5  Diffusion Equation

Herbert F. Wang

5 Diffusion Equation

The study of heat diffusion has a long and storied history in mathematical physics beginning with Fourier in 1822. The rate of heat transport through a material depends, of course, on thermal conductivity, but also storage, because the rate of temperature change in a REV depends on how much heat must be transported in order to change its temperature. For the same thermal conductivity, heat diffusion is slower for a high heat capacity than for a low heat capacity. Thus, heat capacity is something of a foil to thermal conductivity. This inverse role of storage to conductivity in determining the rate of diffusion is captured by taking their ratio. In heat transport, it is called thermal diffusivity, ĸ = K/(????c), where K is thermal conductivity, ρ is density, c is the specific heat capacity, and ????c is specific heat per unit volume. The groundwater analog is that ĸ = T/S for two-dimensional, horizontal flow; and ĸ = K/S_s for one-dimensional linear flow. All diffusivities have units of m²/s. The definitions of hydraulic and thermal diffusivity are shown in Table 3, which is a continuation of Table 1. Theis did not include the diffusion equation in his 1935 paper, probably because it was so well known in mathematical physics, and so it is included in Table 3. Theis did, however, present Lubin’s derivation of the thermal solution analogous to a pumping well. The derivation started with the solution for the “instantaneous line-source coinciding with the axis of z of strength Q”[9] (Table 3). The analogous solution for head, h, is the same if Q is interpreted to be a slug of water added to a well per unit length.

Table 3 – Diffusion equation and Theis’ solution for groundwater flow and heat flow.

	Groundwater Flow	Heat Flow
Diffusivity (ĸ)	Hydraulic diffusivity	Thermal diffusivity
Diffusivity (ĸ)	K/S_s [m²/s]	K/(ρc) [m²/s]
Governing equation for planar flow	$\displaystyle \kappa \left ( \frac{\partial ^{2}h}{\partial x^{2}}+ \frac{\partial ^{2}h}{\partial y^{2}} \right )=\frac{\partial h}{\partial t}$	$\displaystyle \kappa \left ( \frac{\partial ^{2}T}{\partial x^{2}}+ \frac{\partial ^{2}T}{\partial y^{2}} \right )=\frac{\partial T}{\partial t}$
Line source solution	$\displaystyle h=\frac{Q}{4\pi Kt}e^{-(x^{2}+y^{2})/4\kappa t}$	$\displaystyle T=\frac{Q}{4\pi Kt}e^{-(x^{2}+y^{2})/4\kappa t}$
Line source definition	Q is the volume of water per unit length added to the aquifer in a line at the origin	Q is the quantity of heat per unit length added to the plate in a line at the origin
Approximate propagation time to x	t = x²/(2ĸ)

The one-to-one correspondence between columns in Tables 1 and 3 is the physical and mathematical analogy Theis exploited. Both temperature and head are governed by the diffusion equation. In two-dimensional radial flow, a high transmissivity means that fluid is transported quickly in space. On the other hand, high storativity delays the movement because more fluid must be moved into or out of storage to effect a given head change. At an extreme, if an aquifer system has zero storage, then any change of internal or external boundary conditions in flow or pressure will be rapidly accommodated by a new steady state. Hydraulic diffusivity determines how fast a disturbance, such as suddenly injecting a slug into a well or starting to pump a well or changing a boundary condition, will propagate through an aquifer. The nature of diffusion is that an initial spike in head spreads out and diminishes in amplitude. Diffusion “flattens the gradient” so that the system’s approach to steady state slows with time (Figure 10a). Meanwhile, at any point away from the source, the disturbance rises with time, peaks, and then decays (Figure 10b). The time of arrival of the peak occurs at t = x²/(2ĸ) in the case of one-dimensional linear flow. This value of t is a characteristic time for how long it takes for a sudden change at the origin to decay and spread out to a distance x. In the example in Figure 10b, it takes about one hour for the peak to arrive at 200-meters distance for a diffusivity of 6.8 m²/s.

Graph showing solutions for an instantaneous slug injected into a well in an aquifer — **Figure 10 –** Solutions for an instantaneous slug injected into a well for an aquifer with hydraulic diffusivity of 6.8 m²/s. a) Profiles at different times showing how hydraulic diffusion flattens the gradient. b) Time history showing a delay of approximately one hour for the peak head to arrive at 200-meter distance.

[9] The “Q” in Table 3 differs from Theis’ because his definition “Q” included dividing by the specific heat per unitvolume rather than incorporating it explicitly as is done here. Also, Theis, following Carslaw, used “v” instead of “T” for temperature.

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