Analytical predictions for a natural spacing within dyke swarms

Abstract

Dykes often grow next to other dykes, evidenced by the widespread occurrence of dyke swarms that comprise many closely-spaced dykes. In giant dyke swarms, dykes are observed to maintain a finite spacing from their neighbors that is tens to hundreds of times smaller than their length. To date, mechanical models have not been able to clarify whether there exists an optimum, or natural spacing between the dykes. And yet, the existence of a natural spacing is at the heart of why dykes grow in swarms in the first place. Here we present and examine a mechanical model for the horizontal propagation of multiple, closely-spaced blade-like dykes in order to find energetically optimal dyke spacings associated with both constant pressure and constant * 710 Benedum Hall, 3700 O'Hara Street, Pittsburgh, PA, 15261, USA Email address: bunger@pitt.edu (Andrew P. Bunger) Preprint submitted to Earth and Planetary Science Letters May 9, 2013influx magma sources. We show that the constant pressure source leads to an optimal spacing that is equal to the height of the blade-like dykes. We also show that the constant influx source leads to two candidates for an optimal spacing, one which is expected to be around 0.3 times the dyke height and the other which is expected to be around 2.5 times the dyke height.Comparison with measurements from dyke swarms in Iceland and Canada lend initial support to our predictions, and we conclude that dyke swarms are indeed expected to have a natural spacing between first generation dykes and that this spacing scales with, and is on the order of, the height of the blade-like dykes that comprise the swarm.

The Mull Paleocene dykes: some insights into the nature of major dyke swarms

“…Several magma pulses apparently used the same conduit as the Cleveland Dyke. Bunger et al (2013) have developed a mechanical model for the propagation of multiple, closely spaced dykes in a swarm in order to find the optimal, most energy-efficient spacing of dykes for any given stress field, which may have applications in the MDS. The model predicts that the earliest and most widely spaced dykes will be the thickest whereas later, smaller dykes tend to fill in the gaps between them.…”

Constraints on Simultaneous Growth of Hydraulic Fractures From Multiple Perforation Clusters in Horizontal Wells

“…2. These correspond to the two widely spaced limits H(L(h and H(h(L and to the closely spaced limit h(H(L. The estimates of length, width, pressure, and pressure gradient are provided by Nordgren (1972) for the widely spaced cases and by Bunger et al (2013) for the closely spaced limit, and these are summarized in Table 1.…”

“…The expressions for the characteristic power quantities P u , P int , P c , and P f are derived by Bunger (2013) and Bunger et al (2013). As detailed by Bunger (2013), this separation of the total power follows directly as an extension of the classical elasticfracture theory of Griffith (1921) to account for interaction with the stress shadow of neighboring hydraulic fractures and for the dissipation of energy through fluid flow.…”

“…The derivations of these terms are lengthy and technical, particularly regarding the inclusion of the interaction, or "stress-shadow," terms that require the incorporation of asymptotic expressions for the near-field and farfield stresses induced by a pressurized hydraulic fracture into the energy expressions. These derivations will not be repeated here [although the part of the derivation that remains available only in Bunger et al (2013) is outlined in Appendix B for completeness], but instead the expressions for the characteristic power quantities are summarized in Table 2. The input power can therefore be readily obtained for each geometry and regime from Eq.…”

“…This analysis shows that the propensity for multiple hydraulic fractures to grow simultaneously is tied to both fracture geometry and whether the predominant energy-dissipation mechanism is associated with the flow of the viscous fluid that drives hydraulic-fracture propagation (the "viscosity-dominated" regime) or breakage of rock ahead of the leading edge of the hydraulic fractures (the "toughnessdominated" regime). In a complementary paper, Bunger et al (2013) revisit the case of the array of PKN hydraulic fractures growing transversely to the wellbore, but unlike Bunger (2013), who considers the onset of interaction just as the hydraulic fractures become large enough relative to their separation to interact, Bunger et al (2013) consider the power requirements for propagation in the case in which the spacing between the PKN hydraulic fractures is less than their height. However, neither of these contributions accounts for the influence of near-wellbore pressure losses that, under some circumstances, could have a significant impact on the power required to drive hydraulic-fracture growth.…”