An Extended Trajectory Mechanics Approach for Calculating the Path of a Pressure Transient: Derivation and Illustration

• Published in: Water resources research 2018-04, Vol.54 (4), p.2642-2660
• Main Author: Vasco, D. W.
• Format: Article
• Language: English
• Subjects: Aquifers; Asymptotic methods; Asymptotic properties; Boundaries; Boundary layers; Computation; Computer simulation; Dependent variables; Differential equations; Dye dispersion; Dynamics; ENGINEERING; Illustrations; Imaging techniques; Interfaces; Inverse problems; inversion; Mathematical models; Mechanics; Mechanics (physics); modeling; Ordinary differential equations; Parameter sensitivity; Permeability; Piezometric head; Porous media; Pressure; Propagation; Quantum theory; Regions; Simulators; Spatial variations; Trajectories; transient pressure
• ISSN: 0043-1397; 1944-7973
• DOI: 10.1002/2017WR021360

Following an approach used in quantum dynamics, an exponential representation of the hydraulic head transforms the diffusion equation governing pressure propagation into an equivalent set of ordinary differential equations. Using a reservoir simulator to determine one set of dependent variables leaves a reduced set of equations for the path of a pressure transient. Unlike the current approach for computing the path of a transient, based on a high‐frequency asymptotic solution, the trajectories resulting from this new formulation are valid for arbitrary spatial variations in aquifer properties. For a medium containing interfaces and layers with sharp boundaries, the trajectory mechanics approach produces paths that are compatible with travel time fields produced by a numerical simulator, while the asymptotic solution produces paths that bend too strongly into high permeability regions. The breakdown of the conventional asymptotic solution, due to the presence of sharp boundaries, has implications for model parameter sensitivity calculations and the solution of the inverse problem. For example, near an abrupt boundary, trajectories based on the asymptotic approach deviate significantly from regions of high sensitivity observed in numerical computations. In contrast, paths based on the new trajectory mechanics approach coincide with regions of maximum sensitivity to permeability changes. Plain Language Summary I present a new approach for computing the path of a pressure transient in a highly heterogeneous porous medium. The approach utilizes methods developed in quantum dynamics. The trajectories are useful for visualizing pressure propagation in complicated physical models and may form the basis for an efficient tomographic imaging algorithm. Key Points The technique described in this paper is useful for visualization and efficient inversion The trajectory‐based approach is valid for an arbitrary porous medium