Page 222 - Physical Principles of Sedimentary Basin Analysis
P. 222

204                             Subsidence

                 is the thermal expansibility. The density has the reference value   0 at the reference temper-
                                          ◦
                 ature T 0 , which is taken to be 0 C. A typical value for the thermal expansibility of mantle
                 rocks is 3 · 10 −5  K −1 . The temperature difference between the top and the base of the
                                       ◦
                 lithosphere is roughly 1000 C, and the density difference between the base and the top is
                 therefore ∼3%. Although a density difference of only 3% may be considered small it is
                 nevertheless important. The crust and the mantle have different densities, and they are as a
                 function of temperature:
                                 c (T ) =   c,0 (1 − αT ) and   m (T ) =   m,0 (1 − αT )  (7.24)
                 where the subscripts c and m denotes crust and mantle, respectively. The thermal expansi-
                 bility α is taken to be the same for both rock types. We need the temperature as a function
                 of depth before we can use the densities (7.24) in a subsidence calculation. The initial
                 lithospheric temperature is
                                                         z
                                                T 0 (z) = T a                       (7.25)
                                                         a
                 and the temperature after instantaneous and uniform stretching is (see Figure 7.8)
                                                        z          a
                                                 ⎧
                                                 ⎨ T a β  ,  0 ≤ z ≤
                                                 ⎪
                                            +           a          β
                                   T I (z, t = 0 ) =        a                       (7.26)
                                                 ⎩ T a ,      < z ≤ a.
                                                 ⎪
                                                            β
                 The assumption of isostasy states that the pressure at the same depth in the asthenosphere
                 remains the same:
                               c         a
                                 c dz +    m dz =
                             0         c
                                  c/β        a/β
                                                          a

                                     c dz +      m dz + a −  − s I   m (T a ) + s I   s .  (7.27)
                                0          c/β            β
                 The right-hand side is the pressure in the asthenosphere beneath the unstretched litho-
                 sphere, and the left-hand side is the pressure beneath the stretched lithosphere. The initial
                 temperature (7.25) is therefore used in the densities on the left-hand side of equation (7.27),
                 and the transient temperature (7.26) is used on the right-hand side. The contribution   s s
                 from a sedimentary basin of thickness s, where   s is the average density of the sedi-
                 ments, is added on the right-hand side. The sedimentary cover is considered so thin that
                 any temperature dependence can be neglected. Isostatic equilibrium (7.27) gives the basin
                 subsidence

                                                      c      1    c   1
                                       
    m,0 −   c,0  1 − α T a  −   m,0 α T a
                                      1               a      2    a   2
                           s I = a 1 −                
                             (7.28)
                                      β                    1
                                                     m,0 1 − α T a −   s
                                                           2
                 after the integrations over the densities are carried out (see Note 7.1 for details). The
                 subsidence (7.28) accounts for the hot lithosphere that results from the uniform and instan-
                 taneous stretching. Thermal expansion of the lithospheric mantle leads to reduced subsi-
                 dence, which can be seen from equation (7.28). We notice that the last term   m,0 αT a /2 can
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