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Subsurface Fluid Flow: The Hydrology of Geothermal Systems 67

y

σ 2
x
σ 3
σ
1
z

z y
σ 2
σ 3
σ 1 σ 1
x x

FIGUre 4s.1 Representation of a randomly oriented force (bold arrow) acting on a cube face. The force
can be resolved into three orthogonal components, one acting perpendicularly to the cube face (σ x ), and two
(σ y and σ z ) acting parallel to the face. These components can be oriented such that they are parallel to the
maximum and minimum stress directions, thus allowing the stresses to define the stress ellipse for the force
acting on the face. Note that the stress ellipses are drawn at different scales; if scaled equally, σ x would be of
the same length in both ellipses.

third axis defines the orientation of the intermediate stress. These stress directions, defined as σ 1 = σ x , σ 2 = σ y , and
σ 3 = σ z , define the orientations of the maximum, intermediate, and minimum principal stresses to which the rock is
subject. A figure can be drawn in this three-dimensional reference frame that encloses all possible values of stress in
all orientations. Such a figure is called the stress ellipsoid. Two-dimensional sections drawn along two axes of that
ellipsoid are stress ellipses, the dimensions of which are defined by the principal stresses σ x , σ y , and σ z along the
respective axes (see Figure 4S.1).
For real rocks in the Earth’s crust, the state of stress is a complex function of several variables. In the simplest
case, the only stress a rock experiences is due to gravitational forces. Under such circumstances, the stress a rock
experiences only depends upon how deeply buried it is, and the source of the stress comes from the mass of rock
above it. In that case, the vertical stress is the maximum principal stress, which is also called the lithostatic pressure.
An approximate rule-of-thumb is that the lithostatic pressure in the crust increases at the rate of approximately
3.33 × 10 Pa, or 33.3 MPa (mega-Pascals) for every kilometer in depth.
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For a homogeneous rock with sufficient internal strength to prevent plastic flow, the lithostatic pressure is also
equivalent to the confining pressure the rock experiences in all directions imposed on it by its neighboring rock
mass. As a result, the stress ellipsoid is a sphere and σ x , σ y , and σ z are equal. In such a case, there will be no fractures
in the rock, and the only available space to accommodate fluid flow would be the intrinsic rock matrix porosity.
Failure of a rock by fracturing occurs when the difference between the maximum principal stress and the mini-
mum principle stress (also called the stress differential, usually expressed in MPa) exceeds some value characteristic
for that rock. For this to occur, some force must be applied to the rock in addition to that resulting from gravity. Many
geological processes are capable of imposing such forces, such as uplift, settling, down-slope mass movement, dif-
ferential movement of tectonic plates, magma intrusion, and so on. Since these conditions change over geological
time, it is common for a given rock to have experienced a long history of changes in both the orientation and mag-
nitude of stresses. As a result it is likely that a rock will, at some point in its history, experience conditions where its
characteristic strength is less than the stress differential and it will fail by fracturing.
The resulting fractures at the time of failure will be oriented in specific and approximately predictable directions,
relative to the orientation of the principal stress directions. Fractures that form during such an event form what is
called a fracture set. The characteristics of the fractures, such as how long they are in any particular direction, their
planarity, and the spacing between the fractures that form will depend upon the characteristics of the rock, the
pressure, and temperature at the time of fracturing, the magnitude of the stresses, and the rate at which the stress
is applied. Rocks that have experienced multiple failure events are likely to have multiple fracture sets, each with
characteristic properties and orientations.