Page 178 - Physical Principles of Sedimentary Basin Analysis

160 Heat flow

Dike

Sills

Figure 6.26. Sills are tabular igneous intrusions that are fed by near-vertical dikes.

boundary conditions and an initial condition that represents the hot intrusion. The differ-
ence between the temperature equation for sills and dikes is that the temperature of a sill
depends on the z-coordinate and the temperature of a dike depends on the x-coordinate.
◦
We will assume that the country rock has initially the temperature T = 0 C and that the
sill or the dike has a width 2a and an initial temperature T 0 . The thermal diffusivity (κ)of
the intrusion and the country rock do not differ much and can therefore be assumed to be
the same.
Sills and dikes have half the thickness a as a characteristic length, and the sill/dike
temperature at emplacement T 0 as a characteristic temperature. We can scale the length
by a and the temperature by T 0 to make a dimensionless version of the time-dependent
temperature equation (6.226). For sills we get the following dimensionless temperature
equation:
∂T ˆ ∂ T
2 ˆ
− = 0 (6.226)
∂ ˆ t ∂ ˆz 2
2
ˆ
where T = T/T 0 and ˆz = z/a, and where t is scaled with the characteristic time t 0 = a /κ.
We will let ˆz = 0 be the center of the sill, which means that the sill extends from ˆz =−1
to ˆz = 1. Boundary conditions for the temperature equation are
ˆ
ˆ
T (ˆz=−∞, ˆ t) = 0 and T (ˆz=∞, ˆ t) = 0 (6.227)
which say that there is no thermal impact on the country rock “far” away from the sill. The
initial condition is the temperature T = 1 inside the sill and T = 0 outside the sill. The
ˆ
ˆ
solution of the temperature equation (6.226) with these boundary conditions and the initial
condition is
1 ˆ z + 1
ˆ z − 1
ˆ
T (ˆz, ˆ t) = erf √ − erf √ , (6.228)
2 2 ˆ t 2 ˆ t
as explained in Note 6.8. The solution is plotted in Figure 6.27, which shows how the sill
cools. For ˆ t = 1 the temperature has decreased to T ≈ 0.5 at the center of the sill, and for
ˆ