Archive | 2021
Generalized Analytical Solutions for Shale Gas Production in Compressible Porous Media Including a New Scaling Time
Abstract
Summary Shale gas has been established as a key energy source over the last decades. Tight permeability, non-Darcy flow mechanisms, adsorption and compressibility make shale gas formations more challenging to model than conventional reservoirs. The aim of this work is to present analytical solutions overcoming those challenges. We model shale gas production from a 1D porous medium, where a well or fracture with constant pressure is assumed at one side and a noflow boundary at the other (e.g. due to symmetric geometry between multiple fractures). The system is depleted accordingly. Gas is stored as mobile phase in the pores and adsorbed phase on the matrix surface as modeled by a Langmuir isotherm. The gas and rock are both compressible; when pressure is reduced, gas expands (according to a real gas equation of state), while porosity is reduced. The porosity reduction reduces intrinsic permeability. Non-Darcy flow, which is important for shale gas flow in tight porous media, is accounted for via apparent permeability depending on the Knudsen number. The partial differential equation describing this system can be formulated as a nonlinear diffusion equation in terms of the conserved property of mass per bulk volume of free and adsorbed gas. The well pressure boundary condition corresponds to a fixed value of as boundary condition. Similarly a uniform initial pressure corresponds to a uniform initial . This system is comparable to the form described by McWhorter and Sunada (1990) for spontaneous imbibition where they derived universal analytical solutions regardless of the shape of the diffusion coefficient which could depend arbitrarily as function of the conserved property, in their case fluid saturation. Analytical solutions are thus obtained that can give spatial profiles, at given times, of pressure, adsorption, porosity and apparent permeability, in addition to time profiles of gas recovery. In accordance with the analytical solution, it is shown that gas recovery follows a square root of time profile at early times before the no-flow boundary is encountered. Late time behavior and validation is investigated using numerical solutions. We further adapt the work by Schmid and Geiger (2012) who suggested a time scale for the analytical solution during spontaneous imbibition. Adapted to our case, a similar scaled time results in the same depleted fraction of recoverable mass under the given pressure conditions and scaling different cases results in full overlap of recovery profiles at early time, typically to obtainable recoveries of 35–50%. The late time behavior profiles deviate as individual cases may leave the square root profile at different recovery values and with different trends vs time. The role of system length, adsorption isotherm, rock compressibility, porosity-permeability relations and non-Darcy effects are examined to see how they contribute to increase or reduce production rate and time scale, affect profile shapes and the amount produced when the no-flow boundary is met. To our knowledge, no analytical solution yet exists for shale gas production which is able to account for either nonlinear adsorption, non-Darcy flow, compressible porous media or permeability reduction. Keywords: shale gas production; compressible low permeable porous media; non-Darcy flow; generalized time scale; analytical solutions.