Geophys. J. Int. (1995) 123, 149-160

Numerical models of a subsidence mechanism in intracratonic basins: application to North American basins

Boris M. Naimark and Ali T. Ismail-Zadeh International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Warshavskoye shosse 79, kor 2, Moscow 113556. Russia

Accepted 1995 April 17. Received 1995 March 10; in original form 1993 November 11

SUMMARY McKenzie's model of sedimentary basin evolution and its modification, widely used in geophysics, sometimes fails to explain discrepancies between predicted and observed values of extension, thinning and subsidence of the Earth's crust, as for the North Sea. We develop a numerical model of sedimentary basin evolution based on the mechanism suggested by Lobkovsky. In the course of rifting, accompanied by thinning of lower parts of the lithosphere, the roof of the underlying asthenosphere moves upward. The material of the mantle lifts and partially melts owing to the reduction of pressure. The density difference between the melt and the crystalline skeleton results in the filtration of the lighter melt and its accumulation in the form of a magmatic lens. Due to changed P-T conditions, the material of the lens undergoes the gabbro-eclogite phase transform- ation. The resultant anomalously heavy eclogite lens sinks in the surrounding material. This induces a viscous flow, changing the surface topography and forming a sedimentary basin. We construct a 2-D numerical model describing a viscous flow induced by subsidence of a heavy body and compute changes of surface topography. To compute the flow we employ the Galerkin-spline approach, with modifications allowing for density discontinuities and time dependence of the phase transformation. We apply the model to the cases of the Illinois, Michigan and Williston basins. The computed and tectonic subsidence curves agree well for these cases. The proposed model is compatible with the seismic structure of the crust and upper mantle below these basins. The model is also consistent with gravity data. The approach is applicable to other intracratonic basins.

Key words: bicubic spline, Galerkin's method, intracratonic basin, North America, phase change, subsidence.

INTRODUCTION

The nature of the mechanisms of intracratonic basin formation still attracts considerable attention from geophysicists. There are several mechanisms for driving the subsidence of North American intracratonic basins: thermal decay; subaerial ero- sion; subcrustal erosion; phase changes in the lower crust; and subsidence related to horizontally transmitted stress (Quinlan 1987). Most attempts to explain the subsidence of intracratonic basins have centred on the possible role of thermal contraction: cooling of a lithospheric plate causes contraction, an increase in density, and subsidence of the plate surface. This mechanism of subsidence was proposed for the Williston Basin by Abern & Mrkvicka (1984). They assumed that the intrusion of hot matter beneath the basin cooled, contracted, and acted as a load. Heidlauf, Hsui & Klein (1986) suggested that thermal subsidence of the Illinois Basin was in response to mantle intrusion which formed a rift system.

Another attractive mechanism of subsidence in intracratonic basins is the phase transformation from less dense to more dense rocks in the lower crust. The gabbro-eclogite phase transition, taking place at high temperatures (1000-1200 "C) and pressures (12-13 Kb), was observed in experiments (Yoder & Tilley 1972; Ringwood & Green 1966; Green & Ringwood 1967, 1972; Cohen, Ito & Kennedy 1970). Phase transition of this type leads to the replacement of gabbro (with a density of approximately 3.0 x lo3 kgmW3) by eclogite (with a density of approximately 3.5 x 103kgm-3).

Haxby, Turcotte & Bird (1976) suggested that the first stage of the Michigan Basin evolution involved diapiric penetration of hot asthenospheric mantle rocks to the base of the lower crust. The diapiric intrusion triggered a phase change from gabbro to eclogite and induced the subsidence of a basin under the load of the eclogite. They studied the mechanical flexure of the lithosphere in the development of the Michigan Basin as a response of the basement to a load of such a type

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150 B. M . Naimark and A. T. Ismail-Zadeh

and modelled the behaviour of the lithosphere as a flat elastic plate.

Fowler & Nisbet (1985) proposed that the subsidence of Williston Basin was not controlled by thermal decay, as had been assumed by Ahern & Mrkvicka (1984). They speculated that non-exponential subsidence was caused by phase trans- formation from subcrustal body (gabbro) to eclogite. Hamdani, Mareschal & Arkani-Hamed (1991, 1994) studied the com- bined effect of thermal contraction due to cooling and phase transformation in the lower crust for the cases of the Michigan and Williston basins. They suggested that the delay of the phase change could explain an acceleration of subsidence in the early stages of their evolution.

The detection of anomalously high seismic P-wave velocities within the lower crust and upper mantle can serve as an experimental verification of possible phase changes. Such ano- malies were found beneath the Williston and Illinois basins (Hajnal et al. 1984; Sexton et al. 1986). The anomaly was not found in the Michigan Basin due to a lack of seismic data (Quinlan 1987). In general, heavy eclogite bodies, evolved long ago, can sink deep enough in the asthenosphere that they cannot be detected by seismic methods.

The idea of the gabbro-eclogite phase change in the lower crust is based on the linear extrapolation of the experimental P-T curve to the range of relatively low pressures and tempera- tures typical of the lower crust. However, such extrapolation has not been justified theoretically or experimentally until now. Moreover, it has been found experimentally that phase-trans- ition curves are significantly non-linear for rocks that are dry or have a low water content; when the temperature of the phase transition attains the values found in the lower crust, the pressure remains high, as in the uppermost mantle (Carswell 1990).

Lobkovsky et al. (1993) suggested an alternative mechanism of sedimentary basin evolution. In the course of active rifting, accompanied by thinning of lower parts of the lithosphere, the roof of underlying asthenosphere, usually identified with the

‘ isotherm approaching the solidus of mantle rocks, moves upward. The material of the mantle lifts with the roof of the asthenosphere and partially melts owing to the reduction in pressure. The density difference between the melt and the crystalline skeleton ranges from 0.1 x lo3 to 0.4 x 103kgm-’, which results in the filtration of the lighter melt within the bulge. When the material above the bulge is impenetrable to the melt, it accumulates and forms a magmatic basalt lens. After the extension of the lithosphere has terminated, the melt crystallizes due to changed P-T conditions and then turns into an anomalously dense eclogite lens. The heavy eclogite lens sinks in the surrounding material, which results in a viscous flow and the formation of a basin. The quantitative models of the proposed subsidence mechanism were analysed by Lobkovsky et al. (1993), Ismail-Zadeh, Naimark & Lobkovsky (1994), Naimark & Ismail-Zadeh (1994).

Previously, Haxby et al. (1976), Brunet & Le Pichon (1982) and Ahern & Mrkvicka (1984) suggested models of intracra- tonic basin evolution where the crust was considered as a perfectly elastic plate underlain by a fluid substrate (upper mantle) and bending under loads. However, it is difficult to imagine the crust remaining perfectly elastic over time-scales of 100-300Ma. Creep effects must be significant in such time intervals. Evidently, the rheology of the crust is not Newtonian,

but more complex. We assume Newtonian rheology as an approximation.

We develop an advanced model of sedimentary basin evol- ution based on the subsidence of heavy bodies in the astheno- sphere. The improved Galerkin-spline approach is Jsed for numerical study of the model where density discontinuities ?re included. The model is applied to the Illinois, Michigan and Williston basins.

FORMULATION OF THE MODEL

The model suggested is briefly sketched in Fig. 1. The melt, a light component of the partially melted asthenospheric mate- rial, concentrates in the asthenospheric bulge and undergoes the gabbro-xlogite phase transformation due to changed P-T conditions. The total mass and volume of the transformed material grow with time, as shown in Fig. 1. The filtration of melt and its gabbro-eclogite phase transformation are not necessarily separated in time: when a part of the gabbro melt is turned into eclogite, another part of melt is added from below. The time-scale of eclogitization depends on the rate of cooling when the latent heat of phase transformation is removed from the transforming material.

The rate of eclogitization can be derived from the Stefan problem, with the time-scale t , = h2/(~A2) where his the distance from the transforming material to the surface, u is the thermal diffusivity, and 1 is a value obtained from the problem param- eters. In the simplest case of a plane phase boundary separating a layer from a half-space, 1 is obtained from the equation [Turcotte & Shubert 1982, eq. (4.134)]

where c, is the specific latent heat of the transformation, cp is the specific heat at constant pressure, 8,h is the temperature of the phase boundary (the temperature of the transforming material is assumed to be equal to $,h), go is the temperature at the surface, and erf is the error function. The analysis shows that 1+ 00 as the left-hand side of the equation tends to zero. For various simple cases of the Stefan problem, I in the range 0.8-2.0. Taking h = 6 x 104m, u = 10-6m2s-’ and A = 1, we obtain t , = 36 x 1014 s = 114 Ma. In addition to conductive cooling, some kind of convective cooling can occur, thus reducing t , .

The gabbro-eclogite reaction time has strong effects on the temporal and spatial properties of a basin’s subsidence. Ahrens & Schubert (1975) estimated the time-scale as 107-108 years in the case of transformation of gabbros of the lithosphere when it spreads from a ridge. For the transformation of eclogite upon subduction, times are of the order of 105-106 years. Moreover, the kinetics of the gabbro-eclogite transition are controlled by the presence of volatiles and minor hydrous minerals, and by the temperature history of the rock (Fowler & Nisbet 1985).

To model, however roughly, the rate of eclogitization, we use the relation

4 t ) = dini + ( L a x - dini)(l -ex~(-V))

where d( t ) is the density discontinuity across the boundary of a hypothetical eclogite lens, and d,,, and dini are maximum

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Numerical models of a subsidence mechanism 151

A POSSIBLE REALITY IDEALIZATION

-VISCOUS FLUID A HEAW LENS-

t 1

Figure 1. An illustration of the physical mechanism and its idealization accepted for the model. The melt, a light component of the asthenospheric material, moves upward and accumulates in the bulge. A magmatic lens is formed there if the roof of the bulge is impenetrable to the melt. The material of the lens, gabbro, gradually turns into denser eclogite, so that the excessively heavy eclogite body grows with time, and some quantity of melt is probably added from below. The flow induced by the subsidence of the heavy eclogite body results in a depression of the surface. In the idealization accepted for the model an initially elliptic body of a constant volume and variable mass stands for a structure of eclogite and melt. The mass of this idealized body grows with time, the body sinks in the lighter material, and the resultant flow changes the surface topology. The denser the material, the heavier the shading on the figure.

and initial density discontinuities (parameters of the model). The time constant y is related to the time-scale t,. This time comtant is chosen to obtain the best fit of modelled and observed tectonic subsidence curves.

The idealization of the process discussed above is shown in Fig. 1. Instead of a structure composed of eclogite and gabbro in the asthenospheric bulge, we consider an elliptic body (a lens) of space-independent density growing with time. This idealization violates the conservation of mass: the volume of the lens remains constant and its density, hence its mass, grows with time. To avoid this difficulty we assume an additional flow of material from below or, roughly the same, a change of the model height with time. Let us estimate this change of height. We introduce the following notation: the volume of the lens Kn; the volume of the model vmd; the model height H, the change of height dH; and the characteristic value of density ptyp. Obviously, the change of mass in the lens must equal the change of mass at the bottom of the model. This condition implies dH/H = (~n/t(md)(dmax/ptyp). The values of the intro- duced variables are presented below in the discussion of the numerical results. These values yield (v,,/?&) x 0.01 and (dmax/ ptyp) z 0.13. Therefore, dH/H x 0.0013, which changes the aspect ratio, and hence the numerical solution, only negligibly. This idealization, even if in a way non-physical, is suitable for assuming div u = 0 where u is the fluid velocity in the model.

The 2-D model suggested for the analysis is illustrated in Fig. 2. We assume that all initial distributions are symmetric

h a .- - u)

h e .- - v)

d 0-

L ~ 0 , L) ,y /=O (no slip) ~ 0 , f )22p0 (perfect slip)

Figure 2. An illustration of the 2-D model. Curves Pl and 9, divide the right half of 8 : 0 5 x1 I L, -H1 I x2 5 0 into three subdomains. Boundary conditions are shown at respective edges. Obviously, one out of two conditions must be used in calculations.

. o

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152 B. M . Naimark and A. T. Ismail-Zadeh

relative to the vertical axis x2 = 0 and show in Fig. 2 the right half of the domain. The following variables are introduced in the domain R (0 I x1 I L, - H 5 x2 < 0): pressure p , velocity u = (ul,uz), density p(x1,x2, t), and viscosity p(xl,x2, t). Density and viscosity are assumed to be continuous, except for the density across the curves Y,, q = 1,2,. . . , Q, where it can have constant jumps. Two curves, Yl and Y2, are shown in Fig. 2, though only one curve Yl is used in the present model. It is assumed that the curves Y q have no individual or mutual intersections and are either closed, or begin and end at the boundary of a. L and H are horizontal (length) and vertical (depth) dimensions of the domain R.

Hereinafter we assume summation over repeated subscripts and denote D, = a/at, Dj = ajax,, Dkl = DkDl, etc.

A slow flow of an incompressible viscous fluid is governed by equations ( i , j = 1,2)

Dip = D,p(Dj~i + D i ~ j ) + pghi,, Diui = 0 (1)

D , p + ~ , D j p = o , D,p+ujDjp=0

where 6ij is the Kronecker delta and g is the acceleration due to gravity. The first of eqs (1) is the Stokes equation for the 2-D case, the second of eqs (1) represents div u = 0 discussed above, and the third equation is used in the case where density depends on chemical composition only. We neglect this equa- tion in the case where density depends on depth only. The fourth equation of eqs (1) states that the viscosity is transferred with the flow. No-slip boundary conditions take the form u1 = u2 = 0 at the boundary of R, and, in the case of perfect slip, normal velocity u, and shear stress D,u2 -t D,u, vanish at this boundary.

We use a stream function approach and define a numerical solution to eq. (1) in a weak sense. In terms of the stream

function $, eq. (1) takes the form

4D12@12$ + (Dzz - D I M D Z Z - D I N = gDlp

DIP + ( ~ z * ) ( D l P ) - ( D l * ) ( ~ 2 P ) = o

DIP + (D,$)(DlP) - (Dl*)(DZP) = 0 and the boundary conditions are as follows:

$ = 0, Dl$ = 0 at x1 = 0 and x1 = L (no slip),

$ =0, D2$ = O at x 2 = 0 and x2= -H (no slip),

$ = O , Dll$=Oat x,=Oand x,=L(perfectslip),

$ = 0, D,,$ = 0 at x2 = 0 and x2 = - H (perfect slip). (3) The functions p and p satisfy natural boundary conditions at x2 = 0 and x2 = - H (the second derivative vanishes), symmetry conditions at x1 = 0, and periodic conditions at x1 = L (the first derivative vanishes):

DZ2p=O, D, ,p=Oatx ,=Oandx,=-H

D,p = 0, Dlp = O at x1 = O and x1 = L . Unknowns p and p satisfy the following initial conditions. At t = 0,

dp,=dp: (4=1 ,2 ,..., Q),

P=Po(xl,x2), P:(xl,x2)+ P; ( 5 )

(4)

where po(x,,xz), p:(xl, x,), and p: are given functions. Initial distributions po and p: are continuous and pi is a piecewise constant with discontinuities across 2,. Curves dpi are initial positions of curves 2,; coordinates (xlq,xZq) of points on 9, satisfy the set of two ordinary differential equations

dx,,/dt = D2$, dx2,ldt = - Di $ (6)

Figure 3. Location map of the United States showing outlines of the states, sedimentary basins under study, and gravity features. IB, Illinois Basin, MB, Michigan Basin; WB, Williston Basin (modified from Kane & Godson 1985).

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Numerical models of a subsidence mechanism 153

Point 1 2 3 4 5

Values of CI/ptyp at selected points 0.8

Depth. km CI/ptvn 0 1 .o

44 1 .o 96 0.835 148 0.175 200 0.01

0.2 0.85

0.0 0.80 0 50 100 150 200

depth, km

Figure 4. The initial distributions of dimensionless viscosity and continuous component of density. The viscosity distribution (solid line) is a spline drawn through five selected points, and the density distribution (dashed line) is defined tabularly.

lo3, Dens'ty kg.n-3 -2

Depth Topography kn 300 600

(b)

-2 O I U

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-156-

-200 -'-I Topography Depth ToposraDhr Oath

kn 300 600 k n 9 300 600

-22 or---l -22 O 1 n

- 67 -89

-111-

-133-

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-178-

-200-

-67

-89

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Figure 5. Evolution of the Illinois Basin. At time (a) 3 Ma; (b) 36 Ma; (c) 78 Ma; (d) 112 Ma

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154 B. M . Naimark and A. T. Ismail-Zadeh

0 400 a00

with initial conditions

1200

The Illinois Basin

The Illinois Basin is an oval-shaped cratonic basin (Fig. 3) where 6 km of sediments accumulated during Palaeozic time (Heidlauf et at. 1986).

We assume the following typical parameter values for the Illinois Basin: ptypc = 2 x 1223 Pas (the viscosity of the crust), L = 600 km (half-width), din, = 0.35 x lo2 k g ~ n - ~ , d,,, = 0.42 x lo3 kgm-3, y = 0.1 (typical time of eclogitization, TTE, is 10 Ma), a = 0.17 (102 km), and b = 0.021 (4.2 km).

Several stages of evolution are presented in Fig. 5. The figure illustrates the position of the lens, distribution of density, and surface topography at times 3 Ma, 36 Ma, 78 Ma and 112 Ma.

Thus to solve the problem we must find the functions t), p, p, and curves -49, satisfying eqs (2) and (6) with boundary conditions (3) and (4) and initial conditions (5) and (7). The numerical approach based on the Galerkin method and bicubic spline approximations of the unknown functions is described in the Appendix.

NUMERICAL ANALYSES A N D DISCUSSION

We apply the mechanism of subsidence suggested by Lobkovsky et al. (1993) to three North American basins: the Illinois, the Michigan, and the Williston basins (see Fig. 3). We assume that heavy eclogite bodies evolve under these basins. Subsidence of these heavy bodies induces depressions of the Earth's surface. According to stratigraphic data, all these basins have subsidence patterns with rapid and slow phases, though subsidence starts at different times.

We use the numerical approach described in the Appen- dix to model the evolution of North American basins. Dimensionless variables were used in the calculations. The time-scale ttyp was taken as t,,, = ptyp/(ptypgH) where ptyp and ptyp = 3.5 x lo3 kgm-3 are typical values of viscosity and den- sity, respectively, and H = 200 km. In accordance with post- glacial rebound data, the viscosity of the upper mantle for the North American continent is of the order of 102'-1022Pas (Peltier 1984, 1986). This estimate of viscosity was used in the model.

The initial distributions of dimensionless viscosity p0(x2)/ptyp and density pY(xz)/ptyp are illustrated in Fig. 4. These distri- butions are accepted for all three basins. The density in the models is assumed to depend on depth only. We see that the dimensionless viscosity changes from 1.0 at the surface to 0.01 at 200 km depth.

Dimensionless spatial coordinates were chosen to transform the domain Q into the square 0 < x1 5 1,0 5 x2 5 1. We divide Q into rectangular elements: 75 in x1 and 25 in x,; which corresponds to M x N points where M = 76 and N = 26. The curve Y o bounding the lens is initially an ellipse with semi- major and semi-minor axes a and b, respectively, and centre (0.0 0.7) (the depth is 60km).

To calculate the surface topography we used the method described by Ismail-Zadeh et al. (1994). We compute the gravity anomaly Ag at a point (x?,x!) above the upper boundary of models from the equation

x [(XI - x:)' + ( ~ 2 - xi)' + ( ~ 3 - x;)~] -3 '2d~1 dx , dx3

tectonic subsidence - - _ i computed subsidence

Figure 6. Curves of the Illinois Basin subsidence.

120 1 I

1 .f' U (3 E W

x 0

0 C U

x >

- E

-u .- 2 c-7

80

60

40

20

0

MB' WB IB

- - - - - - - -

where G is the gravity constant. Figure7. Curves of present-day gravity anomaly for the models of three basins under study. See abbreviations in Fig. 3.

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Numerical models of a subsidence mechanism 155

These stages correspond to geological times of the Late Cambrian (- 520 Ma BP), the Early Ordovician (- 487 Ma BP), the Late Ordovician (- 445 Ma BP), and the Silurian (-411 Ma BP).

We see that the lens, being initially an ellipse at depth 56-64km, sinks about 9km in 112Ma. This induces a depression of the surface of 1.2km maximum depth. The tectonic subsidence curve (Heidlauf et al. 1986) and the subsid- ence curve calculated from the model for the Illinois Basin are shown in Fig. 6. It shows that the subsidence curve calculated in the suggested model agrees with the observed tectonic subsidence. Inspecting the gravity anomalies over the Illinois Basin (Society of Exploration Geophysicists 1982) and Fig. 7, we see that the computed anomaly agrees with the gravity high. The excessive density of the lens can explain the positive

Depth Topography km 0 400 800

I

(a)

Depth Topography kn 400 800

"I ( C )

-2 Dl -22 O 1 n

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-89 - -111-

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gravity anomaly observed over the Illinois Basin (McGinnis et al. 1979; Kane & Godson 1985).

The Michigan Basin

The Michigan Basin is a circular-shaped cratonic basin (Fig. 3) with the oldest strata of Middle and Late Ordovician age (Haxby et al. 1976). The subsidence history of the Michigan Basin is well documented. It contains 3.5 km of sediments that accumulated between 460 and 400Ma (Hamdani et al. 1991).

A reasonable horizontal dimension of the model for the Michigan Basin is L = 800 km: The following typical para- meter values are assumed: ptyp = 1.5 x loz3 Pa s, dini = 0.35 x 10' kgm-3, d,,, = 0.47 x lo3 kgm-3, y = 0.1 (TTE is about 8 Ma), a = 0.375 (300 km), and b = 0.021 (4.2 km).

Depth Topography km 400 800

1

(b) I - II

-2 OI-

07 I 1

-4 I

Depth Topography kn 400 800

"I (d)

-2 D j n -22

Figure 8. Evolution of the Michigan Basin. At time (a) 1 Ma; (b) 28 Ma; (c) 100 Ma; (d) 151 Ma.

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156 B. M . Naimark and A . T. Ismail-Zadeh

Fig. 8 illustrates stages of evolution of the Michigan Basin: positions of the lens, density distributions, and surface topogra- phies at times (a) 1 Ma, (b) 28 Ma, (c) 100Ma and (d) 151 Ma. These stages correspond to geological times: the Early Silurian (-439Ma BP), the Late Silurian (-412Ma BP), the early Carboniferous (- 340 Ma BP), and the Early Permian (-289 Ma BP). Fig. 8 shows that the lens which initially occupied the depth range 56-64 km moves about 30 km down in 151 Ma, producing a maximum surface subsidence due to phase changes of about 1.12km.

The phase change inducing the subsidence of the basin began 20Ma after setting of the basement had been started (Hamdani et al. 1991). The tectonic subsidence curve (Angevine, Heller & Paola 1990) and the subsidence curve calculated from the model for the Michigan Basin are shown in Fig. 9. The part of the tectonic subsidence curve that seems to be related to the phase changes is compared with the computed curve.

Positive gravity anomalies over the Michigan Basin of about 100mGal (Hinze, Bradley & Brown 1978; Society of Exploration Geophysicists 1982; Kane & Godson 1985) agree with the anomaly obtained from the present model (Fig. 7). Fig. 8(d) shows that the eclogite body sinks deep enough in the upper mantle to be hardly detectable by seismic methods.

The Williston Basin

The Williston Basin is a large circular cratonic basin in North America (Fig. 3) containing about 5 km of sediments ranging in age from Cambrian to Tertiary (Ahern & Mrkvicka 1984; Hamdani et al. 1994). The subsidence history of the Williston Basin is more complex than that of other North American basins under study. The stratigraphic record in the basin is broken by several gaps. An eclogite lens is assumed to evolve under the basin during the Ordovician (-480Ma BP), and a subsidence of the lens leads to the depression of the sur-

o m I

- E Y v

\ \\ M i c h i g a n Basin

tectonic subsidence computed subsidence

0 40 80 120 160

-2

Age (Ma)

Figure 9. Curves of the Michigan Basin subsidence.

face until the Middle Devonian ( - 380 Ma BP). In the late Devonian a new phase of fast subsidence starts. We do not model this phase of setting.

We suggest that in the case of the Williston Basin ptyp= 1.1 x 1023Pas, L= 1200km, dini=0.35 x 102kgm-3, d,,,,,= 0.42 x lo3 kgm-3, y = 0.02 (TTE is about 28 Ma), a = 0.35 (420km), and b = 0.021 (4.2 km).

Positions of the lens, density distributions, and surface topography are presented in Figs 10(a)-(d) at times 2Ma, 55Ma, 140Ma and 480Ma. These correspond to geologi- cal times of the Middle Ordovician (- 478 Ma BP), the Early Silurian (-425 Ma BP), the Early Carboniferous ( - 340 Ma BP), and the Quaternary. Fig. 10(d) shows that the lens sinks about 3 km down in 480 Ma. Such small subsidence is corroborated by the results of Ahern & Mrkvicka (1984) who demonstrated that the location of a possible inhomogen- eity in the upper mantle has not changed significantly in the last 450Ma. Hamdani et al. (1994) have shown that the subsidence of the Williston Basin took place 40Ma earlier than the phase transformation. The tectonic subsidence curve (Angevine et al. 1990) and the subsidence curve calculated from the model for the Williston Basin are represented in Fig. 11. The computed curve is compared with the segment of the tectonic subsidence curve that is probably related to the stage of the phase transformation.

Hajnal et al. (1984) have determined the P-wave velocity beneath the central part of the Williston Basin (at depths 55 km and below) to be 8.3-8.5 kms-'. The layer of elevated velocity can be explained as a layer of eclogites (It0 & Kennedy 1970). Thus the density distribution accepted in the model agrees with seismic data. The model is also consistent with gravity data (Society of Exploration Geophysicists 1982; Kane & Godson 1985). The density of the eclogite body explains the positive gravity anomaly observed over the Williston Basin (Fig. 7).

REMARKS

Numerical results show that depressions induced by subsidence of heavy lenses tend to become less pronounced after suffic- iently long periods. For indeed, lenses sink deeper into the mantle and fall within layers of lesser viscosity, hence the interaction between lenses and uppermost layers becomes weaker; at the same time isostatic adjustments take place. Numerical tests show that basins diminish much more slowly than they evolve (see Fig. 12). Tectonic subsidence curves in Figs 6, 9 and 11 are backstripped by Heidlauf et al. (1986) and represented by Angevine et al. (1990) to show how the basin would have subsided had there been no sedimentary infill.

We neglected the loads due to sediments. The basement must subside deeper under such loads, and values of gravity anomalies obtained from the models should be reduced by lower-density sediments. When the lens is deep enough and weakly affects the uppermost layers, the sediment infill can move upward and become exposed to erosion. This can explain gaps in geological records.

It is interesting to compare, at least roughly, the subsidences due to a flow induced by a heavy lens with the subsidence determined only by the contraction of the lens in the course of the phase transformation. Suppose that the lens is an oblate spheroid with axes 2a > 2b where a = Bb. The mass M of the lens is M = pV where V = 4na2b/3 = 4np2b3/3 is its

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Numerical models of a subsidence mechanism 157

-44-

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Depth Topography kn 0 600 1200

"11 (a)

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7 -111-

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Depth ToPosraphr kn ? 600 1200

( C )

-2 O j t " - - l

-22 O 1 i

Depth Topography kn 9 600 1200

I

"1 (b)

O i C - - - - l

-44

-671

-89

-111-

-133-

-156-

-178-

-200-

Depth Toaosraphhr kn 600 1200

-2

-22 oll----l

Figure 10. Evolution of the Williston Basin. At time (a) 2 Ma; (b) 55 Ma; (c) 140 Ma; (d) 480 Ma.

volume and p is its density. When p changes and the mass M is constant, dplp = - d V / V This implies db = - bdpJ(3p). Taking b = 4.2 km, dp = 0.42 x lo3 kg m-3 and p = 3.5 x lo3 kgm-3, we obtain dh = 0.17 km. Thus the thermal contraction in the direction of the semi-axis b is 0.17 km, as compared with 3 km which is the depth change of the lens beneath the Williston Basin.

CONCLUSION

We have suggested the mechanism of basin formation based on the subsidence of heavy eclogite bodies in the asthenosphere. This mechanism is applied to study subsidence histories of the

Illinois, Michigan and Williston basins. The improved Galerkin-spline approach is used for quantitative analysis of models of three North American basins. We assume density discontinuities and time-dependent basalt-eclogite phase trans- formation. The numerical results indicate that modelled subsid- ence curves are in a good agreement with tectonic subsidence curves obtained by the backstripping analysis. The models suggested. are compatible with the seismic structure of the crust and upper mantle below these basins. The models are also consistent with the gravity data. Thus, the Illinois, Michigan and Williston basins could have similar tectonic histories, although the subsidence of each basin started at a different time.

0 1995 RAS, G J l 123, 149-160

158 B. M. Naimark and A. T. Ismail-Zadeh

'\\, Wil l i s ton B a s i n

\ \ \

tectonic , , subsidence , I ,, , computed subsidence

-2

Age (Ma)

Figure 11. Curves of the Williston Basin subsidence.

n

Y W

Q a, a

-2 0 50 100 150 200 250

Age (Ma) Figure 12. The Michigan Basin subsidence computed for the time interval of 250Ma. A segment of this curve for 160Ma is shown in Fig. 9. It is seen that the subsidence changes from 1.12 km to 0.9 km in the time interval 150-230 Ma.

ACKNOWLEDGMENTS

This research was supported in part by ISF grant MHEOOO and INTAS grant 94-1099. We are grateful to I. A. Abramovich (PMD Engineering Co., USA) for supplying computing facili- ties. Constructive and thoughtful comments from H. Schmeling, Yu. Podladchikov and an anonymous reviewer were greatly appreciated.

REFERENCES

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Angevine, C.L., Heller, P.L. & Paola, C., 1990. Quantitative Sediment- ary Basin Modeling, The American Association of Petroleum Geologists, Tulsa.

Brunet, M. & Le Pichon, X., 1982. Subsidence of the Paris Basin, J. geophys. Rex, 87, 8547-8560.

Carswell, D.A., 1990. Eclogites and the eclogite facies: definitions and classification, in Eclogite Facies Rocks, pp. 1-13, ed. Carswell, D.A., Chapman & Hall, New York.

Cohen, L.H., Ito, K. & Kennedy, G.C., 1967. Melting and phase relations in an anhydrous basalt to 40 kilobars, Am. J. Sci., 265,475-518.

Fowler, C.M.R. & Nisbet, E.G., 1985. The subsidence of the Williston Basin, Can. J . Earth Sci., 22, 408-415.

Green, D.H. & Ringwood, A.E., 1967. An experimental investigation of the gabbro to eclogite transformation and its petrological appli- cations, Geochim. cosmochim. Acta, 31, 767-834.

Green, D.H. & Ringwood, A.E., 1972. A comparison of recent exper- imental data on the gabbrc-garnet granulite-eclogite transition, J . Geol., 80, 277-288.

Hajnal, Z., Fowler, C.M.R., Mereu, R.F., Kanasewich, E.R., Cumming, G.L., Green, A.G. & Mair, A., 1984. An initial analysis of the Earth's crust under the Williston Basin; 1979 CO-CRUST experiment, J. geophys. Res., 89, 9381-9400.

Hamdani, Y., Mareschal, J.C. & Arkani-Hamed, J., 1991. Phase changes and thermal subsidence in intracontinental sedimentary basins, Geophys. J. Int., 106, 657-665.

Hamdani, Y., Mareschal, J.C. & Arkani-Hamed, J., 1994. Phase changes and thermal subsidence of the Williston basin, Geophys. J . Int., 116, 585-597.

Haxby, W.F., Turcotte, D.L. & Bird, J.M., 1976. Thermal and mechan- ical evolution of the Michigan Basin, Tectonophysics, 36, 57-75.

Heidlauf, D.T., Hsui, A.T. & Klein, G.D., 1986. Tectonic subsidence analysis of the Illinois Basin, J. Geol., 94, 779-794.

Hinze, W.J., Bradley, J.W. & Brown, A.R., 1978. Gravimeter survey in the Michigan Basin deep borehole, J. geophys. Res.. 83, 5864-5868.

Ismail-Zadeh, A.T., Naimark, B.M. & Lobkovsky, L.I., 1994. Hydrodynamic model of sedimentary basin formation based on development and subsequent phase transformation of a magmatic lens in the upper mantle, Computational Seismology, 26, 139-155 (in Russian).

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Naimark, B.M., 1987. Existence and uniqueness of the solution of the Rayleigh-Taylor problem, Computational Seismology, 18, 32-41.

Nai.mark, B.M., 1989. Existence and uniqueness in the small of the solution of the Rayleigh-Benard problem, Computational Seismology,

Naimark, B.M. & Ismail-Zadeh, A.T., 1994. Numerical model of intracratonic sedimentary basin formation, Doklady Rossiiskoi AN, 334, 97-99 (in Russian).

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Numerical models of a subsidence mechanism 159

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3, 342-352.

APPENDIX A: THE NUMERICAL APPROACH

Hereinafter, subscripts i, k, rn change in the range 1,2,. . . , M and j , I , n in the range 1,2,. . . , N . To define approximations to the exact solution we use the Galerkin-spline approach. Unknowns are represented as finite linear combinations of basic functions, which are bicubic splines concentrated on small supports:

$ = cij(t)eij(xl, x2),

P =bij(t)fij(X1,X2)r

P =aij(t)hj(xl,xz) + PZ. (Al)

Here the functions IG, and p are continuous and p has jumps across the curves 9,. We assume that p is represented as the sum of two terms, continuous p1 = aij( t)fj(xl, x2) and piecewise constant pz. The function pz assumes constant values in domains bounded by 8,. It is shown later than the disconti- nuities d, of density p2 across the curves Y, are needed in computations, rather than the values of pz.

Basic bicubic splines are products of basic cubic splines:

eij(x,,xz)=s:(~,)i,z(x,), fij(xl>x2)= t!(xl)t?(x2).

The functions c(x,) , [?(x2), t:(xl), and t!(x2) are basic cubic splines chosen to satisfy no-slip or perfect-slip boundary con- ditions (3) for IG, and (4) for p and p. The choice was described in detail by Naimark (1987, 1989) and Naimark & Malevsky (1986). M and N are the numbers of cubic splines in the x1 and x2 directions, respectively.

Substitute representations ( A l ) into eq. (2), multiply the first of (2) by eij(xl, x2) and the second and third of (2) byfij(xl, x2) and integrate the results by parts using the boundary con- ditions. Then the integrals along the boundry vanish and we arrive at a set of linear algebraic equations and two sets of

To verify this form assume for simplicity that one curve 8 : x1 = f(x2) is present; the generalization for Q curves is quite simple. The curve 9 divides !2 into two parts, !2, and R2, with constant densities pcl and pcz. Integrating by parts the right-hand side term in the first of (2) we have

ykl='g J'jp2Dle(xl~x2)dxldx2 R

=gSlPzD1e(x1,x2)dX1~x2 +g ss Pz~le(x l ,xZ)dxl~xz

0 1 a2

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160 B. M . Naimark and A. T . Ismail-Zadeh

Points ( x ~ q s , x ~ q s ) are spaced on 2’: sufficiently densely to obtain an accurate approximation of these curves.

We define a numerical solution as a combination of the sets

1,2, ..., Sq, which satisfy eqs (A2), (A3) and (A5) and initial conditions (5) and (A6). Conditions (5) are used to obtain aij(0) and bi j (0) from the representations

cij(t), aij(t), bi j ( t ) , Xlqs ( tL and xzqs(t), q = 1,2, . . . ,Q, S =

Clearly, we use a mixed description of unknowns; the functions $, p and p1 are represented in the space of variables cij, aij and bij, and pz directly in coordinates x1 and x2.

Introducing dimensionless variables we easily obtain the domain 0 : 0 I x1 I R, 0 I x2 I 1, where R = L/H is the aspect ratio, instead of R : 0 5 x1 I L, - H x2 I 0. Consider scales: distance H, density ptyp, viscosity ptyp, time t,,, and stream functions $,,,. Putting $ = (gpH3)/(ptyp$typ)q and ttYp = ptyp/(ptypgH) we arrive at eq. (2), where $ is replaced by q and the factor g is absent, being included in dimension- less parameters. Thus we obtained eq. (2) and the domain R : O < x l I R , O I x 2 < 1. The coefficients A, B, V , V, G, P, Q, d and B are now standard and can be computed prior to all program runs. Coefficients are now multiplied by R1-(*+”).

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