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In hydraulic fracture stimulation of conventional reservoirs (e.g. tight gas and deep water unconsolidated sands), the use of sophisticated design models is almost indispensable. These hydraulic fractures are typically single fracture treatments and executed from near vertical wellbores. It is well understood that the post-fracture productivity is directly linked to achieving an optimum hydraulic fracture conductivity, which is governed largely by propped fracture length and width. From an engineering and operation execution point-of-view, the goal is to pump into the fracture the desired (large) volume of proppant without encountering pre-mature â screen-outâ . Therefore, the prediction of fracture geometry and the design of pad volume become critical for propped fracture design. Models are calibrated on-site with dedicated mini-frac tests prior to main propped fracture treatments. Two important calibration parameters are fluid efficiency (leak-off behavior) and minimum in-situ stress (stress profile/contrast). The injection pressure during fracturing, which is readily available, is a valuable source of information and often analyzed and compared with model prediction for fracture diagnostics. In practice, a wide range of models have been employed successfully. However, such considerations do not appear to be important for unconventional resources where multiple fractures are pumped from a long horizontal well. In fact, multi-fracced horizontal well technology has advanced through field trials and experimentation without much help from modelling or understanding of multiple-fracture mechanics. Perhaps one reason is the lower risk of screening-out. This could be due to 1) the extreme low permeability of unconventional shales, which renders the use of high proppant concentration unnecessary and 2), the treatment of multiple fractures in one stage of pumping allows for one or two of the fractures to screen-out without causing an unacceptable rise of pumping pressure. In fact, with the pumping of tens and even a hundred fractures in one horizontal well, the â systemâ appears to tolerate some â non-performingâ fractures without impairing the ultimate production. Conventional wisdom has it that fracture length should be maximized, but in the development of onshore unconventional resources, the horizontal wells are spaced evermore closer to each other, and consequently, the fracture length may not need to be long in order to access the reserves. Operators have successfully fractured and produced from unconventional reservoirs without the use of advanced modelling technology. This begs the questions of what areas of research and model design parameters should we focus on Can we avoid the â detailsâ while dealing with the â big pictureâ such as fractures spacing, horizontal well length/direction, the wellâ s landing depth, and their impact on cost and production Are research and model development sufficiently guided and tested by field data/observations
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An exploration of how and why local BC Punjabi singer, GS Hundal, and producer, Intesense, decided to root themselves in the Punjabi music industry. This short film highlights the various influences impacting these individuals, as well as how they themselves impacted the Punjabi music industry.
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Let $M$ be a compact oriented $d$-dimensional manifold with boundary $N$. A natural geometric boundary value problem is to find an asymptotically hyperbolic Einstein metric $g$ on (the interior of) $M$ with prescribed `conformal infinityâ on $N$. A little more precisely, the problem is to find (Einstein) $g$ with the boundary condition $x^2g$ tends to a metric $h$ on $N$ as $x$ goes to $0$, $x$ being a boundary defining function for $N$. The freedom to rescale $x$ by an arbitrary smooth positive function means that only the conformal class of $h$ is naturally well defined. Hence the terminology `conformal infinityâ in this boundary problem. Since the pioneering work of Graham and Lee (1991) the problem has attracted attention from a number of authors. If the dimension $d$ is $4$, there is a refinement, asking that $g$ be anti-self-dual as well as Einstein (satisfying the same boundary condition). If $M$ is the ball, this is the subject of the positive frequency conjecture of LeBrun (1980s) proved by Biquard in 2002. In this talk, which is based on joint work with Joel Fine and Rafe Mazzeo, I shall explain a gauge theoretic approach to the ASDE problem which is readily applicable for general $M$ and the currently available results.
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"Semi-discrete optimal transport between a discrete source and a continuous target has intriguing geometric properties and applications in modelling and numerical methods. Unbalanced transport, which allows the comparison of measures with unequal mass, has recently been studied in great detail by various authors. In this talk we consider the combination of both concepts. The tessellation structure of semi-discrete transport survives and there is an interplay between the length scales of the discrete source and unbalanced transport which leads to qualitatively new regimes in the crystallization limit."
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Cancer is an evolutionary process driven by somatic mutations that accumulate in a population of cells that form a primary tumor. In later stages of cancer progression, cells migrate from a primary tumor and seed metastases at distant anatomical sites. I will describe algorithms to reconstruct this evolutionary process from DNA sequencing data of tumors. These algorithms address challenges that distinguish the tumor phylogeny problem from classical phylogenetic tree reconstruction, including challenges due to mixed samples and complex migration patterns. Joint work with Mohammed El-Kebir and Gryte Satas
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An important and challenging class of two-stage linear optimization problems are those without relative-complete recourse, wherein there exists first-stage decisions and realizations of the uncertainty for which there are no feasible second-stage decisions. Previous data-driven methods for these problems, such as the sample average approximation (SAA), are asymptotically optimal but have prohibitively poor performance with respect to out-of-sample feasibility. In this talk, we present a data-driven method for two-stage linear optimization problems without relative-complete recourse which combines (i) strong out-of-sample feasibility guarantees and (ii) general asymptotic optimality. Our method employs a simple robustification of the data combined with a multi-policy approximation. A key contribution of this work is the development of novel geometric insights, which we use to show that the proposed approximation is asymptotically optimal. We demonstrate the benefit of using this method in practice through numerical experiments. This is a joint work with Dimitris Bertsimas and Brad Sturt.
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Let $L$ be a fixed link. Given a link diagram $D$, is there a sequence of crossing exchanges and smoothings on $D$ that yields a diagram of $L$ We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, and Taniyama that if $L$ is a prime link with crossing number at most five, then there is an algorithm that answers this question in polynomial time. We show that the same holds for all torus $T_{2,m}$ and all twist knots.
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Analytically computing the spectrum of the Laplacian is impossible for all but a handful of classical examples. Consequently, it can be tricky business to determine which geometric features are spectrally determined; such features are known as geometric spectral invariants. Weyl demonstrated in 1912 that the area of a planar domain is a geometric spectral invariant. In the 1950s, Pleijel proved that the n-1 dimensional volume of a smoothly bounded n-dimensional Riemannian manifold is a geometric spectral invariant. Kac, and McKean & Singer independently proved in the 1960s that the Euler characteristic is a geometric spectral invariant for smoothly bounded domains and surfaces. At the same time, Kac popularized the isospectral problem for planar domains in his article, ``Can one hear the shape of a drum'' Colloquially, one says that one can ``hear'' spectral invariants. In this talk I will not only discuss my work with many collaborators (Rafe Mazzeo, Zhiqin Lu, Clara Aldana, Klaus Kirsten, David Sher, Medet Nursultanov) but also highlight the works of many other colleagues, who share a similar interest in ``hearing singularities.''
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