MOHR-COULOMB FAILURE ENVELOPE I. MOHR-COULOMB FAILURE ENVELOPE A. Another modification to the failure envelope is that experimental studies indicate that the actual strength of a rock in tension is lower than predicted by the linear failure envelope. B. st = tensile strength II. SYMMETRY: A. There are two equivalent directions (planes) in a rock which are equally favorable for failure, called "conjugate fractures" B. There is an important geometric / algebraic relationship between the angles f and q on the Mohr plot 1. q = "angle of internal friction" C. Thus, in the real world, f is determined by the slope of the failure envelope which is dictated by the experimentally determined orientation of the fracture, and vise versa 1. So: if we know q1 we can find f 2. If we know f, we can find q D. The slope of tc = ms + t0 is always positive therefore, 1. m>0, and 0<f<90 E. Remember, 2q=90+f, or q=45+f/2 So, 1. Extreme values of f give constraints on q F. Let's look at 3 cases: 1. f=0, q=45 2. f=90, q=90 3. f=30, q=60 G. The third case illustrates Hartmann's rule - conjugate shear fractures, such that: 1. s1 bisects acute angle between fractures 2. s3 bisects obtuse angle between fractures 3. s2 parallels intersection between fractures III. PORE PRESSURE - an additional consideration in fracture formation A. Pore pressure - some rocks have pores, which are interconnected such that fluids (mainly water and petroleum) and gases can move from pore to pore B. There are 2 resulting mechanical effects 1. Grain-to-grain contacts = Ps = pressure supported by solids 2. Pore pressure = Pp = hydrostatic pressure exerted by fluids in pores C. This is also known as Pf, "fluid pressure", which expresses the resistence to compression that a fluid has D. There is a very simple, and important, relationship between Pp and rock behavior first reported by Don Secor (1965, A.J.S. v. 263 p. 633-646). IV. EFFECTIVE STRESS (s): A. Normal stresses exerted on a rock minus Pf 1. s = s - Pf     s1 t Case: s1 = 3 kb = s1 =2 3-Pf 0   s3 = 1 kb = s3 = 0 t s3   Pf = 1 kb = Pf = 0 0 1- Pf s   Pf   Effective stress 3kb 0kb - 1kb 0kb = 2kb 0kb 0kb 1kb   0kb 1kb   0kb 0kb B. Overall effect: 1. s's decrease 2. t stays the same 3. Deviatoric stress stays the same 4. Mean stress decreases, driving center to the left C. All the rock "feels" is the effective stress (Not the external value of the real stress state). D. What happens if we consider the effects of Pf and the failure envelope? 1. Fractures are induced simply by increasing Pf. Don't have to do anything with deviatoric stress. V. TENSILE FRACTURES - (i.e., movement direction at a high angle to fracture plane), rather than shear fractures, may similarly be induced A. Conditions under which tensile fractures occur depend on the confining pressure (Pc, how deep the rock is buried). 1. Remember 4 km depth = 1 kb pressure a. s vertical = Pc (Pc = confining pressure) = rzg r = density z = height of column g = gravity s1 > rzg s1 > 0 rzg > 0 b. @ shallow depths, rzg and s1 are small, and the tensile point can be reached c. @ great depths, rzg and s1 are large, and mainly shear fractures develop. * Tensile fractures can develop, however, if Pf is jacked up. B. Let's consider the effects of elevated Pf. 1. Pf £ rzg is the general case 2. Pf ³ rzg = "over pressure" - unstable situation, but common in shales a. Clays are: * Porous * Impermeable * Build pressure * Form seals for petroleum traps 3. Anticlinal trap 4. Stratigraphic trap a. Barite is added to drill mud to prevent "blowouts", Pf > rzg; b. Rock explosions C. Another consideration for the Mohr-Coulomb-Griffith diagram for brittle failure is that brittle behavior should not be expected at depths below the "brittle-ductile transition" D. Sibson (1977, Journal of the Geological Society of London, v. 133, p. 191-213) first described the "brittle-ductile transition" to explain the lack of deep-focus earthquakes along the San Andreas fault E. Crystal-plastic, rather than crystal-brittle, behavior at depth requires a modification of the Mohr-Coulomb-Griffith failure envelope, known as the McLintock-Walsh modification F.Question? But then how do we explain deep focus earthquakes (~700 km) along subduction zones? e.g., Japan and western South America VI. HYDROFRACTURES A. With relatively small differential stresses but high Pf, tensile fractures can form at great depths. Tensile fractures formed under these conditions are called Hydrofractures. Thus, deep-focus earthquakes at subduction zones could result from hydrofracturing due to high Pf derived from water-rich trench sediments B. Types of hydrofractures: 1. Natural hydrofractures a. As described above for subducted slabs b. Quartz veins - sedimentary and low grade metamorphic rocks (greenschist facies) * Sediments contain water * Metamorphic processes, especially increasing temperature, produce a lot of fluid, mainly H2O and CO2, which can increase Pf enough to cause fracture * H2O + SiO2 + CaCO3 commonly fill these fractures * dehydration reaction: Al2Si2O5(OH)4 à Al2O3 + 2SiO2 + 2H2O kaolinite corundum * decarbonation reaction: CaMg(CO3)2 + 2SiO2à CaMgSi2O6 + 2CO dolomite diopside c. Dikes and sills - tabular igneous intrusions. * Tensile stresses can be generated by liquid magma e.g., Triassic diabase dikes of eastern U.S. * These dikes occur in mainly one orientation throughout a large region 2. Artificial hydrofractures - developed by petroleum companies a. Rubber gaskets isolate producing unit b. Inject water and sand c. Tensile fractures form, allowing recovery of more petroleum d. "Exploitation"