Bottomhole Stress Factors Affecting Drilling Rate at Depth
Authors: Warren T.M. , Smith M.B. 
Published: December 2007

Summary
The mean formation stress near the bottom of a borehole is reduced by strain relaxation when a well is drilled. This causes a PV increase that can significantly reduce the local pore pressure of impermeable rocks, such as shales, but pore pressure of impermeable rocks, such as shales, but does not affect the pore pressure of permeable rocks. Since the penetration rate is strongly affected by the difference between the local pore pressure and the borehole fluid pressure, impermeable formations drill slower than pressure, impermeable formations drill slower than adjacent permeable formations.

Introduction
The rate of penetration (ROP) obtained while a well is drilled generally shows a steady decline as the well depth increases. This reduction of ROP with depth is often attributed to increasing "differential pressure," increasing hydrostatic head, increasing in-situ stresses, decreasing porosity with depth, and chip hold-down. porosity with depth, and chip hold-down. The causes of the reduction in ROP with depth can be divided into two general categories:processes that affect the unbroken rock, andprocesses that act on the rock once it is broken into chips.
While other authors have discussed in considerable detail the chip removal process, our discussion is limited to the factors that process, our discussion is limited to the factors that affect the unbroken rock. The chip removal process is probably more important in terms of total effect on ROP, but probably more important in terms of total effect on ROP, but the strengthening of the unbroken rock is not negligible.
Numerous laboratory tests have demonstrated the severe reduction in ROP with roller-cone bits as the borehole pressure increases. For example, Fig. 1 from Ref. 9 shows the decrease in ROP for Mancos shale as the borehole pressure increases from 500 to 4,000 psi [3.5 to 27.6 MPa]. For these tests the pore pressure was atmospheric. This causes the differential pressure to equal the total borehole pressure. (The differential pressure is defined as the difference between borehole pressure and pore pressure.) It is not clear how these results relate to pore pressure.) It is not clear how these results relate to field drilling because the total hydrostatic borehole pressure in a field well is always greater than the pressure in a field well is always greater than the differential pressure.
A borehole pressure greater than 2,000 psi [13.8 MPa] is needed to reduce the ROP in these tests to a value as low as that expected when drilling Mancos shale at a depth of 10,000 ft [3048 m]. When the equivalent circulating density of the borehole fluid is 1 lbm/gal [120 kg/m3] greater than the pore fluid, the differential pressure at 10,000 ft [3048 m] is only 520 psi [3.6 MPa]. This is obviously insufficient pressure in Fig. 1 to account for the slow penetration rate in a real well.
The pressure in Fig. 1 may also be interpreted as the total hydrostatic head. The hydrostatic head for a 10,000-ft [3048-m] well with 9.3-lbm/gal [1114-kg/m3] mud is 4,800 psi [33.1 MPa]. Although a pressure of 4,800 psi [33.1 MPa] would cause a sufficient reduction ROP to agree with field experience, this interpretation leads to an inconsistency with field experience that also makes it questionable. It is known that the ROP is affected by changes in the pore pressure when areas are drilled where the pore-pressure gradient increases with depth. This is the basis of detecting pore pressure changes with d-exponent plots. The ROP would be unaffected by a change in pore pressure if the hydrostatic head were the only pressure that controlled the ROP. pressure that controlled the ROP. Additionally, normally pressured shales adjacent to normally pressured sandstones have the same pore pressure and the same hydrostatic head, yet the permeable sands drill much faster than the shales. In many cases the sandstones are the stronger rock.
To clarify some of these questions, it is necessary to define the stress environment that exists at the bottom of a well. Several published studies 11–13 of the stresses around the bottom of a borehole are based on both photoelastic methods and finite-element calculations. None photoelastic methods and finite-element calculations. None of these studies consider the effect of a localized pore-pressure change that could be induced during the drilling pore-pressure change that could be induced during the drilling of the borehole.

Basic Rock Mechanics Principles
The stress environment at the bottom of the hole greatly influences the apparent strength and ductility of the rock being drilled. Several rock mechanics principles used in the analysis of the bottomhole stress are reviewed.
Any stress field can be resolved into three mutually perpendicular principal stresses. Principal stresses act perpendicular principal stresses. Principal stresses act normal to planes that have no shear stresses. The principal stresses are denoted by sigma 1, sigma 2, and sigma 3, with sigma 1 the greatest and sigma 3 the smallest.
The compressive strength of a particular rock increases as the minimum principal stress increases, as shown by the data for Mancos shale in Fig. 2. In these tests the confining pressure is the minimum principal stress. At zero minimum principal stress, the failure strength is 11,000 psi [75.8 MPa]. It increases to 20,000 psi [137.9 MPa] as the minimum principal stress is increased to 6,000 psi [41.4 MPa].
The ductility of the shale, defined as the strain at failure, also increases as the confining pressure increases.
JPT
P. 1523