Numerical solutions of stationary flow resulting from immersion of a single body in simple shear flow are reported for a range of Reynolds numbers. Flows are computed using finite-element methods. Comparisons to results of asymptotic low-Reynoldsnumber theory, experimental study, and other numerical techniques are provided. Results are presented primarily for isotropic bodies, i.e. the circular cylinder and sphere, for both of which the two conditions of a torque-free (freely-rotating) and fixed body are investigated. Conditions studied for the sphere are 0 < Re < 100, and for the circular cylinder 0 < Re < 500, with the shear-flow Reynolds number defined as Re =γ a 2 /ν;γ is the shear rate of the Cartesian simple shear flow u = (γ y, 0, 0), a is the cylinder or sphere radius, and ν is the kinematic viscosity of the fluid. In the torque-free case, the rotation rate of the body decreases with increasing Re. Qualitative dependence, seen in the Re = 0 fluid flow field, upon whether the body is fixed against rotation or torque-free vanishes as Re increases and the fluid flow is more similar to that around the fixed body: the influence of rotation of the body and the associated closed streamlines are confined to a narrow layer about the body for Re > O(1). Separation of the boundary layer is observed in the case of a fixed cylinder at Re ≈ 85, and for a fixed sphere at Re ≈ 100; similar separation phenomena are observed for a freely rotating cylinder. The surface stress and its symmetric first moment (the stresslet) are presented, with the latter providing information on the particle contribution to the mixture rheology at finite Re. Stationary flow results are also presented for elliptical cylinders and oblate spheroids, with observation of zero-torque inclinations relative to the flow direction which depend upon the aspect ratio, confirming and extending prior findings.

Abstract. A posteriori error estimates have had a major impact on adaptive error control for the finite element method. In this paper, we review a relatively new approach to a posteriori error estimation based on residuals and a variational analysis. This approach is distinguished by a direct attempt to account for the effects of stability on the propagation of error. We illustrate properties of this approach using several examples.Key words. a posteriori error estimate, adaptive error control, computational error estimation, dual problem, finite element method, multi-scaled problems, residual, stability, stability factor, variational analysis 1. Introduction. The search for reliably accurate numerical tools for multi-scaled differential equations has become increasingly urgent in recent years. Not the least because multi-scaled problems arise in a wide range of applications and computing accurate numerical solutions often strains the limits of computational resources. One way to obtain accurate solutions of multi-scaled problems is through computational error estimation and adaptive error control. Computational error estimation is directed towards determining the kind of information that can be accurately obtained from a particular computation and estimating the accuracy of said information. This is an important goal both in general scientific terms as well as in terms of adaptive error control, that is for deciding how to use computational resources to achieve a desired accuracy.Over the last two decades, there has been significant progress in computational error estimation and adaptive error control arising out of developments in a posteriori error analysis. In a posteriori analysis, the error of a numerical solution is estimated as much as possible in terms of computable quantities that depend on the numerical solution and in particular the estimate is computed after the numerical solution has been computed. The progress in a posteriori analysis has resulted in important advances in the reliably accurate solution of multi-scaled problems. As a consequence, a posteriori techniques have found widespread use in engineering and mathematics.There are several different approaches to a posteriori error analysis. In this paper, we review a relatively new approach based on residuals and variational analysis involving the dual, or adjoint, problem to the original equation. This approach is distinguished by a direct attempt to account for the effects of propagation and accumulation of errors by computational means. We present a formal description of this approach and describe the application to elliptic and parabolic differential equations. We also illustrate various aspects of this approach using a set of numerical examples.

Waterfloods serve many purposes and their performance can have major economic impact on the drilling, production and management of hydrocarbon reservoirs. The primary purpose of the Mars (Mississippi Canyon 807) waterflood in the deepwater of the GOM is to increase recovery efficiency in three main reservoirs. In addition, the waterflood helps maintain reservoir pressure in selected sands, which minimizes compaction and subsequent well failure. Surveillance of the waterflood through carbon/oxygen logging, of the formation through compaction logging, and of the individual reservoir layers through multi-rate production logging is critical to the success of the Mars field. The program has consisted of obtaining time-lapse logging and reservoir pressures combined with reservoir modeling. Data has been obtained in the injection, producing, and monitor intervals of several Mars wells and proven to be vital to the evaluation of waterflood efficiency and the prediction of the waterflood front. A monitoring logging program has been conducted in the Mars area since 1996, with initial baseline surveys, to the present day monitoring surveillance program. The logging program has had two purposes. The primary purpose of the program has been to monitor the "sea water" waterflood saturation fronts with carbon/oxygen logging and integrate this data into the reservoir models. A second purpose has been to monitor strain as a result of reservoir compaction and to monitor the effects of the waterflood on the rate of strain. This data is used to help determine wellbore integrity and ultimately to predict wellbore failure. It also provides calibration data for the compaction model used in reservoir simulation. Selected Mars well examples are described in detail to highlight the results of time-lapse monitoring of the waterflooded reservoirs. A comparison of log data with simulation modeling predictions demonstrates the benefit of the data acquisition and evaluation methods. Discussion is focused on best practices learned during the 12 year program, how log responses have helped verify modeling parameters, and on justification of future activities. Introduction The Mars Field (Figure 1) consists of six OCS leases in the Mississippi Canyon Area - Blocks 762, 763, 806, 807, 850 and 851 - located in the Gulf of Mexico about 130 miles southeast of New Orleans. The leases were acquired in 1985 and 1988 and the first discovery well was drilled on Mississippi Canyon Block 763 in 1989. Shell and their partner BP announced plans in 1993 to develop Mars utilizing a 24-slot tension leg platform (TLP) to be installed on Block 807. The TLP was installed in May 1996 in a water depth of 2,940 feet. Production began July 8, 1996 and peaked in June 2000 at 208,000 BOPD and 217 mmcfd. Mars consists of a series of Miocene to Pliocene age turbidite sands deposited within a minibasin bounded by the deeply rooted Venus salt body to the southeast and the more tabular Antares salt body to the north and west. The northeastsouthwest trending basin becomes narrow and more confined within the deeper Miocene interval. The geologic age of the Mars formations above 14,000 feet are Pliocene and the deeper reservoirs are Miocene. Exploratory and appraisal drilling encountered 14 major and 10 minor moderately geopressured pay bearing sands between 10,000 feet and 19,000 feet sub sea.

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