Estimates of heat flow and heat production and a thermal model of the São Francisco craton

Abstract

An updated analysis of geothermal data from the highland area of eastern Brazil has been carried out and the characteristics of regional variations in geothermal gradients and heat flow examined. The database employed includes results of geothermal measurements at 45 localities. The results indicate that the Salvador craton and the adjacent metamorphic fold belts northeastern parts of the study area are characterized by geothermal gradients in the range of 6–17°C/km. The estimated heat flow values fall in the range of 28–53 mW/m², with low values in the cratonic area relative to the fold belts. On the other hand, the São Francisco craton and the intracratonic São Francisco sedimentary basin in the southwestern parts are characterized by relatively higher gradient values, in the range of 14–42°C/km, with the corresponding heat flow values falling in the range of 36–89 mW/m². Maps of regional variations indicate that high heat flow anomaly in the São Francisco craton is limited to areas of sedimentary cover, to the west of the Espinhaço mountain belt. Crustal thermal models have been developed to examine the implications of the observed intracratonic variations in heat flow. The thermal models take into consideration variation of thermal conductivity with temperature as well as change of radiogenic heat generation with depth. Vertical distributions of seismic velocities were used in obtaining estimates of radiogenic heat production in crustal layers. Crustal temperatures are calculated based on a procedure that makes simultaneous use of the Kirchoff and Generalized Integral Transforms, providing thereby analytical solutions in 2D and 3D geometry. The results point to temperature variations of up to 300°C at the Moho depth, between the northern Salvador and southern São Francisco cratons. There are indications that differences in rheological properties, related to thermal field, are responsible for the contrasting styles of deformation patterns in the adjacent metamorphic fold belts.

Appendix: Theoretical basis for the combined solution of heat conduction equation, involving Kirchoff and generalized integral transform techniques

The two dimensional heat conduction equation for a medium with heat sources may be written as:

$$ \frac{\partial}{{\partial z}}{\left[{\lambda {\left(T \right)}\frac{{\partial T}}{{\partial z}}} \right]} + \frac{\partial}{{\partial x}}{\left[{\lambda {\left(T \right)}\frac{{\partial T}}{{\partial x}}} \right]} = - A{\left({x,z} \right)} $$

(14)

Consider now the following boundary conditions:

$$ T{\left({x,z = 0} \right)} = T_{0} (x) $$

(15a)

$$ \frac{{\partial \,T\,{\left({x,\,z = L_{V}} \right)}}}{{\partial \,z}} = \,f_{1} (x)\, = \,\frac{{q_{L} (x)}}{{\lambda \,(T)}} $$

(15b)

$$ T{\left({x = 0,z} \right)} = f_{2} {\left(z \right)} $$

(15c)

$$ T{\left({x = L_{H}, z} \right)} = f_{3} {\left(z \right)} $$

(15d)

The function T ₀ (x) represents the temperature distribution in the domain 0 > x > L _H . The functions f ₂(z) and f ₃(z) are respectively, solutions of 1D problems at x = 0 and x = L _H . All functions are known, with the exception of f ₁(x) which determines the heat flux at the base of the model crust. It is for this reason that a combination of two transform techniques is necessary:

• (a) Kirchoff transform, that removes the non-linearity in thermal conductivity, when the temperature dependence is written in the form:

  $$\lambda {\left(T \right)} = \lambda _{0} {\left[{1 + BT} \right]}$$

  (16)

• (b) The Generalized Integral Transform Technique, that removes the non-homogeneity of the differential equation and the boundary conditions. Assuming that the vertical distribution of radiogenic heat production in the first 10 km of the crust obeys the relation (Lachenbruch 1970):

  $$ A{\left(z \right)} = A_{0} \exp (- z/D) $$

  (17)

We adopt a strategy composed of the following steps:

• (a) Divide the domain 0 > x > L _H , by the number of points for which heat production data are available;

• (b) For each position x use Eq. (17) to determine heat production in the domain 0 > z > L _V ;

• (c) Setup the function A(x, z) using the multiple regression theory (or polynomial):

  $$A{\left({x,z} \right)} = a_{1} + a_{2} x + a_{3} z + a_{4} x^{2} a_{5} xz + a_{6} z^{2}$$

  (18)

After the conclusion of steps a, b and c we apply the Kirchoff transform in the Eqs. (14) and (15) and we have the transformsT→ U:

$$ \frac{{\partial ^{2} U}}{{\partial z^{2}}} + \frac{{\partial ^{2} U}}{{\partial x^{2}}} + \frac{{A{\left({x,z} \right)}}}{{\lambda _{0}}} = 0 $$

(19)

$$ U{\left({x,z = 0} \right)} = 0; \frac{{\partial \,U{\left({x,z = L_{V}} \right)}}}{{\partial \,z}} = f_{1} ^{*} (x); U{\left({x = 0,z} \right)} = f_{2} ^{*} {\left(z \right)}; U{\left({x = L_{H}, \,z} \right)} = f_{3} ^{*} {\left(z \right)} $$

The solution of Eqs. (19) may be obtained using the technique of Generalized Integral Transform (Cotta 1993). The solution of the problem in U allows us return to the domain in T, using the relation:

$$ T(x,z) = \frac{{{\left[{BT_{0} - 1 + {\sqrt {1 + 2BU{\left({x,z} \right)}}}} \right]}}}{B} $$

(20)

FORTRAN subroutines available in IMSL (1989) were used in numerical computation of the transforms.