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PEH:Thermal Recovery by Steam Injection

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Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume V – Reservoir Engineering and Petrophysics

Edward D. Holstein, Editor

Chapter 15 – Thermal Recovery by Steam Injection

Jeff Jones, SPE, Berry Petroleum

Pgs. 1309-1365

ISBN 978-1-55563-120-8
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The most common method used to enhance oil production over primary rates is water injection, commonly referred to as secondary oil recovery. Common practice in the industry is to refer to all other methods as tertiary enhanced oil recovery. According to Prats,[1] thermal enhanced oil recovery (TEOR) is a family of tertiary processes defined as "any process in which heat is introduced intentionally into a subsurface accumulation of organic compounds for the purpose of recovering fuels through wells." By far, the most common vehicle used to inject heat is saturated steam. Hot water and heated gasses have been tried, but none are as effective as quality steam. According to a 2000 Oil and Gas Journal survey,[2] steam enhanced oil recovery projects account for 417,675 barrels of oil per day (BOPD), or 56% of the total for all tertiary enhanced recovery methods. That production rate has been essentially flat for more than 15 years. Hydrocarbon gas injection and CO2 gas injection are the only other significant contributors and amount to only 17 and 24%, respectively. This chapter refers to the general process as steam enhanced oil recovery (SEOR).

Properties of Saturated Steam

Like other substances, water can exist in the form of a solid (ice), as a liquid (water), or as a gas (loosely called steam). SEOR processes are concerned with the liquid and gas phases, and the change from one phase to the other. The phase change region, in which water coexists as liquid and gas, is where our interest lies when considering steam for use in the oil field. The term "steam" is an imprecise designation because it refers to a water liquid/gas system that can exist from 32°F to any higher temperature; from 0.1 psia to any higher pressure; and from nearly all liquid to 100% gas. Steam quality refers to the phase change region of liquid to gas and is defined as


Heat capacity is expressed in units of Btu/(lbm-°F). A "Btu" is defined as the amount of heat required to raise 1 lbm of water from 60 to 61°F. All liquids and solids are compared to pure water, which has the highest heat capacity of any substance at 1 Btu/(lbm-°F). By calculating a ratio of the heat capacity of water divided by that of another substance, a convenient fraction called "specific heat" is obtained. Notice that petroleum has a specific heat of 0.5, or half that of water, and sandstone is only 20% of water on a per pound basis. No other liquid or gas carries as much heat per pound as water. Also, the temperature range at which this high heat carrying performance is achieved, 34 to 700°F, is ideal for many processes, including SEOR.

Enthalpy is a useful property defined by an arbitrary combination of other properties and is not a true form of energy. The absolute value of enthalpy is of no practical value. Changes in enthalpy are extremely useful, however, and are the basis for SEOR energy calculations. The total enthalpy held by each pound of liquid water at any temperature is called sensible heat, hf. The heat input, which produces a change of state from liquid to gas without a change of temperature, is called the "latent heat of evaporation" and is shown by hfv. The total heat, hv, in each pound of 100% quality or saturated steam is the sum of these two, hv = hf + hfv.

In the phase change or "saturation" region, steam can only exist at one temperature for a given pressure regardless of quality or latent heat content, as shown in Fig. 15.1 . Steam increases in volume as latent heat increases, as in Fig. 15.2. This is a useful property in displacing oil in an SEOR process. The volume occupied by 1 lbm of steam at any pressure is its specific volume in ft3/lbm and is represented by υs. Values for these thermal properties of water are published widely.[3][4] Fig. 15.3 is a chart of the phase change region. Equations have been derived that approximate the values to acceptable accuracy for most SEOR calculations.

Simple versions that are accurate to within a few percent in the normal pressure ranges encountered in SEOR projects are









These are recommended for hand calculations or simple analytical equations. There are more precise versions in the literature[4], but for most purposes, Eqs. 15.2 through 15.8 are more accuate than any other available information that goes into SEOR calculations and are more than adequate for most calculations. The more accurate equations are cumbersome and lend themselves to computer applications. However, in a computer application, a lookup table is easy to create and is much more accurate than even the advanced analytical equations.

Reservoir Heating

The basis for reservoir heat-transfer calculations is traced back to the published solution[5] to a mechanical engineering problem. Marx and Langenheim[6] were the first to publish an adaptation of this solution to an SEOR process. They assumed that the equations for temperature response in a thin plate, backed in perfect contact to a semi-infinite solid after sudden exposure to constant-heat input, were analogous to steam injection into an oil-bearing reservoir.

Fig. 15.4 shows the temperature distribution anticipated in this model. The entire flood zone is at steam temperature upstream of the steam front, and the reservoir downstream of the steam front is at initial reservoir temperature. The true temperature profile is much less defined.

Their equation for the heated area, At, over time, t, is


G(tD) from the mechanical engineering problem is a function of dimensionless time, tD.


where tD is time of injection, t, multiplied by a few reservoir properties.


and the complementary error function, erfc(x), is common in heat conduction calculations. Several publications have tables of values for G(tD) vs. time, but van Lookeren[7] offers a simple equation with sufficient accuracy for most calculations, which is written as


Other useful relationships derived from the Marx and Langenheim equations are heat loss to the adjacent formations,


rate of heated zone growth,


heat remaining in the reservoir,


cumulative heat loss to the adjacent formations,


and reservoir efficiency or fraction of injected heat remaining in the reservoir,


Two constants appear in the equations that need definitions. MR is the volumetric heat capacity of the composite formation including rock and fluids.


Note that there are two gas components: inert gas represented by the volumetric heat capacity, Mg, and steam represented by two terms, one owing to the latent heat of vaporization and one to the sensible heat.

Thermal diffusivity, α, is the ratio of the thermal conductivity to the volumetric heat capacity,


Farouq Ali[8] showed it is appropriate to use average values for M and α when the thermal properties of various layers of formation and over/underburden differ significantly.

Ramey[9] and Prats[10] showed that Eq. 15.17, reservoir heating efficiency, Eh, is independent of reservoir and steam zone geometry. Further the relationships apply to any form of heat transport (convection, conduction, or even radiation) in the plane of the reservoir and when the heat transport to adjacent formations is three-dimensional (3D). Assuming the reservoir properties are constant and vary little with changing temperature, Eh is solely a function of dimensionless time.

Steam Zone Growth

Reservoir heating by steam injection translates into the formation of a swept zone of some shape and an oil bank consisting of a migrating zone containing the displaced oil. Two general concepts have been proposed to describe this process.

Viscous Displacement

This is an extension of early water flooding concepts in which the displacement front is considered to be an advancing vertical front, the plane of which is normal to the bedding plane and extending vertically from the top to the bottom of a homogeneous flow section. Displacement of oil in this model is piston-like and is directly proportional to injection rate. The equations in the Reservoir Heating section, Sec. 15.2, have been used to calculate steamfloods as frontal advance floods.

Mandl and Volek,[11] followed by a slight modification by Myhill and Stegemeier,[12] contributed the concept of a critical time beyond which the vapor front stagnates and heat is carried only by hot water traveling through the condensation front. Dimensionless critical time, tcD, is defined by


where fhv is the fraction of heat injected as latent heat.


and fs and hfv are at reservoir conditions.


The Marx and Langenheim equation (Eq. 15.10) is an upper bound to Eq. 15.22, as shown in Fig. 15.5.


These models address the severe buoyancy forces that pertain when steam vapor, a gas, is injected into a liquid filled reservoir. In bypass models, the fronts are not vertical; the steam migrates quickly to the top of the reservoir. Van Lookeren[7] describes the severity of this override with a buoyancy factor.


Average steam zone thickness is


The shape of the steam zone is described by


Note that steam zone thickness is not dependent on net zone thickness in Eqs. 15.24 and 15.25.

Fig. 15.6 shows the calculated injector liquid level as a function of time and five values of ARD. ARD is normally smaller than 0.6 in field projects; thus, the steam zone does not cover the vertical zone except for very thin reservoirs.

Neuman[13] addresses the bypass or steam override concept by basically assuming that injected steam quickly rises to the top of the zone and spreads out evenly. Thickness of the steam zone, hs, is related to



Field projects usually include features of both frontal advance and bypass processes. Owens and Ziegler[14] propose an analytical model that calculates the contribution of each process. They calculate total oil rate, qo, as the sum of gravity drainage oil rate, qog, after Clossman,[15] and viscous oil rate, qov, as











This series of equations can be solved in a computer spreadsheet. The authors report that the viscous component of oil rate in the Kern River Ten-Pattern is significant and varied from 66% early in the project to 53% later. The method is also used to show the effect of operating with (supporting qov) and without (inhibiting qov) a casing vapor recovery system. Fig. 15.7 shows the impact of shutting in the casing vapor recovery on a selected project. This is not to say that shutting in casing vapor recovery will have this profound negative impact on every steamflood. Notice that this method does not account for steam injection rate. A project that is significantly over-injected (high-pressure drawdown) would be expected to respond as in Fig. 15.7, while a project with more controlled steam rate (modest-pressure drawdown) may show little, if any, reduction in production rate.

Kimber et al.[16] found with a physical model that there is a viscous component of steamflooding that has potential to add significant oil recovery. Fig. 15.8 shows that there is an optimal steam quality for a steamflood and that some amount of overinjection may be beneficial also. Ward and Sharpe[17] studied the subject at the Kern River field using a numerical simulator and found that shutting in the casing vents resulted in a permanent 18% loss in oil production rate as shown in Fig. 15.9. Oil production could be restored if either the vents were reopened or a depleted zone in the wellbore were perforated, allowing the excess steam to vent by crossflow.

Steamflood Design

Candidate Selection

Screening criteria for identification of steamflood candidates have been published for many years. Table 15.1 shows the screening guides from five different sources.[18][19][20][21][22]

It is obvious from Table 15.1 that there is a finite envelope of properties that define successful candidates. However, within that envelope there is a relatively wide spread of values for the indicators. The reason for this is that each reservoir is unique and success is a function of a combination of all of the screening criteria plus a myriad of other considerations. The authors of the papers[18][19][20][21][22] typically offer linear regression equations to generate an indicator for a specific reservoir. The most recent version is by Donaldson,[23] which is written as


Doscher and Ghassemi[24] showed that there is an upper oil viscosity barrier that makes economic recovery of very viscous oils very difficult by conventional SEOR methods. Using scaled physical models, they found that for oils with a viscosity much greater than that in the Kern River or Midway Sunset fields in California, the steam/oil ratio for conventional steamflood or cyclic steam stimulation was uneconomic. The 14 and 12° API lines on Fig. 15.10 show the general upper bound range for oil viscosity in a successful conventional steamflood. They proposed a correlation equation to estimate the limiting viscosity. The correlation is a function of steam and oil viscosities.


Other critical requirements exist such as reservoir continuity between injector and producer and barriers to contain steam from uncontrolled migration to nonsteam-bearing intervals. Qualitatively, one looks for relatively shallow, low pressure, low temperature, thick, high porosity, highly permeable reservoirs with medium to high saturation of high viscosity liquid oil.

All of the following applies to oil sands with mobile oil. The SEOR process in reservoirs that are fractured during the steam injection process are too complicated to be calculated with analytical models.


The simplest way to design an SEOR project is by analogy. If there is an analogous project in the same field or in a field with enough similarities to the one in question, simply using the design and results from the former may be adequate. Fig. 15.11 from a paper by Greaser et al.[25] in the Kern River field, California is an example of the use of analogy. In that project there are thousands of steamflood injection zones. Texaco made good use of the huge amount of steamflood performance data by calculating an "average pattern response" and using that as the standard to design and monitor future flood patterns.

Recovery Mechanisms

Waterflooding, successful in reservoirs with low viscosity/high mobility crude oil, is extremely inefficient in reservoirs with low mobility/high viscosity crude oil. Three dimensionless flow parameters help us understand the problem.


These are, respectively, the ratio of gravity forces to applied pressure; the ratio of capillary forces to applied pressure; and the ratio of applied pressure to viscous forces. The ability to modify the variables in the above relationships is indicative of the potential success of extracting oil from the reservoir.

In the gravity ratio, the main controllable parameter is the distance between the injector and producer. The only other potential variable is density difference between injectant and the crude oil, which is large but does not change appreciably.

σ cosθc in the capillary pressure-applied pressure ratio represents the interfacial tension of the crude oil in the rock pore and can be varied. The obvious goal is to reduce it to zero. Several approaches have been tried, including adding emulsified chemicals in the steam to form a single-phase water/oil emulsion, which has lower viscosity and lower interfacial tension than the crude alone. Many field trials have been tried over the years but have yet to prove economical.

The viscosity parameter in the third ratio has been the most important in designing a successful SEOR project. Other methods of reducing oil viscosity, such as miscible solvent injection, have been tried but have proven to be far inferior to heat injection. Fig. 15.10 shows the viscosity vs. temperature relationship for a few representative crude oils. Raising the temperature of the oil from the typical reservoir temperature of less than 100°F to approximately 300°F gives a viscosity reduction of orders of magnitude. Of all of the potential heat-carrying media, water-based steam is inexpensive and universally available and has the highest heat-carrying capacity of any compound; it exists at the ideal 300 to 500°F temperature range to achieve optimal oil viscosity reduction.

Texaco[26] published a correlation that estimates the residual oil saturation to steam based on the huge database generated over the decades of steamflood in the Kern River field. They found the residual saturation is a function of gross zone thickness and presteam oil viscosity. There appears to be an optimal zone thickness, above and below, where residual oil is less attractive. Residual oil saturation continuously increases for increasing viscosity, reaching a point of immobile oil at some high value. Their data was based on an average initial oil saturation of 55%. Fig. 15.12 shows their correlation, normalized for any initial oil saturation. Short of actual core or laboratory data, that figure can be used to estimate residual oil saturation by finding the value of the factor for the desired initial oil saturation and zone thickness and then multiplying that factor by initial oil saturation. The equations presented next provide an alternative method for obtaining the same information.







This is instructive as to the qualitative relationships between the parameters, but caution should be used when applying it quantitatively outside of the Kern River field. Also, the authors explain that they had no information for zones thicker than 120 ft or oil viscosity greater than 16,000 cp.

Other mechanisms include formation compaction, in-situ steam distillation generated solvent banks, and formation and fluid heat-induced swelling, but they are rarely necessary to supplement the previous ones in adequately describing the SEOR process. Of these secondary mechanisms, compaction has had the most significant impact on SEOR. The most notable regions to benefit are the Bolivar Coast, western Venezuela, and the Long Beach area near Los Angeles, California. The Bolivar Coast has been using cyclic steam stimulation (CSS) since the early 1960s. Rattia and Farouq Ali[27] published a study that concluded that formation compaction enhanced process efficiency in CSS but harmed process efficiency in steamflood.

Design Calculations: Viscous Displacement Models

The Marx-Langenheim[6] and Mandl-Volek[11] (M-V) Models

These steam zone growth models have often been used to calculate cumulative oil recovery and steam zone size over time. They are a simple way to get a quick estimate of project viability. Volume of steam zone is proportional to the fraction of heat remaining in the steam zone, Eh.


It follows that oil displaced from the steam zone is


Ec is an arbitrary "capture" factor that is inserted to "scale" invariably optimistic oil volume to realistic values. This factor is best determined by history matching the equation to field project and normally has a value from 0.7 to 1.0. This represents a serious limitation in calculating oil recovery because the calculation predicts the highest oil production rate at the beginning of the project.

This method is most useful in calculating steam zone size and extent. It is less useful for calculating oil rates and recovery because it does not account for the terms in the ratios in Eq. 15.36.

The calculation allows any value for steam injection rate and calculates oil rate. Information on practical steam rates must be found before practical results can be derived. Either method can be done by hand or in a computer spreadsheet.

The Jones Model

The Jones model [28]extended the M-V model by accounting for the third dimensionless factor in Eq. 15.2 and by honoring the oil in place. The model modifies the former model by calculating the delayed oil response to a growing steam chest using the third term in Eq. 15.36, which shows that well spacing and oil viscosity are important parameters. Also addressed is the often-present depletion in the form of in-situ gas and depletion gas cap, both of which must be filled with steam before oil can be displaced. M-V steam zone growth rate is converted to an oil production rate by multiplying by three dimensionless factors.










The method can be calculated in a spreadsheet by first calculating a displaced oil rate from Eq. 15.44 for a time period, Δt; 1 ∕12 year is convenient. That volume is then multiplied by the dimensionless modifiers in Eqs. 15.45 through 15.47.

AcD accounts for the viscosity of the oil and the size of the well spacing and has these restrictions: 0 ≤ AcD ≤ 1.0, and AcD = 1.0 at μoi ≤100 cp.

VpD accounts for reservoir fill-up that must occur because of in-situ gas saturation or depleted gas zones before steam zone growth can begin to displace oil. Restrictions are 0 ≤ VpD ≤ 1.0, and VpD = 1.0 at Sg = 0.

The M-V method allows the steam zone to grow to an indefinite size for an indefinite time. Oil recovery can amount to more than original oil in place. VoD is used to limit predicted oil recovery to some fraction of the original oil in place with this limit; 0 ≤ VoD ≤ 1.0.

Fig. 15.13 shows the results of the M-S viscous displacement model with and without the Jones corrections for a hypothetical steamflood. In this example Kern River California reservoir properties were used. A small pattern size was chosen to illustrate that the M-V displacement rate is independent of oil in place and pattern size. At 75 months, the M-V displacement was 132% of original oil in place, while the Jones model corrected it to 91%. This calculation was done to an unrealistic terminal point, but if the flood were terminated at 48 months when instantaneous oil/steam ratio had fallen to below 0.2 then recovery calculated by Jones would be 77%.

This method is very good for estimating oil rates, especially early in a steamflood and for incremental recovery. It has been used in many projects worldwide with success. As with the M-V method, this calculation allows any value for steam injection rate and calculates oil rate. Information on practical steam rates must be found before practical results can be derived. Because this method produces best results if calculated in small time steps (i.e., 10 days), it is best programmed on a computer.

Design Calculations: Gravity Drainage Models

The Neuman[13] Model

The Neuman method is very useful for calculating post-steam breakthrough performance and for heat management in calculating required steam injection rates. It does not address the presteam breakthrough period. Van der Knaap[29] shows that the Neuman method can be derived from the Mandl-Volek method, and they are completely compatible. Note that Neuman’s paper[13] contains typographical errors in Eqs. D-2 andD-3. They are corrected next.

Steam-zone thickness is calculated as a function of the time lapse between the time of prediction, t, and the time when the location reached steam temperature, τ.


An estimation of the steam-injection rate and the time required to achieve the required complete coverage of the project area with a steam zone is given by


Because everything in Eq. 15.51 is a constant except time (t), an estimate of required rate is easy to derive by simply changing the time. Caution should be used because this time/rate is divorced from reservoir reaction, primarily to viscous forces during these early stages of a flood; therefore, impractical results are easily obtained.

The steam-zone volume is


Oil displaced from both the steam zone and the adjacent hot-water zone, heated by condensate convection and by conductive heat lost from the steam zone, is


where oil rate from the hot-water zone is expressed as a fraction (fb) of the oil from the steam zone.




Once steam covers the entire area, steam injection can be reduced by




This method can be done by hand but is easier to use if programmed on a computer spreadsheet.

The Vogel Model

Similar to the Neuman method, the Vogel method[30] is very useful for post-steam breakthrough heat-management calculations. It does not address early steamflood performance. Vogel first postulates that a producing well in a California-type reservoir with thick, dipping zones and gas caps that are incapable of maintaining pressure will produce at a maximum rate regardless of excess steam injection over some optimal rate. Thus, the steam rate can be calculated to maintain a steam chest by replacing lost heat and the voidage left by produced fluids. To use this method, a good estimate of a hot well producing rate is necessary, but it is very useful for optimizing existing projects with established producing rates.

A modified interpretation of the M-V method is used to calculate heat necessary to overcome reservoir heat loss. Vogel uses the following equations to calculate the steam injection rate in barrels per day.






RTENOTITLE is a value added that accounts for surface and wellbore and miscellaneous heat losses.

Fig. 15.14 shows the main benefit provided by the Neuman and Vogel methods. They are very useful in managing steam-injection requirements that continuously reduce as the project matures.

These equations apply only after steam has overlaid the reservoir and oil production has peaked because oil rate is known and steam rate is unknown in the equations. This method is easily done by hand but can also be programmed on a computer spreadsheet.

Discussion of Steam Rate Reduction

Kumar and Ziegler[31] investigated the issue of steam rate reduction schedule using a numerical reservoir simulator. The Neuman and Vogel methods previously described are based on analytical heat balances and do not directly predict oil production rates. Fig. 15.15 shows the effect on oil production rate determined using the simulator as a result of three steam-injection schedules: constant rate, linear constant reduction, and Neuman reduction schedule. Fig. 15.16 shows the steam injection for each of these schedules. The constant rate schedule results in the highest oil-production rate but at the cost of high late-steam rates. The linear reduction schedule yields nearly the same oil for a lot less steam. Neuman results in arrested steam-zone growth, but the severe early reduction in steam results in an equally severe loss in oil rate.

Table 15.2 shows that the linear rate-reduction compromise is economically superior to either of the other two schedules. Because the injection rate reduction changes net salable oil at various times in the project life, the authors used discount factors to properly value the oil production stream for each case. They termed this time-weighted oil volume "discounted net present barrels of oil."

Design Calculations: Other Models

The Gomaa Model

Gomaa[32] approached the problem by first doing a history match on a Kern River field steamflood using a numerical simulator then extracting an analytical solution from the simulator results.

Because the method is totally a function of a history match to the Kern River field, caution must be used in applying it to other projects. However, because it is a graphical method with few calculations, it offers ease in estimating project performance. After achieving a history match, Gomaa did several parametric simulations that allowed generation of several graphs that are, in turn, used to estimate project performance. Important parameters for the method are initial reservoir pressure (< 100 psia); steam injection pressure (< 200 psia); reservoir thickness (10 to 300 ft); mobile oil saturation (0.05 to 0.60); porosity (0.21 to 0.35); and heat injection rate (0.05 to 0.6 MM Btu/D/acre-ft). The steps for the procedure are outlined next.

  • Provide downhole steam quality, pressure, and injection rate and determine enthalpy, hfs.
  • Calculate the heat injection rate,


  • Determine vertical heat loss, Ql, from Fig. 15.17.
  • Determine the heat utilization factor from


For a series of time steps (t, days), calculate the effective heat injection rate and the cumulative heat injected with




  • Estimate the mobile oil saturation,


  • Use Fig. 15.18 to determine the oil recovery factor, fR.
  • Calculate the initial mobile oil in place with


  • Calculate the oil recovery at time (t),


  • Calculate the cumulative oil/steam ratio,


Repeat these steps for a period equal to the expected life of a steamflood in convenient time steps (i.e., 365 days) to estimate steamflood performance. This calculation method is easily done by hand but can be programmed in a computer spreadsheet.

Cyclic Steam Stimulation Design

Prats[1] defines stimulation as "any operation (not involving perforating or recompleting) carried out with the intent of increasing the post-treatment production rate without changing the driving forces in the reservoir." Periodic injection of steam into a producing well, alternating with a production cycle, has many features of this definition but also has many features that distinguish it as a true enhanced recovery mechanism. The primary benefit of the process is true stimulation—near wellbore reduction of flow resistance, viscosity reduction. However, there are EOR benefits of high-temperature gas dissolution, wetability changes, and relative permeability hysteresis (water flows into the reservoir easier than it flows out). Fortunately, calculating the temperature history of the wellbore, tracking the water/oil saturation history and the oil viscosity reduction is adequate to estimate the oil production response to the process.

Design Calculations

Steamflood design is simple compared to cyclic steam stimulation (CSS) design. Whereas steamflood reaches equilibrium and can be represented by a set of steady-state equations for much of its life, the CSS process is one of constantly changing conditions. First there is the injection phase, which is relatively so short that it is a total transition period. Then during the soaking period, steam vapor condenses and temperature begins to fall. The producing period is in a constant state of flux as testified by the constantly changing producing rates. Relative permeability curves, which can typically be ignored in steamflood calculations, become very important to CSS.

In spite of these problems, there are several desktop calculations that give a good representation of what can be expected from CSS. Probably the simplest representation of the process is by Owens and Suter,[33]


This simply indicates the productivity ratio resulting from steam temperature-induced oil-viscosity reduction. No attempt was made to calculate how the reservoir got the peak temperature, but once the well is steamed and placed on production, the authors propose that the operator can simply watch leadline temperature and accurately predict the production history of the production period prior to the next cycle.

The Boberg and Lantz Method

The referenced paper describes the definitive work that serves as the basis of virtually all subsequent analytical analyses of CSS. They first calculate the reservoir temperature distribution resulting during the injection period. Eq. 15.9 is used to calculate the area of the processed zone that is heated to Ti. Then, the well is placed on production and temperature of the heated volume, which is assumed to remain constant and begins to fall by conduction to the surrounding cold reservoir rock and by hot fluid production. The average temperature in the hot zone is


where fVr and fVz are unit solutions of component conduction in the radial and vertical directions, respectively. They can be estimated from Fig. 15.19 or from




The term fpD accounts for heat removed with produced fluids.




The subscript, h, indicates that the properties should be for fluids from the hot zone at the sand face. The model does not predict steam, gas, or water producing rates, which must be estimated from some other source. Oil production rates are given by a method similar to Eq. 15.69, which is written as




F1 and F2 are radial flow factors for which Boberg and Lantz give expressions in Table 15.3. Note that the production rate is a function of only two variables—oil viscosity and the heated radius.

The method can be calculated by hand for a very few time steps, but it is much easier to use if programmed into a spreadsheet.

The Towson and Boberg Model

The Boberg and Lantz [34]method assumes that there is significant reservoir energy to produce oil under primary conditions. Because many CSS candidates have only gravity forces and initial viscosity is high, there is no significant primary production. Many California reservoirs have free liquid surfaces in the oil zones with a gas oil interface at atmospheric pressure. Towson and Boberg[35] extended the former work to cover this situation. Eq. 15.70 is used to calculate the heated zone temperature from which oil viscosity is estimated. Then, gravity drainage oil rate may be calculated.


hh must be computed for each time step during the production cycle by first calculating the average hot-zone fluid level.


Now the fluid level at the heated zone radius is


This procedure can be hand calculated but is much easier to use if a computer spreadsheet is used.

The Jones Method

Jones[36] took a similar approach to Towson and Boberg[35] in calculating oil rates as a function of gravity forces alone. He extended the model by also calculating heated-zone water rate. Information on relative permeability is necessary to accomplish this. Further, recognizing that Towson and Boberg and other similar models commonly over-predict oil production, he limited the vertical size of the zone that is invaded with steam using a version of Eq. 15.24. This phenomenon is easily demonstrated by running a downhole temperature survey following a steam cycle. Then, because cold oil sand is still exposed in the wellbore, another set of equations similar to Eq. 15.77 is used to calculate oil and water from the cold zone. Using this modification, fluid rates can be matched quite well without need of a scaling factor to reduce predicted oil rates to realistic levels.

A convenient parameter to track, when trying to history-match a field steam cycle with this model, is produced fluid temperature that represents a combination of cold/hot oil and water.


This method does not lend itself to hand calculation and should be programmed on a computer.

Because steam only enters a small fraction of the sandface in a thick interval as in California oil fields, there is opportunity to improve performance of a steam cycle by using packers or other methods to divert steam into more of the oil zone.

Process Optimization

There are always the operational questions of how much steam should be injected during a cycle; what rate should steam be injected; when should a well be resteamed; etc. Jones[36] reported the results of the use of the model previously described to history-match a massive 20-year, 1,500-well cyclic steam project in the Potter Sand in the Midway Sunset field, California. He then used the history-match information to do a long-life parametric study of the process. Table 15.4 lists the conclusions for this particular application. This is, however, not a common practice. There are so many variables that the results from a single well or even a small group of wells cannot be used for a meaningful history match. Further, cyclic steam is easy to apply in the field and is relatively inexpensive, so most operators simply start immediately with a field trial. Very little is published on optimizing CSS.

It is generally true of CSS that soak time should be as short as possible and that steam quality should be as high as possible. Further, efforts should be made to divert steam out of depleted zones and gas caps and into as much good oil-saturated sand as possible.

There are generally two reasons to apply CSS. First, there is the obvious stimulation of economic oil production immediately from the well. Second, because of the time delay in oil response from the initiation of steam injection into a continuous steam injector in a steamflood project, CSS concentrated in the steamflood zone is often used to accelerate project response.

Cumulative Average Daily Profit Method

Because process optimization is ultimately an economic decision, a resteaming decision can be based on the Rivero and Heintz[37] cumulative average daily profit (CADP) method. Fig. 15.20 shows a graphical representation of how to use this method. When steam is injected into a producer, profits of the cycle are driven negative because of the cost of the steam, costs to prepare the well for steaming, and lost production as a result of the well being shut down. Once the well is put back on production, the oil rate will peak, and daily cash flow will be at a relative high. Concurrently, CADP for the cycle will begin to increase as the daily production begins to pay for the injection costs. As the well continues to produce, the oil rate gradually falls, as does daily profit. CADP hopefully soon becomes positive, then continues to increase until it reaches a value equal to the daily cash flow. It is at this point that the well should be recycled because cash flow for the next day’s production will fall below the CADP.

Although instructive as a concept for picking resteaming time, actual field application of the method is practically impossible because of ever-present problems in gathering precise enough well production gauges and in collecting all of the necessary economic data in a timely manner. Also, because the method is divorced from the reservoir process, it may lead to short-term economic decisions that damage the reservoir.

Sequential CSS Method

In a large CSS project, one needs a way to decide which well to steam and in what sequence. McBean[38] and Jones and Cawthon[39] presented a sequential CSS method that ensures that all wells will be stimulated in a timely manner and takes advantage of the interwell stimulation often observed.

By steaming wells in a sequential manner from downdip to updip as shown in Fig. 15.21, they observed not only the oil response from the steamed wells but also some response from offset wells caused by a mini-steamflood. Kuo et al.[40] found in numerical simulations that small cycles in closely spaced wells are preferable in this process. Field experience in the sequential CSS project confirmed that finding with wells drilled on 5/8 acre (0.25 ha) spacing.

Steam Delivery Systems

Heat Loss in Surface Distribution Piping

For any SEOR process, no matter how efficient, the major cost is always that of generating the process steam. Whether the product of oilfield steam generators, industrial boilers, or electrical/steam cogeneration plants, steam must be delivered through a network of pipes and through pipes down a wellbore to the oil-bearing formation. It is imperative that the unavoidable heat losses in this distribution system be minimized with some type of insulating system.

The basic equation for heat loss is


The rate of heat loss in surface lines is usually calculated at steady-state conditions because transients disappear quickly in surface pipes. Thermal resistance to heat loss for that system is


The terms in Eq. 15.82 are the coefficients of heat transfer for each of the layers of an insulated pipe as shown in Fig. 15.22. They are, from left to right: conduction in the laminar layer in contact with the pipe wall; conduction in the scale or other solid coating on the inside pipe wall; conduction in the pipe wall; conduction in the scale or other coating on the outside pipe wall; conduction in the insulation; and convection and radiation from the outer surface of the insulation. Table 15.5 shows thermal conductivity for various materials. Refer to appropriate textbooks[41][42] for more in depth information.

In the previous calculation, hr is the coefficient of radiant heat transfer for the outermost surface of the system; in this case, it is for the insulation. It is common practice to cover insulation with a thin sheath of bright aluminum, mainly for protection from weather and from mechanical abuse. A side benefit is that the bright surface has low radiant emissitivity that, combined with low surface temperature, results in negligible radiant heat loss, so this term is often ignored. If the pipe is uninsulated, the term applies.

Note that for every system there is an optimum insulation thickness. Adding more insulation above this optimum will not result in more heat savings. This is because there are two competing effects; the rate of heat loss decreases with increasing insulation thickness, but heat loss increases as the exposed surface area increases.

Coefficients of Heat Transfer'''. The following are useful in most cases, but the reader should refer to appropriate[41][42] texts for more complicated systems.

Value of hf. For condensing steam, the coefficient is large, and it is generally adequate to use


when flow is turbulent (which is most of the time) and is determined by




Because μs ~ 0.018 cp for typical oilfield steam temperatures, turbulence will prevail at


Value of hpi and hpo. These are seldom known, and the terms are usually ignored for steam distribution lines. Actually, these deposits outside and, if not too thick and firmly attached, inside the pipe are desirable because they result in resistance to heat conduction. If they are present but no values are known, McAdams[41] recommends a value of 48,000 Btu/(ft2 -D-°F).

Value of hfc. McAdams[41] offers the next equation to calculate the coefficient of forced convection at the outer surface of a pipe system in air.


for 1,000 < NRe < 50,000, where


Value of hr. In the following relationship for the coefficient of radiant heat transfer, temperatures must be expressed in °Rankine, which is°F + 460.


Emissitivity, ε, of various materials is listed in Table 15.5.

Buried Lines. A special case of insulated lines is pipes buried in the earth. See Fig. 15.23. Eq. 15.82 applies, except for two modifications,[43] which are




Heat loss rate is very high for short-term injection for buried pipes, even in dry soil, so this is not recommended in cyclic steam projects. If the soil contains moisture, the losses are even greater.

Heat Loss in Wells

Heat loss in wells never reaches a steady-state condition. It begins at a very high rate when the well casing is suddenly heated by initial steam injection, then continually decreases in rate as the surrounding earth is heated. For long term continuous steam injection over a period of years, wellbore heat loss becomes relatively small. Conversely, for intermittent cyclic steam injection, the heat-loss rate will always be relatively high because the surrounding earth is never appreciably heated. Eq. 15.82 still applies but is complicated by the ambient (earth) temperature increasing with depth because of geothermal gradient and by the "insulation," earth again, having high conductivity and practically infinite thickness. The latter property results in the thermal resistance being time dependent. Fig. 15.24 is a schematic depiction of typical elements that contribute to the resistance to heat flow which is described by


where the first five terms are similar to those in Eq. 15.82. The last five terms are the resistances in radiation and convection in the casing annulus, in the casing, in the cement, in the altered earth zone (dried earth because of high temperatures), and in the time dependent loss to the earth. If other resistance zones can be identified, such as coatings in the casing or scale deposits, etc., terms should be added for them. Every system should be analyzed according to the elements included, such as wells with no insulation on the tubing, wells with no tubing at all, or simply injection down the casing. All of the additional terms can be determined with equations previously presented except for the coefficient of heat transfer in the annulus, hrc,an and ftD.

Heat loss is a serious problem in cyclic steam stimulation because the wellbore and surface lines are never heated to steady-state conditions. Fig. 15.25 shows the results of Eq. 15.92 for several steam-injection rates in a typical Kern River field producer. It demonstrates that because steam is at a relatively constant temperature heat loss rate, it is a constant and injection should be done at the highest practical rate. Fig. 15.26 shows the benefit of insulating the casing from contact with steam for short duration injection as in a steam cycle. Conversely, Fig. 15.27[44] shows that for long-term injection, as in a steamflood injector, and for shallow wells, as encountered in the San Joaquin Valley oil fields, there is no benefit from insulating the casing. Thus, it is possible to drill inexpensive slimhole injectors, completed simply with a tubing string, and not appreciably increase heat loss over the life of a project.

Value of hrc,an. In an air-filled annulus operating under free convection, the coefficient of heat transfer for radiation and convection is given by Willhite.[43]




The Grashof number is


the Prandtl number is


and the temperature function is


The temperatures for use in Eq. 15.97 are proportional to the fractional thermal resistance between the outer tubing surface, in this case the insulation, Tins, and the casing inner wall, Tci. They can be estimated by




Value of f(tD'''''').This is a function of dimensionless time.


The radius in the denominator is the radius of the outermost element in contact with the reservoir, which is the heat altered zone in this case. Willhite[43] gives a table of values for f (tD), for tD <100. A reasonable estimate can be derived from


Ramey[44] gives a calculation of f (tD), for tD ≥100.


Because Rh, Tins , Tci , and f (tD) are interrelated by nonlinear expressions, they must be solved by an iterative trial-and-error procedure.[43]

Heat Management

While always an implicit goal in SEOR processes, overall process heat management became a topic in the literature in the mid-1980s. The growth of the discipline has closely followed the development of the personal computer and computer applications. Fig. 15.28[45] is a graphical representation of the major components of a heat balance that must be performed to properly manage a SEOR process. Ziegler et al.[46] published a very good summary of a method of implementing the principle. In essence, the operator must establish an iterative process that continues for the duration of the project and that continuously collects and analyzes pertinent data. Based on that analysis, the operator then makes appropriate midcourse adjustments to optimize the project. The process is actually a complete project optimization method but has adopted the name "heat management" because steam generating costs tower over any other cost and are even several times larger than the initial substantial capital investment in a steamflood. There are three basic parts of the method, as shown in Fig. 15.29,[46] which are data gathering, data monitoring, and adjustments to the process.

Data Gathering

As its name implies, data gathering consists of compiling regular and typically large amounts of information on the operation. Data from producing wells, injection wells, observation wells, and surface facilities are compiled and stored in computer databases. Table 15.6 was derived from Ziegler[46] and indicates the information currently considered necessary for California steamfloods. In addition to the types of data collected, the operator must also specify a regular schedule for collection.

All of the data are from existing and necessary components of the system except for the observation wells. Although it is tempting to economize by eliminating them, that is false economy. Information they provide on the change in temperature and gas/oil saturation over time is critical in maintaining process efficiency. The data must be stored in computer databases that are accessible to desktop PCs.

Data Monitoring

As with any oil production operation, there are daily and weekly tasks that must be done. However, in heat management, the operator must schedule formal project reviews on a much longer term such as a semiannual or annual basis. Table 15.7[45] is an example review schedule. During these reviews, the project performance data is compared to the original design (using methods described in previous sections in this chapter), and adjustments are made accordingly.

Key calculations are heat and mass balances to quantify the terms shown in Fig. 15.28.




Table 15.8[46] is an example of a list of metrics that should be established to aid in making heat management decisions. Note the use of the term "produced oil/fuel" ratio, Fof, in the metrics table. This is in place of the more common "produced oil/steam injected" ratio, Fos. The reason for using the former is that steamfloods often are operated at steam qualities that vary significantly from the normal 70 to 80%. In these cases, Fos would be a misleading indicator, and an in/out energy factor would be a better benchmark to use.[47] Either of these is useful as a project metric, but the user must be aware of the limitations. Fig. 15.30 shows the Fof for three different steam injection schedules.[31] Using only Fof as the criteria, The Neuman steam-injection schedule (Fig. 15.15) seems to be far superior to the other two. However, Fig. 15.14 shows that the linear steam rate reduction probably gives the best overall economic performance. The Neuman reduction rate is designed to optimize Fof, but severe early steam reduction results in lower oil rates and cumulative production.

Adjustments to the Process

Once the project has been reviewed and compared to the various benchmarks, it is either on or ahead of schedule, thus requiring no changes, or is not performing as expected and is in need of a midcourse correction. Table 15.7 lists common changes. There are only a few operational areas that can be changed.

Fig. 15.31[30] is an example of a Midway Sunset, California, steamflood that benefited from heat management. The operator determined that the wells were producing at their maximum rate but that steam injection was much higher than required. Based on this analysis, the steam rate was redirected and then reduced with improved oil production, thus greatly improving project economics and ultimate recovery.

Fig. 15.32[46] is another example of heat management techniques being used to improve an in-progress steamflood. This Kern River, California, project was found to be overinjected, and some producers were not completed in the proper zones. Steam rate was redirected in new injectors, and some producing wells were recompleted; later, steam rate was reduced, resulting in lower steam cost, increased oil production, and reduced water production. At the time of publication, the operator was faced with an apparent opportunity for another heat management study and adjustment to the process.

Horizontal Well Applications in Steam Injection—S. M. Farouq Ali, Consultant and Jeff Jones

Horizontal wells are being employed in innovative ways in steam injection operations to permit commercial exploitation of reservoirs that are considered unfavorable for steam, such as very viscous oils and bitumen and heavy oil formations with bottomwater, etc. Numerous papers have explored steam injection using horizontal- vertical-well combinations by use of scaled physical models or numerical simulators. For example, Chang, Farouq Ali, and George[48] used scaled models to study five-spot steamfloods, finding that for their experimental conditions, a horizontal steam injector and a horizontal producer yielded the highest recovery. Fig. 15.33 shows a comparison of oil recoveries for various combinations of horizontal and vertical wells and for four different cases: homogeneous formation, 10% bottomwater (% of oil zone thickness), 50% bottomwater, and homogeneous formation with 10% pore volume solvent injection before steam. Huang and Hight[49] carried out numerical simulations of a variety of hypothetical situations involving horizontal and vertical wells. A few field-tested applications of horizontal wells are briefly described next.

Vertical Injectors, Multilateral Producer: Kern Hot Plate Test

Dietrich[50] described the Kern Horizontal-Well Pilot, in a high viscosity (about 5,000 cp) oil reservoir, shown in Fig. 15.34. Key data are given in Table 15.9. Eight vertical injectors were used to inject steam at an average rate of 1,900 B/D (CWE). The producer consisted of 8 branches of ultra-short radius horizontal wells. The project operated for about 14 months. The production response from the wells averaged 120 B/D. The project was not economic (steam-oil ratio of 15) because of unfavorable placement of the horizontal producer and other factors.

Vertical Injectors and Horizontal Producer With Bottomwater: Tangleflags Project

Jespersen and Fontaine[51] described the Tangleflags project that utilizes a system of vertical injectors and horizontal producers. There are 11 producers. Table 15.9 gives the key data for the project. Fig. 15.35 shows the first well, together with the estimated sizes of the steam zones around the vertical injectors at three different times. The original horizontal well (400 m [1,312 ft] in length, but only 107 m [351 ft] were open to flow) was drilled for primary production, but the water cut became prohibitive within months, at which time it was decided to inject steam into the offset vertical wells. As a result, the water cut declined rapidly and stayed at a low value. The well produced almost 2.5 million bbl of oil. It should be noted that a small, discontinuous primary gas cap is present over the field. Temperature surveys showed that the injected steam migrated into the gas cap and during the early part of the project, exerted a downward drive, restricting the advance of the oil/water contact. At a later stage, steam injection rate was decreased and fluid flow was gravity controlled.

There are several other steam injection operations like Tangleflags. If the vertical permeability is high, and gas cap absent or small, an operation utilizing horizontal producers and vertical injectors may be viable.

Steam-Assisted Gravity Drainage (SAGD)

SAGD is an outstanding example of a steam injection process devised for a specific type of heavy oil reservoir utilizing horizontal wells. Several variations of the basic process have been developed, and are being tested. The original process, as developed by Butler, McNab, and Lo[52] in 1979, utilizes two parallel horizontal wells in a vertical plane: the injector being the upper well and the producer the lower well (Fig. 15.36, taken from Butler[53]). If the oil/bitumen mobility is initially very low, steam is circulated in both wells for conduction heating of the oil around the wells. Steam is then injected into the upper well, while producing the lower well. As a result, steam rises forming a steam chamber with oil flowing at the sides of the chamber and condensate flowing inside the chamber, as shown in Fig. 15.36. This is an idealized situation. Other flow regime may occur depending on the oil and formation properties. For the simplest case, the oil production rate qoh, in B/D per ft well length, is given by (multiplier "2" indicates flow from two sides of the steam chamber)


where the kinematic viscosity of oil (in centistokes) at the steam temperature, Ts, is given by νs, and that at any other temperature, T, is given by


where m is derived from the viscosity-temperature relationship of the oil.

Eq. 15.106 predicts rates of 0.1 to 0.7 B/D per ft for a horizontal well for an oil viscosity of 100,000 cp. For example, a 2,000-ft long well may be expected to produce about 800 B/D at a steam temperature of 400°F. The theory has been verified by laboratory experiments. Field results to-date have been encouraging. One commercial project (EnCana’s Foster Creek Project), consisting of 22 well pairs, has been in operation since October 2001. The current steam/oil ratios are averaging 2.5 bbl oil/bbl of steam. Earlier field tests of SAGD in Athabasca tar sands were successful at a depth of about 600 ft, which is too deep for surface mining and not deep enough for high-pressure steam injection.

SAGD is a complex process because gravity flow strongly relies on a high vertical permeability. The initial oil mobility determines the vertical spacing of the two wells. In a million cp tar sand, the spacing would be 5 to 6 m [16 to 20 ft]. It is also important that the steam chamber be sealed. There is no steam migration to offset vertical wells. In California, SAGD failed to achieve commercial success because of relatively high initial mobility of oil, as well as other reasons.

One variation of SAGD is known as single-well SAGD. Here, insulated tubing is used to inject steam into a single horizontal well, with production from the annulus. This process was successful in a few cases but generally failed. Another variation (Vapex) utilizes a suitable solvent (such as ethane, propane, etc.) instead of steam and is being field tested.

Cyclic Steam Stimulation and Other Uses of Horizontal Wells

Horizontal wells are being used for cyclic steam stimulation, much in the same manner as vertical wells, in Cold Lake, Alberta, and in Eastern Venezuela. In Cold Lake, horizontal well lengths are of the order of 3,000 ft, and steam slugs are very large (100,000 to 200,000 bbl). The resulting steam/oil ratios are 2 to 4, and performance is superior to SAGD continuous steamflood.[54] In Venezuela, wells are around 1,000 ft in length, and the steam slugs are of the order of 50,000 bbl. Response is good, with average stimulation ratios of 2 to 4 over a production period of 6 months.

In California, horizontal wells have been used sparingly in thermal projects, mainly because of sand control problems but also because of the fact that vertical wells are very cost effective, and it is difficult for horizontal wells to be competitive. Horizontal wells have successfully been incorporated into existing thermal projects to supplement conventional vertical-well performance for both Midway Sunset field cyclic steam[55][56] and Cymric/McKittrick field steamfloods.[57] In the Duri field, Indonesia, horizontal wells are also being used to supplement conventional vertical-well steamflood.[58] Table 15.9 shows reservoir data for these three areas. All of these latter applications are accessing bypassed oil in mature projects for which vertical wells would be impractical. Fig. 15.37 is a schematic drawing showing the wedges of oil sand in the Midway Sunset cyclic steam project at the very bottom of the zones that cannot be accessed by the existing vertical wells but will be drained by the horizontal wells. Fig. 15.38[58] shows the result of a seismic study of the mature Duri steamflood. Highlighted on the map are pockets of oil that were bypassed by the flood because of proximity to a fault. The operators are strongly considering a suite of horizontal wells to access these reserves.


A = area, sq ft [m2]
AcD = dimensionless factor defined by Eq. 15.45
ARD = dimensionless buoyancy defined in Eq. 15.23
At = time-dependent heated area, sq ft [m2]
Bo = oil formation volume factor, RB/STB [res m3/stock-tank m3]
C = isobaric specific heat
Can = isobaric specific heat of annular fluid, Btu/(lbm-°F) [kJ/kg•K]
Co = isobaric specific heat of oil, Btu/(lbm-°F) [kJ/kg•K]
Cw = isobaric specific heat of water, Btu/(lbm-°F) [kJ/kg•K]
D = depth below surface, ft [m]
erfc(x) = complementary error function
E = efficiency
Ec = fraction of oil displaced that is produced
Eh = heat efficiency—fraction of injected heat remaining in reservoir
f = volumetric fraction of noncondensable gas in vapor phase
fb = hot-water zone oil-rate factor defined by Eq. 15.54
fh = heat utilization factor defined by Eq. 15.55
fhv = fraction of heat injected as latent heat
fp = fraction of heat injected that is produced
fpD = heat loss factor caused by hot fluid production
fR = oil recovery factor defined by Eq. 15.14
fs = steam quality
f(tD) = function of dimensionless time defined by Eq. 15.102
f(T) = temperature function defined in Eq. 15.56
fVr = conductive heat loss factor caused by radial conduction
fVz = conductive heat loss factor caused by vertical conduction
F1, F2 = constants defined in Table 15.2
Ffo = ratio of fuel burned to produced oil, B/B [m3/m3]
Fof = produce oil/fuel burned ratio, B/B [m3/m3]
Fos = produce oil/injected steam ratio, B/B [m3/m3]
FSor = residual oil factor used in Eq. 15.37
F (Tins, Tei) = temperature function defined by Eq. 15.97
g = gravity acceleration constant, 32.174 ft/sec2 [9.8067 m/s2]
gc = conversion factor in Newton’s second law of motion, 32.174 lbm-ft/lbf-s2 [1.0 kg•m/N•s2]
G (tD) = function defined by Eq. 15.10
h = enthalpy per unit mass, Btu/lbm [kJ/kg]
he = fluid level in reservoir at external boundary, ft [m]
hf = enthalpy of liquid portion of saturated steam, Btu/lbm [kJ/kg]
hfc = forced convection coefficient of heat transfer, Btu/(sq ft-D-°F) [kJ/m2•d•K]
hfs = enthalpy of < 100% quality saturated steam, Btu/lbm [kJ/kg]
hft = film coefficient of heat transfer, Btu/(sq ft-D-°F) [kJ/m2•d•K]
hfv = enthalpy of vapor portion of saturated steam, Btu/lbm [kJ/kg]
hh = fluid level in stimulated reservoir, ft [m]
hn = net reservoir thickness, ft [m]
hpi = film coefficient of heat transfer at pipe inner radius, Btu/(sq ft-D-°F) [kJ/m2•d•K]
hpo = film coefficient of heat transfer at pipe outer radius, Btu/(sq ft-D-°F) [kJ/m2•d•K]
hr = coefficient of radiant heat transfer for the outermost surface, Btu/(sq ft-D-°F) [kJ/m2•d•K]
hrc,an = radiant/convection heat transfer coefficient in well annulus, Btu/(sq ft-D-°F) [kJ/m2•d•K]
hs = steam zone thickness, ft [m]
ht = gross reservoir thickness, ft [m]
hv = enthalpy of 100% quality (saturated) saturated steam, Btu/lbm [kJ/kg]
hw = fluid level in cold wellbore, ft [m]
iw = cold water equivalent steam injection rate, B/D [m3/d]
J = productivity of a cold well, B/psi-D [m3/kPa•d]
Jh = productivity of a stimulated well, B/psi-D [m3/kPa•d]
k = reservoir permeability, md [μm3]
kro = relative permeability to oil
krs = relative permeability to steam
L = distance between wells, ft [m]
m = mass, lbm [kg]
m* = exponent in Eqs. 15.33, 15.105, and 15.106
mcasing_blow = mass (gas) extracted from system, lbm [kg]
mi = mass injection, lbm [kg]
minflux = mass exiting system, lbm [kg]
ml = mass of liquid, lbm [kg]
m(o/w)influx = mass flowing into system, lbm [kg]
m(o/w)prod = mass (fluid) extracted from system, lbm [kg]
mv = mass of vapor, lbm [kg]
mZ(accum) = mass accumulating in system, lbm [kg]
Mg = volumetric heat capacity of gas, Btu/(ft3-°F) [kJ/m3•K]
Mo = volumetric heat capacity of oil, Btu/(ft3-°F) [kJ/m3•K]
MR = volumetric heat capacity of the reservoir, Btu/(ft3-°F) [kJ/m3•K]
Ms = volumetric heat capacity of steam zone, Btu/(ft3-°F) [kJ/m3•K]
MS = volumetric heat capacity of surrounding formation, Btu/(ft3-°F) [kJ/m3•K]
Mw = volumetric heat capacity of water, Btu/(ft3-°F) [kJ/m3•K]
Mσ = volumetric heat capacity of reservoir rocks, Btu/(ft3-°F) [kJ/m3•K]
n = index of time increment
N = initial oil in place, B [m3]
Nd = oil displacement rate, B/D [m3/d]
NGr = Grashof number
Nm = initial mobile oil in place, B [m3]
Np = cumulative oil produced, B [m3]
NPr = Prandtl number
NRe = Reynolds number
p = atmospheric pressure, psia [kPa]
pe = external boundary pressure, psia [kPa]
ps = steam pressure, psia [kPa]
pw = wellbore pressure, psia [kPa]
RTENOTITLE = steam injection rate to make up for surface heat losses, B/D [m3/d]
qgh = hot gas production rate, Mcf/D [std m3/d]
qis = reproduced steam rate, B/D [m3/d]
qiso = initial steam injection rate, B/D [m3/d]
qls = steam injection rate to make up for reservoir heat losses, B/D [m3/d]
qoc = cold oil production rate, B/D [m3/d]
qoD = displaced oil rate defined in Eq. 15.49, B/D [m3/d]
qog = oil production rate owing to gravity displacement, B/D [m3/d]
qoh = hot oil production rate, B/D [m3/d]
qoi = initial oil production rate, B/D [m3/d]
qot = total oil production rate, B/D [m3/d]
qov = oil production rate because of viscous displacement, B/D [m3/d]
qps = steam rate to replace reservoir volume of produced oil, B/D [m3/d]
qwc = cold water production rate, B/D [m3/d]
qwh = hot water production rate, B/D [m3/d]
Q = amount of injected heat remaining in reservoir, Btu [kJ]
Q casing_blow = heat removed with produced gas, Btu [kJ]
Qi = total heat injected, Btu [kJ]
Q influx = heat leaving system, Btu [kJ]
Ql = heat lost in reservoir, Btu [kJ]
Qls = surface piping heat loss/unit length, Btu/ft [kJ/m]
Qot = cumulative oil recovery at time (t), B/D [m3/d]
Q (o/w)influx = heat flowing into system, Btu [kJ]
Q(o/w)prod = heat removed with produced liquids, Btu [kL]
Qz(accum) = heat accumulating in system, Btu [kJ]
RTENOTITLE = heat injection rate, Btu/D [kJ/d]
RTENOTITLE = heat loss rate, Btu/D [kJ/d]
RTENOTITLE = heat removed with produced fluids, Btu/D [kJ/d]
RTENOTITLE = volumetric heat injection rate, MMBtu/D/acre-ft [kJ/m3]
r = radius of reservoir, ft [m]
rci = casing internal radius, ft [m]
rco = outer casing radius, ft [m]
re = external radius of heated zone, ft [m]
rEa = altered radius in earth around wellbore, ft [m]
rh = radius of heated or steam zone, ft [m]
ri = inside pipe radius, ft [m]
rins = insulation external radius, ft [m]
ro = outside pipe radius, ft [m]
rw = radius of well, ft [m]
Rh = overall specific thermal resistance, °F-ft-D/Btu [K•m•d/kJ]
S = skin factor before stimulation
Sg = gas saturation fraction
Sh = skin factor after stimulation
So = oil saturation
Soi = initial oil saturation fraction
Som = mobile oil saturation fraction
Sor = residual oil saturation fraction
Sora, Sorb = terms used in Eq. 15.40
Sors = residual oil saturation to steam fraction
Sorw = residual oil saturation to water fraction
Sw = water saturation fraction
t = time, D [d]
t* = time at which steam injection rate reduction is to begin, D [d]
tcD = critical dimensionless time
tD = dimensionless time
T = average temperature in heated reservoir, °F
T* = temperature above which oil saturation is reduced to Sorw, °F
Ta = air temperature, °F
TA = ambient temperature, °F
Tb = bulk fluid temperature, °F
TBHF = bottomhole fluid temperature, °F
Tci = temperature of casing wall, °F
TE = temperature of earth, °F
TFW = steam generator feedwater temperature, °F
Th = temperature in stimulated zone, °F
Ti = influx water temperature, °F
To = temperature of outer surface, °F
Tp = produced fluid temperature, °F
TR = unaffected reservoir temperature, °F
Ts = steam temperature, °F
TSZ = steam zone temperature, °F
u = volumetric flux, ft3/sq ft-D [m/d]
U = unit function equals 1 for tDtcD > 0, 0 for tDtcD < 0 in Eq. 15.22
VoD = dimensionless factor defined by Eq. 15.42
VpD = dimensionless factor defined by Eq. 15.43
Vs = steam zone volume, acre ft [m3]
Vsz = steam zone volume per Neuman in Eq. 15.52, B [m 3]
wst = mass flow rate of dry steam, lbm/D [kg/d]
x = distance along the x ordinate
X = factor defined in Eq. 15.30
Y = factor defined in Eq. 15.31
α = thermal diffusivity of reservoir, ft2/D [m2/d]
αE = thermal diffusivity of earth, ft2/D [m2/d]
αs = thermal diffusivity of surrounding formation, ft2/D [m2/d]
β1 = thermal volumetric expansion coefficient, 1/°F [1/K]
βan = volumetric thermal expansion coefficient of gas in annulus, 1/°F [1/K]
γ = specific gravity
Δ = increment or decrement
Δhh = change in stimulated zone fluid level, ft [m]
Δt = time steps, D [d]
ΔT = steam temperature/reservoir temperature, Ts/TR , °F
Δγ = oil/steam specific gravity difference
Δρ = density difference between water and oil, lbm/ft3 [kg/m 3]
ε = emissivity
εci = radiant emissivity of casing wall
εins = radiant emissivity of insulation outer surface
θc = wetting contact angle, deg (°) [rad]
θ = formation dip angle, deg (°) [rad]
λ = thermal conductivity, Btu/(ft-D-°F) [kJ/m•d•K]
λa = thermal conductivity of air, Btu/(ft-D-°F) [kJ/m•d•K]
λa,a = thermal conductivity of air in well annulus, Btu/(ft-D-°F) [kJ/m•d•K]
λE = thermal conductivity of unaltered earth, cp [Pa•s]
λEa = thermal conductivity of altered earth, cp [Pa•s]
λins = thermal conductivity of insulation, cp [Pa•s]
λp = thermal conductivity of pipe, cp [Pa•s]
λS = thermal conductivity of surrounding formation, cp [Pa•s]
μ = viscosity, cp [Pa•s]
μa = viscosity of air, cp [Pa•s]
μan = viscosity of well annulus gas, cp [Pa•s]
μoh = hot oil viscosity, cp [Pa•s]
μoi = initial oil viscosity, cp [Pa•s]
μs = steam viscosity, cp [Pa•s]
π = constant pi, 3.141
ρ = density, lbm/ft3 [kg/m3]
ρa,sc = density of air, lbm/ft3 [kg/m 3]
ρan = density of well annulus gas, lbm/ft3 [kg/m3]
ρo = density of oil, lbm/ft3 [kg/m3]
ρs = density of dry steam, lbm/ft3 [kg/m3]
ρw = density of water, lbm/ft3 [kg/m3]
ρw,sc = density of water at standard conditions, 62.4 lbm/ft3 [662.69 kg/m3]
σ = interfacial tension, oil/water, dyne/cm [mN/m]
τ = time since location in reservoir reached steam temperature, D [d]
υs = steam specific volume, ft3/lbm [m3/kg]
υw = wind velocity, miles/hr [km/h]
Ф = porosity


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General References

Prats, M. 1982. Thermal Recovery, Vol. 7. Richardson, Texas: Monograph Series, SPE.

Green, D.W. and Willhite, G.P. 1998. Enhanced Oil Recovery, Vol. 6, 301. Richardson, Texas: Textbook Series, SPE.

Hong, K.C. 1994. Steamflood Reservoir Management Thermal Enhanced Oil Recovery. Tulsa, Oklahoma: PennWell Books.

Thermal Recovery Processes, Vol. 7. 1965. Richardson, Texas: Reprint Series, SPE.

Thermal Recovery Processes, Vol. 7. 1985. Richardson, Texas: Reprint Series, SPE.

Thermal Recovery Techniques, Vol. 10. 1972. Richardson, Texas: Reprint Series, SPE.

Lake, L.W. 1989. Enhanced Oil Recovery. Indianapolis, Indiana: Prentice Hall.

SI Metric Conversion Factors

acre × 4.046 856 E − 01 = ha
acre-ft × 1.233 489 E + 03 = m3
°API 141.5/(131.5 + °API) = g/cm3
B × 1.590 E − 01 = m3
B/D × 1.590 E − 01 = m3 /d
Btu × 1.055 056 E + 00 = kJ
Btu/(ft-D-°F) × 6.231 E + 00 = kJ/m•d•K
Btu/hr × 2.930 711 E − 01 = W
Btu/lbm × 2.326 E + 01 = kJ/kg
Btu/(lbm-°F) × 4.187 E + 00 = kJ/kg•K
cp × 1.0* E − 03 = Pa•s
D × 1.0 E + 00 = d
ft × 3.048* E − 01 = m
ft2 × 9.290 304* E − 02 = m2
ft3 × 2.831 685 E − 02 = m3
°F (°F − 32)/1.8 = °C
lbm × 4.536 E − 01 = kg
lbm/ft3 × 1.062 E + 01 = kg/m3
MM Btu/D/acre-ft × 8.556 E + 02 = kJ/m3
psi × 6.894 757 E + 00 = kPa
psia × 6.895 E + 00 = kPa


Conversion factor is exact.