# Modeling and simulation of an acoustic well stimulation method

## TL;DR

This paper develops a mathematical and numerical model to simulate acoustic well stimulation, aiming to optimize sound wave transmission for enhancing rock permeability around oil and gas wells.

## Contribution

It introduces a novel Helmholtz equation-based model with impedance boundary conditions and non-reflecting boundaries, enabling efficient simulation and optimization of AWS.

## Key findings

- Model accurately predicts acoustic energy transmission.
- Optimal emission frequencies can be effectively identified.
- Numerical examples demonstrate the method's effectiveness.

## Abstract

This paper presents a mathematical model and a numerical procedure to simulate an acoustic well stimulation (AWS) method for enhancing the permeability of the rock formation surrounding oil and gas wells. The AWS method considered herein aims to exploit the well-known permeability-enhancing effect of mechanical vibrations in acoustically porous materials, by transmitting time-harmonic sound waves from a sound source device---placed inside the well---to the well perforations made into the formation. The efficiency of the AWS is assessed by quantifying the amount of acoustic energy transmitted from the source device to the rock formation in terms of the emission frequency and the well configuration. A simple methodology to find optimal emission frequencies for a given well configuration is presented. The proposed model is based on the Helmholtz equation and an impedance boundary condition that effectively accounts for the porous solid-fluid interaction at the interface between the rock formation and the well perforations. Exact non-reflecting boundary conditions derived from Dirichlet-to-Neumann maps are utilized to truncate the circular cylindrical waveguides considered in the model. The resulting boundary value problem is then numerically solved by means of the finite element method. A variety of numerical examples are presented in order to demonstrate the effectiveness of the proposed procedure for finding optimal emission frequencies.