# A Condensation Heating Model for Evaluating Early-Period SAGD Performance

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## Abstract

In SAGD, it is important to obtain steam chamber conformance along the horizontal wellbore to shorten the ramp-up time and maximize economics. However, reservoir heterogeneity, wellbore undulation, and operation conditions make it hard to achieve this objective. This paper proposes a new method using a condensation temperature model to evaluate the steam chamber conformance of early-period SAGD by interpreting temperature falloff data. The initial temperature distribution of the model is categorized into three zones: a hot-zone of steam temperature, a cold-zone of reservoir temperature, and a transition-zone from steam temperature to reservoir temperature. All three zones are assumed to have circular shapes. This assumption is valid since the model focuses on the early stages of SAGD. The model takes into account the steam condensation on the chamber edge during shut-in, thereby being able to calculate the condensation-front (chamber edge) movement. A semi-analytical approach is developed to solve this model. Sensitivity analysis indicates that the sizes of the steam zone and transition zone, the observing location, and the thermal diffusivity of the formation directly affect the temperature behavior. The synthetic case study shows that the temperature falloffs from the condensation model and from simulation are in good agreement and suggests that the condensation model has the potential to estimate the chamber size during the early stages of SAGD, if calibrated to the field data. Because of the ready-to-use temperature data and the semi-analytic solution, the condensation model proposed in this paper can provide a quick and reliable estimation of the steam chamber size to help the engineers monitor and optimize the chamber development thereafter.

## Keywords

SAGD Early-period Transient temperature analysis Condensation model## Abbreviation

- SAGD
Steam-assisted gravity drainage

- TTA
Temperature transient analysis

## List of symbols

- \(T\)
Temperature in the cylinder as a function of time and space (\(^{\circ }\)C)

- \(T_\mathrm{R}\)
The initial reservoir temperature (\(^{\circ }\)C)

- \(T_\mathrm{s}\)
Steam temperature (\(^{\circ }\)C)

- \(t\)
Time, day

- \(R_\mathrm{w}\)
Wellbore radius

- \(r\)
Radius from the location to the hot-zone center (m)

- \(R_1\)
Outer radius of steam-zone (m)

- \(R_2\)
Outer radius of the liquid-zone (m)

- \(R_\mathrm{o}\)
Observing distance (m)

- \(R_\mathrm{B} \)
The radius of condensation front (m)

- \(C_\mathrm{{tS}}\)
Reservoir overall volumetric heat capacity in steam-zone (J/m\(^3\) \(^{\circ }\)C)

- \(C_\mathrm{{tL}}\)
Reservoir overall volumetric heat capacity in liquid-zone (J/m\(^3\) \(^{\circ }\)C)

- \(C_\mathrm{o} \)
Volumetric heat capacity of oil (J/m\(^3\) \(^{\circ }\)C)

- \(C_\mathrm{w} \)
Volumetric heat water of oil (J/m\(^3\) \(^{\circ }\)C)

- \(C_\mathrm{r} \)
Volumetric heat water of rock (J/m\(^3\) \(^{\circ }\)C)

- \(K_\mathrm{{tS}}\)
Overall thermal conductivity of steam-zone (J /m day \(^{\circ }\)C)

- \(K_\mathrm{{tL}} \)
Overall thermal conductivity of liquid-zone (J /m day \(^{\circ }\)C)

- \(K_\mathrm{{L-r}} \)
Thermal conductivity of the mixture of rock and liquid (J /m day \(^{\circ }\)C)

- \(K_\mathrm{{G-r}} \)
Thermal conductivity of the mixture of rock and gas (J /m day \(^{\circ }\)C)

- \(K_\mathrm{o} \)
Thermal conductivity of oil (J /m day \(^{\circ }\)C)

- \(K_\mathrm{w} \)
Thermal conductivity of water (J /m day \(^{\circ }\)C)

- \(K_\mathrm{r} \)
Thermal conductivity of rock (J /m day \(^{\circ }\)C)

- \(k_\mathrm{{tS}} =\frac{K_{tS} }{C_{tS} }\)
Overall thermal diffusivity of the steam-zone (m\(^{2}\)/\(^{\circ }\)C)

- \(k_\mathrm{{tL}} =\frac{K_\mathrm{{tL}} }{C_\mathrm{{tL}} }\)
Overall thermal diffusivity of the liquid-zone (m\(^{2}\)/\(^{\circ }\)C)

- \(q_\mathrm{1}\)
Heat flux on the inter-boundary in steam-zone (J /day)

- \(T_\mathrm{1} \)
Temperature in the steam-zone

- \(q_\mathrm{2}\)
Heat flux on the inter-boundary of liquid-zone (J /day)

- \(T_\mathrm{2} \)
Temperature in the liquid-zone

- \(\overline{T} \)
Temperature in Laplace space

- s
Laplace space variable

- \(I_n (x)\)
Modified bessel function of the 1st kind of the

*n*th order- \(K_n (x)\)
Modified bessel function of the 2nd kind of the

*n*th order- \(\rho _\mathrm{w} \)
Water density (kg/m\(^{3}\))

- \(\rho _\mathrm{s} \)
Steam density (kg/m\(^{3}\))

- \(L_\mathrm{s} \)
Latent heat of steam (J/kg)

- \(\phi \)
Reservoir porosity

- \(S_\mathrm{g} \)
The average gas saturation inside the steam chamber

## Subscript

- D
Dimensionless

- p
Particular solution

## Notes

### Acknowledgments

The authors appreciate the financial support from the Petroleum Technology Research Centre and NSERC to Dr. Fanhua (Bill) Zeng.

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