Assignment #5 - Heat Flow by Spreadsheet

In this laboratory you will use an Excel spreadsheet to solve a heat flow problem.

1. Heat Flow Concepts

 The fundamental relationship is known as Fourier's Law:   

 Q_d = [dT/dz]_d . k_d                   (1)                 

which states in words "The heat flow (Q) measured within a substance is the product of the thermal gradient (dT/dz) multiplied by the thermal conductivity (k)". The subscript (d) in equation (1) refers to the depth in the Earth at which the quantities are measured. T is the temperature (degrees Celsius or Kelvin).

Generally within the Earth, the thermal conductivity k depends on the physical properties of the rocks, i.e. their chemical composition, porosity, compaction etc. In this problem, the thermal conductivities refer to the in situ values, i.e. the values you would measure in the Earth at a particular depth.

When we have more than one layer (1, 2, ...), the average conductivity is the harmonic mean conductivity of the individual layers, i.e.

(k_av/z_av) = [(Dz₁/k₁) + (Dz₂/k₂) + ...]^-1           (2)

We also refer to the individual heat flows for each layer as the interval heat flow.

The thermal resistance R of a section (i) down to a certain depth z_i is defined as the integral (or summation) of the inverse of the thermal conductivity down to that depth, or:

R = S_i (Dz_i/k_i)                                   (3)

The Bullard plot (after Sir Edward Bullard, in 1939) is the plot of temperature versus R for each depth in the model. In a purely conductive steady state, vertical heat flow with no internal heat production produces a straight line on the Bullard plot. Any deviation from a straight line signifies a change in these assumptions.

2. Heat Flow Exercise

Start a new Excel Spreadsheet called assn5.xls. You are given data from a well that has been drilled through a geological section of 4 layers to a depth of 3.7 km. The thermal conductivity and temperature has been determined at various depths. Enter the headings and columns of each layer into the spreadsheet as follows

Note that mK refers to milli Kelvin. You are also told that the surface temperature is 10°C.  Make new columns in the spreadsheet as you need them to answer the questions below.

Questions:

(1) What is the average heat flow in the well?

(2) Plot the temperature in the well as a function of depth.

(3) Calculate and plot the interval heat flow for each of the 4 temperature intervals in the well

(4) Comment on where you think there is a significant change in the thermal properties of the well.

(5) Now construct a Bullard plot of the temperature versus the thermal resistance

(6) Using the TREND function in Excel, fit two straight lines (you decide which two) to the Bullard plot data (x=R, y=T).

(7) Display these two straight lines on the Bullard plot in (5).

(8) Comment on where the heat in the section is being generated or removed, at what depth and how much.

Hand in your two plots (2) and (7), together with answers to questions (1), (4), and (8).

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