Geology 222b Problem Geothermometry 1. Show the following on a single plot of Temperature (horizontal axis -- increasing to the right) versus Depth (vertical axis -- increasing downward from the surface of the earth): a) a linear geothermal gradient of 15 K/km ( Blueschist metamorphic geotherm ) b) a linear geothermal gradient of 30 K/km ( Barrovian metamorphic geotherm ) c) a linear geothermal gradient of 60 K/km ( Buchan metamorphic geotherm ) d) the stability fields of the aluminosilicate minerals according to Michael Holdaway (triple point: 3.76 kb, 501 C; 1 bar intercepts: Kyanite-Andalusite = 200 C, Andalusite- Sillimanite = 770 C; Kyanite-Sillimanite curve passes through 10 kb, 810 C) e) a "wet" granite melting curve that passes through the following points P (GPa) T( C) 0.00 985 0.15 725 0.30 688 0.45 645 0.60 630 0.70 625 0.80 610 Use graph paper or a plotting program such as Excel or Kaleidagraph please. Assume that the density of the crust is uniform at 2.85x10 3 kg/m 3. Use g=9.78 m/s 2. The three geothermal gradients should pass through the normal conditions at the surface (0 C or 273 K,and 1 bar). You may use either C or K for the temperature axis. 2. On the same Temperature-Depth graph, show a steady-state geotherm for a 30 km thick crust with the following properties: a thermal conductivity of 2.5 W/mK, an average heat production of 2.0 x 10-6 W/m 3, and heat flux from the mantle into the base of the crust of 0.010 W/m 2 as derived in the attached model calculation. 3. Calculate a metamorphic temperatures using the Ferry-Spear geothermometer (see xerox of article) for the following garnet-biotite pair compositions. Plot your results on the graph from (1). Assume that the pressure is 2 kb. (a) (X Alm ) Gar = 0.89, (X Ann ) Bio = 0.63 (b) (X Alm ) Gar = 0.82, (X Ann ) Bio = 0.52 where (X Alm ) Gar stands for the mole fraction of almandine in the garnet and (X Ann ) Bio stands for the mole fraction of annite in the biotite.
A Crustal Geostatic Gradient Pressure increases with depth in the earth due to the increasing mass of the rock overburden. Computing the pressure as a function of depth in a homogeneous crust is a straightforward calculation. In SI units, pressure (Pascals) is the force (Newtons) per unit area (meters 2 ) such that 1 Pa = 1 N/m 2. You may also see pressure written as bars or atmospheres with 1 bar = 1 x 10 5 Pa = 0.9872 atm. To see how the pressure would increase with depth in the crust (the geostatic gradient), consider the pressure beneath a one meter cube of granite (density = 2.8x10 3 kg/m 3 ). The force applied by the 2.8x10 3 kg of this cube to the rocks beneath it is given by force = mass x acceleration = (2.8x10 3 ) (9.8 m/s 2 ) = 2.7x10 4 N. where (9.8 m/s 2 ) = g, the acceleration of gravity at the surface of the earth. Because this force is distributed across the 1 m 2 area of the base of the cube, the pressure beneath the cube is pressure = 2.7x104 N 1m 2 = 2.7x10 4 Pa. If another cube is placed on top of the first one, the pressure under the two cubes will be 5.4x10 4 Pa. As more cubes are stacked, the pressure at the base rises at the rate of 2.7x10 4 Pa/m = 2.7x10 7 Pa/km = 27 MPa/km = 270 bars/km where MPa (=10 6 Pa) stands for megapascals. Alternatively, this pressure distribution may be expressed as 3.7 km/kbar = 37 km/gpa where GPa (=10 9 Pa) stands for gigapascals. Remember that these numbers are only correct for a uniform crustal density of 2.8x10 3 kg/m 3. Higher densities will yield higher pressure gradients. The geostatic gradient changes with depth as the density increases. Our procedure may be generalized to the earth with the following differential equation: dp(r) dr = g(r)ρ(r) where r is the radial distance from the center of the earth. By integrating this equation, pressure can be found for any depth if density and gravity are known. Density, gravity, and therefore pressure vary with depth as shown in the following graphs found in Tromp (2001):
Our linear approximation predicts a pressure of 54 GPa at a depth of 2000 km, whereas the model shown in the graph predicts a pressure of 87 GPa at that depth.
Steady-State Geotherm Problem : Calculate the setady-state geotherm for a 30 km thick crust with a uniform distribution of heat producing elements. Assume that the average heat production (A) is 2.0 x 10-6 W/m 3, that the steady mantle heat flux into the base of the crust is 1.0 x 10-2 W/m 2, that the thermal conductivity (k) of the crust is 2.5 W/mK, that the volumetric heat capacity (ρc P ) of the crust is 2.5 x 10 6 J/m 3 K, and that the temperature (T) at the surface is 0 C. Let the depth z=0 at the surface and z=- 3.0 x 10 4 m at the base of the crust. The required heat conduction equations are: heat flux = k T z t and T t z = k ρc P 2 Τ z 2 t + Α ρc P where t (s) is the time, ρ (Kg/m 3 ) is the density of the crust, and C P (J/KgK) is the specific heat capacity of the crust. The second equation assumes (1) that the thermal parameters for the crust are uniform throughout the crust and (2) that the symmetry of the problem permits a one-dimensional solution. In the steady-state, T/ t = 0. Therefore, the heat conduction equation reduces to d 2 Τ = A dz 2 k, which is a comparatively simple differential equation. The solution is of the form Τ= A 2k z2 + αz + β with dt dz = A k z+α where α and β are constants. At the surface, z=0 and T=0; therefore, β=0. At z=-30,000 m, heat flux = k dt dz = k A k z kα = 0.01 W/m2, which may be solved for α to yield The solution is then α= Az k 0.01 = 2.0x10-6 3.0x10 4 k 2.5 0.01 2.5 = 0.028 K/m. T= 4.0x10-7 z 2 0.028 z. The heat flow at the surface for this model is given by heat flux = k dt dz z=0 = kα = (2.5) (0.028) = 0.07 W/m2.
-2.0-1.8 T( C) 800 750 700 650 600 550 lnk D = -2109 + 0.782 T(K) ln KD -1.6-1.4-1.2-1.0 9.0 10.0 11.0 12.0 10,000/T(K) Mg Fe K D = Mg Fe garnet biotite
K D = Mg Fe garnet Mg Fe biotite
14 12 P kbar 10 8 6 KD = 0.10 KD = 0.15 KD = 0.20 KD = 0.25 KD = 0.30 4 2 Ky And Sil 0 400 500 600 700 800 900 T C
T 500 C K D = 0.14 T 800 C K D = 0.30 Al2O3 Al2O3 (a) (b) Garnet Garnet FeO MgO FeO MgO Biotite Biotite Mg/(Fe+Mg) garnet 0.0 0.2 0.4 0.6 0.8 1.0 (c) Mg/(Fe+Mg) garnet 0.0 0.2 0.4 0.6 0.8 1.0 (d) 0.0 0.2 0.4 0.6 0.8 1.0 Mg/(Fe+Mg) biotite 0.0 0.2 0.4 0.6 0.8 1.0 Mg/(Fe+Mg) biotite