g] ([[rho].sub.o] - [[rho].sub.g])g 1/[DELTA][S.sub.o][phi]fSOR. (16)

which reveals that the lower the fSOR (which means the better the thermal efficiency), the higher the finger rise rate at the symmetry plane.

Heat Balance around a Steam Finger

Equations (15) and (16) reveal that the rise rate depends on the volume-based finger steam-to-oil ratio, fSOR. There are two heat sinks that absorb heat from steam that enters the finger. First, heat is absorbed by rock, steam, and residual oil within the steam finger, [H.sub.f]. Second, energy is conducted into the layer of oil and condensate at the edges of the finger, [H.sub.o]. The total energy contained in the steam finger is given by:

[H.sub.f] = [[rho].sub.c][c.sub.c]u([T.sub.s] - [T.sub.r]) [x.sub.i] (17)

where [[rho].sub.c] and [c.sub.c] are the density and heat capacity of the material in the finger, respectively. The heat content of condensate and mobilized bitumen that leaves the steam finger is given by:

[H.sub.o] = u[phi][DELTA][S.sub.o]([T.sub.m] - [T.sub.r]) [x.sub.i][[rho].sub.o][c.sub.o] + [[rho].sub.w][c.sub.w]) (18)

where [c.sub.o] and [c.sub.w] are the oil and condensate heat capacities, respectively, and [T.sub.m], is the temperature of the mobile fluid layer at the edge of the finger is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

which can be simplified by using Equation (7) to yield:

[T.sub.m] - [T.sub.r]/[T.sub.s] - [T.sub.r] = 1/2 [A.sub.m2] (20)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is plotted in Figure 4b. The total heat supplied to the finger from the steam chamber, [H.sub.T], equals the sum of heat terms given by Equations (17) and (IS):

[H.sub.T] = u [[[rho].sub.o][c.sub.o] + [[rho].sub.w][c.sub.w]) ([T.sub.s] - [T.sub.r])[x.sub.i] + [A.sub.m2]/2 [x.sub.i][[rho].sub.o][c.sub.o][phi][DELTA][S.sub.o] ([T.sub.m] - [T.sub.r])] (21)

Thus, the volumetric finger steam-to-oil ratio is then:

fSOR = ([T.sub.s] - [T.sub.r])/[[rho].sub.g][lambda][phi][DELTA][S.sub.o] [[[rho].sub.c][c.sub.c] + [A.sub.m2]/2 [([[rho].sub.o][c.sub.o] + [[rho].sub.w] [c.sub.w])[lambda][phi][DELTA][S.sub.o]] (22)

where [lambda] is the latent heat of steam. After substituting Equation (22) into Equation (15), the finger interfacial velocity at position [X.sub.i} is then given by:

u = [A.sub.m1]k[k.sub.ro]/[[mu].sub.o][k.sub.rg/ [[micro].sub.g] ([[rho].sub.o] - [[rho].sub.g])g x {[[rho].sub.g][lambda] (1 - [X.sub.i])/([k.sub.rg]/ [[micro].sub.g])[[rho].sub.g][lambda][DELTA][S.sub.o][phi][X.sub.i] + [A.sub.m1]([k.sub.ro]/[[mu].sub.o])([T.sub.s] - [T.sub.r] [[[rho].sub.c][c.sub.c] + ([A.sub.m2][phi][DELTA][S.sub.o]/2) ([[rho].sub.o][c.sub.o] = [[rho].sub.w][c.sub.w])](1 - [X.sub.i])}. (23)

Equation (23) can be simplified to:

u = [A.sub.m1]k[k.sub.ro]/[[mu].sub.o][k.sub.rg/[[micro].sub.g] ([[rho].sub.o] - [[rho].sub.g])g [[rho].sub.g][lambda] {(1 - [X.sub.i]/[A.sub.1][X.sub.i] + [A.sub.2](1 - [X.sub.i])} (24)

where

[A.sub.1] = [k.sub.rg]/[[micro].sub.g] [[rho].sub.g][lambda][DELTA][S.sub.o][phi]

and

[A.sub.2] = [A.sub.m1][k.sub.ro]/[[mu].sub.o] ([T.sub.s] - [T.sub.r) [[[rho].sub.c][c.sub.c] + [A.sub.m2][phi][DELTA][S.sub.o]/2([[rho.sub.o] [c.sub.o] + [[rho].sub.w][c.sub.w]].

The average steam finger rise rate is then:

[u.sub.avg] = [A.sub.m1][k.sub.ro]/ [[mu].sub.o][k.sub.rg]/[[mirco].sub.g] ([[rho].sub.o] - [[rho].sub.g) g[[rho].sub.g][lambda][[A.sub.1]/[A.sub.1] - [A.sub.2]ln[A.sub.1] [A.sub.2] - 1/([A.sub.1] - [A.sub.2])]. (25)

At the centreline of the steam finger, [X.sub.i] = 0, the maximum value of rise rate is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

This represents a fundamental result: the first bracketed term in Equation (26) is the gas (read steam) phase Darcy velocity and the second bracketed term is the ratio of latent heat delivered to the finger to the total heat content of the finger and flowing steam condensate and oil. Equation (26) states that the larger the density difference between the oil and steam phases, the higher the gas phase relative permeability, the greater the amount of latent heat delivered to the finger, the higher is the rise rate. The formula also states the higher the steam temperature, the lower the rise rate. This is because of the greater amount of heat stored in the finger rock and fluids. It is important to note that the maximum rise rate is not directly dependent on the oil phase viscosity but does depend on the viscosity coefficient in. The higher the value of in, which implies greater change in oil viscosity with temperature, the lower the maximum rise rate.

Equation (26) can be further modified to account for a pressure difference between the steam finger and the oil sand beyond. The result is an additional term that adds to the gravity drainage term:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where [P.sup.*] is the pressure gradient at the interface of the steam finger. If the steam chamber was operated at pressures higher than the original reservoir pressure, then the induced pressure gradient between the chamber and virgin oil sand would promote gas flow into the oil sand.

Characteristic Dimensions of a Steam Finger

Providing the shape of the finger is known, the characteristic length and width of the steam finger can be estimated. From Equation (27), the maximum rise rate is equal to the rate of change of the finger height:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

Thus, the finger height is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

A mass balance about a finger gives:

[[rho.sub.o][Q.sub.o] = d/[d.sub.t]([[rho].sub.g][V.sub.f]) (30)

where [V.sub.f] is the volume occupied by steam finger per unit width. From a similar analysis that Butler (1997) applied along the SAGD chamber interface, the volumetric oil rate flowing out at the base of a steam finger, if the oil viscosity follows Equation (7), is:

[Q.sub.o] = [square root of 2[phi][DELTA] [S.sub.o][kk.sub.ro][[rho].sub.o]g[alpha][h.sub.f]/[ m[micro].sub.o]] (31)

If steam properties do not change with time, then the rate of change of steam finger volume is given by:

d[V.sub.f]/dt = [[rho].sub.o]/[[rho].sub.g][square root of 2[phi][DELTA][S.sub.o][kk.sub.ro][[rho].sub.o] g[alpha][h.sub.f]/[m[micro].sub.o]] (32)

The volume of the finger per unit distance, perpendicular to the page, is also given by the integral under the height versus time. This is determined from the average speed of the finger interface as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

After combining Equations (33) and (32) and substituting Equation (24) yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

After integration and rearrangement, the height of finger is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

Equation (35) reveals that finger height is directly proportional to permeability.

RESULTS

Table 2 lists input data used in the calculations conducted in this study. Table 3 compares Butler's (1987) tabulated values of the steam rise rate, values determined from his theory as calculated in this study, and values determined from the theory derived in this study given by Equation (26) at a series of different relative permeability characteristics and viscosity parameter in as was done in Butler (1987). The pressure gradient across the interface of the steam finger, [P.sup.*], has been set to zero in all cases. Viscosity parameter m = 2 is representative of Lloydminster heavy oil whereas m = 4 represents Athabasca bitumen. Within Table 3, in Case A, the difference between initial and residual oil saturation is equal to 0.8 and the effective permeabilities with respect to the oil and gas phases are 0.2 and 0.04, respectively. For Lloydminster heavy oil (m = 2), the oil-to-gas mobility ratio equals 8.2178 x [10.sup.-3]. In Cases B and C, the oil saturation change, [DELTA][S.sub.o], is 0.65, reflecting a reduced amount of mobile oil in the oil sand. In Case B, the oil and gas phase effective permeabilities are 0.25 and 0.2, respectively, whereas in Case C, they are 0.5 and 0.4, respectively. For Lloydminster heavy oil (m = 2), the oil-to-gas mobility ratios, in Cases B and C, are both equal to 2.054 x [10.sup.-3]. For Athabasca bitumen (m = 4), the oil-to-gas mobility ratios are equal to 4.9679 x [10.sup.-3], 1.227 x [10.sup.-3], and 1.227 x [10.sup.-3], for Cases A, B, and C, respectively.

The results in Table 3 reveal that there are significant discrepancies between values tabulated in the article by Butler (1987) for Athabasca bitumen (m = 4), values calculated from Butler's maximum steam rise rate, Equation (la) of this paper, and the theory developed in this study, Equation (26). Figure 5a and b compares the rise rates predicted by Butler's theory (Equation la) and the theory developed here (Equation 26). The reasons for the differences in the values in Butler's (1987) paper and those calculated from his theory (Equation la) with the same parameters are unclear but may result from a simple calculation error in Butler's paper.

There are several reasons for the difference between the new theory developed here and Equations (1a) and (1b). First, the new theory includes the effect of the density of the gas phase. Second, the theory derived here has maximum steam rise rate at the mid-symmetry plane of the finger. In Butler's theory, as found by Equation (1b), the maximum steam rise rate is located some distance away from the mid-symmetry plane of the finger. This result of Butler's theory is counter-intuitive. Third, Butler theory gives indeterminate value of fSOR at dimensionless distance [X.sub.i] = 0, which is not realistic. Fourth, in this study, unlike Butler where a single term of the truncated Maclaurin series approximation to the error function was used, greater than five terms were found to be required to obtain sufficient accuracy as represented in the [Am1], and [A.sub.m2] terms. Physically, this implies that the viscosity behaviour with distance (really the viscosity dependence on temperature which in turn depends on distance), represented by the constant m, at the edge of the chamber is not adequately represented by a single term of the error expansion.

Table 1 lists field rise rates calculated from temperatures obtained from thermocouple measurements. The re-calculated values from Butler's theory (Equations 1a and 1b) tend to be lower than typical rise rates obtained from field data as listed in Table 1. For example, Butler's theory gives rise rates in Athabasca reservoirs ranging from 0.2173 to 1.3712 cm/day whereas field thermocouple data indicates between 1 and 6 cm/day. The maximum rise rates calculated from the theory developed in this paper (Equation 26) are between 1.7510 and 18.1011 cm/day and fall in the range of rise rates determined from field thermocouple data. This is because the new theory also accounts for the full viscosity dependence with distance given by the error function expansion term [A.sub.m1,] whereas Butler's single term expansion does not capture the behaviour adequately. Moreover, the new theory predicts that the maximum front velocity is located at [X.sub.i] = 0, which means the steam finger is symmetrical about y-axis. This is a significant difference between the new theory and Butler's (1987) theory. Also listed in Table 3 are the average rise rates. For the Lloydminster oil (m = 2), the average rise rate ranges from about 1 to 7 cm/day for Cases A, B, and C. For the Athabasca oil (m = 4), the average rise rate is between about 0.2 and 1 cm/day. As expected, the higher the oil viscosity, in other words the higher the value of m, the lower the average rise rate.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Figure 6a and b is the plots of the maximum steam finger rise rate versus distance from the mid-symmetry plane of the finger for viscosity parameter values equal to m = 2 and 4, respectively. The results show that the lower the values of m, the more gradual the steam finger interfacial velocity profile. In the Athabasca oil (m = 4) for Case C, the relatively rapid change of the slope of the velocity profile reflects the effect of higher gas to oil mobility ratio and higher dimensionless parameter, m. Also, the higher the value of m, the greater the change of viscosity with temperature and the steeper the velocity profile of the finger. Figure 7a and b gives maximum steam rise rate versus steam saturation temperature for two different [k.sub.rg] values. The results show that the steam rise rate increases with lower m values. That is, the lighter the oil, the higher the steam rise rate. The results also show that the higher the temperature, the greater the steam rise rate. The plot reveals that there is a temperature where the rise rate achieves a maximum value. This is due to a maximum of the ratio of the latent heat of steam versus temperature and the amount of heat stored in the finger. There is a temperature where the combination of latent heat, stored heat, and heat losses yield a maximum rise rate.

[FIGURE 7 OMITTED]

Table 3 also lists finger heights calculated from Equation (35) for both the m = 2 and 4 cases. The results reveal that the higher the gas-to-oil mobility ratio, the greater the finger height. However, the absolute value of the oil and gas effective permeabilities also impacts the results: the larger the permeabilities, the larger the height of finger. Also, the more viscous the oil, that is, the higher the value of m, the shorter the finger height.

Another comparison is made between theory and a two-dimensional SAGD physical model experiment conducted by Sasaki et al. (2001). Table 4 lists experimental values used in the SAGD experiment. The oil sand consisted of motor oil (998 kg/[m.sup.3]) and glass beads (0.21 mm diameter). The viscosity of the oil was roughly one-fifth that of Athabasca bitumen: at 20[degrees]C, the viscosity was 93 000 cP whereas at 106[degrees]C, it was 120 cP. The steam injection pressure was set equal to about 121 kPa with corresponding saturation temperature equal to 106[degrees]C. As steam was injected into the physical model, a depletion chamber grew and from the images of the steam chamber at various times, the rise rates are estimated to be between 30 and 50 cm/day. Table 4 compares the rise rate value predicted by Butler's theory (Equations 1a and 1b) and the theory developed in this study. The results reveal that the rise rate predicted from Butler's theory is 0.27 cm/day whereas from the theory derived in this paper is 39 cm/day. The results show that the new theory provides an improved estimate of the rise rate. The difference between the time evolution of the processes, displayed in Figures 8 and 9, is also displayed in the cross-section visualizations of the steam chamber. From Equation (35), the finger height in the SAGD experiment is 0.130 mm which is of the order of the size of the pores. Figure 9 supports that the fingers are of this order of magnitude.

DISCUSSION

The results of the new theory provide significantly improved estimates of the rise rate of the steam chamber over that of Butler's (1987) theory. The derivations between the two theories are nearly the same except that the new theory presented here contains more terms of the error function expansion, thus capturing the dependence of viscosity with distance and the change of viscosity with temperature more accurately through the parameters [A.sub.m1] and [A.sub.m2]. Butler's derivation with a single term of the expansion is not sufficient to represent this behaviour and leads to the counter-intuitive result that the maximum rise rate is not at the symmetry plane of the finger. This is not the case in the theory derived here; the maximum rise rate is at the symmetry plane of the finger.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

There are two distinct modes of heat transfer at the edges of a steam chamber where there is no mobile water. The first is simple heat conduction from the chamber edge into the oil sand. This is controlled by the overall thermal diffusivity of the oil sand. The second is by convection of the steam, as fingers, into the cool oil sand. As shown by the theory derived above, this is controlled by the gas phase effective mobility, density difference between the gas and oil phases, and inversely proportional to the oil content per unit reservoir rock volume and the finger steam-to-oil ratio, a measure of the thermal efficiency of heat transfer within the finger. The conductive length scale can be estimated from the thermal diffusivity of oil sands and is found to be between 1 and 3 m. The length scale of the fingers determined in the theory presented here reveals that the fingers range from pore-scale to tens of centimetres long which is shorter than the conductive heat length scale. This implies that between either conductive heat transfer or convective steam transfer, the dominant one, in reservoirs where there is no mobile water, is conductive heat transfer. If there is mobile water, then this suggests that there could be a mobile hot water flood ahead of the steam chamber edge that also conveys heat to the oil sand.

The theory derived here shows that the finger length is directly proportional to the permeability, gas relative permeability, density difference between the gas and oil, and inversely proportional to the change of oil saturation as gravity drainage proceeds. Equation (35) highlights dependency of permeability on finger height. The height linearly varies with permeability, which supports Ito and Ipek's (2005) observation.

CONCLUSIONS

This study derives a new analytical theory for steam fingers that are at the edges of the steam chamber that arises in the steam-assisted gravity drainage (SAGD) process. These fingers extend from the chamber into the cool oil sand. In previous research, see for example Butler (1987), Ito (1984), Ito and Ipek (2005), it has been suggested that steam fingers extend up to several metres from the steam chamber into the oil sand in heavy oil and bitumen reservoirs. Butler's (1987) theory predicts significantly smaller rise rates than values estimated from field thermocouple data and predicts that the maximum rise velocity does not occur at the centreline of the steam finger. The theory derived in this study is new and provides significantly improved estimates for the rise rate. Also, the theory has the maximum rise velocity at the centreline of the steam finger. The new theory reveals that the maximum rise rate is proportional to the gas phase mobility and density difference between the oil and gas phases and inversely proportional to the oil content per unit reservoir rock volume and the finger steam-to-oil ratio, a measure of the thermal efficiency of heat transfer within the finger. The results of the study indicate that the steam fingers do not appear to be penetrating several metres into the oil sand as previously suggested by other authors. This implies that conductive heating of the oil sand, which has length-scale equal to about 1-3 m, dominates the heat transfer process at the edges of the steam chamber in the case where there is no mobile water.
NOMENCLATURE

[A.sub.m1], [A.sub.m2],
A, B, [A.sub.1],
[A.sub.2] constants

[f.sub.D] characteristic dimension of steam finger
 (m)

fSOR volume based steam to oil ratio

g gravitational acceleration ([m.sup.2]/s)

[h.sub.f] height of finger (m)

[H.sub.f] heat content of steam finger per unit
 width (J/m s)

[H.sub.o] heat content of heated oil and condensate
 flowing on inner edge of steam finger per
 unit width (J/m s)

[H.sub.T] total heat content of steam entering
 finger per unit width (J/m s)

k absolute vertical permeability (mz)

[k.sub.rg] relative permeability for steam flow

[k.sub.ro] relative permeability for oil flow

m viscosity-temperature relationship
 parameter

[P.sup.*] induced pressure gradient at interface of
 steam finger (MPa)

[Q.sub.g] total steam flow rate flowing into steam
 finger per unit width ([m.sup.3] /m s)

[Q.sub.io] oil flow rate at distance [x.sub.i] from
 origin (at point P in the interface)
 ([m.sup.3]/m s)

[Q.sub.o] total oil flow rate flowing into the steam
 finger ([m.sup.3]/m s)

[R.sup.'] mass based steam to oil ratio

[DELTA][S.sub.o] difference in oil saturation of reservoir

T temperature at point P ([degrees]C)

[T.sub.m] temperature of heated oil and condensate
 mixture ([degrees]C)

[T.sub.r] initial temperature of reservoir
 ([degrees]C)

[T.sub.s] temperature of saturated steam
 ([degrees]C)

u, V steam finger rise rate (m/s)

[u.sub.max] maximum steam finger rise rate (m/s)

[u.sub.avg] average steam finger rise rate (m/s)

[x.sub.1] half spacing between two consecutive steam
 fingers (m)

[x.sub.i] distance reference to steam finger
 interface at point P (m)

[X.sub.i] dimensionless horizontal distance,
 [x.sub.i]/[x.sub.1]

[X.sup.max.sub.i] horizontal location where steam finger
 rise rate is maximum

Greek Symbols

[THETA] angle of interface with horizontal surface and
 also angle between the normal plane and
 vertical as shown in Figure 3 ([degrees])

[[THETA].sub.f] angle of steam finger with horizontal surface
 as shown in Figure 2 ([degrees])

[x] distance measured from point P to plane of
 symmetry (m)

[[xi].sub.1] distance between point P and plane of symmetry
 at x = [x.sub.1] (m)

[[micro].sub.g] dynamic viscosity of steam (kg/m s)

[[micro].sub.o] dynamic viscosity of oil (kg/m s)

[[rho].sub.g] density of steam at saturation temperature
 (kg/[m.sup.3])

[[rho].sub.o] density of oil at saturation temperature
 (kg/[m.sup.3])

[alpha] thermal conductivity of reservoir
 ([m.sup.2]/s)

[phi] porosity of the reservoir

[lambda] latent heat of condensation of saturated steam
 at [Psub.inj] (kJ/kg)

[[rho].sub.c][c.sub.c] volumetric heat capacity of steam chamber
 (kJ/[m.sup.3] [degre]C)

[[rho].sub.o][c.sub.o] volumetric heat capacity of flowing bitumen
 (kJ/[m.sup.3] [degre]C)

[[rho].sub.r][c.sub.r] volumetric heat capacity of initial reservoir
 (kJ/[m.sup.3] [degre]C)

[[rho].sub.w][c.sub.w] volumetric heat capacity of flowing water
 (kJ/[m.sup.3] [degre]C)


ACKNOWLEDGEMENTS

The authors wish to acknowledge the financial support from Natural Sciences and Engineering Council of Canada (NSERC) for this research. We also wish to thank the Institute for Sustainable Energy, Environment and Economy (ISEEE) of University of Calgary for support.

Manuscript received 20 February 2008; revised manuscript received 14 July 2008; accepted for publication 27 August 2008.

REFERENCES

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Birrell, G., "Heat Transfer ahead of a SAGD Steam Chamber: A Study of Thermocouple data from Phase B of the Underground Test facility (Dover Project)," Paper 2001-88 presented at the Canadian International Petroleum Conference, Calgary, Alberta, Canada, June 12-14, 2001.

Birrell, G. E. and P. E. Putnam, "A Study of the Influence of Reservoir Architecture on SAGD Steam Chamber Development at the Underground Test Facility," Northeastern Alberta, Canada, Using a Graphical Analysis of Temperature Profiles. Paper 2000-104 presented at Canadian International Petroleum Conference, Calgary, Alberta, Canada, June 4-8, 2000.

Braun, A., S. Youn, T. Boyle, S. Solanki, S. Gupta, K. Easton and M. Pittman, "Performance report. EnCana Christina Lake In Situ Oil Sands Scheme 2006 Update," Alberta, Canada, May 25, 2006.

Butler, R. M., "New Interpretation of the Meaning of the Exponent "m" in the Gravity Drainage Theory for Continuously Steamed Wells," Alberta Oil Sands Technical Res. Authority J. Res. 2(1), 67-71 (1985).

Butler, R. M., "Rise of Interfering Steam Chambers," J. Can. Pet. Technol. 26(3), 70-75 (1987).

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Butler, R. M., "GravDrain's Blackbook: Thermal Recovery of Oil and Bitumen". GravDrain Inc., Calgary, Alberta (1997), ISBN 0-9682563-0-9.

Carslaw, H. S. and J. C. Jaeger, "Conduction of Heat in Solids," 2nd ed., Clarendon Press, Oxford (1959), p. 58.

Closmann, P. J. and R. A. Smith, "Temperature Observation and Steam-Zone Rise in the Vicinity of a Steam-Heated Fracture," Soc. Pet. Eng. J. 35, 575-586 (1983).

Collins, P. M., "Geomechanical Effects on the SAGD Process," SPE 97905 presented at the 2005 SPE International Operations and Heavy Oil Symposium. Calgary, Alberta, Canada, November 1-3, 2005.

Eric, W W., "Erf." From MathWorld--A Wolfram Web Resource," http://mathworld.wolfram.com/Erfc.html, January 2006.

Fung, G., "ConocoPhillips Steam Chamber Observations. Surmount SAGD Pilot Performance Presentation to the EUB," Canada May 18, 2004.

Gates, I. D. and N. Chakrabarty, "Optimization of Steam-Assisted Gravity Drainage (SAGD) in Ideal McMurray Reservoir," J. Can. Pet. Technol. 45(9), 54-62 (2006).

Good, W K., C. Rezk and B. D. Felty, "Possible Effects of Gas Caps on SAGD Performance," Report to the Alberta Environment and Alberta Energy Utilities Board, Canada, March 1997.

Gupta, S. C. and S. D. Gittins, "Effect of Solvent Sequencing and other Enhancements on Solvent Aided Process," Paper 2006-158 presented at the Canadian International Petroleum Conference, Calgary, Alberta, Canada, June 13-15, 2006.

Ito, Y, "The Introduction of the Micro-Channeling Phenomena to Cyclic Steam Stimulation and its Application to the Numerical Simulation (Sand Deformation Concept)," Soc. Pet. Eng. J. 24, 417-430 (1984).

Ito, Y and T. Hirata, "Numerical Simulation Study of a Well in the JACOS Hangingstone Steam Pilot Projects near Fort McMurray," J. Can. Pet. Technol. Special Edition 38(13), 29-36 (1999).

Ito, Y and G. Ipek, "Steam-Fingering Phenomena during SAGD Process," SPE 97729 presented at the 2005 SPE International Operations and Heavy Oil Symposium. Calgary, Alberta, Canada, November 1-3, 2005.

Ito, Y and A. K. Singhal, "Reinterpretation of Performance of Horizontal Well Pilot No. 1 (HWPI)," J. Can. Pet. Technol. Special Edition 38(13), 29-36 (1999).

Ito, Y and S. Suzuki, "Numerical Simulation of the SAGD Process in the Hangingstone Oil Sands Reservoir," J. Can. Pet. Technol. 38(9), 27-35 (1999).

Ito, Y, M. Ichikawa and T. Hirata, "The Growth of the Steam Chamber during the early Period of the UTF Phase B and Phase I Projects," J. Can. Pet. Technol. 40(9), 29-36 (2001x).

Ito, Y, T. Hirata and M. Ichikawa, "The Effect of Operating Pressure on the Growth of the Steam Chamber Detected at the Hangingstone SAGD Project," J. Can. Pet. Technol. 43(1), 47-53 (2001b).

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Sasaki, K., S. Akibayashi, N. Yazawa, Q. T. Doan and S. M. Farouq Ali, "Experimental Modeling of the SAGD Process-Enhancing SAGD Performance with Periodic Stimulation of the Horizontal Producer," Soc. Pet. Eng. J. 41, 89-97 (2001).

Sasaki, K., A. Satoshi, Y. Nintoku and K. Fuminori, "Microscopic Visualization with High Optical-Fiber at Steam Chamber Interface on Initial Stage of SAGD Process," SPE 75241 presented at the 2002 SPE/DOE Improved Oil Recovery Symposium. Oklahoma, April 13-17, 2002.

Dharmesh R. Gotawala and Ian D. Gates * Department of Chemical and Petroleum Engineering and Alberta Ingenuity Centre for In Situ Energy (AICISE), Schulich School of Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1 N4

* Author to whom correspondence may be addressed. E-mail address: ian.gates@a ucalgary.ca
Table 1. Rise rate data for SAGD steam chambers from field operations

Project oil Average steam API
sand reservoir temperature ([degrees]C)

Dover (UTF) Phase A 190 8-10
Athabasca Reservoir

Dover (UTF) Phase B 190 8-10
Athabasca Reservoir

Hangingstone 265 8-10
Athabasca Reservoir

Surmount Athabasca 240 8-10
Reservoir

Christina Lake 250 8-10

Cold Lake Bitumen 250 11

Burnt Lake 250 10
(Cold Lake bitumen)

Lloydminster 250 13
Heavy Oil

Project oil Approximate dead oil Average
sand reservoir viscosity at initial estimated
 conditions/steam rise rate
 temperature (cP) (cm/day)

Dover (UTF) Phase A 1.7 million/7 2
Athabasca Reservoir

Dover (UTF) Phase B 1.7 million/7 1-6
Athabasca Reservoir

Hangingstone 2 million/4 5
Athabasca Reservoir

Surmount Athabasca 5 million/7 4
Reservoir

Christina Lake 1 million/5 13-17

Cold Lake Bitumen 278000/4 8-10

Burnt Lake 100000/4 2
(Cold Lake bitumen)

Lloydminster 1000/<3 4-13
Heavy Oil

Project oil References
sand reservoir

Dover (UTF) Phase A Birrell and Putnam (2000)
Athabasca Reservoir

Dover (UTF) Phase B Birrell (2001) and
Athabasca Reservoir Ito et al. (2001 a)

Hangingstone Ito and Ipek (2005),
Athabasca Reservoir Collins (2005)

Surmount Athabasca Fung (2004)
Reservoir

Christina Lake Braun et al. (2006)

Cold Lake Bitumen Jiang (2006)

Burnt Lake Jiang (2006)
(Cold Lake bitumen)

Lloydminster Miller et al. (1991)
Heavy Oil

Table 2. Data used for comparisons to Butler (1987)

Data Value

Reservoir temperature, [T.sub.r] ([degrees]C) 4
Steam temperature, [T.sub.s] ([degrees]C) 200
Pressure (kPa) 1553
Latent heat of vapourization of steam, 1941 ([10.sup.2])
[lambda] (J/kg)
Porosity, [phi] 0.39
Density of oil, [[rho].sub.o] (kg/[m.sup.3]) 1005
Density of steam, pg (kg/m3) 7.8522
Volumetric heat capacity of steam 0.373 ([10.sup.6])
chamber excluding condensate, [c.sub.c]
(J/[msup.3] [degrees]C)
Volumetric heat capacity of oil, [c.sub.o] 1.8796 ([10.sup.6])
(J/[m.sup.3] [degrees]C)
Absolute permeability, k (D) 0.5
Kinematic viscosity of steam/gas at 2.044 ([10.sup.-6])
steam temperature ([m.sup.2]/s)
Thermal conductivity of reservoir, [alpha] 7.0602 ([10.sup.-7])
([m.sup.2]/s)
Dynamic viscosity of steam, [[micro].sub.g] 0.00034
(Pa s)
Kinematic viscosity of oil at steam 11 ([10.sup.-6])
temperature ([m.sup.2]/s) (m=2)
Kinematic viscosity of oil at steam 18.425 ([10.sup.-6])
temperature ([m.sup.2]/s) (m=4)

Table 3. Rise rates for Lloydminster and Athabasca oils

Reservoir data Case A

Initial oil saturation, [S.sub.oi] 0.8
Change in oil saturation, [DELTA][S.sub.o] 0.8
Relative permeability of oil flow, [k.sub.ro] 0.2
Relative permeability of steam flow, [k.sub.rg] 0.04

Lloydminster oil (m=2)
 [[k.sub.ro]/[[mu].sub.o]]/[[k.sub.rg]/ 8.2178 x [10.sup.-3]
 [[mu].sub.g]] at steam temperature
 Butler's maximum rise rate as calculated 0.37
 in this study (Equations 1 a and 1 b)
 (cm/day)
 This work, maximum rise rate (Equation 26) 1.8
 (cm/day)
 Field data, maximum rise rate (cm/day)
 Height of finger (mm) 0.29
 Average rise rate (Equation 25) (cm/day) 0.98

Athabasca oil (m=4)
 [[k.sub.ro]/[[mu].sub.o]]/[[k.sub.rg]/ 4.9679 x [10.sup.-3]
 [[mu].sub.g]] at steam temperature
 Maximum rise rate reported in Butler (1987) 0.47
 (cm/day)
 Butler's maximum rise rate as calculated 0.22
 in this study (Equations 1 a and 1 b)
 (cm/day)
 This work, maximum rise rate (Equation 26) 1.8
 (cm/day)
 Field data, maximum rise rate (cm/day)
 Height of finger (mm) 0.10
 Average rise rate (Equation 25) (cm/day) 0.21

Reservoir data Case B

Initial oil saturation, [S.sub.oi] 0.8
Change in oil saturation, [DELTA][S.sub.o] 0.65
Relative permeability of oil flow, [k.sub.ro] 0.25
Relative permeability of steam flow, [k.sub.rg] 0.2

Lloydminster oil (m=2)
 [[k.sub.ro]/[[mu].sub.o]]/[[k.sub.rg]/ 2.054 x [1.sup.-3]
 [[mu].sub.g]] at steam temperature
 Butler's maximum rise rate as calculated 1.2
 in this study (Equations 1 a and 1 b)
 (cm/day)
 This work, maximum rise rate (Equation 26) 9.2
 (cm/day)
 Field data, maximum rise rate (cm/day) 4-13
 Height of finger (mm) 2.6
 Average rise rate (Equation 25) (cm/day) 3.3

Athabasca oil (m=4)
 [[k.sub.ro]/[[mu].sub.o]]/[[k.sub.rg]/ 1.227 x [10.sup.-3]
 [[mu].sub.g]] at steam temperature
 Maximum rise rate reported in Butler (1987) 1.5
 (cm/day)
 Butler's maximum rise rate as calculated 0.69
 in this study (Equations 1 a and 1 b)
 (cm/day)
 This work, maximum rise rate (Equation 26) 9.1
 (cm/day)
 Field data, maximum rise rate (cm/day) 1-6
 Height of finger (mm) 0.42
 Average rise rate (Equation 25) (cm/day) 0.47

Reservoir data Case C

Initial oil saturation, [S.sub.oi] 0.8
Change in oil saturation, [DELTA][S.sub.o] 0.65
Relative permeability of oil flow, [k.sub.ro] 0.5
Relative permeability of steam flow, [k.sub.rg] 0.4

Lloydminster oil (m=2)
 [[k.sub.ro]/[[mu].sub.o]]/[[k.sub.rg]/ 2.054 x [10.sup.-3]
 [[mu].sub.g]] at steam temperature
 Butler's maximum rise rate as calculated 2.3
 in this study (Equations 1 a and 1 b)
 (cm/day)
 This work, maximum rise rate (Equation 26) 18.40
 (cm/day)
 Field data, maximum rise rate (cm/day)
 Height of finger (mm) 5.1
 Average rise rate (Equation 25) (cm/day) 6.5

Athabasca oil (m=4)
 [[k.sub.ro]/[[mu].sub.o]]/[[k.sub.rg]/ 1.227 x [10.sup.-3]
 [[mu].sub.g]] at steam temperature
 Maximum rise rate reported in Butler (1987) 3.1
 (cm/day)
 Butler's maximum rise rate as calculated 1.4
 in this study (Equations 1 a and 1 b)
 (cm/day)
 This work, maximum rise rate (Equation 26) 18.1
 (cm/day)
 Field data, maximum rise rate (cm/day)
 Height of finger (mm) 0.84
 Average rise rate (Equation 25) (cm/day) 0.93

Table 4. SAGD physical model data from Sasaki et al. (2001) and
experimental and predicted values of the steam rise rate

Data Sasaki et al.
 (2001) value

Reservoir temperature, [T.sub.r] ([degrees]C) 20
Steam temperature, [T.sub.s] ([degrees]C) 106
Pressure (bar) 1.2507
Latent heat of condensation of steam, 2241 ([10.sup.3])
[lambda]) (J/kg)
Porosity, [phi] 0.38
Dimensionless parameter, m 0.3
Density of oil, [[rho].sub.o] (kg/[m.sup.3]) 998
Density of steam, [[rho].sub.g] (kg/[m.sup.3]) 0.7262
Volumetric heat capacity of steam chamber 4.028 ([10.sup.6])
excluding condensate, [[rho].sub.c][C.sub.c]
(J/[m.sup.3] [degrees]C)
Volumetric heat capacity of oil, [[rho].sub.o] 6.0144 ([10.sup.6])
[C.sub.o] + [[rho].sub.w][C.sub.w]
(J/[m.sup.3] [degrees]C)
Absolute permeability, k ([m.sup.2]) 115 ([10.sup.-12])
Kinematic viscosity of steam/gas at steam 1.708 ([10.sup.-5])
temperature ([m.sup.2]/s)
Kinematic viscosity of oil at steam 120 ([10.sup.-4])
temperature ([m.sup.2]/s)
Initial oil saturation, [S.sub.oi] 1.0
Change in oil saturation, [DELTA][S.sub.o] 1.0
Relative permeability of oil flow, [k.sub.ro] 0.4632
(Good et al., 1997)
Relative permeability of steam flow, 0.0214
[k.sub.rg] (Good et al., 1997)
[[k.sub.ro]/[[mu].sub.o]]/[[k.sub.rg]/ 0.0308
[[mu].sub.g]] at steam temperature

Butler's maximum rise rate as calculated in 0.27
this study (Equations 1 a and 1 b) (cm/day)
This work, maximum rise rate (Equation 26) 38.9
(cm/day)
Experimental value (cm/day) 30-50
Height of finger (mm) 0.130
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Author:Gotawala, Dharmesh R.; Gates, Ian D.
Publication:Canadian Journal of Chemical Engineering
Date:Dec 1, 2008
Words:9264
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