Building geomechanical models

The elements of the geomechanical model that form the basis for analysis of wellbore stability are:

• The state of stress (the orientations and magnitudes of the three principal stresses)
• The pore pressure
• The rock properties, including strength (which can be anisotropic, particularly in consolidated shales)

This section outlines methods for determination of the stress magnitudes and the pore pressure.

Overburden, S_v

Overburden pressure, or S_v, is almost always equal to the weight of overlying fluids and rock. Thus, it can be calculated by integrating the density of the materials overlying the depth of interest (see Fig. 1).

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  Fig. 1—(a) Density logs for a subsea well beneath 1,000 ft of water, extrapolated to the mud line using an exponential curve. Density within the water column is 1.04 gm/cm. (b) Integration results in a plot of overburden (S_v) vs. depth. (c) When converted to an equivalent density, overburden can be displayed in psi/ft, lbm/gal, or specific gravity (SG) (courtesy GeoMechanics Intl. Inc.).

....................(1)

Here, G is the gravitational coefficient. The best measurement of density is derived from well logs. However, density logs are seldom acquired to ground surface or to the sea floor. If good seismic velocities are available, a velocity to density transformation can be used to estimate density where it has not been measured directly. A number of transformations from velocity to density are available (see Eq. 2 and Table 1). In the absence of good velocity or density data, densities must be extrapolated from the surface to the depths at which they are measured. Shallow density profiles can take many forms, and thus they ideally should be calibrated against in-situ log data or measurements of sample densities. A good resource for information about shallow density profiles is the archives of the Deep Sea Drilling Project and the Ocean Drilling Program. In the absence of good calibrations or data, reasonable mudline densities for clean sands are between 2.0 and 2.2 gm/cm³, and for fine-grained shales are between 1.4 and 1.8 gm/cm³.

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  Table 1 Empirical Constants For Eq. 2 For Various Lithologies (Data From Mavko et al.)

and

....................(2)

where a, b, c, d, and e are constants that vary with lithology. The values in Table 1 are for velocity in km/s and density in gm/cm³.

Pore pressure, P_p

The only accurate way to determine pore pressure is by direct measurement. Such measurements are typically done in reservoirs at the same time fluid samples are taken with a wireline formation-testing tool. Recently, advances in while-drilling measurements make it possible to measure in-situ pore pressure while drilling. However, it is difficult (if not impossible) to measure pore pressure in shales because of their very low permeability and small pore volume. In addition, because of their low permeability, pore pressure in shales adjacent to permeable reservoirs may be different from pore pressures in the reservoir. However, there are a number of methods that can be used to estimate pore pressure in shales based on other measurements. Because pore pressure (and the derived fracture gradient, which will not be discussed) is often the only geomechanical parameter on which mud weights are based, we will take some time to review standard and new methods for its prediction. Keep in mind that these prediction methods are intended for use only in shales.

Pore-pressure-prediction methods fall into a few general categories. In the first category are normal compaction trend (NCT), ratio, and equivalent depth methods, which are all more or less empirical. The second category includes methods that explicitly utilize relationships between measured values and the effective stress. These first two methods assume that the in-situ material is either normally compacted or undercompacted. In the third category are models that are also applicable to overcompacted rock. All of these methods require measurement of one or more physical properties that are functions of effective stress. These physical properties include:

• Resistivity
• Density
• Seismic or sonic velocity

In most cases, the only measurement that is available prior to drilling is (P-wave) seismic velocity. After the first well has been drilled, or during drilling (using logging while drilling (LWD)) log data are acquired that make it possible to improve predrill pore-pressure estimates. Using LWD, and adding a pressure while drilling (PWD) measurement, pore-pressure analysis can be carried out in real time. Typically, in shale sections above a target, only LWD resistivity and gamma ray are acquired, but deeper in the well, additional measurements are often made, including density and velocity.

One additional measurement that has been used to predict pore pressure is the drilling exponent, D_c, which defines the rate of drill-bit penetration as a function of depth. Because ease of drilling is related to strength, which in turn is a function of porosity (and therefore of effective stress), the rate of penetration should be a function of effective stress, provided that it is corrected for changes in any other drilling parameters. Therefore, D_c can be used to determine pore pressure using the same analyses used to compute pore pressure from physical properties like resistivity or velocity.

Although all shale pore-pressure-prediction methods rely on the fact that rock physical properties depend on the effective stress, σ = (S – αP_p), equivalent depth and NCT methods use depth as a proxy, and in ratio methods even depth is implicit. Effective stress methods work by

1. Measuring the total stress (S).
2. Using either an explicit relationship or an implicit function to derive the effective stress (σ) from a measured parameter.
3. Computing the pore pressure as the difference between the effective stress and the total stress, divided by alpha, the Biot coefficient [P_p= (S – σ)/α].

In relatively young, unconsolidated shales α = 1, but values of 0.9 or less may be more appropriate for older, more highly compacted sediments.

Because overburden (S_v) can be computed as the integral of the density of the rock and fluid overlying the depth of interest, pore-pressure-prediction methods were developed initially using P_p= (S_v – σ). This is more reasonable than it might seem because properties such as velocity depend most strongly on stress in the direction of propagation, which for near-offset seismic data is nearly vertical. In some cases, the vertical stress is replaced by the mean stress. This results in adjustments to the pore pressure computations based on differences in the magnitudes of the horizontal stresses in different regions.

The relationship between the measured quantity and the effective stress is derived either using explicit functional relationships or by so-called trend-line methods. Trend-line methods require the existence of a depth section, over which the pore pressure is hydrostatic, to derive the NCT.

Methods used to determine pore pressure

The methods for determining pore pressure are:

• Equivalent depth methods
• The ratio method
• Eaton's method
• Effective stress methods

Undercompaction can cause complications in evaluating pore pressure. Centroid and buoyancy effects can also affect the evaluation of pore pressure.

Least principal stress, S₃

The least principal stress can be measured directly, using either extended leakoff tests or minifrac tests. These tests are similar to casing integrity tests or standard leakoff tests, except that the test procedure is slightly modified. Fluid is pumped into the wellbore to pressurize a short interval of exposed rock until the rock fractures and the fracture is propagated a short distance away from the well by continued pumping. In either case, pumping is carried out at a constant rate, and pressure and the volume of fluid pumped are recorded as a function of time. Pressure-time curves typically look like those in Fig. 2.

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  Fig. 2—This figure shows an idealized pressure vs. time plot for an extended leakoff test (modified after Gaarenstroom) (courtesy GeoMechanics Intl. Inc.).

The theory behind using these tests to measure S₃ is that a fracture created during the test will, to minimize the energy required for its propagation, grow away from the well in an orientation that is perpendicular to the far-field least principal stress. Therefore, the pressure required to propagate the fracture will be equal to or higher than the least stress. Fracture propagation will stop when leakoff of fluid from the fracture and wellbore and into the formation occurs faster than the fluid is replaced by pumping. If pumping stops entirely, fluid leakoff will continue from the walls of the fracture until it closes, severing its connection to the wellbore. The fracture will close as soon as the pressure drops below the stress acting normal to the fracture (which is the least principal stress). The change in flow regime after pumping stops, from one in which the fracture contributes to fluid losses to one in which all fluid losses occur through the walls of the well, can be seen in pressure-time and other plots of pressure after shut-in (for example, pressure vs. the square root of time, Fig. 3). The least principal stress is taken to be the pressure at which the transition in flow regime occurs. This pressure is a clear indication of the least stress regardless of whether the test created one or a multitude of subparallel fractures, as modeling suggests sometimes occurs.

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  Fig. 3—Pressure vs. square root of time plots are often used to detect fracture closure. On these plots, the closure pressure is the pressure at the inflexion point of the pressure decay curve (modified after Nolte and Economides).

Recently, and with evidence based on the ability to control pressure flowback in microfrac tests, it has been suggested that fracture closure can overestimate the least principal stress, and that choked flowback at various rates is required to determine that stress accurately. However, until such techniques become available for use in leakoff tests, the approach outlined below, based on the above theory, is the best for measuring S₃ in practice.

As for casing integrity tests or standard leakoff tests, extended leakoff tests are conducted after cementing a casing string and drilling out a short section (often between 20 and 50 ft) below the casing shoe. The conduct of the test should be as follows (refer to Fig. 2):

1. Perform a pretest pumping cycle with the formation isolated from flow using a very low flow rate. Pump until the pressure reaches a predefined upper limit for the casing. Record pressure and flow rate as a function of time, and draw a pressure-volume curve. This gives you a plot of pressure vs. volume (the slope of which is the system stiffness) if no fracture is initiated in the formation. The pressure vs. volume plot may initially be slightly concave up, indicating that the mud is compliant, possibly owing to entrained gas that is being forced into solution. Alternatively, refer to tables for the particular wellbore fluid and plot the appropriate pressure vs. volume curve by hand.
2. Open the formation to the well, pump at a low rate (1/4 to 1/2 bbl/min), and overlay a plot of pressure vs. volume pumped on the curve from the casing test. The initial inflation should be approximately parallel to (or a little less steep than) the casing test curve. A concave down curve may be an indication of losses into the formation or a shoe integrity problem. If the former, it is possible to overcome this problem by stopping, flowing back, and starting again with a higher flow rate. If the latter, the problem must be dealt with before proceeding.
3. Pump until one of two things happens: either you have pumped a fixed volume of fluid above what was required to reach a given pressure (in which case there is a fluid loss problem to be dealt with), or the inflation pressure curve will break over, indicating the creation of a hydraulic fracture.
4. In a leakoff test, the formation would be shut in as soon as the slope of the pressure vs. volume curve begins to flatten, which is defined as the leakoff point. The fracture gradient at the shoe would be set equal to the pressure at the leakoff point. This is (in general) an upper bound on the least stress, and can, in the absence of better data, be used with caution in geomechanical models. However, to determine the least stress more accurately, continue pumping until the pressure stabilizes or begins to drop, and then shut in by stopping the pumps.
5. Record the pressure after shut-in until the pressure stabilizes. The value to which the pressure drops immediately after the pumps are shut off is typically called the instantaneous shut-in pressure (ISIP).
6. The least stress is determined using a variety of analysis methods. One method that is commonly used is to plot pressure vs. the square root of time after shut-in. The fracture closure pressure is defined by a change in the curvature of the line, as shown in Fig. 3.
7. Ideally, a second cycle should be performed. In this cycle, the previously created fracture is re-opened and extended, and then shut in again. Either or both of the following “step-rate tests” can be employed to refine the least principal stress measurement (Fig. 4).

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  Fig. 4—Plots of pressure vs. flow rate showing that at low flow rates, before fracture opening, pressure increases rapidly with flow rate, but once the fracture opens, a large increase in flow rate can be accommodated with only a small increase in pressure. Sometimes the transition between the two regimes is abrupt (top); sometimes it is more gradual and requires a wide range of flow rates to delineate (bottom)(courtesy GeoMechanics Intl. Inc.).

(a) Re-open the fracture, starting with very low flow rates, and increasing the flow rate in discrete steps until the fracture opens and starts to grow. Maintain a fixed flow rate at each step until the pressure equilibrates. A plot of pressure vs. flow rate will have two slopes. At low pressure, fluid losses into the formation will result in a radial flow pattern in which flow rate increases systematically with pressure. Once the fracture is open, fracture growth and losses from the fracture walls will cause the pressure to increase much more slowly with flow rate. The intersection of lines fit to these two trends provides an upper bound on S3. For a viscous fluid, extrapolation of the latter fit to zero flow gives a lower bound. The benefit of this procedure over 7b is that there is no need to extend the fracture as far.

(b) Open the fracture, pumping at the same rate as in the first cycle. The pressure at which a plot of pressure vs. volume deviates from the first cycle is the fracture reopening pressure. The difference between that pressure and the leakoff pressure is a measure of the formation tensile strength. Once the fracture begins to extend, decrease the pump rate in fixed increments, recording flow rate and pressure after the pressure has equilibrated at each step. Analyze this data using the same technique as described for step-rate reopening. The benefit of this procedure over 7a is that a measure of tensile strength can be obtained.

Methods used to estimate state stress and orientation

Several methods are available for estimating stress and orientation:

• Ballooning to estimate least principal stress
• Using wellbore failure to constrain the magnitude of S_Hmax

• Constraining the magnitude of S_Hmax in deviated wellbores

• Constraining the stress state in the visund field.

Nomenclature

a, b, c, d, e	= constants used in Eq. 2 and tabulated in Table 1
G	= acceleration of gravity, m/s2
Pp	= pore pressure, MPa, psi, lbm/gal
S3	= least principal stress, MPa, psi
Vp	= compressional-wave velocity, km/s
z	= Cartesian coordinate system
Zo	= depth, ft, m
Sz	= axial stress along a wellbore, MPa, psi
σ	= Terzaghi effective stress, MPa, psi

Noteworthy papers in OnePetro

A. Azad and R.J. Chalaturnyk 2010. A Mathematical Improvement to SAGD Using Geomechanical Modelling. 141303-PA. http://dx.doi.org/10.2118/141303-PA.

J.T. Fredrich, J.G. Arguello, Sandia National Laboratories; G.L. Deitrick, E.P. de Rouffignac 2000. Geomechanical Modeling of Reservoir Compaction, Surface Subsidence, and Casing Damage at the Belridge Diatomite Field. 65354-PA. http://dx.doi.org/10.2118/65354-PA.

Online multimedia

Blunt, Martin, and Herman Lemmens. 2012. Pore-scale Imaging and Modeling of Rocks. https://webevents.spe.org/products/pore-scale-imaging-and-modeling-of-rocks

External links

Ocean Drilling Program

See also

Hydraulic fracturing

Models for wellbore stability

Subsurface stress and pore pressure

Handling uncertainty in geomechanical design

PEH:Geomechanics_Applied_to_Drilling_Engineering

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Rene Alcalde

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