Mathematics
http://hdl.handle.net/2429/19748
Fri, 27 Feb 2015 04:32:33 GMT2015-02-27T04:32:33ZOn the Hilbert-Pólya and Pair Correlation Conjectures (Issued:2014-01-01)
http://hdl.handle.net/2429/50314
The Hilbert-Pólya Conjecture supposes that there exists an operator in a Hilbert space whose eigenvalues are the zeroes of the Riemann Zeta function ζ(s). This conjecture, if true, would very likely expedite the proof of the Riemann Hypothesis, namely that the non-trivial zeroes of ζ(s) have real part 1/2. In this thesis we summarize work by Berry, Keating and others in constructing such an operator. Although the work so far has not yet yielded such an operator, some have been found that have properties very close to what is desired. We also summarize a (partially proven) conjecture by Montgomery that motivates the search for this operator. He conjectures that the pair correlation function for the spacing between the imaginary parts of the Riemann zeroes is the same as the correlation function for the spacing between eigenvalues of random Gaussian unitary matrices.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/2429/503142014-01-01T00:00:00ZSlurry flow, gravitational settling, and a proppant transport model for hydraulic fractures (Issued:2014-08-13)
http://hdl.handle.net/2429/50000
The goal of this study is to analyze the steady flow of a Newtonian fluid mixed with spherical particles in a channel based on a continuum model, where the constitutive behaviour of the slurry is approximated
by an empirical formula. In order to account for the gravitational settling of particles, two-dimensional
flow needs to be considered as the pressure gradient and gravity may not always be collinear. It is shown
that the problem under consideration features a boundary layer, whose size is on the order of the particle
radius. The expressions for both the outer (i.e. outside the boundary layer) and inner (i.e. within the
boundary layer) solutions are obtained in terms of the particle concentration, particle velocity, and fluid velocity. Unfortunately, these solutions require numerical solution of an integral equation, depend on the ratio between the pressure gradient and the gravity force, and the orientation of the pressure gradient relative to the gravity. Consequently, the development of a proppant transport model for hydraulic fracturing based on these results is not practical. For this reason, an approximate solution is introduced, where the effect of gravity is accounted for in an approximate fashion, reducing the complexity of the slurry flow solution. To validate the use of this approximation, the error is estimated for different regimes of flow. The approximate solution is then used to calculate the expressions for the slurry flux and the proppant flux, which are the basis for a model that can be used to account for proppant transport with gravitational settling in a fully coupled hydraulic fracturing simulator.
Wed, 13 Aug 2014 00:00:00 GMThttp://hdl.handle.net/2429/500002014-08-13T00:00:00ZA new paradigm for proppant schedule design (Issued:2014-02-27)
http://hdl.handle.net/2429/46110
This study introduces a novel methodology for the design of the proppant pumping schedule for a hydraulic fracture, in which the fi nal proppant distribution along the crack is prescribed. The method is based on the assumption that the particles have relatively small impact on the fracture propagation, unless they reach the tip region. This makes it possible to relate the proppant velocity to the clear fluid velocity inside the fracture, which is calculated assuming no proppant. Having the history of the clear
fluid velocity distribution, the prospective proppant motion can be computed. Then, volume
balance is used to relate the final concentration at some point inside the fracture to the corresponding input concentration at a speci fic time instant, which helps to avoid solving an inverse problem. One exceptional feature of the approach lies in the fact that it is applicable to multiple fracture geometries and can be implemented using various hydraulic fracturing simulators. To verify the technique, two fracture geometries are considered - Khristianovich-Zheltov-Geertsma-De Klerk (KGD) and pseudo-3D (P3D). It is shown that the developed approach is capable of properly estimating the pumping schedule for both geometries. In particular, the proppant placement along the fracture at the end of the pumping
period, calculated according to the adopted proppant transport model, shows close agreement with the design distribution. The comparison with Nolte's scheduling scheme shows that the latter is not always accurate, and cannot capture the essential di fferences between the schedules for the fracture geometries considered.
Thu, 27 Feb 2014 00:00:00 GMThttp://hdl.handle.net/2429/461102014-02-27T00:00:00ZProppant transport in hydraulic fracturing : crack tip screen-out in KGD and P3D models (Issued:2014-02-27)
http://hdl.handle.net/2429/46109
The aim of this study is to develop a model for proppant transport in hydraulic fractures capable of
capturing both gravitational settling and tip screen-out effects, while prohibiting the particles from
reaching the crack tips by imposing a width restriction based on the particle size. First, the equations
that govern the propagation of hydraulic fractures and the proppant transport inside them are formulated.
They are based on the solution for the steady flow of a viscous fluid, mixed with spherical particles, in a
channel, which is obtained assuming an empirical constitutive model. This proppant transport model is
applied to two fracture geometries – Khristianovich-Zheltov-Geertsma-De Klerk (KGD) and pseudo-3D
(P3D). Numerical simulations show that the proposed method makes it possible to capture proppant
plug formation and growth, as well as the gravitational settling for both geometries. A dimensionless
parameter, whose magnitude reflects the intensity of the settling, is introduced for the P3D fracture.
[The previous file was updated 2014-08-13 to reflect changes to publication].
Thu, 27 Feb 2014 00:00:00 GMThttp://hdl.handle.net/2429/461092014-02-27T00:00:00Z