Analytical Solution for Fractured Well Performance
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Solution Overview
Problem
Conventional methods lack an analytical solution for pseudo-steady state flow in vertically fractured wells with finite fracture conductivity, relying on time-consuming numerical simulations and empirical relations, which are inefficient and inaccurate.
Innovation Solution
An analytical solution for pseudo-steady state flow is derived, expressed in elementary functions, providing a simple expression for the pseudo-steady state constant and dimensionless productivity index, applicable to hydraulically fractured wells with finite fracture conductivity in nearly circular reservoirs, and extendable to other geometrical shapes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional numerical simulations are used to determine pseudo-steady state flow in vertically fractured wells with finite fracture conductivity, then the solution can be obtained, but the computational time is excessively long (hours or days)
Solution Approach 1:
The patent creates a simplified analytical model that copies the essential physics of pseudo-steady state flow in fractured wells, replacing complex numerical simulations with a tractable mathematical formulation that yields the same physical insights but computes results in seconds rather than hours
Solution Approach 2:
The patent transforms the problem by introducing dimensionless parameters and applying coordinate transformations that simplify the governing partial differential equations into forms suitable for analytical solutions, changing the mathematical representation to enable rapid computation
2Productivity
If empirical relations are used to estimate pseudo-steady state constant, then the computation is fast, but the accuracy is insufficient
Solution Approach 1:
The patent replaces empirical correlation methods with a first-principles analytical solution based on mass conservation and Darcy's law, substituting approximate statistical relationships with exact mathematical derivations that maintain high computational speed while ensuring physical accuracy
3Reliability
If numerical simulations are used for fractured wells with finite fracture conductivity, then the solution accounts for complex flow patterns, but the computational complexity increases significantly
Solution Approach 1:
The patent segments the flow problem into distinct regions (fracture and reservoir) with different governing equations, applying appropriate boundary conditions at the interface, which allows analytical treatment of each region while maintaining overall accuracy for finite conductivity cases
Solution Approach 2:
The patent introduces an intermediary analytical solution that bridges the gap between simple infinite-conductivity models and complex numerical simulations, providing a middle ground that captures finite conductivity effects through closed-form expressions without requiring iterative numerical methods
Data Source
AI summary
An analytical solution is obtained for a pseudo-steady state production from a vertically fractured well with finite or infinite fracture conductivity. The analytical solution may be used to compute a pseudo-steady state constant for the reservoir. Subsequently, performance parameters relating to the reservoir may be derived from the pseudo-steady state constant. For example, parameters such as production decline rate, total hydrocarbon reserves, and economically recoverable reserves for the reservoir may be computed. The disclosed analytical solution, instead of a conventional numerical simulation, can significantly speed up analysis and improve the accuracy of the calculation of these parameters.