SUMMARYA computational procedure is developed for solving the problem of a circular hydraulic fracture propagating under the action of frac-fluid being pumped in at a central wellbore. The crack is modelled as continuous distributions of ring dislocations and the resulting elasticity singular integral equation is solved numerically. The fluid flow equations are approximated by local and global interpolation finite difference schemes. The coupling between elasticity and fluid flow is handled numerically, by two different algorithms: one iterates on crack tip velocity whereas the other varies the time step size until it agrees with the chosen increment in crack length. Sample results are given; it is found that the velocity algorithm is computationally more efficient and more stable. The model allows detailed tracing of pressure distribution and fluid flow in the fracture, even under complex conditions of cyclic injection and fluid rheology. It may serve as a stand-alone model of (horizontal) hydrafracs-especially at shallow depths-or it may be used as a reference frame to test the various numerical formulations/algorithms required for the ongoing development of a fully 3-D hydrafrac simulator.
Range and Doppler tracking data from Mariners 6 and 7 have been used to obtain values for the ratio of the mass of the earth to that of the moon which are in substantial agreement with those determined from other Mariner and Pioneer spacecraft. There is an inconsistency of about 0.004 percent in values for the mass of the moon determined from lunar trajectories. A gravitational constant for Mars of 42,828.48 +/- 1.38 cubic kilometers per second per second, obtained on the basis of data collected during the 5 days prior to the closest approach of Mariner 6 to Mars, is in excellent agreement with the result obtained by Null from tracking data of Mariner 4.
During the Pioneer Saturn encounter, a continuous round-trip radio link at S band ( approximately 2.2 gigahertz) was maintained between stations of the Deep Space Network and the spacecraft. From an analysis of the Doppler shift in the radio carrier frequency, it was possible to determine a number of gravitational effects on the trajectory. Gravitational moments ( J(2) and J(4)) for Saturn have been determined from preliminary analysis, and preliminary mass values have been determined for the Saturn satellites Rhea, Iapetus, and Titan. For all three satellites the densities are low, consistent with the compositions of ices. The rings have not been detected in the Doppler data, and hence the best preliminary estimate of their total mass is zero with a standard error of 3 x 10(-6) Saturn mass. New theoretical calculations for the Saturn interior are described which use the latest observational data, including Pioneer Saturn, and state-of-the-art physics for the internal composition. Probably liquid H(2)O and possibly NH(3) and CH(4) are primarily confined in Saturn to the vicinity of a core of approximately 15 to 20 Earth masses. There is a slight indication that helium may likewise be fractionated to the central regions.
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