We consider the problem of choosing a set of k sensor measurements, from a set of m possible or potential sensor measurements, that minimizes the error in estimating some parameters. Solving this problem by evaluating the performance for each of the m k possible choices of sensor measurements is not practical unless m and k are small. In this paper we describe a heuristic, based on convex optimization, for approximately solving this problem. Our heuristic gives a subset selection as well as a bound on the best performance that can be achieved by any selection of k sensor measurements. There is no guarantee that the gap between the performance of the chosen subset and the performance bound is always small; but numerical experiments suggest that the gap is small in many cases. Our heuristic method requires on the order of m 3 operations; for m = 1000 possible sensors, we can carry out sensor selection in a few seconds on a 2 GHz personal computer.

SPE Member Abstract This paper reviews the state-of-the-art of horizontal well and drainhole technology, and provides guidelines for the initial evaluation of horizontal well and drainhole drilling prospects. The primary emphasis of the paper is reservoir engineering. Since drilling and completion methods influence a horizontal well's performance, a brief review of these aspects of technology is also included. The paper also reports the field production histories available in the literature. Introduction During the last decade, significant advances in drilling technology have made it possible to drill horizontally. Although the major emphasis of this paper is a reservoir engineering review, for completeness a brief description of drilling and completion of horizontal wells is also included. Moreover, for a successful field operation, a drilling method should be chosen based upon reservoir considerations. Horizontal wells are normally new wells, 1000 to 3000 ft long, which are drilled from the surface. Drainholes are generally drilled from the existing vertical wells and are 100 to 700 ft long. One can drill either a single drainhole or multiple drainholes through a single vertical well. In this paper the term horizontal well refers to both new horizontal wells and drainholes, unless noted otherwise. Horizontal wells and drainholes represent wells with limited fracture height, where fracture height is equal to the wellbore diameter. A properly designed horizontal well would be equivalent to a vertical well with a fully penetrating fracture. A horizontal well represents a long, controlled vertical fracture. In most fracture jobs it is difficult to obtain infinite conductivity and, moreover, fracture conductivity decreases over time. In contrast, a horizontal wellbore offers an almost permanent infinite conductivity fluid flow path. Additionally, in reservoirs where the bottom water or top gas cap renders fracturing difficult, a horizontal well offers an alternative to obtain high production rates without gas and water coning. A horizontal well offers a viable completion option and will compete with fracturing in the future. In general, horizontal wells are found effective in thin reservoirs, some naturally fractured reservoirs, tight reservoirs, and in reservoirs with gas and water coning problems. In this paper these horizontal well characteristics and applications are discussed. In addition to a literature review, the paper provides guidelines for oil production forecasting. Moreover, available field data are also discussed. DRILLING AND COMPLETION As listed in Table 1, the presently available horizontal drilling methods can be classified in four broad categories, depending upon the turning radius required to turn from a vertical to a horizontal direction. Figure 1 compares turning radii of different drilling methods. A detailed discussion of various drilling methods is given elsewhere, but a brief description is given below. Ultra Short Turning Radius (1 to 2 ft) This method utilizes water jets to drill 100–200 ft long drainholes with a turning radius of 1 to 2 ft. The process involves underreaming the vertical wellbore and then drilling several radials from the underreamed zone. The first drilling system requires a 48 inch diameter underreamed zone while the improved second system requires a 24 inch diameter zone. The underreamed zone length varies from 6 to 10 ft depending on the system utilized. The drainhole diameter varies from 1-1/2 to 2-1/2 inches. Usually two or more drainholes are drilled. For sand control, the drainholes are completed using either slotted liners or gravel packing. After completing the drainhole, the pipe is severed. P. 613^

This paper presents a technique to forecast the production from horizontal wells in rectangular drainage areas. The technique is useful for wells located either centrally or offcentrally in the areal drainage plane. Analytical pressure transient solutions were used to calculate the shape factor, CA, and the corresponding equivalent skin factor, sCA. Moreover, dimensionless time at which the pseudo-steady state begins is reported for centered and off-centered horizontal wells and fully penetrating infinite-conductivity vertical fractures. These results are presented in the form of easy to use correlations. Furthermore, these correlations were tested by analyzing the relevant pressure transient data reported in the literature. The pseudo-steady state shape factor, CA, and skin term, sCA, for a horizontal well approaches fully penetrating infinite-conductivity vertical fracture for very large values of dimensionless horizontal well length, LD. Knowledge of shape factor, CA, and of corresponding beginning of pseudo-steady state time, tDA, pss would facilitate computation of the horizontal well production forecast from a corresponding infinite-conductivity, fully penetrating vertical fracture forecast. The methods for prediction of the inflow performance relationship for horizontal wells is also outlined, using example problems representative of horizontal well applications in on-shore and off-shore areas. Introduction In recent years, horizontal wells have been increasingly used in field applications. Currently, various commercial techniques are available to drill, complete and test horizontal wells. The up-to-date technology of drilling, completion, testing, as well as production forecasting for horizontal wells and related vertical well technology has been reviewed by various authors. A horizontal well pressure analysis has shown that the horizontal well behaves as a vertical fracture with a fracture height equal to the wellbore diameter. The insignificant pressure drop observed in the horizontal wells indicates an infinite-conductivity for the flow in the wellbore. On the other hand, it is difficult to obtain infinite fracture conductivity in a conventional fracture stimulation. Thus, horizontal wells would provide an alternative for conventional fracture stimulation. The state of the art review of the reservoir engineering aspects of horizontal well production has been presented by Giger et al. and others. The pressure transient solutions for horizontal wells in finite and infinite reservoirs have been discussed by various authors in the literature. However, there seems to be a lack of information regarding the influence of the drainage area and its shape on horizontal well performance. The purpose of this paper is to address this issue. The shape factors indicate the influence on well productivity of well location within the drainage boundary.

Abstract-We consider the problem of choosing the gate sizes or scale factors in a combinational logic circuit in order to minimize the total area, subject to simple RC timing constraints, and a minimum-allowed gate size. This problem is well known to be a geometric program (GP), and can be solved by using standard interiorpoint methods for small-and medium-size problems with up to several thousand gates. In this paper, we describe a new method for solving this problem that handles far larger circuits, up to a million gates, and is far faster. Numerical experiments show that our method can compute an adequately accurate solution within around 200 iterations; each iteration, in turn, consists of a few passes over the circuit. In particular, the complexity of our method, with a fixed number of iterations, is linear in the number of gates. A simple implementation of our algorithm can size a 10 000 gate circuit in 25 s, a 100 000 gate circuit in 4 min, and a million gate circuit in 40 min, approximately. For the million gate circuit, the associated GP has three million variables and more than six million monomial terms in its constraints; as far as we know, these are the largest GPs ever solved.

Abstract-We first consider the problem of determining the doping profile that minimizes base transit time in a (homojunction) bipolar junction transistor. We show that this problem can be formulated as a geometric program, a special type of optimization problem that can be transformed to a convex optimization problem, and therefore solved (globally) very efficiently. We then consider several extensions to the basic problem, such as accounting for velocity saturation, and adding constraints on doping gradient, current gain, base resistance, and breakdown voltage. We show that a similar approach can be used to maximize the cutoff frequency, taking into account junction capacitances and forward transit time. Finally, we show that the method extends to the case of heterojunction bipolar junction transistors, in which the doping profile, as well as the profile of the secondary semiconductor, are to be jointly optimized.