Some Fractional Extensions of the Temperature Field Problem in Oil Strata

This survey is devoted to some fractional extensions of the incomplete lumped formulation, the lumped formulation and the formulation of Lauwerier of the temperature field problem in oil strata. The method of integral transforms is used to solve the corresponding boundary value problems for the fractional heat equation. By using Caputo’s differintegration operator and the Laplace transform, new integral forms of the solutions are obtained. In each of the different cases the integrands are expressed in terms of a convolution of two special functions of Wright’s type.* Partially supported by National Science Research Fund - Bulgarian Ministry of Education and Science, under Grant MM 1305/2003

Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function

2000 Mathematics Subject Classification: 26A33, 33C45This paper refers to a fractional order generalization of the classical Jacobi polynomials. Rodrigues’ type representation formula of fractional order is considered. By means of the Riemann–Liouville operator of fractional calculus fractional Jacobi functions are defined, some of their properties are given and compared with the corresponding properties of the classical Jacobi polynomials. These functions appear as a special case of a fractional Gauss function, defined as a solution of the fractional generalization of the Gauss hypergeometric equation.* Partially supported by Project MM 1305 - National Science Fund, Bulgarian Ministry of Educ. Sci

The Grünwald–Letnikov method for fractional differential equations

AbstractThis paper is devoted to the numerical treatment of fractional differential equations. Based on the Grünwald–Letnikov definition of fractional derivatives, finite difference schemes for the approximation of the solution are discussed. The main properties of these explicit and implicit methods concerning the stability, the convergence and the error behavior are studied related to linear test equations. The asymptotic stability and the absolute stability of these methods are proved. Error representations and estimates for the truncation, propagation and global error are derived. Numerical experiments are given

In Situ Determination of Capillary Pressure and Relative Permeability Curves Using Well Logs.

A pressure profile obtained from the formation tester defines the wettability, free water level, and hydrocarbon and water densities as well as the capillary pressure above the free water level. Correlating the pressure values to the water saturation values determined from the resistivity logs results in a capillary pressure/water saturation, \rm P\sb{c}(S\sb{w}), curve characteristic of the reservoir. A relative permeability curve then can then be derived from this \rm P\sb{c}(S\sb{w}) curve using empirical relationships. This approach was tested in the laboratory using an eight-foot vertical sandstone core to simulate the formation. The core was fitted with electrode arrays, and resistivity measurements were used to construct the water saturation profile. The capillary pressure values were calculated from both the densities and the height above the free water level values. The free water level was indicated by a tube connected to the core setup. Chapter III documents the laboratory details of this experimental work together with its results and conclusions. A technique that can be used to extrapolate existing core data to cases where such data is absent or not representative of in-situ conditions is of interest. Chapter IV of this dissertation documents a new approach that has been developed and is based on using log data to derive a water saturation versus depth profile in the transition zone of the formation of interest. The log derived water saturation distribution is then correlated to generalized capillary pressure curves typical of the formation studied. This curve matching yields, by comparison, a capillary pressure curve specific to the formation of interest. The capillary pressure type curves are generated from already available core data and other petrophysical information. Relative permeability curves are subsequently generated using correlations based on Purcell\u27s model. The technique is successfully applied to several field examples. Special attention is given to cases of tight sands where relative permeability measurements on core samples are very complex, time consuming, and inaccurate due to the very small pore space available to the fluid to move through the tight sand cores. In Chapter V, the above mentioned technique is extended to tight sand cases where a special relationships characteristic of tight sands are developed and mathematically manipulated to adapt already existing relative permeability equations

Secondary Natural Gas Recovery: Reservoir Heterogeneity and Potential for Reserve Growth through Infield Drilling: An Example from McAllen Ranch Field, Hidalgo County, Texas

Integrated engineering, geological, geophysical, and petrophysical analyses of McAllen Ranch field have delineated several controls on secondary recovery of natural gas. Barriers to the flow of natural gas within laterally continuous lower Vicksburg sandstone reservoirs can be demonstrated through finite-element modeling. These barriers are probably diagenetic in origin. In the B area of McAllen Ranch field, faults are unlikely to be the primary barriers to gas flow because faults were not inferred from analysis of high-quality three-dimensional seismic images between the key wells used in this study (Hill and others, 1991). Barriers result in incremental reserve additions when some reservoir domains contain no well completions. Areas containing potential incremental gas resources, identified through this analysis, were confirmed by subsequent recompletions in 1991. Three recompletions proposed by this project have proved successful. Our analysis of public domain production data indicates that new infield wells in the Vicksburg S reservoir have increased reserves 69 percent above an estimate made from analysis of 1980 public domain data. Additionally, more than 100 barrels per day of reserves has been added through new wells drilled between 1988 and 1991. Most of the McAllen Ranch Vicksburg S reserve increases are due to a geological reinterpretation that has stimulated infield step-out development of the Vicksburg S reservoir. Distributary-channel-fill sandstones are the most likely candidates to contain incremental reserves because they are laterally discontinuous and are predominant in areas where numerous reservoir sandstones are stacked.Bureau of Economic Geolog

Application of Frontal Drive Principles to Stratified Reservoirs

Mechanical Engineerin

Automation of the longwall mining system

Cost effective, safe, and technologically sound applications of automation technology to underground coal mining were identified. The longwall analysis commenced with a general search for government and industry experience of mining automation technology. A brief industry survey was conducted to identify longwall operational, safety, and design problems. The prime automation candidates resulting from the industry experience and survey were: (1) the shearer operation, (2) shield and conveyor pan line advance, (3) a management information system to allow improved mine logistics support, and (4) component fault isolation and diagnostics to reduce untimely maintenance delays. A system network analysis indicated that a 40% improvement in productivity was feasible if system delays associated with all of the above four areas were removed. A technology assessment and conceptual system design of each of the four automation candidate areas showed that state of the art digital computer, servomechanism, and actuator technologies could be applied to automate the longwall system

Major Structures of the Rocky Mountains of Colorado and Utah

This paper describes the major structures of Colorado and Utah and presents a theory of origin based on new knowledge of the layering and constitution of the upper mantle and lower crust. It proposes that the Ancestral Rockies and the more modem ones of Cretaceous and early Tertiary age of both the shelf of Colorado and eastern Utah and the miogeosyncline of western Utah are the result of vertical uplifts of the silicic crust. The uplifts are caused by the rise, from the upper mantle, of basalt in scattered places to the base of the silicic crust. This rise domed the silicic crust and the overlying sedimentary veneer. The surficial structures are viewed as gravity-caused mass movements along the flanks of the uplifts and in addition, particularly in the miogeosyncline of western Utah, as synclinoria of Paleozoic strata between uplifts where folding and considerable bedding plane thrusting occurs. The theory also relates widespread magmatism to tectonism in a reasonable fashion. It recognizes that the Rocky Mountains have been a region of Cenozoic regional uplift in which possibly more energy was required than for the building of individual uplifts. This, too, is related to a transformation of the uppermost mantle into a lighter density state. And finally, the Basin and Range faulting of western Utah is viewed as the result of the rise of sufficient basalt from the mantle to form a continuous layer under the silicic crust of the miogeosyncline thus bringing into existence a new framework of forces. The surficial structures are framed about the primary uplifts, but the entire silicic crust becomes attenuated toward the Pacific as it is activated by a component of gravity on a mobilized lower crust