Analytical Solution for SAGD with Consideration of Temperature Variation along the Edge of a Steam Chamber

Steam-assisted gravity drainage (SAGD) is a widely-used method for heavy-oil and bitumen recovery. Analytical models were presented in the literature for bitumen-production rate and steam-to-oil ratio (SOR) for SAGD. They often overestimate bitumen-production rate substantially. Various attempts were made to correct for simplifying assumptions made in their derivations. However, no research has been conducted to solve for temperature at the edge of a steam chamber. Although bitumen-production rate and SOR depend significantly on temperature near the chamber edge in SAGD, previous analytical models assumed the injected-steam temperature to uniformly distribute along the edge of a steam chamber. The main objective of this research is to investigate temperature variation along the edge of a steam chamber. Local material balance and Darcy’s law are applied to each cross section perpendicular to the edge of a steam chamber. Then, they are coupled with the global material balance for the chamber geometry that is an inverted triangle. New analytical equations are presented for bitumen-production rate and SOR, in addition to variables as functions of elevation from the production well, such as oil flow rate, temperature, and composition along the linear-chamber edge. Bitumen-production rate and SOR can be calculated for a given temperature at a certain elevation from the production well. The new analytical model is validated on the basis of numerical flow simulations. Comparison of the analytical model with numerical simulations shows that bitumen-production rate and SOR can be accurately estimated when the new model is used with the temperature taken from the midpoint of the edge of a steam chamber. The temperature that gives accurate results is 60%-90% of the injected-steam temperature in the cases tested. Hence, the analytical model presented in this research requires a representative temperature (i.e., temperature at the midpoint of the chamber edge) for a given time for a given SAGD operation, unlike previously-proposed models. This is plausible because the assumption of one-dimensional heat conduction on a moving chamber edge is expected to be less accurate near the top and bottom of a steam chamber, in which multi-dimensional heat transfer is significant owing to heat losses to the over and underlying formation rocks. Numerical simulations show that such heat losses are necessary for a steam chamber to have a linear edge. In addition, multidimensional flow near the bottom of the reservoir causes substantial heat convection, and makes the one-dimensional conduction equation inaccurate. Hence, the previous assumption of the injected-steam temperature at the chamber edge is simplistic, and gives inaccurate results for oil-production rate in SAGD. Among widely-used assumptions for analytical SAGD models, most simplistic assumptions are identified, such as single-oleic phase flow, one-dimensional flow along the edge of a steam chamber, and one-dimensional heat conduction ahead of the chamber edge. The new analytical model is also applied to estimate bitumen-production rate and SOR for three SAGD projects, although there are various uncertainties in actual field data, such as reservoir heterogeneity.