The Effect of Heat Transfer To A Nearby Layer on Heat Efficiency
- Michael Prats (Michael Prats and Assocs. Inc.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 2001
- Document Type
- Journal Paper
- 262 - 267
- 2001. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 5.4.6 Thermal Methods, 4.1.2 Separation and Treating
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This paper provides an analytic solution in Laplace space for the heat distribution in two nearby layers undergoing heat injection at variable rates. The heat content in the layers undergoing thermal injection, divided by the cumulative net-injected heat, is known as the heat efficiency. In this treatment, the thermal properties of the layers, the thickness of the layers undergoing heat injection, and the intervening layer can have any values. The overburden and underburden are semi-infinite.
Examples are given for two sets of thermal properties and three values of the intervening layer's thickness. Only a constant rate of heat injection is considered in the examples.
Results indicate that significant thermal interaction occurs at an intervening layer thickness as great as 45 ft, with the effect being faster and more pronounced the thinner the intervening layer.
Marx and Langenheim1 developed equations showing how the heat content of a single reservoir layer undergoing steam injection increases with time. The results took into account the heat losses to the adjacent formations and considered the entire heated volume to be at the injection temperature. The fraction of the injected heat within the injection layer was called the heat efficiency. Prats2 later showed that, although the parameters appearing in the equation depend on the process, the equation for the heat efficiency is independent of the process used to introduce heat into the reservoir. These results make use of vanishing vertical temperature gradients within the heated zone, an approximation first made by Lauwerier3 for hot fluid injection.
Heat efficiencies provide a quick indication of the attractiveness of potential steam-injection projects and are useful in estimating steamflooding performance.4,5
In practice, and especially in steamflooding operations, steam is injected into nearby sands either simultaneously or sequentially.6,7 This paper examines how the heat losses from one layer affect the heat content of a second layer. Closmann8 developed an analytical model that describes the volumes of the heated zones for simultaneous and equal injection into an infinite number of equal and equally spaced layers. Because of the symmetry of his model, Closmann only considers two different sets of thermal properties - one in the reservoir layers and another in the intervening layers. Here, all five formations (two injection layers, an overburden, an underburden, and a center layer) may have arbitrary properties.
Steam is injected into a well open to two sands, Layers 1 and 2, which are separated by an impermeable center layer of nonzero thickness. More generally, heat is injected or generated in the two layers by any means, but this discussion is in terms of steam injection. The thickness and thermal properties (volumetric heat capacity and thermal conductivity) of the five formations may differ, but are constant and uniform within a formation. Fig. 1 is a schematic of the system considered. Heat transfer within the injection layers is by both convection and conduction.
The temperature within Layer 1, resulting from steam injection, is independent of the vertical position within the layer (the Lauwerier assumption). This temperature varies with time and within the lateral extent of the layer. The same is true for temperatures in Layer 2 that result from steam injection.
Heat losses to the overburden and underburden are considered to behave as though they were semi-infinite. No assumptions are made about the direction of the heat flow within the overburden, underburden, and center layer. Some of the heat lost from Layer 1 is conducted through the center layer into Layer 2 and beyond, and vice versa.
Before the breakthrough of heat at the producing wells, the fluid flow within the layers does not affect the overall heat balance. After heat breakthrough, the heat balances are still correct if the heat-injection rate is considered to be that due to the difference between the heat injected and that produced. Of course, the heat production rate is difficult to predict for arbitrary well configurations, so the analyses and results have more significance before heat breakthrough.
Because fluid flow does not affect the overall heat balance before breakthrough, gravity plays no role during this period. Gravitational effects would, however, affect the time at which heat breakthrough occurs, and the subsequent heat-production rate. With this proviso, the layers may be tilted at any angle to the horizontal plane. Analyses of the results are based on horizontal layers, but the term "vertical" can be generalized to mean normal to the layers, with corresponding comments for horizontal.
The solution has two steps. In the first, steam injection is into Layer 1 only. The total heat content of Layer 1 is determined as a function of time, taking into consideration the differences in the thermal properties of Layer 2, the center layer, the overburden, and the underburden. This heat content is denoted by H11. The heat transferred from Layer 1 to Layer 2 is also determined as a function of time and is denoted by H21. In the second step, steam injection is only into Layer 2. In a similar manner, the heat content in Layer 2 owing to injection in Layer 2, H22, and that in Layer 1 owing to injection in Layer 2, H12, are determined. Because the systems are linear, the heat contents during simultaneous steam injection are additive, so the total heat content in Layer j is Hj(t)=Hj1+Hj2, with j=1 or 2.
A full discussion of the solution is provided in the Appendix. Laplace transforms are used extensively, employing time solutions obtained with the Stehfest inversion algorithm.9
Examples illustrate the thermal interaction between nearby layers undergoing steam injection. Two sets of thermal properties are used (Table 1). In set P1, volumetric heat capacities and thermal conductivities are the same everywhere. In set P2, the properties of Layers 1 and 2 are the same, but those of the other formations reflect possible values of shales, sands, and lower porosities. Note that the properties are not symmetric in set P2.
Three sets of geometries are used (Table 2). Layers 1 and 2 are 25 ft thick, with the thickness of the center layer at 25, 35, and 45 ft. The overburden and underburden are semi-infinite.
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