The earth and moon are considered as a two‐body system in gravitational isolation from the sun and other planets. The lunar orbit is taken as circular, and the solid earth is assumed to be a rigid sphere (with no tidal deformation) so that there are no precessional torques other than those arising from the tidally deformed ocean. Numerical solutions to Laplace's tidal equations, with dissipation by linear bottom friction, are used to obtain ocean‐wide distributions of tidal amplitude for two idealized continentalities: a single circular continent (spherical cap) centered at the north pole, and a single spherical cap centered on the equator. Calculations are made for two different values of the frictional resistance coefficient, thus giving rise to four sets of solutions. The computed tidal amplitudes are used to calculate the oceanic tidal torque, which in turn is used to integrate the orbital equations backward in time for solution of the two‐body problem. The Coriolis parameter and the tidal frequencies change with time, thus requiring that the tidal equations be solved several times during the course of each orbital integration. In this manner, the earth's rotational and the moon's orbital histories are determined on a geologic time scale for each of the four models. The calculated position of the lunar orbit at 4.5 billion years ago is found to range from 38 to 53 earth radii in the four models and corresponds to a sidereal month of 330 to 550 hours. The sidereal day would have been 12 to 18 hours, with a relative inclination of 3° to 22° between the terrestrial and lunar poles. These results are in sharp contrast to those from previous studies of the earth‐moon system, most of which indicated a Roche limit approach of the two bodies roughly 1 to 2 billion years ago and presented therefore a time scale difficulty in theories of lunar origin. This contrast arises mainly from the fact that previous modelers avoided solution of Laplace's tidal equations by prescribing a constant frictional phase lag angle between the angular position of the moon and the major axis of the second‐degree harmonic of the tidally deformed surface of the earth. The amount of phase lag was established from the present astronomically deduced rate of tidal dissipation, but this precluded dynamic variations in tidal torque over geologic time, which are critical for determination of the orbital time scale. The present rate of oceanic tidal dissipation is evidently anomalously high because of the near resonance of the oceanic response in both frequency and shape to the tidal forcing. For the earth and moon to have evolved to their present separation from a distance of less than about 35 earth radii solely on the basis of oceanic tidal dissipation would have required the M2 response to have been near resonance for a significant portion of geologic history, a rather implausible scenario. The calculations reported here show that, on the contrary, frictional coupling between the earth and moon was much weaker than at present throughout most of ...
Summary Finite-element models of depletion-induced reservoir compaction and surface subsidence have been calibrated with observed subsidence, locations of surface fissures, and regions of subsurface casing damage at South Belridge and used predictively for the evaluation of alternative reservoir-development plans. Pressure maintenance through diatomite waterflooding appears to be a beneficial means of minimizing additional subsidence and fissuring as well as reducing axial-compressive-type casing damage. Introduction The South Belridge field, Kern County, CA, produces from oil accumulations in both the shallow Tulare formation, an unconsolidated to loosely consolidated sandstone 400 to 600 ft thick, and the underlying diatomite reservoir, a highly compressible formation averaging some 1,000 ft in thickness. In the mid-1980's, operators in the field experienced casing damage and an increasing number of well failures, and in Jan. 1987, after a heavy rainstorm, they observed surface fissures at the north end of the field just outside its eastern and western boundaries. The fissures were oriented north-south to north northeast-south southwest, approximately parallel to the orientation of the greatest horizontal principal stress measured in the formation subsurface. Suspicions were that the well failures and surface fissures were related to reservoir depletion and resulting compaction and subsidence in the diatomite formation. A major program of numerical modeling was implemented to help monitor the compaction/subsidence process and to optimize field development in order to minimize future adverse effects of compaction and subsidence on casing damage and fissuring. Finite elements had been used for modeling reservoir compaction and surface subsidence at a number of fields throughout the world, including the Lost Hills field in California, a diatomite/porcelanite reservoir similar to South Belridge. For our Belridge analysis, we adopted the Geomechanical Finite-Element Program (GEOFEP) used in the study of the Groningen gas field in the Netherlands.
Summary Earth stress magnitudes in the South Belridge oil field, determined from integrated density logs and microhydraulic fracturing tests, indicate that the vertical stress is generally the intermediate principal stress, except possibly at the deepest zone tested (2,100 ft [640 m]), where it is approximately equal to the lesser compressive horizontal stress. Azimuth of the greater horizontal stress and of induced hydraulic fractures, as measured or inferred by several different techniques, is N15 degrees E 15 degrees. Introduction Wells completed in the diatomite/porcelanite reservoir of the North and South Belridge oil fields (Fig. 1) usually are hydraulically fractured at initial completion. Current expectations are that primary recovery will be relatively low and that supplemental recovery techniques may provide a significant portion of the ultimate production from this major oil accumulation. The success of any supplemental recovery technique will depend on the geometry of the extensive hydraulic fracture system induced during primary production, as well as that of the natural fractures present. Because fracture geometry is determined by earth stresses acting at the time of fracture formation, we attempted to determine as precisely as possible the stress state in the diatomite/porcelanite reservoir at Belridge. The three principal earth stresses may reasonably be assumed to be oriented vertically and horizontally in this reservoir. A complete description of the stress state therefore requires determination of the magnitudes of the three stresses and the azimuth of one of the two horizontal stresses. The various methods used for determining stress state at Belridge are based on geological observation and borehole mechanics. Geological methods allow orientation and possibly relative magnitudes of the three principal stresses to he inferred. For example, in areas of recent or active faulting, the following stress regimes are believed to prevail. For high-angle normal faulting, SV greater than SH greater than SH, with the greater horizontal stress, SH, oriented along the direction of surface strike. For near-vertical strike/slip faulting, SH greater than SV greater than SH, with SH Oriented at about 30 to 45 degrees with respect to the fault. In the case of low-angle thrust faulting, SH greater than SH greater than SV, with SH perpendicular to the surface strike of the fault. Borehole- mechanics methods examine stress state at the borehole wall under conditions of tensile failure (hydraulic fracturing) or compressive shear failure (borehole sloughing). When a fracture is induced at a borehole wall and then extended into the formation, it tends to align itself in a plane perpendicular to the least principal earth stress. If, as is assumed here, the principal earth stresses are oriented vertically and horizontally, and one of the two horizontal principal stresses is the least of the three, then a hydraulically induced fracture will leave a vertical wellbore in a vertical plane and will maintain that orientation as it propagates outward into the formation. The induced fracture will align with the direction of SH. Thus, a measure of fracture azimuth gives the direction Of SH and vice versa. On the other hand, if the vertical stress is the least of the three principal stresses, then the fracture may still leave the well in a vertical orientation, but will tend to turn over and to become horizontal as it enters a region in which the earth's stresses are essentially undisturbed by the presence of the borehole. By the time a fracture has been extended 5 to 10 borehole diameters into the formation, it should be responding to earth stresses, which, at this distance from the borehole, are distorted by less than 1 %. 3 The closure pressure of the fracture should then very nearly equal the least principal earth stress, either SV or SH-The instantaneous shut-in pressure (ISIP) recorded following fracture initiation and propagation is sometimes a reasonably good measure of the fracture closure stress and in this Belridge study is considered to be a direct measure of the least earth stress. If there is sufficient imbalance between the two horizontal stresses, compressive shear failure of the borehole wan may occur, causing sloughing and hole enlargement along a preferred cross-sectional axis. In a vertical hole, enlargement generally will be greater along an axis aligned with the direction Of SH (see the Appendix). If there are known instances of out-of-round boreholes that can reasonably be attributed to stress-related shear failure, then oriented four-ann caliper surveys or borehole televiewer logs can be used to determine azimuths of the horizontal stresses. Stress Magnitudes During Oct. and Nov. 1980, 11 small-volume hydraulic fractures ("microfracs") were induced to determine the magnitudes of the principal horizontal stresses. The fractures were formed between straddle packers set in open hole at two or three different elevations in each of four closely spaced wells. Location and Procedure. The four wells selected for this study penetrated the contact between the Belridge diatomite and the overlying Tulare sandstone near the crest of the structure (Fig. 2). Three intervals were fractured in Wells A, B, and C and two intervals in Well D. Correlation of electric logs ensured that corresponding intervals were fractured in all four wells. The 8-ft [2.4-m] intervals between straddle packers in which the small fractures were induced fell within larger intervals that were later hydraulically fractured through perforated casing during routine completion operations for the four wells. Preparatory to microfracturing in open hole, the straddle-packer assembly was run in on drillpipe with the hole and pipe filled with the "standard" mud used fieldwide for drilling the diatomite. Mud density was nominally 8.9 lbm/gal [1066 kg/m3]. After the packers were set, fresh water was pumped in on top of the mud at a constant rate of 3 gal/min [11 L/min]. Pressure was recorded as a function of time both at the surface and in the interval between the packers by a surface-recording Amerada pressure gauge. A digital memory readout (DMR) pressure recorder was hung below the bottom packer to detect any pressure communication that might occur during the fracturing operation. Breakdown of the formation was followed by a succession of five or more periods during which mud was pumped into the induced fracture for 1 to 5 minutes, after which the system was shut in for 2 to 10 minutes. Fig. 3 shows the three pressure-vs.-time records obtained in Well B. The continuous solid lines are recordings of the downhole Amerada pressure gauge, and the dotted lines are the DMR records adjusted to the elevation of the Amerada gauge. Vertical Stress. As stated above, the principal earth stresses at Belridge are assumed to be oriented vertically and horizontally. SV was determined from formation-density-compensated (FDC) logs run in Wells B and D. Differences of less than 20 psi [140 kpa] were observed over the intervals of interest, so averages of measurements from these two wells were used to provide SV as a function of depth for all four wells. SPEFE P. 541^
Summary Pore-pressure (PP) and fracture-gradient (FG) predictions were prepared for Prelude development wells in the Browse basin in offshore northwest Australia. The PP forecasts were based on resistivity- and sonic-based models calibrated with pressure measurements and drilling events, such as kicks from existing wells. FGs were based on leakoff tests and loss events from offset wells and were not necessarily equal to either the minimum compressive principal stress (often considered a lower bound to FG) or the formation-breakdown pressure (often considered an upper bound to FG that includes effects of formation tensile strength and near-wellbore hoop stress). The minimum compressive horizontal stress was calculated from lithology-dependent effective-stress ratios. Maximum horizontal stress was inferred from observed breakouts. PP and stresses were combined with formation properties from well logs and laboratory rock-mechanics tests to provide input for elastoplastic (shales) and poroelastic (sands) borehole-stability (BHS) models. These techniques are applicable to exploration, appraisal, or early-development wells that have potential for encountering geopressured formations in high-angle well sections requiring good predrill estimates to adequately plan the casing and drilling programs and determine BHS. The predrill studies can be extended to provide integrated real-time PP and BHS while drilling, and the models can be recalibrated after each well to provide updated predictions for subsequent wells. There are only minor deviations in the predicted PP and FG among the different well locations considered. Common features include potential loss zones in the shallow overburden, pressure ramp within the Jamieson, pressure regression below the Aptian, and near-hydrostatic pressure within the Upper Swan and below. The BHS models indicate that minimum-required mud weight in deviated sections could be up to 20% higher than that required to balance formation PP. In one well that would cross a suspected fault, the risk of fault reopening or reactivation is low.
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