The Effect of Heat Transfer Between Nearby Layers on the Volume of the Steam Zones
- Michael Prats (Michael Prats and Assocs. Inc.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2002
- Document Type
- Journal Paper
- 221 - 230
- 2002. Society of Petroleum Engineers
- 1.2.3 Rock properties, 5.4.6 Thermal Methods, 4.1.2 Separation and Treating, 4.1.5 Processing Equipment, 5.2.1 Phase Behavior and PVT Measurements
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The work of Mandl and Volek1 is extended analytically to include the effect on the volume of the steam zone in one layer owing to heat transfer from a second layer undergoing steam injection. The contribution of the heat transfer delays the onset of heat convection across the advancing condensation front (the critical time) and significantly increases the volume of the steam zone even where 30 or more feet separate the layers. The increase in the heat content of the steam zone (or volume) increases with increasing ratio of the sensible to latent heat of the injected steam, and is larger than the increase in the total heat content. The interpretation for this behavior is that heat transfer delays the rate of condensation of the steam vapor, this being more important at poor steam qualities. Results of the simple approach presented here have been confirmed qualitatively through numerical simulation.
In 1967, Mandl and Volek1 developed upper and lower bounds for the heat content and volume of the steam zone undergoing variable steam injection in a single layer. In 1981, Yortsos and Gavalas2,3 refined these bounds. This paper extends the work of Mandl and Volek to two nearby layers undergoing steam injection. The necessary information on the heat transfer between the two layers is taken from Prats.4,5 Because the volume of the steam zone is closely related to the amount of oil displaced, the results provide a simple approach that may be used to screen reservoirs with multiple layers for subsequent commercial steamflood evaluation.
The working equations developed by Mandl and Volek are based on the following assumptions:
The temperature in the steam zone is that of the steam injected into the reservoir, which remains constant.
All saturations, temperatures, and rock properties are uniform within the steam zone. Gravitational effects are neglected.
The heated layer is of uniform thickness.
The velocity of the condensation front is the same everywhere.
Pressure gradients are considered to be negligible or nonimportant.
The rates of injection of sensible and latent heat may vary with time.
Thermal properties are uniform within and outside the layer undergoing steam injection.
No heat is produced from the heated zones.
In considering the effects of heat transfer between nearby layers, all but one of the above assumptions are retained. This paper specifically considers constant injection rates of both sensible and latent heat. The manner in which the heat transfer between layers is handled is discussed with reference to Fig. 1. Steam is injected into a well open to two sands, Layers 1 and 2, separated by an impermeable center layer of nonzero thickness. First, consider steam injection only into Layer 1. This results in heat transfer by conduction into Layer 2. If we now inject steam into Layer 2, its heat content would be higher than normal by the amount of heat transferred from Layer 1.
Mandl and Volek developed the size of the steam zone based on the rate of heat lost to surrounding layers. An additional assumption is made here, that the rate of heat lost from a layer undergoing steam injection is offset by the total rate of heat transfer from an adjacent layer undergoing heat injection. With this one and sole additional assumption, the development parallels that of Mandl and Volek.
The solution has two phases. In the first, the distribution of the total heat between the two layers is determined based on a method of Prats.4,5 The total heat content of Layer 1 under steam injection is determined as a function of time, taking into consideration the differences in the thermal properties of the other layer to be injected, the center layer, and the overburden and underburden. This is denoted by H11. The heat transferred by conduction from Layer 1 to Layer 2 is also determined as a function of time, and this is denoted by H12. In a similar manner, the total heat content in Layer 2 under steam injection is denoted by H22, and that transferred by conduction from Layer 2 to Layer 1 by H21. Because the systems are linear, the heat contents during simultaneous steam injection are additive, so that the total heat content in layer j (for j=1,2) is Hj=Hj1+Hj2 The total rates of heat transfer from one layer to another are also obtained.4,5
The second phase is the determination of the steam zone volume in each of the two layers. This is done following the work of Mandl and Volek1 and the refinements of Myhill and Stegemeier.6 Mandl and Volek introduced the concept of a critical time tc, at which heat is first transferred across an advancing steam condensation front. For t tc, the steam zone is the entire heated zone. For t>tc, the steam zone is smaller than the equivalent volume of a heated zone at the steam temperature. The critical time is obtained based on the Hj(t) defined above.
The volume of the steam zone is controlled by the steam condensation rate, which is partly controlled by the rate of heat loss from the layers undergoing steam injection. Here, the rate of heat lost from a layer undergoing steam injection is offset by the total rate of heat transfer from an adjacent layer undergoing heat injection. Eq. A-9, an extension of Ref. 1 to account for the net rate of heat loss when steam is injected into a nearby layer, is the basis for the determination of the volume of the steam zone. Two important elements affecting the rate of heat transfer from the nearby layer are the rate of heat injected into that layer and the distance between the layers.
Somewhat expanded outlines of the procedures used are provided in Appendix A. Laplace transforms are used extensively in the development, with solutions in time obtained using the Stehfest7 inversion algorithm.
For the constant rates of heat injection considered here, the most important parameters affecting heat transfer between two nearby layers are the properties of the intervening center layer: thickness, thermal conductivity, and volumetric heat capacity.
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