Another reasonable assumption is to consider the change in atmospheric pressure to be sinusoidal. Actual weather systems do give a fair approximation to this, with the change from high to low pressure and back again occurring typically with a time scale of around five days. In addition wind effects can be represented by the superposition of different sinusoidal changes in pressure with time scales of order seconds, and this was considered by Kimball and Lemon [Kimball 71]. Some work by Hintenlang and Al-Ahmady
[Hintenlang 92] suggests that atmospheric tides, with a 12 hour cycle like ocean tides also affect soil gas flow.
Hence we see that there is the possibility of atmospheric or surface pressure cycling over a whole range of time scales, making a general result of considerable value.
Although we wish to represent a sinusoidal variation, it is most easily done in the
complex exponential form. We assume that the surface pressure varies as p(t) = e*^, then we expect a solution in the form P = + Q (z ). e*^. In equation (3.1) this gives for Q
iio t f g Ü _ _ 2 dz"
= . i ( ù . . Q{z) . (3.22)
This gives the general solution for Q as
Q = A.exp(az/îw) + B.exp(-az/fw ) . (3.23)
But the root of i is (l+i)/\/2, and the solution for Q must be finite as z tends to infinity, so that A must be 0. This gives Q, and hence P, as:
P = Pq + P , . e x p ( - , | ^ a z ) . e x p ( - i ^ y a z ) = Pq ^ P , ■ e x p ( - , J ^ a z ) . c o s ( w t - ,
(3.24) where P^is the amplitude of the surface pressure fluctuation. Generally the complex exponential would be replaced by cos(cot - az/(co/2)), ie the real part of the result. This result reproduces that of Fukuda. It can then be differentiated to find the velocity at any point from Darcy's Law.
— az) . exp(-z
2 \ 2 (3.25)
Taking the real part of this and using it in Darcy's Law, and substituting C for a\/(w/2), and then taking the real part of the expanded expression (3.25) gives
kCP, e
V = --- . [cos((or - Cz) - sin(o) r - Cz)] . (3.26) P
When z=0 this simplifies to
kCP,
y(0) = . [coscor - sinw f] . (3.27)
But C = a\/(a)/2), and a^ = ep/kP^ so that C = / (coep / 2kPo) and the permeability k and viscosity p appear in the full expression as root terms, equation (3.27) becoming
V(0) = p , .
\
ekixi r 1
. [coscof - smwf] . (3.28)
This result means that the permeability has a square root impact, ie less than linear, on the velocity of soil gas flow from the soil surface. This is of essentially the same form as that for the linear pressure rise considered earher. The finding that the permeability has only a square root impact on the velocity in equation (3.28) is interesting, since it is not the same as in the Darcy Law (3.13). It reflects the fact that the soil gas pressure follows the surface pressure, and more permeable soils follow more closely than less permeable ones.
The pressure field decays with depth exponentially, with the decay constant as C. Hence, as expected, the pressure field extends further for larger permeabihties. It is also
predicting that the pressure extends further for lower frequencies of atmospheric pressure fluctuation.
Examples
As with the other two problems the analytical result to this problem lends itself to a visual representation of the pressure developing with time and depth. Figure 3.5 shows the pressure against time at different depths. The pressure at any depth lags behind that at the surface, and is damped so that its absolute value is lower. Figure 3.6 shows the velocity developing with time with the same parameters as Figure 3.5.
Graph of pressure against time at different depths 150
S.
© 3 (A (A (1) C L C o CO -50 ■c 0) -100 CL -150 0 50 100 150 200 T im e (hours) -B- D epth = 0 (m) ^ 1 -A-2 -0- 3 4 -6-5F igu re 3.5: G raph o f p re ssu re a g a in st tim e f o r the sin u so id a l p r e s s u r e p r o b le m
Graph of velocity against time at different depths
-52 E 2=' o 0 1 S o 0.04 0.02
ss
-0.02 - 0 50 100 Tim e (h ou rs) 150 200 •B D epth = 0 (m ) ^ 1 6 - 2 © 3 -©■ 4 -6-5F igure 3.6: G raph o f ve lo city a g a in st tim e f o r the sin u so id a l p r e s s u r e p r o b le m
An interesting feature of the plots against time is that there is generally a depth in the soil where the pressure is in the opposite phase to that at the surface. This is because of the time lag for the changing pressure signal to reach the lower levels of the soil. The radon modelling work by Hintenlang and Al-Ahmady in Florida [Hintenlang 92] is based on
this concept. The atmospheric pressure fluctuations they took to occur due to
Atmospheric Tides, with a time cycle of around 12 hours. The fact that the inside of the basement of a house was in step with the external pressure then allowed radon-laden air to enter from the soil next to it, which is at a higher pressure.
To examine how significant this effect can be, substitute for a in (3.24) and look at the phase effects on the pressure:
P = Pq + . exp \ c o e p COS (ùt - z N (3.29) The exponential term indicates the damping effect of the soil on the overall size of the pressure, and so has no impact on the phase of the pressure wave. However if w is large and k small then the wave will be damped fairly quickly.
The expression in the cosine term defines the phase of the pressure at any point. If the second part, which includes the z term, is equal to t z then the pressure at that depth z will
have the opposite phase to that at the surface. Using the same values for the parameters in the expression that were used before:
Porosity € 0.5 (),
Viscosity p 1.83 e-5 (Pa.s), Atmospheric pressure P^ 1 e+5 (Pa),
the expression for the first depth, where the soil pressure is first of opposite sign to the surface pressure is
^opposiie = 4.64e5 = 1.85e5 . (3.30)
where
o) defines the rate of change in surface pressure, and is given by 2 % /t, T is the time for one complete cycle.
Thus to find a typical value of the depth at which the pressure is first in the opposite phase we need to choose an appropriate value for o). Wind gustiness can be considered to be a change with time scales of order 10 seconds, while the atmospheric tides have a cycle of 12 hours. These give values for w of 0.63 and 1.45x10 '* s'* respectively.
Substituting in (3,30) gives simple expressions for the depth value as a function of the permeability as follows
10 second 12 hour
(3 31) t '" 3 . 8 x 1 0 '.
Hence if we are most interested in a depth of 1 metre, the permeability at which it is first exactly out of phase is
10 second 12 hour
(3 32)
1x 1 0 - ' ^ .
These results only indicate a feature of a soil type but help to explain the atmospheric pumping effect over a half day time scale, which can result in increased entry rates. When the atmospheric pressure is lower than its mean value, flow could occur into a basement. The expression here for phase would only matter when greater accuracy was needed, or in very permeable soil.
Conclusions to chapter 3
The results in this chapter show how soil gas pressure responds to atmospheric pressure changes, and allows us to understand some aspects of the gas flow produced by these atmospheric driving forces.
If there is a steady change in atmospheric pressure, due to a passing weather front, it can produce a flow as large as that due to the pressure which has built up in a landfill site. As a result the weather conditions need to be taken into account when considering the measurements taken on site.
When a pressure is switched on instantly, for example with a fan, then there is a time lag before the effect reaches soil at greater depth. This would be seen on site with a delay before the concentration of gas changed significantly.
When a sinusoidal variation of pressure is assumed, different depths in the soil will be at different phases of the cycle. This could help to cause the entry of soil gas into houses, because the soil at depth is out of phase with that in the house.