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Models for wellbore stability

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In many cases, wellbore stability analysis can be carried out with very simple models that are time-independent and relate stress and pore pressure, only through the effective stress law. These do not account for the fact that stress changes induce pore pressure changes, and vice versa. Nor do these models account for thermal and chemical effects and their relationships to pore pressure and stress. In this section, we briefly discuss each of these issues and how they affect wellbore stability analysis. We start with a discussion of failure caused by anisotropic rock strength, which is a characteristic of consolidated shales that can cause considerable problems in wells drilled at oblique angles to bedding.

While the examples shown here demonstrate that it is possible to quantify uncertainties in the minimum safe mud weight, it is also possible to quantify uncertainties in the maximum safe mud weight. In that case, the likelihood of success decreases with increasing mud weight, and the two edges of the field defining the most stable mud weights form a possibly skewed bell-shaped curve.

Anisotropic strength

In many rocks (lithified shales in particular), the elastic properties are anisotropic. In other words, they are a function of the orientation of the applied stress with respect to bedding planes (in general, shales are stiffer along the bedding planes than perpendicular to bedding). At the same time, the rock strength is also anisotropic. In both cases, the anisotropy is caused by a preferred orientation of shale particles that generally becomes more pronounced with compaction.

A well that is drilled perpendicular to shale bedding is generally not affected by bedding-parallel weakness planes. However, when a well is drilled at an oblique angle to bedding, bedding-parallel weakness planes can become very important. Fig. 1 shows an acoustic wellbore image of breakouts that occur along oblique bedding planes intersecting a well, demonstrating that this mode of failure does occur. In fact, in this well failure associated with weak bedding caused severe instabilities, necessitating a sidetrack.

Fig. 2 is an example plot that shows the required mud weight as a function of wellbore orientation for wells drilled through dipping beds that are highly anisotropic and illustrates the two angles required to define the orientation of a well with respect to bedding. The first angle is the attack angle, which is simply the angle between the well axis and the normal to the bedding plane. The larger the attack angle, the more likely it is that failure will occur because of bedding-parallel weakness planes. The second is the angle between the dip direction and the projection of the well axis onto the bedding plane. While in general it is found that wells drilled updip or downdip are more stable than those drilled along strike, the relationship depends critically on the orientations and magnitudes of the in-situ stresses.

Both of these effects can be seen in the lower right stability plot shown in Fig. 2, which indicates the mud weight required to maintain stability as a function of wellbore orientation.

  • The bedding normal is shown in this plot as a white dot.
  • The lighter gray shading close to the dot indicates that lower mud weights are required for wells drilled nearly perpendicular to the bedding planes.
  • Darker colors show that high mud weights are required for wells drilled obliquely to bedding.

The highest mud weights are required for wells drilled with moderate to high deviations to the East Northeast (ENE). The asymmetry in the plot is a characteristic of the effect of strength anisotropy, and it is caused by the complex interplay between the stress field concentrated around the well and the weak bedding planes. For comparison, the lower left stability plot shows recommended mud weight if there were no weak bedding planes. The difference between the two is the affect of bedding, which requires raising the mud weight if those planes are active. Where bedding planes are not active, similar mud weights are recommended. Notice that the relative stability of the wells shown on the two plots is quite different when bedding weakness is taken into account from when it is not.

It is very important to realize that, contrary to cases in which wellbore instability is caused by failure of the intact rock, raising the mud weight past a certain point usually exacerbates failure in anisotropic shales. Because failure in these rocks often involves slip along discrete planes, the result is that irregular chunks of rock are often produced, and when cross-cutting fractures are present, the pieces are often spindle-shaped. Raising the mud weight when this type of failure is observed often causes an increase in fluid pressure along the weak planes, reducing their resistance to slip, thereby making failure worse. This problem is often addressed in part by adding fluid-loss-control agents to the drilling mud.

Poroelasticity and thermoporoelasticity

Poroelasticity theory describes the coupling between pore pressure and stress in rocks. When pore pressure and stress are coupled, fluid diffusion plays an important role, and stability becomes time-dependent. To use the poroelasticity equations developed by Biot[1] to model this process requires knowledge of more rock properties than are required for elastic analyses. These include:

  • The elastic moduli
  • The porosity
  • The permeability
  • A pore pressure-stress coupling term

Even without modeling the problem, however, it is obvious that when a well is overbalanced, fluid diffusion into the rock is likely to cause instability to increase over time. This is because diffusion causes the initial overbalance required to support the wellbore to decrease with time as the near-wellbore pore pressure increases, leading to a decrease in wellbore support and increased failure of the rock. This time-dependent weakening is reduced by development of a mud cake. Thus, it is often observed in wells with strong, brittle shales and weak, high-porosity, high-permeability sands that the strong shales break out, whereas the weaker sands appear more intact. An additional reason for the apparently anomalous stability of the sands is discussed briefly in the section on plasticity.

Effect of fluid diffusion

The effect of fluid diffusion is illustrated in Fig. 3.

  • On the top is shown a wellbore cross section.
  • Superimposed on the cross section is a series of contours that define the volume of rock in which the stresses exceed the rock strength as a function of time.
  • The heavy gray curves show the boundary of the breakout zones after 1 minute.
  • The other curves show its shape at 10, 100, 1,000, 10,000, and 100,000 minutes.

Although the amount of failure gets larger with time, the width of the failed zone at the wellbore does not change. This is because, in this example, it is assumed that no mudcake forms, and there is perfect communication between the wellbore fluid and the pore fluid. However, away from the well, the amount of failure increases with time.

  • The lower plot shows the total angular coverage of the failure zones as a function of time and mud weight, at a radial distance from the center of the well that is 20; larger than the drilled radius.

Although higher mud weights do reduce the amount of failure at short times after drilling, there is a slow but systematic increase in the amount of failure with time, regardless of the mud weight used.

  • The stars in the lower plot show conditions corresponding to each breakout drawn on the well cross section.

Thermal energy transfer

Thermal energy transfer obeys the same diffusion law as does the movement of pore fluid. Hence, it is straightforward to model the time-dependent effects of wellbore cooling using the same equations as are used for poroelasticity. This is potentially quite important because cooling a well reduces the circumferential stress and thereby temporarily decreases the likelihood of breakout formation. Simply modeling the pore pressure and temperature independently is not enough, however, because thermal energy transfer occurs both by conduction (heat transfer) and by convection (motion of warm or cold fluids). Thus, a fully-coupled thermoporoelastic theory is required.

Fig. 4 shows analysis of the effect of a 30°F reduction in mud temperature for the same parameters used to generate Fig. 3. It is clear that failure is much less pronounced when the mud has been cooled. In fact, the analysis indicates that not only can cooling increase the length of time this well remains stable, it may also allow a significant decrease in mud weight. This is because of the contributions of two effects:

  1. Cooling the wellbore reduces the circumferential stress that leads to failure.
  2. Cooling the fluid reduces the pore pressure, increasing the effective strength of the rock.

Mud/rock interactions

From the perspective of wellbore stability, shales are the most problematic lithologies to drill through. Evidence abounds that the shale sections of wells drilled with water-based mud are significantly more rugose than the same sections of similar wells drilled with oil-based mud. The primary reason for these observations is that chemical interactions that occur between shales and water-based drilling muds cause a significant reduction in the effective strength of the shales. Two effects contribute to this problem:

  • Osmotic diffusion - the transfer of water from regions of high salinity to regions of low salinity, which causes water in low-salinity mud to diffuse across the membrane formed at the mud/rock interface
  • Chemical diffusion - the transfer of specific ions from regions of high concentration to regions of low concentration

These two effects both change the internal pressure of water in the shale and also affect its strength. Each occurs at a different rate, which in some cases can lead first to weakening and then to strengthening of a wellbore. Example of modeling failure in a poroelastic-plastic material. Failure is assumed to occur at a plastic strain of 3×10–3 (3 millistrain). This produces a breakout defined by the white area at the side of the well. The rock properties are shown in Table 1.

Salinity of drilling mud water phase

When the salinity of the drilling mud water phase is lower than the salinity of the pore fluid, osmotic diffusion causes shales to swell and weaken because of elevated internal pore pressure caused by uptake of water into the shale. Consequently, one solution to shale instabilities is to increase the salinity of the water phase of the mud system, and this works in some cases. However, if the salinity is increased too much, it can cause microfracturing to occur.

Water phase activity

In calculating the magnitude of the pressure generated by osmotic diffusion, the parameter that is used to select the appropriate salinity is the water phase activity. Activity (which is explicitly the ratio of the vapor pressure above pure water to the vapor pressure above the solution being tested, and can be measured at the rig with an electrohygrometer) varies from zero to one. Typical water-based muds have activities between 0.8 and 0.9. Typical shales in situ have pore-fluid activities between 0.75 and 0.85, based on extrapolations of laboratory data. The use of typical muds in typical shales thus causes an increase in the pore pressure within the shale, leading to shale swelling, weakening, and the development of washouts. Mody and Hale[2] published Eq. 1 to describe the pore pressure increase owing to a given fluid activity contrast.


If ΔP is negative, it indicates that water will be drawn into the shale. Here, R is the gas constant; T is absolute temperature, and V is the molar volume of water (liters/mole). Decreasing the mud activity often alleviates shale swelling because ΔP is positive if Ap (the pore fluid activity) is larger than Am (the mud activity), and water will be drawn out of the shale for this condition. The parameter Em is the membrane efficiency, which is a measure of how close to ideal the membrane is. Explicitly, it is the pressure change across an ideal membrane owing to a fluid activity difference across the membrane, divided into the actual pressure difference across the membrane in question.

Membrane efficiency is affected both by mud chemistry and by the properties of the shale. In particular, the ionic radius and the pore throat size of the shale appear to play a strong role. Oil-based mud has nearly perfect efficiency. Although water-based mud generally has very low efficiency, some recently developed water-based synthetics have been designed to have high efficiencies approaching those of oil-based mud. Fig. 5 shows the relationship between membrane efficiency, mud fluid activity, and degree of failure (quantified in terms of the widths of the failed regions) for shale with a nominal pore fluid activity of 0.7. Higher mud activities than the shale pore fluid cause an increase in breakout width, whereas predicted breakout width is less for muds with lower activities. The effect decreases for lower membrane efficiencies.

The model described by Eq. 1 is implicitly time-independent, and diffusion is a time-dependent process. Time-dependent models have been developed that predict variations in pore pressure due to chemical effects as a function of time and position around the hole. These are explicitly both chemo-elastic and poro-elastic (that is, they account for interactions between the pore pressure and the stress as well as the chemical effects on the pore pressure). The results allow selection of mud weights for specific mud activities, or mud activities for specific mud weights. Fig. 6 (top) shows a plot of failure vs. time and mud weight for a shale with a pore-fluid activity of 0.8, for a mud activity of 0.9. As can be seen, failure gets worse over time, and even a mud weight as high as the fracture gradient of 16 lbm/gal maintains hole stability for less than one day. On the other hand, for a mud activity of 0.7 (Fig. 6, bottom), the time before failure begins to worsen is extended, and it is possible to select a mud weight below the fracture gradient and yet still provide several days of working time.

Wellbore failure in plastic rock

As previously discussed, young, weak rocks that are still undergoing compaction behave plastically. The same can be said of high-porosity reservoir sands. One consequence is that these materials “fail” with only a small reduction in strength. Therefore, wellbore stability modeling can be done more accurately in young rocks using plastic models. But, because the current state of a plastic material is a function of its stress/strain history, fully 3D plastic models require numerical methods. While plastic models are not necessary for extremely brittle rocks, it is not always clear which model is the most appropriate when a rock has intermediate properties.

The simplest way to decide whether it is important to use plasticity is to look at the stress-strain curve of the rock of interest. If it has large strain at failure and it has a detectable yield point, and/or it fails without total loss of strength, plastic models should be considered. That said, no one has yet published a definitive study that demonstrates that an elastic-plastic model is a better predictor of required mud weight for drilling than an equivalent elastic-brittle model. In fact, in cases in which both approaches are used, it is often found that predicted mud weights to avoid excess wellbore instabilities using the two techniques are within 0.1 lbm/gal.

Fig. 7 is the output from a strain-hardening, poroelastic-plastic analysis of failure around a balanced well. The stresses, pore pressure, and mud weight are shown in the upper right of the figure. The other parameters, obtained from measurement of the core the properties of which were used in this analysis, are shown in Table 1. With a failure model calibrated by laboratory tests, which predicts the onset of failure after 3 plastic millistrain (0.3%), a failed zone is predicted to have a half-angle of approximately 55 degrees, as shown in white on the side of the well. Fig. 8 presents a similar analysis, using the same stresses, pore pressure, and mud weight, using a poroelastic-brittle model for failure. In this case, there is a remarkable similarity between the width of failure (the elastic model describes only the initial zone which will deepen with time, as discussed previously) of these two analyses, indicating that it is not necessary to use a plastic model to describe the material.


Am = mud activity, ratio
Ap = pore fluid activity, ratio
Em = membrane efficiency, ratio
R = gas constant, J/mol/°K
To = tensile strength, MPa, psi
V = molar volume of water, liters/mole
ΔP = difference between the pressure of fluid in a well and the pore pressure


  1. Biot, M.A. 1941. General theory of three-dimensional consolidation. J. Appl. Phys. 12 (2): 155–164.
  2. Mody, F.K. and Hale, A.H. 1993. Borehole-Stability Model To Couple the Mechanics and Chemistry of Drilling-Fluid/Shale Interactions. J Pet Technol 45 (11): 1093–1101. SPE-25728-PA.

See also


Predicting wellbore stability

Building geomechanical models

Rock mechanical properties

Subsurface stress and pore pressure

Noteworthy papers in OnePetro

D. Lee, V. Singh, and T. Berard, 2009. Construction of a Mechanical Earth Model and Wellbore Stability Analysis for a CO2 Injection Well, SPE International Conference on CO2 Capture, Storage, and Utilization, 2-4 November, San Diego, California, USA, 126624-MS,</ref>

Choi, S.K. and Tan, C.P. 1998. Modelling of Effects of Drilling Fluid Temperature on Wellbore Stability, SPE/ISRM Rock Mechanics in Petroleum Engineering, 8-10 July, Trondheim, Norway, 47304-MS,

External links

Page champions

Fersheed Mody, Ph.D., P.E.