Propagation of Coupled Pressure Waves in Borehole with Drillstring
Authors: Lea Svein-Helge , Kyllingstad Age 
Published: December 2007

Abstract
This paper focuses on coupling effects between pressure waves propagating inside and outside the drillstring in a borehole. Because of the radial elasticity of the drillstring and the formation, waves in the three domains or layers - inner mud, drillstring and annular mud - can not be treated independently or decoupled. Instead, the borehole with the drillstring must be treated as a cylindrical multi-layered wave guide having three distinct wave modes. Each mode has a characteristic wave propagation speed and involves pressure or stress variations in all layers.
The analysis included in this paper shows that the coupling effects can be strong in drillpipe sections, especially if the formation is stiff. A consequence of this coupling effect is that pressure pulses from a downhole telemetry tool can be partially reflected and converted to other modes at a discontinuity in the formation stiffness or in the wellbore diameter.

Introduction
Pressure pulses propagating inside the drillstring are widely used for transmitting information through the drill string. Measurement While Drilling (MWD) tools normally use coded pressure pulsed generated downhole to transmit downhole information to the surface. Such information can be directional survey data, weight on bit, torque and formation data, such as resistivity or radioactivity. MWD tools are routinely used in drilling, and are highly appreciated by drillers and formation evaluation people.
The relative high attenuation at high frequencies seriously limits the bandwidth or the maximum data rate, especially in long wells. In order to optimise the transmission rate and avoid errors in the decoding of the weak signals reaching the surface, it is important to understand the physics of the pressure waves and the mechanisms that tend to hinder a good transmission.
This paper describes a mechanism that can affect mud pulse transmission substantially, but which is not well known. This mechanism is related to the coupling between the various layers. Because of the radial flexibility of the drillstring and the formation, pressure waves inside the string communicate with the annulus. This coupling effect, which is normally neglected, has implications not only in mud pulse telemetry but also in other areas, such as drillstring dynamics and acoustic gas kick detection.
This paper represents a condensed version of a recent thesis work but the model and its basic equations are, to a great extent, based on the PhD work by Lee. The model presented here is more general than the model of Lee because it allows for different inner and outer mud properties and it needs no assumption that the string must be concentric with the borehole.
Derivation of coupled wave equations
The system to be studied is visualised in figures 1 and 2. It can be regarded as a multiple wave guide consisting of the mud inside the drillstring, the drill string, the mud in the annulus and the formation.
Assumptions. To simplify the treatment of this theory the following assumptions are made:–The layers are uniform–The wellbore is open with no casing–The formation is isotropic and impermeable–All materials are assumed to behave linearly with small disturbances–No damping effects are included–The steady mud velocity is low compared to the axial wave propagation speed–The waves are considered monochromatic–The wavelength is long compared to the borehole diameter
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