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Fig. 1. Variation of equiaxed zone with superheat of liquid steel in continuously cast high carbon steel billets.

Abstract. We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.

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A two dimensional Computational Fluid Dynamics (CFD) model was developed to simulate the fluid flow and heat transfer of the molten steel in a ladle during the holding time, with gas purging from the bottom. Transient analysis of the temperature and the velocity distribution of the liquid steel during ladle standing and subsequent gas stirring was conducted, by employing a pressure-based fully-implicit finite volume approach. Stratification, which can adversely affect the quality of steel products, was seen to develop due to natural convection. Therefore, particular attention was paid to study the effect of bottom gas stirring in minimizing the thermal stratification. This was accomplished by introducing a novel approach of coupling the effects of natural convection and axisymmetric bottom gas injection. Various parametric studies was undertaken to examine the effects of standing time, gas flow rate and geometry of the ladle on the resultant thermal field. It was observed that bottom purging situation induces a strong recirculatory flow in the molten steel bath, with an increase in the order of turbulence giving rise to thermal homogenization. The results indicate that the thermal stratification can be effectively eliminated by a relatively gentle agitation. Homogenization takes place at a faster rate with an increase in the amount of bottom gas flow. The effect of ladle size was found to be inconsequential, in comparison to other parameters.

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Summary Chromium acetate-hydrolyzed polyacrylamide gel systems are applied in fractured reservoirs for conformance control. A portion of the gelant leaks off into the adjoining matrix during placement of the gelant in the fracture. This paper describes an experimental study on the effect of fluid leakoff on the performance of a gel treatment. The stability of a gel that is placed in a fracture and is subjected to a pressure gradient was also investigated. Physical models of a fracture were developed to conduct displacement experiments. The models were fractured Berea sandstones designed to permit leakoff of the gelant into the matrix on the sides of the fracture. A polyacrylamide-chromium acetate gelant was injected into the fracture under conditions in which there was leakoff and no leakoff into the matrix. A gel did not form, and the gelant was easily displaced from the fracture by subsequent brine injection when the gelant was placed without leakoff. When the gelant was placed with leakoff, a gel formed in the fracture after placement and provided significant flow resistance. It is hypothesized that the lack of gelation in the absence of leakoff was caused by diffusion of chromium from the fracture to the matrix. Diffusion reduced the chromium concentration in the gelant to levels at which gelation would not occur when the gelant was placed without leakoff. It was discovered that gels that were formed in a fracture ruptured when a brine pressure was applied at the inlet. The pressure where rupture occurred was determined for gels placed in tubing of various lengths and diameters. The rupture pressure was proportional to a length/diameter ratio. Introduction Fractures occur in hydrocarbon reservoirs for various reasons and at a variety of scales. In addition to natural fractures, the reservoir around oil wells may be hydraulically fractured to improve production. Over pressuring and thermal stresses during water injection also cause fractures. During waterflooding of a fractured reservoir, most of the water may bypass the matrix and preferentially flow through channels consisting of a single fracture or an interconnected network of fractures. The consequence is additional cost for handling of excess water, and an incomplete recovery of oil from the reservoir. One remedy is injection of a gelant into the fracture that reacts to form an immobile gel. The immobile gel then diverts injected water to previously unswept portions of the reservoir. Investigators have studied the application of gelled polymer treatments to fractured systems. Sydansk1 presented detailed laboratory testing and evaluation of a Cr(III)-partially hydrolyzed polyacrylamide gel system for fracture applications. Seright2 studied the performance of several immature, preformed, and mechanically degraded gels by displacing them through fractured cores. Immediate breakthrough of the tracer after treatment of the core with immature gel was observed, indicating poor performance of the gel to divert flow into the matrix. Similar experiments with mature gel resulted in delayed breakthrough of the tracer, indicating sweep improvement. Seright concluded that superior diversion can be obtained by injecting mature gel (rather than gelant) in the fracture. Seright3 also studied the placement of preformed Cr(III)- partially hydrolyzed polyacrylamide gel in fractured Berea core. He investigated the pressure drop, the gel dehydration, and the delay in gel propagation that occur during the flow of preformed gel through fractured cores. Dehydration and leakoff of water concentrated the gel in the fracture. The gel injected at a later stage of extrusion made wormholes through the thickened gel to reach the fracture outlet. During placement of an immature gel (referred to here as a gelant) through a fracture, some of the gelant leaks off to the adjoining matrix. The effect of leakoff on the strength of gel placed in a fracture was not studied previously. The objectives of this research were to study (a) the role of leakoff on the performance of a gel placed in a fracture and (b) the ability of a gel formed within a fracture to resist failure when subjected to an imposed brine pressure. Experimental Details Fracture Models. Three types of fracture models were used for displacement experiments: fractured slabs, fractured cores, and slots. The slabs and cores contained one fracture and were constructed to provide for leakoff from both sides of the fracture to the adjoining matrix. The slot was constructed from a saw-cut rock and an acrylic wall. A schematic of the fractured slab and associated fluid ports is shown in Fig. 1. Slabs cut from Berea sandstone were 10 in. long, 2 ft wide, and 1 in. thick. Each slab was fractured along the 10 in. length using a Hydrasplit (Park Industries Inc., St. Cloud, Minnesota) rock splitter. Spacers were placed between the two symmetric halves to establish a fracture aperture of known width. The top and bottom of the fracture were then sealed with an epoxy coating. Acrylic fluid ports were installed across the fracture aperture at the front and back faces to allow flow through the fracture. Acrylic side plates were installed on each side of the matrix opposite the fracture. The side plates were milled to provide an aperture between the plate and the smooth face of the slab. Ports at each end of the side plates provided for fluid withdrawal from each side of the slab. The slab was sealed by coating the top, bottom, front, and back surfaces with epoxy. Pressure ports were installed at the inlet and outlet ends of the fracture. Details of the slab preparation process are found in Ref. 4. Each slab was initially saturated with 1% NaCl brine. Permeability of the fracture was determined by flowing brine at constant flow rates through the fracture with the side outlets closed. Aperture width was estimated using the theory of flow between two parallel plates. Matrix permeability was determined by injecting brine into the fracture and measuring the flow rate at the matrix (side) outlets and pressure differentials across the matrix sections. Pore volume of the matrix was determined from tracer runs. A step change in concentration of potassium iodide was introduced into the fracture, and the effluent concentrations from the matrix (side) outlets were monitored using a UV spectrophotometer. The slabs had permeabilities of about 200 md and an average porosity of 0.17. Some slabs were cleaned and reused in other runs.

We show that a cuspidal normalized Hecke eigenform g of level one and even weight is uniquely determined by the central values of the family of RankinSelberg L-functions L(s, f ⊗ g), where f runs over the Hecke basis of the space of cusp forms of level one and weight k with k varying over an infinite set of even integers.