Unrestricted Pressure-Transient Solutions for Homogeneous and Composite Elliptical Flow Models with Limited and Full Height Inner Boundary

Abstract

Abstract Pressure-transient solutions for homogeneous and composite elliptical flow models have been presented by several authors over the past 40 years, but all have computational limitations, either with respect to time range or degree of eccentricity of elliptical boundaries. This paper removes these limitations on time and eccentricity with a new fully flexible computational scheme for unbounded and bounded homogeneous and composite elliptical flow also extended to cases with a limited-height inner boundary, thus also covering horizontal well scenarios in composite elliptic models. Kuchuk and Brigham's (1979) Laplace-space solution is fully flexible with respect to the shape of ellipses, but has a lower limit on time that does not reach purely linear early data unless slow extended precision computations are used. However, this limitation can be removed with the approach of Riley et al. (1991) using asymptotic expansions for early data and Kuchuk and Brigham's (1979) solution for late data. This works for a fracture as inner boundary but has restrictions on the outer boundary. Another asymptotic expansion was used by van den Hoek (2016) to derive a more flexible but not fully flexible solution, since limitations remain for highly eccentric scenarios. This paper extends results from these references with other asymptotic expansions to achieve full flexibility on time and eccentricity. A wide range of examples are included to highlight the flexibility and utility of the new solutions, with cases ranging from damage-zone dimensions to long-time injection-falloff scenarios, but only as stationary cases with emphasis on the dimensions of the scenarios. Contrasts between exact degenerate elliptical models and standard solutions for infinite-conductivity fractures based on uniform-flux models and an equivalent pressure point are also discussed briefly. This concerns a solution artifact that might be overlooked. With a limited-height fracture used as a horizontal well replacement, cases are also included for horizontal wells in composite elliptical models.