Summary
This paper presents the consistency of thermal effects (i.e., the
thermal-induced pore pressure and rock stresses) between two available models
for inclined boreholes. Thermal effects for both a permeable and an impermeable
boundary are studied. The solutions of pore pressure and stresses are provided,
and the singularity is solved in the solutions for the condition of hydraulic
diffusivity being equal to thermal diffusivity. The analytical solutions in the
Laplace domain and in the real-time domain are verified using the
finite-difference solutions. The collapse failure index and critical mud
weights, as well as thermally induced pore pressure, are presented and analyzed
under the circumstances of heating and cooling the wellbore. Results show that
heating the wellbore can destabilize the near-wellbore region by raising both
the collapse and the fracturing mud weight. The model presented in this paper
can be applicable to deepwater drilling, where a narrow mud-weight window often
occurs.
Introduction
When a wellbore is drilled, such as in oil and gas operations, the rock is
suddenly replaced by a drilling fluid that applies a certain pressure in the
wellbore. This pressure is normally less than the in-situ stresses that are
acting on the well. This results in an immediate stress concentration near the
wellbore, especially at the wellbore wall. 1 There are two time-dependent
poroelastic effects to consider. 2 First, if the wall of the wellbore is
permeable, fluid pressure diffuses from the well into the formation (Mode 2
effect 2 ). Second, the stress concentration causes an immediate increase in
pore pressure (the undrained loading effect or Mode 3 effect 2 ), which
dissipates with time. In addition, however, the wellbore fluid temperature is
usually different from the formation temperature for both a permeable and an
impermeable wellbore boundary. Thus, thermal effects must also be considered
for both low-permeability shales and high-permeability formations. For a
low-permeability material such as shale, temperature variations not only result
in direct thermally induced stresses but also in transient pore pressure
changes.
For a permeable wellbore wall, thermal diffusion can be fully coupled with
hydraulic diffusion. 3,4 For a low-permeability shale (with or without a
permeable wellbore wall), the diffusivity equations can be decoupled by
ignoring the effect of pore pressure changes on temperature variations. By
including the undrained loading effect, 2 thermal effects can be included in
the fully coupled poroelastic solutions for low-permeability formations. 5
Thermally induced pore pressure and rock stresses can be expressed in the
Laplace domain 5 and in the real-time domain 4 for a permeable boundary
condition (PBC). The two solutions are found to be consistent and they match
the solutions for the coupled equations calculated using the finite-difference
approach (this paper).
An impermeable boundary condition (IMPBC) at the wellbore wall is also
presented in this paper. An impermeable boundary can occur when an oil-based
fluid is used to drill water-wet shales, due to the high capillary entry
pressure. It can also occur if a filter cake forms on the wellbore wall, which
prevents fluid pressure invasion into the formation. Temperature distributions
remain the same regardless of the wellbore boundary condition. The pore
pressure will depend only on the temperature field for an IMPBC, whereas it is
dependent upon both the wellbore pressure and the temperature change for a PBC.
In reality, the boundary condition may be "partially" permeable, which
allows some pressure penetration between the wellbore wall and the formation.
In this case only a "partial" pressure differential, instead of the
complete pressure differential ( p w -- p 0 ), will be driving the fluid flow.
For example, the driving force could be the hydraulic differential less the
capillary threshold pressure (STABView, Version 2.0). 6
For simplicity, temperature is assumed to be constant both at the wellbore
wall and in far-field formations. In fact, the wellbore wall temperature is not
truly constant at different times as the drilling fluid circulates through the
drillpipe and the annulus. Time-dependent boundary conditions (i.e., variable
fluid circulating temperature 7 ) can be used to accurately calculate formation
temperature profiles using a superpositional approach. 8
For low-permeability formations such as shale, thermal diffusion is faster
than hydraulic diffusion, and the former can dominate the pore pressure and
stress changes. The distribution of temperature and pore pressure is controlled
by the diffusivity equations, which can be solved using the following three
approaches: (1) the finite-difference method, (2) numerical solution in the
real-time domain, and (3) numerical solution using Laplace inversion of the
closed-form solution in the Laplace domain. The first method can solve coupled
diffusivity equations, while the latter two approaches can only be applied to
decoupled diffusivity equations.
Theory Background
The fully coupled poroelastic analysis is decomposed into three different
loading modes. 2 The solutions for each loading mode can then be superposed to
form a complete solution for an inclined wellbore under either permeable or
impermeable boundary conditions. 9,10 The undrained loading effect can be
significant for a very low-permeability formation such as a compacted shale,
especially at short times. 11 A vertical wellbore subjected to equal horizontal
stresses is investigated herein in order to deactivate the undrained loading
effect.
There is no undrained loading effect in this condition because the mean
stress after creation of the wellbore is the same as the mean stress prior to
creation of the wellbore.
© 2005. Society of Petroleum Engineers
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History
- Original manuscript received:
12 October 2004
- Manuscript approved:
23 March 2005
- Version of record:
15 June 2005
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