Search

A highly accurate and efficiently computable analytical solution to the diffusivity equation is presented for modeling fluid flow into a 3D, arbitrarily oriented plane sink within a box-shaped, anisotropic medium with Neumann boundary conditions. The plane sink represents a gathering system for a well stimulated by means of hydraulic fracturing. Our plane-source Neumann function arises from analytic double integration of the point-source solution to the diffusivity equation along two vectors, forming a parallelogram. A Neumann boundary condition is achieved by means of the method of images, resulting in triple infinite summations that are reduced with mathematical identities to a combination of closed-form expressions and infinite sums with exponential damping. Our solution forecasts time-dependent behavior of fractured wells, useful in interpreting field experiments for the characterization of fracturing efficacy, reservoir size, and matrix fluid-transport properties. We demonstrate our model with two applications. One is pressure-transient analysis with identified flow regimes from a pressure vs. time plot. The other is pseudosteady-state (PSS) pressure mapping, simulating inflow from multiple fractures along the trajectory of a single horizontal well, which is achieved with superposition theory and adjustment of flux strength of each plane source to achieve a common pressure at each well/fracture intersection. [ABSTRACT FROM AUTHOR]

The article looks at the study which is conducted to demonstrate the usefulness of analytic methodology with several time-dependent, discrete fracture problems for shale gas production with typical Barnett conditions. Topics discussed include allow the direct use of complex hydraulic fracture patterns which is interpreted from micro seismic mapping, problems with horizontal and slanted-well productivity problems with unprecedented speed and accuracy and modeling of unconventional reservoirs.

A semi-analytical mathematical solution to the Heat Equation has been derived to address the issue of rate forecasting from hydraulic fractures in unconventional oil reservoirs. The fracture is assumed to have a time-independent pressure profile along its length with a terminally declining flow rate. The analytical model makes use of the point solution to the heat equation for a constant rate in a bound system of a rectangular geometry. The constant pressure solution is constructed from the constant rate solution through convolution performed in the Laplace domain, then analytic inversion to the time domain. The line source solution is then acquired through analytic integration of the point source solution along a well trajectory. Results are generated using a simulator based on the mathematical model for arbitrarily oriented fractures in a 2D domain. Since the physical problem involves inverse modeling of flow rate data for reservoir characterization, diagnostics techniques for probing the fracture and drainage geometry are presented using derivative analysis of rate transient behavior of hydraulically-fractured media. [ABSTRACT FROM AUTHOR]