Mathematicshttp://hdl.handle.net/2429/197482015-03-03T07:48:06Z2015-03-03T07:48:06ZAn enhanced pseudo-3D model for hydraulic fracturing accounting for viscous height growth, non-local elasticity, and lateral toughness (Issued:2015-04-27)Donsov, EgorPeirce, Anthonyhttp://hdl.handle.net/2429/522892015-02-28T10:02:51Z2015-04-27T00:00:00ZThe goal of this paper is to develop an enhanced pseudo-3D (P3D) model for hydraulic fracturing (HF), whose predictions are more accurate compared to that of the original P3D model, but which requires significantly less computational resources than a fully planar HF simulator. We show that the lack of viscous resistance in the height growth and the local approximation in the computation of elastic interactions, which precludes the incorporation of lateral toughness, are the primary weaknesses of the original P3D model that considers symmetric stress barriers and no leak-off. To account for the viscous resistance, an apparent fracture toughness is introduced. The apparent toughness is calibrated using a one-dimensional HF model resulting in an approximate expression that captures all regimes of propagation. To incorporate non-local elastic interactions, the fracture opening in every vertical cross-section is approximated by a plane-strain solution, and then the 2D elasticity interaction integral is evaluated. To increase the computational efficiency, this 2D integral is further approximated by two one-dimensional integrals. The use of non-local elasticity allows us to include the asymptotic solution in the tip element, and, in particular, to include the effect of lateral fracture toughness. To further increase the accuracy of the P3D model, the flat fracture tip is replaced by its curved counterpart. This also permits us to capture radial behaviour at early times before the fracture has reached the stress barriers. To evaluate the accuracy of the model we have developed, the results are compared to the predictions calculated using a recently developed fully planar HF simulator, which is able to capture viscous, toughness, and intermediate propagation regimes. It is shown that the enhanced P3D model is able to approximate the propagation of hydraulic fractures accurately for various regimes of propagation, as well as for different fracture aspect ratios.
2015-04-27T00:00:00ZAn enhanced pseudo-3D model for hydraulic fracturing accounting for viscous height growth, non-local elasticity, and lateral toughness (Issued:2015-04-27)Donsov, EgorPeirce, Anthonyhttp://hdl.handle.net/2429/522892015-02-28T10:02:51Z2015-04-27T00:00:00ZThe goal of this paper is to develop an enhanced pseudo-3D (P3D) model for hydraulic fracturing (HF), whose predictions are more accurate compared to that of the original P3D model, but which requires significantly less computational resources than a fully planar HF simulator. We show that the lack of viscous resistance in the height growth and the local approximation in the computation of elastic interactions, which precludes the incorporation of lateral toughness, are the primary weaknesses of the original P3D model that considers symmetric stress barriers and no leak-off. To account for the viscous resistance, an apparent fracture toughness is introduced. The apparent toughness is calibrated using a one-dimensional HF model resulting in an approximate expression that captures all regimes of propagation. To incorporate non-local elastic interactions, the fracture opening in every vertical cross-section is approximated by a plane-strain solution, and then the 2D elasticity interaction integral is evaluated. To increase the computational efficiency, this 2D integral is further approximated by two one-dimensional integrals. The use of non-local elasticity allows us to include the asymptotic solution in the tip element, and, in particular, to include the effect of lateral fracture toughness. To further increase the accuracy of the P3D model, the flat fracture tip is replaced by its curved counterpart. This also permits us to capture radial behaviour at early times before the fracture has reached the stress barriers. To evaluate the accuracy of the model we have developed, the results are compared to the predictions calculated using a recently developed fully planar HF simulator, which is able to capture viscous, toughness, and intermediate propagation regimes. It is shown that the enhanced P3D model is able to approximate the propagation of hydraulic fractures accurately for various regimes of propagation, as well as for different fracture aspect ratios.
2015-04-27T00:00:00ZOn the Hilbert-Pólya and Pair Correlation Conjectures (Issued:2014-01-01)Hudoba de Badyn, Mathiashttp://hdl.handle.net/2429/503142014-09-12T21:12:03Z2014-01-01T00:00:00ZThe 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.
2014-01-01T00:00:00ZSlurry flow, gravitational settling, and a proppant transport model for hydraulic fractures (Issued:2014-08-13)Dontsov, EgorPeirce, Anthonyhttp://hdl.handle.net/2429/500002014-08-15T09:36:37Z2014-08-13T00:00:00ZThe 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.
2014-08-13T00:00:00Z