SUMMARYResearch investigations on three-dimensional (3-D) rectangular hydraulic fracture configurations with varying degrees of fluid lag are reported. This paper demonstrates that a 3-D fracture model coupled with fluid lag (a small region of reduced pressure) at the fracture tip can predict very large excess pressure measurements for hydraulic fracture processes. Predictions of fracture propagation based on critical stress intensity factors are extremely sensitive to the pressure profile at the tip of a propagating fracture. This strong sensitivity to the pressure profile at the tip of a hydraulic fracture is more strongly pronounced in 3-D models versus 2-D models because 3-D fractures are clamped at the top and bottom, and pressures in the 3-D fractures that are far removed from the fracture tip have little effect on the stress intensity factor at the fracture tip. This rationale for the excess pressure mechanism is in marked contrast to the crack tip process damage zone assumptions and attendant high rock fracture toughness value hypotheses advanced in the literature. A comparison with field data is presented to illustrate the proposed fracture fluid pressure sensitivity phenomenon. This paper does not attempt to calculate the length of the fluid lag region in a propagating fracture but instead attempts to show that the pressure profile at the tip of the propagating fracture plays a major role in fracture propagation, and this role is magnified in 3-D models.

A three-dimensional hydraulic fracture simulator (HYFFIX) is reformulated using finite element methodology and a newly adapted fixed grid. The numerical procedures for the coupled equations governing the fracture width, fluid pressure, and evolution of equilibrium planar crack in layered media are summarized. Fixed grid mesh control algorithms for the efficient tracking of the moving crack/fracture fluid front are detailed. The introduction of these novel algorithms in the simulator makes it numerically efficient and stable, in comparison to previously reported models which utilize migrating mesh techniques. Due to the enhanced numerical efficiency and compactness of the refined code, the model can also be readily implemented on a workstation or microcomputer.

The three-dimensional geometry of a solder joint associated with a surface mount electronic packaging process is modeled by employing a height function, z = ζ(x, y), and applying the finite element solution procedure. A rigorous formulation based on the variational methodologies is derived in two-dimensional integral form so that the standard numerical techniques for plane problems can be applied. The appropriate finite element formulation and corresponding solution procedures are devised. Numerical examples representing circular and rectangular pads are tested for validation of the developed simulator and illustration of its overall capability. The method presented in this paper shows a remarkable alternative method to the complete three-dimensional finite element approach. The developed method eliminates the difficulties associated with setting up three-dimensional brick meshes or adjusting the shape of the joint body to determine the equilibrium geometry of a soldering joint.

A broad class of engineering problems in fracture mechanics, thermal/fluid transport and electromagnetic theory involve the evaluation of two-dimensional finite part integrals of the form A method for evaluation of such integrals is developed by deriving an equivalent integral using Fourier transformation. This equivalent integral does Got involve a kernel with singular behaviour. Consequently, standard numerical integration methodologies with conventional analytical evaluation techniques can be used in the finite element computations. The accuracy and convergence of the developed numerical procedure are successfully demonstrated by numerical examples for planar fracture geometries.