Numerical inversion of the Laplace transformation and the solution of the viscoelastic wave equations

In the numerical inversion of Laplace transform two general techniques of inversion are analyzed; a study of the procedures based on an expansion of f̄(s) by a polynomial in 1/s (Salzer\u27s method) provides accurate results for certain ranges of t. A curvature criterion for the selection of the range of points for polynomial interpolation, and a geometric distribution in this interval markedly improves the range and the degree of accuracy of inversion. The results may be further improved by multiplying f̄(s) by 1/s of 1/s². Other procedures are based on approximation of f̄(t). This investigation includes a general review of the theory, study of the Papoulis method (using a Fourier sine series expansion of f(t)) and the application of a curvature criterion for determination of a distribution factor, Δ. Other techniques, such as those based on expansion of f(t) by other trigonometric sets, Legendre polynomials, Jacobi polynomials, and Laquerre polynomials are briefly analyzed. Papoulis method is adequate for numerical inversion of the L-transform of the viscoelastic wave equations. Numerical solutions are made for complex viscoelastic wave equations. Utilizing the correspondence principle for dynamic problems the generalized equations for plane, spherical and cylindrical viscoelastic wave equations are formulated. The study of the three-and five-element models reveals that their attenuation for steady-state response is proportional to frequency squared for low frequencies and is constant for high frequencies. Transients in three- and five-element models do not predict accurately the behavior of real earth materials, although both give better approximation than the theory of elasticity --Abstract, page ii-iii

Complex Fluids and Hydraulic Fracturing

Nearly 70 years old, hydraulic fracturing is a core technique for stimulating hydrocarbon production in a majority of oil and gas reservoirs. Complex fluids are implemented in nearly every step of the fracturing process, most significantly to generate and sustain fractures and transport and distribute proppant particles during and following fluid injection. An extremely wide range of complex fluids are used: naturally occurring polysaccharide and synthetic polymer solutions, aqueous physical and chemical gels, organic gels, micellar surfactant solutions, emulsions, and foams. These fluids are loaded over a wide range of concentrations with particles of varying sizes and aspect ratios and are subjected to extreme mechanical and environmental conditions. We describe the settings of hydraulic fracturing (framed by geology), fracturing mechanics and physics, and the critical role that non-Newtonian fluid dynamics and complex fluids play in the hydraulic fracturing process

The role of rock joint frictional strength in the containment of fracture propagation

The fracturing phenomenon within the reservoir environment is a complex process that is controlled by several factors and may occur either naturally or by artificial drivers. Even when deliberately induced, the fracturing behaviour is greatly influenced by the subsurface architecture and existing features. The presence of discontinuities such as joints, artificial and naturally occurring faults and interfaces between rock layers and microfractures plays an important role in the fracturing process and has been known to significantly alter the course of fracture growth. In this paper, an important property (joint friction) that governs the shear behaviour of discontinuities is considered. The applied numerical procedure entails the implementation of the discrete element method to enable a more dynamic monitoring of the fracturing process, where the joint frictional property is considered in isolation. Whereas fracture propagation is constrained by joints of low frictional resistance, in non-frictional joints, the unrestricted sliding of the joint plane increases the tendency for reinitiation and proliferation of fractures at other locations. The ability of a frictional joint to suppress fracture growth decreases as the frictional resistance increases; however, this phenomenon exacerbates the influence of other factors including in situ stresses and overburden conditions. The effect of the joint frictional property is not limited to the strength of rock formations; it also impacts on fracturing processes, which could be particularly evident in jointed rock masses or formations with prominent faults and/or discontinuities

Horizontal-Well Fracturing: Why Is it So Different?

Technology Update Advances in drilling and fracturing horizontal wells have been two of the main contributors to economic production from very-low-permeability oil and gas reservoirs. In spite of much attention to the subject, several critical questions regarding horizontal-well fracturing remain unanswered. Among these are higher fracturing pressures, large pressure fluctuations, and more frequent screenouts, which has prompted use of much finer proppant at very low concentrations. This article answers some of these questions. Fracture Initiation From Horizontal Wells The state of stress in a borehole gives it a natural tendency for longitudinal (axial) fracture initiation (Fig. 1a). This is a geometrical effect and independent of the in-situ stress orientations (Daneshy 1973). It can occur even when the least in-situ principal stress is parallel with borehole axis. Since the fracture is not perpendicular to the least in-situ principal stress, it takes higher pressure to extend it. As the axial fracture grows, it will eventually reorient itself to become perpendicular to the least in-situ principal stress and the fluid pressure gradually decreases with it. The location of fracture reorientation depends on location and orientation of planes of weakness in the formation. This leaves open the possibility that the fracture may reorient itself more than once, and depending on the orientation and size of the plane of weakness, may temporarily grow in directions other than perpendicular to the least in-situ principal stress. Initiation of a transverse fracture (perpendicular to borehole axis) requires axial forces and stresses (parallel with borehole axis), which is shown in Fig. 1b. In open holes, such stresses can be induced inside open natural fractures, at the packer seats, or at the bottom of the well (if exposed to fluid), each with its own characteristics. This is shown in Fig. 2. Natural fractures are randomly located and distributed along the well. Initiation from natural fractures results in random location, orientation, and spacing between fractures. One way for partial location control is to reduce the spacing between the packers, thus reducing the length of the pressurized interval. </jats:sec

Dynamics of Innovation in the Upstream Oil and Gas Industry

Guest editorial - No abstract available.</jats:p

Aggressive Fracturing Treatments in Horizontal Wells: Benefits and Pitfalls

Abstract The oil and gas industry has been successful in its use of trial-and-error as the main technique for enhancing the productivity of fracturing treatments in horizontal wells. The lower fracturing costs of the last three years have provided the impetus for moving in the direction of more aggressive treatments; much larger volumes of fluid and proppant, longer horizontal reaches and shorter spacing between adjacent fractures. In general, these changes are claimed to have improved the productivity of fractured horizontal wells, at least in the short term for which data has been available. The justification for these changes requires assuming a different model of fracture propagation than is often assumed by the fracturing community. Closer examination of these treatments indicates that they implicitly assume a more planar fracture shape. Some of the existing data on pressure interaction between offset wells while fracturing shows incidents of fracture shadowing as well as frac/frac intersections. At times the observed data indicates low effective fracture conductivities, which is in-line with the practice of using more proppant. The relatively consistent fracture orientation within a single well and in multiple offset wells indicates larger difference between the two horizontal principal stresses than usually assumed within the fracturing technical community. This condition is needed for the success of the more aggressive treatments. On the negative side, aggressive well and fracture spacing can increase fracture length and the likelihood of interference between fractures in adjacent wells. Taken too far, this can have significant impact on future reservoir development, including damaging the existing fractures, and production re-distribution between offset wells.</jats:p

Fracture Shadowing: Theory, Applications and Implications

Abstract The fluid pressure causing the extension of a hydraulic fracture also compresses the adjacent formation, mainly in the direction perpendicular to the fracture face. If there is a closed-in passive fracture within the compressed region, this causes an increase in its fluid pressure. The magnitude of pressure increase is a function of the distance between the two fractures (passive and active), the net extension pressure in the active fracture, and the overlap area. This pressure increase is defined as fracture shadowing and has been used for estimation of different fracture parameters, including orientation and length. This paper presents the mathematical background of fracture shadowing, including relationship between net extension pressure, distance between fractures, extent of each fracture, and volume of existing passive fracture system. Through actual field data, it shows that shadowing can be used as a very simple and cost effective tool for estimation of different fracture parameters. Shadowing in the same well is used to determine fracture growth pattern as well as a rough indicator of orientation. In offset wells, shadowing can provide estimation of fracture orientation, type, length, and in some cases conductivity. Paper demonstrates and recommends use of fracture shadowing as a simple and inexpensive additional diagnostic tool for determination of fracturing parameters.</jats:p