Find Thermal Diffusivity from Temperature Surveys
Authors: O&apos;Dell Mike 
Published: December 2007

Abstract
Heat losses to the cap rock and base rock account for up to 50% of the energy injected in a steamflood project. It is therefore important to have a good estimate of thermal diffusivity. Often, it can be estimated graphically from temperature surveys. A spreadsheet can iterate to find a best fit value in more complicated cases. Theory, limitations, and field examples are presented.

Introduction
Steam overlay is a common phenomena in steamflood projects. From the viewpoint of a temperature observation well, there is a sudden, rapid heating to steam zone temperature at or near the top of the reservoir. Heat conduction within the cap rock can be considered a linear process. In hot waterflood projects, there is a sudden rise in temperature at or near the bottom of the reservoir.

Heat Conduction
The solution for linear conduction into the cap rock is given by the following expression (see Nomenclature).
(1)
This is the same as Eqn (5.35) of Prats, and related to Eqn (2.1.4) of Carslaw and Jaeger. Only a few assumptions are required for this equation:there is a step change in temperature at the reservoir - cap rock interface,thermal diffusivity within the cap rock is constant and uniform, andthere is no convection in the cap rock.

Graphical Procedure
Since heat conduction is described by the complimentary error function, plot the observed cap rock temperature on probability paper and look for a straight line. The temperature function, = 0.5 (T - T0) / (T1 - T0), normalizes the temperature increase between zero and one half. Plot on the probability scale and distance above the top of the reservoir on the cartesian scale. Draw a straight line through the data, which must pass through = 50%-ile at z = 0. To determine diffusivity, read the value of z where the straight line passes through = 20%-ile. Denote this value z20. Calculate diffusivity as = (z20/1.19)2/t. The constant, 1.19, is proportional to the slope. It is the argument yielding 20%-ile = 1 - F[1.19 / sqrt(2)]. F is the Cumulative Distribution Function of a Standardized Normal Random Variable, which is related to the error function (see Reference 4, or any book of math tables). One can also use z5, corresponding to = 5%-ile by substituting 2.326 for 1.19, above.

Hot Waterflood Field Example
A hot waterflood pilot was conducted in Loco Field, Oklahoma. There were several observation wells in one of the 2 1/2 acre 5-spot patterns. One set of temperature surveys is reproduced as Figure 1. These data were averaged and subjected to the graphical procedure described above. Based on injection rate and reservoir volume, the heated time is estimated as 325 days. The hot water temperature is estimated as 270 F. The initial temperature was 62 F. Data read from Figure 1 and the calculation of is given in Table 1. From Figure 2, z20 = 19 ft, resulting in a diffusivity of = 0.78 ft2/day.

Steamflood Example
Steamflood projects have been active in Midway Sunset Field for many years. Progress has been reported by many authors. A temperature survey (Figure 4 of Reference 6) in a steamflood started in 1971 was evaluated with the graphical procedure. Using the temperature suggested as the normal steam zone temperature, observed data was fit, and a thermal diffusivity of 1.07 ft2/day was calculated.

Discussion
There are limitations to the procedure. It is sensitive to the initial temperature, steam zone temperature, and heated time. In some fields, there are productive zones above and below the zone being heated. The cap rock can be a sequence of sands and shales resulting in non-uniform thermal diffusivity. It should be possible to evaluate such composite media using methods similar to methods used in this paper, but that has not been done, yet. P. 711