**4. Theorem of minimum potential energy**

"At this point, the main constraint is creating sufficient connectivity within the injection and production well system in the stimulated region of the EGS reservoir to allow for high per-well production rates without reducing reservoir life by

Field experiments to extract geothermal energy from rock at depth by hydraulic fracturing were started in 1970 by scientists of the Los Alamos National Laboratory, USA. Two boreholes were drilled into crystalline rock (one 2.8 km deep, rock temperature 195°C; the other 3.5 km rock, 235°C) at Fenton Hill, New Mexico. Hydraulic fracturing was used to develop fractures from the boreholes in order to create a fractured region through which water could be circulated to extract heat from the rock. The experiment was terminated in 1992. Commenting

"The idea that hydraulic pressure causes competent rock to rupture and create a disc-shaped fracture was refuted by the seismic evidence. Instead, it came to be understood that hydraulic stimulation leads to the opening of existing natural joints that have been sealed by secondary mineralization. Over the years additional evidence has been generated to show that the joints oriented roughly orthogonal to the direction of the least principal stress open first, but that as the hydraulic

This is an early indication that pre-existing fractures mass significantly affect how hydraulic

Hydraulic fracturing can be considered as a technique to overcome the strength of a rock mass in situ, initiation and propagation of a crack through a system of pre-existing fractures,

In examining the fracture propagation process, the pioneering work of Griffith (1921, 1924) is a logical point of departure. Griffith had identified planar discontinuities, or flaws, in fabri‐ cated materials as the reason why the observed technical strength of brittle materials was about three orders of magnitude lower than the theoretical inter-atomic cohesive (tensile) strength.

 Using an analytical solution by Inglis (1913) for the elastic stresses generated around an elliptical crack in a plate, Griffith observed that the maximum tensile stress at the tip of the

the ellipse degenerated to a sharp crack or flaw (i.e., as the ratio a/b became very high)5

4 A fractured rock mass is typically about two orders of magnitude lower in strength than the strength of a laboratory

= σ<sup>0</sup> (1+ 2a/b), where a and b are the major and minor semi-axes of the ellipse, and as

could rise to a value high enough to reach the inter-atomic cohesive strength sufficient

, the

**3. Influence of fractures and discontinuities on the strength of brittle**

essentially planar discontinuities (e.g., bedding planes), and intact rock.

on what was learned from the Fenton Hill study, Duchane and Brown (2002) note:

rapid cooling." 3

50 Effective and Sustainable Hydraulic Fracturing

pressure is increased, additional joints open."

fractures propagate in a rock mass.

to cause the original crack to start to extend.

3 Future of Geothermal Energy (2005)Synopsis and Executive Summaryp.1-5 (5).

specimen taken from the rock mass [Cundall (2008); Cundall et al, (2008)].

**materials**

4

crack σ<sup>t</sup>

stress σ<sup>t</sup>

"The stable equilibrium state of a system is that for which the potential energy of the system is a minimum. The equilibrium position, if equilibrium is possible, must be one in which rupture of the solid has occurred, if the system can pass from the unbroken to the broken condition by a process involving a continuous decrease of potential energy."

Although much of classical Fracture Mechanics has emphasized applications to problems of Linearly Elastic Fracture Mechanics (LEFM) it is important to recognize that the theorem of minimum potential applies equally to inelastic problems.
