12.11c), this interaction is significant and the
regions of positive Coulomb stress extend from
the surface to the sill tips, and these regions
extend laterally to a distance of almost 4 km from
the point immediately over the center of the sill.
The overburden out to this distance is susceptible
to the development of bedding-plane faults,
except for the region immediately over the center
of the sill where enhanced vertical compression
would prevent frictional slip. Delamination is predicted to develop above the distal margin of the
sill and not immediately over its center. This distribution of bedding-plane faulting would
promote the formation of laccoliths with flat tops
and monoclinal bending over their periphery
(Koch et al., 1981).
From the elasticity theory we understand that
the stress perturbation associated with the lateral
growth of a sill enhances the shear stresses and
lowers the normal compressive stresses on horizontal bedding planes in such a way that faulting
is likely on weak bedding planes above the advancing sill tip. As the sill approaches a length equal to
the overburden thickness, the delamination
spreads all the way to Earth’s surface and we would
anticipate a transition to laccolithic bending,
accommodated by sliding of the mechanical units
over one another along bedding-plane faults.
Elasticity theory and the Coulomb failure criterion
explain how and where these faults form, and this
prediction is consistent with field observations of
bedding-plane faults at Mt. Holmes in the Henry
Mountains (Fig. 12.9).
12.2.3 Viscous fluid mechanics: how
rapidly can magma flow into sills?
The questions asked in the previous two sections
focus attention on host rock deformation during
sill and laccolith formation. The mechanical role
of the magma is reduced to providing a pressure
on the stack of plates or on the elastic material
surrounding the sill. In other words the mechanical action of the magma is replaced by the appropriate distribution of tractions in the form of a
boundary condition. This is an effective way to
simplify the mechanical system composed of both
injecting magma and deforming host rock, but it
necessarily means that one cannot address questions about the rate of development of the sill or
laccolith (Fig. 4.12). The context, or general boundary conditions, for the analysis of magma flow is
that branch of fluid mechanics devoted to the flow
of fluids with mechanical properties that include
viscosity and strength. Reference textbooks on
fluid mechanics provide the background for
model development and include solutions for
flow of materials with a variety of physical properties in conduits of various shapes (Lamb, 1945;
Schlichting, 1979; Landau and Lifshitz, 1960;
White, 1974).
In Section 4.3.2 we used dimensional analysis
to identify Reynolds Number (4.55) as the scale
factor for viscous flow in conduits (Reynolds, 1883;
White, 1974). This dimensionless group is a ratio of
inertial to viscous forces in the flowing fluid and is
proportional to the conduit width, fluid density,
and characteristic velocity, and inversely proportional to the Newtonian viscosity. Reynolds’ laboratory experiments (Fig. 4.8c) demonstrated that
the flow regime is laminar for numbers less than
about 2000. The apparent viscosities of silicate
liquids (magma) have been measured for a wide
variety of chemical compositions and water contents over the range of melting temperatures
(Shaw, 1963, 1969; Shaw et al., 1968; Murase and
McBirney, 1973; McBirney and Murase, 1984; Ryan
and Blevins, 1987). Given the great viscosity of
most magmas and the modest velocities for flow in
sills the regime is likely to be laminar.
For the purpose of a simple example we consider an isothermal fluid, the model magma,
flowing in a tabular conduit of length 2a and
height 2h, surrounded by rigid host rock (Fig.
12.12a). A feeder dike at the center of the sill supplies the magma. Of course hot magma emplaced
into cold sedimentary rock will loose heat to the
surroundings (Lovering, 1935, 1936; Jaeger, 1957,
1964b), but rock is a good insulator, so it is not
unreasonable to postulate that the temperature
change is insignificant over the time required for
the sill to propagate to the transition length for
laccolith formation. In Section 4.2.2 we reviewed
the solution for conductive heat loss from a
tabular intrusion of magma and showed that
emplacement times on the order of a few days are
consistent with this postulate (Delaney and
Pollard, 1981, 1982). As the previous sections indicate, the sill grows in length and thickness as
12.2 SELECTION OF GENERAL BOUNDARY CONDITIONS
471
regions of positive Coulomb stress extend from
the surface to the sill tips, and these regions
extend laterally to a distance of almost 4 km from
the point immediately over the center of the sill.
The overburden out to this distance is susceptible
to the development of bedding-plane faults,
except for the region immediately over the center
of the sill where enhanced vertical compression
would prevent frictional slip. Delamination is predicted to develop above the distal margin of the
sill and not immediately over its center. This distribution of bedding-plane faulting would
promote the formation of laccoliths with flat tops
and monoclinal bending over their periphery
(Koch et al., 1981).
From the elasticity theory we understand that
the stress perturbation associated with the lateral
growth of a sill enhances the shear stresses and
lowers the normal compressive stresses on horizontal bedding planes in such a way that faulting
is likely on weak bedding planes above the advancing sill tip. As the sill approaches a length equal to
the overburden thickness, the delamination
spreads all the way to Earth’s surface and we would
anticipate a transition to laccolithic bending,
accommodated by sliding of the mechanical units
over one another along bedding-plane faults.
Elasticity theory and the Coulomb failure criterion
explain how and where these faults form, and this
prediction is consistent with field observations of
bedding-plane faults at Mt. Holmes in the Henry
Mountains (Fig. 12.9).
12.2.3 Viscous fluid mechanics: how
rapidly can magma flow into sills?
The questions asked in the previous two sections
focus attention on host rock deformation during
sill and laccolith formation. The mechanical role
of the magma is reduced to providing a pressure
on the stack of plates or on the elastic material
surrounding the sill. In other words the mechanical action of the magma is replaced by the appropriate distribution of tractions in the form of a
boundary condition. This is an effective way to
simplify the mechanical system composed of both
injecting magma and deforming host rock, but it
necessarily means that one cannot address questions about the rate of development of the sill or
laccolith (Fig. 4.12). The context, or general boundary conditions, for the analysis of magma flow is
that branch of fluid mechanics devoted to the flow
of fluids with mechanical properties that include
viscosity and strength. Reference textbooks on
fluid mechanics provide the background for
model development and include solutions for
flow of materials with a variety of physical properties in conduits of various shapes (Lamb, 1945;
Schlichting, 1979; Landau and Lifshitz, 1960;
White, 1974).
In Section 4.3.2 we used dimensional analysis
to identify Reynolds Number (4.55) as the scale
factor for viscous flow in conduits (Reynolds, 1883;
White, 1974). This dimensionless group is a ratio of
inertial to viscous forces in the flowing fluid and is
proportional to the conduit width, fluid density,
and characteristic velocity, and inversely proportional to the Newtonian viscosity. Reynolds’ laboratory experiments (Fig. 4.8c) demonstrated that
the flow regime is laminar for numbers less than
about 2000. The apparent viscosities of silicate
liquids (magma) have been measured for a wide
variety of chemical compositions and water contents over the range of melting temperatures
(Shaw, 1963, 1969; Shaw et al., 1968; Murase and
McBirney, 1973; McBirney and Murase, 1984; Ryan
and Blevins, 1987). Given the great viscosity of
most magmas and the modest velocities for flow in
sills the regime is likely to be laminar.
For the purpose of a simple example we consider an isothermal fluid, the model magma,
flowing in a tabular conduit of length 2a and
height 2h, surrounded by rigid host rock (Fig.
12.12a). A feeder dike at the center of the sill supplies the magma. Of course hot magma emplaced
into cold sedimentary rock will loose heat to the
surroundings (Lovering, 1935, 1936; Jaeger, 1957,
1964b), but rock is a good insulator, so it is not
unreasonable to postulate that the temperature
change is insignificant over the time required for
the sill to propagate to the transition length for
laccolith formation. In Section 4.2.2 we reviewed
the solution for conductive heat loss from a
tabular intrusion of magma and showed that
emplacement times on the order of a few days are
consistent with this postulate (Delaney and
Pollard, 1981, 1982). As the previous sections indicate, the sill grows in length and thickness as
12.2 SELECTION OF GENERAL BOUNDARY CONDITIONS
471