Patience A. Cowie

Physical explanation for the displacement-length relationship of faults using a post-yield fracture mechanics model

Abstract A plane strain model for a fault is presented that takes into account the inelastic deformation involved in fault growth. The model requires that the stresses at the tip of the fault never exceed the shear strength of the surrounding rock. This is achieved by taking into account a zone, around the perimeter of the fault surface, where the fault is not well developed, and in which sliding involves frictional work in excess of that required for sliding on the fully developed fault. The displacement profiles predicted by the fault model taper out gradually towards the tip of the fault and compare well with observed displacement profiles on faults. Using this model it is found that both (1) the shape of the displacement profile, and (2) the ratio of maximum displacement to fault length are a function of the shear strength of the rock in which the fault forms. For the case of a fault loaded by a constant remote stress, the displacement is linearly related to the length of the fault and the constant of proportionality depends on the shear strength of the surrounding rock normalized by its shear modulus. Using data from faults in different tectonic regions and rock types, the in situ strength of intact rock surrounding a fault is calculated to be on the order of 100 MPa (or a few kilobars). These estimates exceed, by perhaps a factor of 10, the strength of a well developed fault and thus provide an upper bound for the shear strength of the crust. It is also shown that the work required to propagate a fault scales with fault length. This result can explain the observation that the fracture energy calculated for earthquake ruptures and natural faults are several orders of magnitude greater than that for fractures in laboratory experiments.

Displacement-length scaling relationship for faults : data synthesis and discussion

Abstract It is observed that the amount of displacement (d) on a fault is proportional to the mapped trace length L. The exact form of the fault scaling relationship, i.e. d = f(L), is still a subject of some disagreement. A number of workers have interpreted data from individual data sets as indicating a linear relationship between d and L. However, these individual data sets have large scatter and a limited range of scale, so their interpretations are not fully conclusive. Other workers have interpreted combinations of different data sets, taken together, and concluded that the d vs L scaling relationship is non-linear. Fault growth models, however, indicate that the scaling relationship should depend on rock properties so correlations using combined data sets may be questionable. This paper presents a synthesis of published data sets on the displacements and lengths of faults. A summary of each data set is given, including: the geologic setting; the mode of faulting (normal/thrust/strike-slip); and the measurement methods used to obtain the displacement and length data. Sources of scatter in the data due to geologic effects and measurement procedures are reviewed. Our preferred interpretation is that the d-L relationship is linear with a possible cross-over phenomenon between small and large faults, but the unambiguous resolution of this question will require some significant improvements in the existing database.

FAULT GROWTH AND FAULT SCALING LAWS : PRELIMINARY RESULTS

We report progress made in the last few years on the general problem of the mechanism of fault growth and the scaling laws that result. Results are now conclusive that fault growth is a self-similar process in which fault displacement d scales linearly with fault length L. Both this result and the overall nature of along-strike fault displacement profiles are consistent with the Dugdale-Barenblatt elastic-plastic fracture mechanics model. In this model there is a region of inelastic deformation near the crack tip in which there is a breakdown from the yield strength of the unfractured rock to the residual frictional strength of the fault over a breakdown length S and displacement d0. Limited data also indicate that S and d0 also scale linearly with L, which implies that fracture energy G increases linearly with L. The scaling parameters in these relationships depend on rock properties and are therefore not universal. In our prime field locality, the Volcanic Tableland of eastern California, we have collected data over 2 orders of magnitude in scale range that show that faults obey a power law size distribution in which the exponent C in the cumulative distribution is ∼1.3. If the fault is growing within the brittle field, the zone of inelastic deformation consists of a brittle process zone which leaves a wake of fractured rock adjacent to the fault. Preliminary results of modeling the process zone are consistent with observations now in hand both in predicting the preferred orientation of cracks in the process zone wake and the rate of falloff of crack density as a function of distance from the fault. The preferred orientation of these cracks may be used to infer the mode and direction of propagation of the fault tip past the point in question. According to the model, the width of the process zone wake may be used to infer the length of the fault at the time its tip passed the measurement point, but data have not yet been collected to verify this prediction. If the fault displacement has been accumulated by repeated seismic slips, each of these will sweep the fault with a crack tip stress field of a smaller spatial extent than that of the fault tip stress field, producing an inner, more intensely fractured, process zone wake. This may be the mechanism that creates the cataclasite zone, rather than simple frictional wear, as has been previously supposed.

A healing–reloading feedback control on the growth rate of seismogenic faults

Spatial and temporal variations in the growth rates of faults are explained in terms of a stress feedback mechanism operating in the seismogenic upper crust. It is based on the idea that seismic rupture of a fault perturbs the surrounding stress field, advancing the occurrence of future earthquakes on some faults that are optimally oriented while relaxing stress levels on others. If post-slip healing is geologically rapid, then the earthquakes that are thus induced will contribute to reloading along the earlier rupture zone because of the symmetry of the optimal geometry. A positive feedback is set up so that, even in areas that are undergoing uniform tectonic straining, some faults develop higher displacement rates and grow more rapidly while others experience reduced rates or become inactive. Using a thin plate elastic model for lithospheric-scale faulting, it is shown that this healing–reloading feedback mechanism drives rapid localisation and the formation of major through-going faults moving at plate boundary velocities. Enhanced displacement rates (compared to an isolated fault) develop shortly after the onset of deformation along those faults which are optimally positioned in the overall fault population. Thus the formation of a new plate boundary fault zone is predetermined and is a consequence of, rather than the precursor of, preferentially high displacement rates. Also, fault segments located at points of rupture symmetry, e.g. the central portion of a fault zone, are reloaded more frequently and develop higher displacement rates and consequently have longer segment lengths and/or larger displacement to length ratios. Episodic fault movement through time is a general feature of the model. These predictions are consistent with available field observations over a wide range of scales. Thus, elastic–brittle failure and healing appear to be important rheological components of the lithosphere on long time scales (104–106 y), as well as on the time scale of earthquake recurrence.

Growth of faults by accumulation of seismic slip

The accommodation of large strains in the upper crust is largely achieved by the accumulation of displacement on faults. Observation shows that as a fault accumulates displacement, it grows in size, i.e., its surface area and its length increase. Here we address the question: “For an increase in the amount of displacement on a fault, by how much would the length of the fault change?” It is argued by Cowie and Scholz [1992a] that the displacement on a fault is linearly related to the length of the fault. This simple result is expanded upon in this paper by assuming that a fault accumulates displacement by repeated earthquakes. Two different approaches are presented: The first approach considers the balance that must be achieved between the energy available for deformation and the work involved in creating new fault surface area as the fault grows. The available energy is provided by changes in strain energy when the fault slips. The second approach is to construct a geometrical model for fault growth using the scaling relationship between the slip during a single earthquake and the length of the rupture. The total displacement on a fault is the sum of the slips contributed by many earthquakes. The usefulness of these two approaches is that the growth of a fault over geologic time can be described by parameters that can be obtained from earthquake and fault data. The models presented here predict that (1) the maximum amount a fault can grow in a single earthquake that ruptures the entire fault is of the order of 1% of its previous length and (2) the work of faulting may account for 10% of the total energy available during an earthquake. The energy lost from the system is accounted for by work done against friction and seismic radiation. Consequences of fault growth for the segmentation and thus seismogenic potential of a fault over geologic time are discussed using predictions of the theory.

Bedrock channel adjustment to tectonic forcing: Implications for predicting river incision rates

We present detailed data of channel morphology for a river undergoing a transient response to active normal faulting where excellent constraints exist on spatial and temporal variations in fault slip rates. We show that traditional hydraulic scaling laws break down in this situation, and that channel widths become decoupled from drainage area upstream of the fault. Unit stream powers are ∼4 times higher than those predicted by current scaling paradigms and imply that incision rates for rivers responding to active tectonics may be significantly higher than those heretofore modeled. The loss of hydraulic scaling cannot be explained by increasing channel roughness and is an intrinsic response to tectonic forcing. We show that channel aspect ratio is a strongly nonlinear function of local slope and demonstrate that fault-induced adjustment of channel geometries has reset hillslope gradients. The results give new insight into how rivers maintain their course in the face of tectonic uplift and illustrate the first-order control the fluvial system exerts on the locus and magnitude of sediment supply to basins.

A mechanism to explain rift-basin subsidence and stratigraphic patterns through fault-array evolution

Rift-basin stratigraphy commonly records an early stage of slow subsidence followed by an abrupt increase in subsidence rate. The physical basis for this transition is not well understood, although an increase in extension rate is commonly implied. Here, a numerical fault-growth model is used to investigate the influence of segment linkage on fault-displacement-rate patterns along an evolving normal fault array. The linkage process we describe is controlled by a stress feedback mechanism, which leads to enhanced growth of optimally positioned faults. Model results indicate that, even with constant extension rates, slow displacement rates prevail during an initial phase of distributed extension, followed by an increase in displacement rates as strain becomes localized on linked fault arrays. This is due to the dynamics of fault interactions rather than mechanical weakening. Comparison of model simulations with rift-basin subsidence and stratigraphic patterns in the Gulf of Suez and North Sea suggests that the occurrence and timing of rapid basin deepening can be explained by the mechanics of fault-zone evolution, without invoking a change in regional extension rates.

Constraining slip rates and spacings for active normal faults

Abstract Numerous observations of extensional provinces indicate that neighbouring faults commonly slip at different rates and, moreover, may be active over different time intervals. These published observations include variations in slip rate measured along-strike of a fault array or fault zone, as well as significant across-strike differences in the timing and rates of movement on faults that have a similar orientation with respect to the regional stress field. Here we review published examples from the western USA, the North Sea, and central Greece, and present new data from the Italian Apennines that support the idea that such variations are systematic and thus to some extent predictable. The basis for the prediction is that: (1) the way in which a fault grows is fundamentally controlled by the ratio of maximum displacement to length, and (2) the regional strain rate must remain approximately constant through time. We show how data on fault lengths and displacements can be used to model the observed patterns of long-term slip rate where measured values are sparse. Specifically, we estimate the magnitude of spatial variation in slip rate along-strike and relate it to the across-strike spacing between active faults.

FAULT TIP DISPLACEMENT GRADIENTS AND PROCESS ZONE DIMENSIONS

Abstract Finite displacement gradients measured at fault tips appear to contradict the predictions of the post-yield fracture mechanics (PYFM) model for fault tip propagation proposed by Cowie, P. A. and Scholz, C. H. (1992) Journal of Structural Geology , 14, 1133–1148. The results of a high resolution survey of a 3.6 km long normal fault in SE Utah are presented as evidence that the contradiction is real and not simply due to problems of limited resolution. A theoretical explanation for finite tip gradients is then proposed which involves a positive stress feedback between sequential fault slip increments. According to this growth model, strength heterogeneities on a fault surface limit the size of individual ruptures so that only a patch of the fault moves at any one time. Each slipping patch produces a stress perturbation which raises the shear stress on adjacent healed portions of the fault as well as the surrounding rock volume. Healing takes place after each slip event, allowing local strength recovery. Using a simple two-dimensional planar fault model, we show that when the size of the slipping patch is much smaller than the dimensions of the fault plane, and strength recovery is geologically instantaneous, the displacement profile follows an approximately linear decrease towards the tip similar to natural examples. A bell-shaped displacement profile, with tip gradients that tend to zero, is predicted only in the special case where the size of each slip patch equals the fault plane dimensions. Our main modification of the earlier model is that the size of the process zone wake, or frictional breakdown zone, scales with the dimensions of the slipping patch as opposed to the entire fault length. Model results show that the stress field at the tips of faults formed by this mechanism decays rapidly, so the range of significant interaction is small compared to the fault dimensions.

A conceptual model for the origin of fault damage zone structures in high-porosity sandstone

We present a conceptual model to explain the development of damage zones around faults in high-porosity sandstones. Damage zone deformation has been particularly well constrained for two 4-km-long normal faults formed in the Navajo Sandstone of central Utah, USA. For these faults the width of the damage zone increases with fault throw (for throws ranging from 0 to 30 m) but the maximum deformation density within the damage zone is independent of throw. To explain these data we modify a previously published theoretical model for fault growth in which displacement accumulates by repeated slip events on patches of the fault plane. The modifications are based on field observations of deformation mechanisms within the Navajo Sandstone, the throw profiles of the faults, and inferences concerning likely slip-patch dimensions. Zones of enhanced stress are generated around the tips of each slipping patch, raising the shear stress on adjacent portions of the fault as well as potentially causing off-fault damage. A key ingredient in our model for off-fault damage accumulation is the transition from strain hardening associated with deformation band development, to localised strain softening as a slip-surface develops. This transition occurs at a critical value of deformation density. Once a new slip-surface develops at some distance from the main fault plane and it starts to accumulate throw it can, in turn, generate its own damage zone, thus increasing the overall damage zone width. Our approach can be applied to interpret damage zone development around any fault as long as the host-rock lithology, porosity and deformation mechanisms are taken into consideration.