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The recovery of heavy crudes using a special form of steamflooding has become known as Steam-Assisted Gravity Drainage process. The gravity drainage idea was originally conceived and developed by Dr. Roger Butler, in 1970's, as about the same time as the introduction of horizontal well. He tested the concept with Imperial Oil 1n 1980, in a pilot at Cold Lake which featured one of the first horizontal wells in the industry, with vertical injectors.
Alberta Oil Sands Technology and Research Authority (AOSTRA) commenced development of the Underground Test Facility (UTF) in the December 1982. Feasibility studies were completed by January 1984. One of the first processes selected for testing at the UTF was SAGD. Drilling of the wells for a 3 well pair technical pilot (Phase A) of the twin well steam assisted gravity drainage process was initiated in October 1986 and completed and tied in during 1987. The pilot was very successful and lead to a Phase B pre-commercial pilot. The AOSTRA patented SAGD process is being proven to be economically feasible for the Athabasca McMurray Oil Sands with shaft and tunnel access at the UTF site .
Steam assisted gravity drainage a thermal in-situ heavy oil process. The process begins with preheat phase by circulating steam in both of wells so that bitumen is heated enough to flow to the lower production well. The steam chamber heats and drains more and more bitumen until it has overtaken the oil bearing pores between well pair. Steam circulation in the production well is then stopped and injected into the upper injection well only. The cone shaped steam chamber, anchored at the production well, began to develop upwards from the injection well. As a new bitumen surfaces are heated, the oil lowers in viscosity and flows downwards along the steam chamber boundary into the production well by way of gravity as seen on Figure 2-1. 
Figure 2-1: Typical horizontal well pair in Butler SAGD. 
The interface is at steam temperature Ts, beyond the interface, into reservoir successive layers of material are cooler at a distance in perpendicular direction from the interface. Oil drainage rate due to gravity is described by Darcy's law per unit length of the horizontal producer
: kinematics oil viscosity
: gravity acceleration
: inclined angle of the steam interface from horizontal
: absolute permeability.
In this model, reservoir heating was assumed to be due to steady state heat conduction, and the steam zone interface was assumed to move uniformly at a constant velocity U as shown in Figure 2-2. The temperature distribution between the constant velocity steam zone interface and unheated reservoir was thus given by
: specific distance from the interface
: distance measured normal to the advancing front
: velocity of advancing front
: initial reservoir temperature (21 deg C)
: steam temperature (103 deg C).
Figure 2-2: Schematic diagram to calculate fluid displacement in SAGD process in Butler model. 
If the reservoir were unheated, then in Equation (2-1) would be kinematic oil viscosity at reservoir temperature. Increase of flow due to heating then
Interpretation of Equation (2-3) results in Equation (2-4)
To evaluate the integral it is necessary to know viscosity of the oil as a function of distance from the interface. Since Equation (2-2) gives the temperature as a function of distance, it is necessary to know the viscosity only as a function of temperature to evaluate .
The variation of viscosity with temperature depends upon the properties of the particular oil in the reservoir. One arbitrary form of temperature function that corresponds reasonably well to the performance of actual oil over the range of interest is given by Equation (2-5)
In order to use Equation (2-5) it is necessary to specify the viscosity at the steam temperature and a value for the parameter . For heavy crudes, it is found that the parameter should have a value of about 3 to 4.
Kinematic oil viscosity is very high, , then
Combining Equation (2-4) and (2-6) and eliminating the integrals, gives expression shown in Equation (2-7) for the flow.
The property was used in empirical equation of flow rate to allow for the effect of temperature on viscosity. Using relationship of temperature on distance, described by Equation (2-2) it is possible to change the variable of integration in Equation (2-4) from distance to temperature. The expression for given by Equation (2-8) is obtained by differentiating Equation (2-2) and combining the result with Equation (2-2) by eliminating the exponential term.
Substitution of from Equation (2-8) gives the following expression for the integral of Equation (2-4)
Equation (2-9) allows the evaluation of the integral for any specified dependence of viscosity on temperature. In order to continue to use this equation, it is convenient to redefine . Combining Equation (2-6) and (2-9) and solving for results
This defines as a function of the viscosity-temperature characteristics of the oil, the steam temperature, and the reservoir temperature. For specific type of oil, with experimental data on its viscosity temperature curve, the exponent can be calculated accordingly. As such, is mainly a function the characteristics of viscosity-temperature relationship for the oil (or bitumen) being considered, between the steam and reservoir temperatures. is dimensionless number that does not vary rapidly with either or . In many applications it is adequate to consider m as a constant.
Relationship between the flow of oil and the front velocity can be defined by considering the material balance at the interface. If the interface is advancing, then oil must be flowing out of the region at a faster rate than it flowing in; it is the difference in the rates that determines the advance of the interface rather than rate itself. Following oil drainage rate equation is obtained
(one side) (Eq. 2-11)
Equation (2-11) is marked one side. It gives the rate at which oil drains from one side of the steam chamber. For the usual field situation where oil is draining from both sides of the steam chamber, the rate must be doubled. Equation above is a function of the drainage height but is not dependent on the shape of the interface or on its horizontal extension.
It is assumed that steam chamber is initially a vertical plane above the production well, then the horizontal displacement is given as function of time and height by Equation (2-12)
Equation (2-12) may be rearranged to give as a function of and , as Equation (2-13)
Values of dimensionless form of and at different are plotted in Figure 2-3.
Figure 2-3: Calculated interface curves.
It is important to note that the model assumed conduction as the dominant mode of heat transfer as the temperature difference was developed from Equation (2-2), and the entire process was localized at the interface. Convection of heat ahead of steam zone interface was not considered.
It was implicitly assumed that all the heated oil ahead of the interface was produced once it reached the bottom of the steam zone. As such, no horizontal potential gradient was required for the oil production ahead of the steam zone interface, at the bottom of the steam zone. In reality the pressure gradient between the horizontal injector and producer would be required to provide the driving force. 
The dimensionless similarity between experimental and numerical model could be achieved by using the dimensionless time as determined by
: vertical distance from production well to the top of formation
: absolute permeability
: gravity acceleration
: thermal diffusivity of oilsand
: differential oil saturation
: function of the viscosity-temperature characteristics of the oil, the steam temperature, and the reservoir temperature
: oil kinematic viscosity at steam temperature.
Analysis of dimensionless similarity through usage of dimensionless time shows that only condition for dimensionless time not overcomes this problem.
The extent of the rise of the temperature in a solid body being heated by conduction is determined by the dimensionless number
is known as the Fourier number. It may looked upon as the dimensionless time, which compares the depth of the penetration of isotherms into a body that is being heated by conduction to its physical dimensions, For dimensionless similarity between experimental and numerical model, both and should be the same. If they are equal in both models, then it follow that their quotient
will be also equal. This is additional condition for dimensionless similarity.
Extension to the original SAGD theory was described by Butler and Stephens (1981) . A point of concern with the solution derived in original theory was that the oil draining down the interface curves would have to drain horizontally to the well after it reached the bottom. Some of the available head must be used to cause this lateral flow. It was assumed that the lower parts of interface curve of Figure 2-3 can be replaced by tangent drawn from the production well to the curves as shown in Figure 2-4.
Figure 2-4: Calculated interface positions for an infinite reservoir using Tandrain assumption.
Effective head causing rate of drainage of oil reduced from to 75% of . The remainder of the head is used to cause horizontal movement of the draining oil.
Butler and Petela (1989)  studied the growth of steam zone during initial stage of the SAGD process. The growth of steam chamber observed to be downward from injector to producer initially. It was controlled mostly by pressure gradient between wells and also by thermal properties of the reservoir. Development of an equation for breakthrough time was with assumption that steam condensate and oil flow were a single fluid.
Reis (1992) assumed that steam zone shape to be an inverted triangle to develop simple analytical model for predicting recovery performance for the linear SAGD process. The inverted triangular steam zone was anchored to the producer. Steady state solution to 1-d heat conduction ahead of moving steam zone interface was applied (Carslaw and Jaeger, 1959). Due to these assumptions, solutions for the energy balance and SOR were determined. Results were compared to experimental data presented by Butler et al. (1981). Little agreement between results was obtained.
2.2 Numerical Modeling for SAGD 
2.3.1 Sink/Source Model
In a sink/source (SS) model, flow from/to a reservoir is presented by a single term in the reservoir flow equation. A steady state is assumed to be in the wellbore, i.e. there is no wellbore storativity. Only one equation per completion (layer) is solved with a bottom hole pressure as a primary variable. That's means only the pressure distribution due to gravity is known in the wellbore, but not the composition and temperature. Heat conduction between the wellbore and the reservoir is also neglected. Fluid flow from/to the reservoir is calculated from Equation (2-18):
WI is a well index that describes the geometry of a specified wellbore. It may be calculated based on the Peaceman model and takes into account also the reservoir heterogeneity. represents a fluid mobility and has a different meaning for an injector or a producer. When fluid is injected, would be the total mobility of a grid block. When fluid is produced would be the mobility of each phase produced from the grid block.
A simpler Sink/Source well model may be adequate:
For reservoirs with reasonable injectivity where the effect of heat conduction between a wellbore and a reservoir is negligible. Injectivity is very low in heavy oil or tar sands reservoirs without bottom water and therefore oil may be initially mobilized only by heat conduction which is not possible with a Sink/Source model.
For processes with small flow rate or big pipe diameters where frictional pressure drop is almost nonexistent.
For short horizontal wells with a possibility of homogeneous fluid along a wellbore.
For homogeneous reservoirs where wellbore-reservoir communication is uniform.
For vertical wells where fluid segregation is minimal.
For reservoirs which have much higher draw-down than the expected friction pressure drop. One has to keep in mind that the absolute value of the frictional pressure drop is not so important as is the ratio of frictional pressure drop in the wellbore and pressure drop in the reservoir. It means that low frictional pressure drop may affect results when SAGD is used in very permeable thin reservoirs, but may not have a significant effect on thicker reservoirs with lower permeability.
For any other case the DW Model should be used.
2.3.2 Discretized Wellbore Model
The discretized wellbore (DW) model is a fully coupled mechanistic wellbore model. It models fluid and heat flow in the wellbore and between a wellbore and a reservoir/overburden. Wellbore mass and energy conservation equations are solved together with reservoir equations for each wellbore section (perforation).
To be able to solve the wellbore and reservoir equations together, some steps had to be taken to translate the pipe flow equations into Darcy's law equations. Darcy's law equations are used in reservoir simulation for flow in porous media. It means, that properties such as porosity, permeability, etc. must be assigned to the wellbore. For example, permeability may be evaluated by equating pipe flow and porous media velocity: Velocity equation in porous media in x- direction is:
: relative permeability
: potential gradient
Velocity equation for homogeneous flow in a pipe is:
: wellbore radius
: Fanning friction factor
: mass density
When we assume that the relative permeability curves in a pipe are straight lines going from zero to one then for homogeneous fluid and for multiphase flow will equate to saturation. For laminar flow and
Substituting these values into equation Equation (2-20) will give permeability in a laminar mode as
Permeability expression for a turbulent flow is more complex and depends also on friction factor, fluid viscosity and density. It is also evaluated from Equations (2-19) and (2-20) as
Permeability is updated at each time step and its value would depend on the flow pattern and fluid composition. Potential gradient is the sum of frictional, gravity and viscous forces.
Injection or production in respect of heavy oil or tar sands reservoirs may be strongly affected by wellbore hydraulics when the driving forces in a reservoir has magnitude similar to frictional forces in the wellbore. Therefore, one of the major functions of a DW model is to describe reasonably well the frictional pressure drop in a wellbore.
Friction factor for turbulent, single-phase flow is calculated from Colebrook's equation as:
: relative roughness.
When two phase fluid (liquid-gas) is present in the wellbore then liquid hold-up must be also considered in the friction pressure drop calculation. Liquid hold-up represents a slip between gas and liquid phase. Its magnitude depends on the flow regime i.e. the amount of each phase present as well as phase velocities. Liquid hold-up is predicted from Bankoff's correlation as:
The correlation parameter K is a function of Reynolds number, Froude number and a flowing mass void fraction Y. It may attain values from 0.185 to one. Gas phase mobility is altered to account for the difference in liquid and gas phase velocities i.e. gas relative permeability is augmented by the ratio of gas saturation and void fraction .This operation relates the liquid hold-up calculated from pipe flow equations to saturation needed in flow equations in porous media.
Wellbore hydraulics may be used in wells with co-current upward or horizontal flow due to the chosen correlation for the liquid hold-up. In dual stream wells flow through tubing and annulus must be considered. Tubing flow is handled similarly as wellbore flow. For laminar flow the annulus permeability is calculated as:
: annulus radius
: tubing radius.
Velocity and permeability for turbulent flow is calculated with the proper hydraulic diameter for annulus. The same correlations as mentioned above are used to calculate the friction pressure drop and slip between gas and liquid phase. Correct area and hydraulic diameter is applied wherever necessary.
Only conductive heat transfer is allowed between tubing and annulus along the tubing length. Fluid is allowed to flow to the annulus at the end of the tubing. The same equations as mentioned above are used, but the equivalent drainage radius is calculated as
Fluid and energy flow between a wellbore section and a reservoir grid block is handled the same way as between individual reservoir grid blocks. Peaceman's equation is used to calculate transmissibilities (well index) between wellbore sections and the reservoir.
: wellbore or annulus radius
The equivalent drainage radius r is obtained from
The flow term of energy consists of convective and conductive flow. Convective heat transfer uses the same phase transmissibilities as the component flow equations. Conductive transmissibility is expressed as:
with equivalent drainage radius
The inflow and outflow of fluid through each perforation changes properties in each wellbore section because all wellbore conservation equations are solved implicitly. Therefore, the DW model is able to handle backflow (crossflow) between reservoir and a wellbore correctly.
The initial conditions in the wellbore will determine the short behavior of a reservoir in the vicinity of a well and dictate the length of a transient state. When initial pressure, temperature and composition differ considerably from conditions at which fluid is injected or produced, the period of transient behavior may be extended to several days. Simulation of the transient behavior does not affect long term physical results for most of the processes used in EOR simulation. However, transients may be important in cyclic processes where the cycle duration is the same order of magnitude as the transient period. The transient period is generally longer for injectors. It will increase when low mobility fluid is injected or when low mobility fluid is originally in the wellbore. Simulation of wellbore transients is necessary in well test analysis.
The effect of wellbore transients on numerical performance is larger in heavy oils or bitumen reservoirs than in conventional oil reservoirs due to very low oil mobility. It also seems to be more pronounced in injectors than producers. In addition, attempts to simulate the transient period will change the overall numerical performance in comparison with the sink/source approach where pseudo-steady state is assumed. High pressure, temperature or saturation changes occur due to small wellbore volume. Even in an implicit simulator the timestep size will be fairly small (10e-3 to 10e-4 days, probably smaller for high rates). For example, the worst scenario is to inject steam into wellbore containing cold oil, which may be the case after primary production. Thus, the well type may be changed instantaneously, but the condition in the discretized part of the well will take time to change.
When transient behavior is not requested then STARS will do automatic pseudo-steady state initialization in the discretized wellbore at the beginning of simulation and at each time when operating conditions are changed. Operating conditions such as pressure, rate compositions etc. are taken into consideration during pseudo-steady state initialization.