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Stress impact on rock properties

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Deformation of rocks — elasticity

To first order, most rocks obey the laws of linear elasticity. That is, for small strains, the elements of the stress and strain tensors are related through

 

where

 ....................(1)

In other words, the stress required to cause a given strain, or normalized length change (Δlk /ll), is linearly related to the magnitude of the deformation and proportional to the stiffnesses (or moduli), Mijkl. Furthermore, the strain response occurs instantaneously as soon as the stress is applied, and it is reversible—that is, after removal of a load, the material will be in the same state as it was before the load was applied. A plot of stress vs. strain for a laboratory experiment conducted on rock that obeys such a law is a straight line with slope equal to the modulus. However, in real rocks, the moduli increase as a function of effective stress, particularly at low stress. Some of this increase is reversible (nonlinear elasticity), but some of it is irreversible (plasticity or end-cap compaction). To make matters even more complicated, rocks also exhibit time-dependent behavior, so that an instantaneous stress change elicits both an instantaneous and a time-dependent response. These anelastic effects can be seen in laboratory experiments, as shown in Fig. 1.[1]

At the top of Fig. 1 is shown the stress as a function of time applied in the laboratory to two samples of an upper Miocene turbidite. As in most experiments of this type, a cylindrical rock sample is jacketed with an impermeable soft sleeve and placed in a fluid-filled pressure cell. The fluid pressure surrounding the sample is increased slowly, and the fluid pressure (confining stress) and sample axial and circumferential strains are measured. To identify the various deformation processes that occur in this unconsolidated sand, the stress is slowly increased at a constant rate and then held constant until the sample stops deforming. Then the pressure is decreased to approximately half of the previous maximum pressure. After that, the pressure is increased at the same rate until the next pressure step is completed. This process continues until the desired maximum pressure is achieved, and then the sample is slowly unloaded and removed from the pressure cell.

Stress-strain curve

All aspects of typical rock behavior can be seen in the stress-strain curve plotted on the bottom of Fig. 1. At low pressure, the sample is soft, and there is a rapid increase of stiffness with pressure (nonlinear elasticity) owing to crack closure, as well as an increase in stiffness caused by irreversible compaction. Once the pressure increase stops, the sample continues to deform, with deformation rate decreasing with time (time-dependent creep). The sample is stiffer during unloading than during loading, and during this phase of the experiment, it essentially behaves as a linear elastic material; the permanent strains during loading and creep that occurred through plastic/viscous deformation mechanisms are not recoverable. Reloading follows the (purely elastic) unloading path until the maximum previous pressure has been reached, after which additional plastic deformation begins to occur again as the material resumes following the compaction curve. All of these effects can be seen in situ, including the difference between the loading and unloading response.

Measurements of P-wave and S-wave velocity made on this sample during the experiment by measuring the time of flight of pulses transmitted axially along the sample were used to calculate the dynamic shear (G) and bulk (K) moduli with Eq. 2. The implications of the results for pore pressure prediction are discussed later in this chapter.

 ....................(2)

The dynamic bulk modulus calculated from the velocity measurements is higher than the moduli computed from the slopes of the unloading/reloading curves, which in turn are larger than the modulus calculated from the slope of the loading curve. This dispersion (frequency-dependence of the moduli) also is typical of reservoir rocks, and it is the justification for empirical corrections applied to sonic log data to convert from the dynamic moduli measured by the sonic log to static moduli that are used to model reservoir response. However, it is important to realize that there are two different “static” moduli:

  • A “compaction modulus,” the slope of the loading curve, which includes plastic effects
  • A “static elastic modulus” measured by unloading/reloading, which is truly elastic

It is critical when measuring material response in the lab to differentiate between these two and to use the appropriate one for in-situ modeling—the elastic unloading modulus when no compaction is occurring, for example when pore pressure is increased by injection during waterflooding, and the compaction modulus when modeling, for example, very large depletions in weak reservoirs.

These considerations can become very important when modeling and predicting how the wellbore will respond during and after drilling. In the discussion of wellbore stability that follows, however, we will assume that the rock is purely elastic and only briefly discuss the implications of more complicated rheological models.

Compaction and end-cap plasticity

When rocks are loaded past a certain point, they will no longer behave elastically. If the load is approximately isotropic (σ1 ≅ σ2 ≅ σ3), the rock will begin to compact and lose volume, primarily because of a decrease in porosity. This process is referred to as shear-enhanced compaction because, in general, the effect occurs at lower mean stress as the shear stress increases. Fig. 2[2] shows a plot of the shear stress as a function of mean stress for a variety of rocks, labeled for use in a normal faulting regime where Sv is S1. Compaction trends are shown as arcs bounding the data from the right, and they define end caps of the stress regime within which the rock at a given porosity can exist. Values of porosity decrease as the end caps move outward, owing to material compaction that is caused by the increase in confining stress. The shapes of the end caps for any porosity depend on the form of the relationship between the mean stress at the compaction limit and the shear stress. In many studies, the shape of the end-cap is assumed to be elliptical. At any point along an end cap, the porosity is in equilibrium with the state of effective stress.

In unconsolidated materials, shear-enhanced compaction begins at zero confining stress as soon as the material begins to be loaded (see Fig. 1). In situ, this compaction is the primary cause of the increase in stiffness and decrease in porosity of sediments with burial. The assumption inherent in all standard pore-pressure-prediction algorithms that rock properties are uniquely related to the effective stress is equivalent to assuming that the rock in situ lies along a compaction trend defined by an end cap.

If the mean effective stress decreases (for example, because of erosion) or the pore pressure increases, the rock becomes overcompacted. When this occurs, its porosity is no longer in equilibrium with the end cap, and it will behave elastically, as occurred during the unloading stages of the experiment shown in Fig. 1.

When a differential load is applied (e.g., σ1 > σ2 ≅ σ3), eventually the maximum stress (σ1) will get so large that the sample either will begin to yield through a process of distributed deformation or will fail because of shear localization and the creation of a brittle failure surface (a fault). In Fig. 2, the data at the left edge of the plot lie along a limit in the ratio of shear stress to mean stress that is defined by the onset of brittle failure or plastic yielding by shear localization, as discussed next.

Failure models and rock strength

Rock strength models that define stress states at which brittle failure occurs follow stress trajectories that lie along the left edge of the data shown in Fig. 2. It is clear that the ability of a rock to carry differential stress increases with confining stress. To establish the exact relationship, rock strength tests are conducted at a number of confining pressures. In these tests, a jacketed, cylindrical sample is loaded into a pressure vessel, a constant confining pressure is established, and an axial load is applied by means of a hydraulic ram. The load is increased slowly by driving the ram at a constant rate, monitoring axial and circumferential strains and maintaining a constant confining pressure, until the sample fails or yields. An example of an axial stress vs. axial strain plot from a typical triaxial stress experiment is shown in Fig. 3.

One criterion to define the stress state at failure is the 2D linear Mohr-Coulomb criterion. The Mohr-Coulomb criterion defines a linear relationship between the stress difference at failure and the confining stress using two parameters:

  • So, the cohesion (or Co, the unconfined compressive strength)
  • Φ, the angle or μi, the coefficient of internal friction, where μi = tanΦ

The equation that defines the 2D linear criterion is τ = So + μiσn. These parameters can be derived from triaxial strength tests on cylindrical cores by measuring the stress at failure as a function of confining pressure.

Fig. 4 shows graphically how the Mohr-Coulomb parameters are derived. The upper plot shows a series of Mohr circles, with x-intercepts σ1 and σ3 at failure and diameter σ1 – σ3, in a plot of shear stress to effective normal stress. The failure line with slope μi and intercept So that just touches each of the circles defines the parameters of the 2D linear Mohr-Coulomb strength criterion for this material. Co is the value of σ1 for σ3 = 0 of the circle that just touches the failure line. The lower plot graphs σ1 vs. σ3 and can be used to derive Co directly.

Some of the other strength criteria include the following:

Strength Criteria Description
Hoek and Brown (HB) criterion like the Mohr Coulomb criterion, is 2D and depends only on knowledge of σ1 and σ3, but which uses three parameters to describe a curved failure surface and, thus, can better fit Mohr envelopes than can linear criteria.
Tresca criterion A simplified form of the linearized Mohr-Coulomb criterion in which μi = 0. It is rarely used in rocks and is more commonly applied to metals, which have a yield point but do not strengthen with confining pressure.
Modified Lade criterion Has considerable advantages, in that it, too, is a 3D strength criterion but requires only two empirical constants, equivalent to Co and μi. Thus, it can be calibrated in the same way as the simpler 2D Mohr-Coulomb failure criterion, but because it is fully 3D, it is the preferred criterion for analysis of wellbore stability.
Other failure criteria, such as Drucker Prager (inscribed and circumscribed, both extensions of the von Mises criterion) Incorporate the dependence of rock strength on the intermediate principal stress, σ2, but require true polyaxial rock strength measurements that have σ1 > σ2 > σ3 and are difficult to carry out

Single-sample testing

Because triaxial tests are so difficult and time-consuming to carry out, and because of the amount of core required and the difficulty in finding samples that are similar enough to be considered identical, it is common to attempt to reduce the number of tests requiring core preparation. One method is simply to carry out a uniaxial strength test in which the confining pressure is zero. This requires a much simpler apparatus; in fact, the sample does not even have to be jacketed, although this is recommended. By definition, the axial stress at failure in a uniaxial test is a direct measure of Co. Unfortunately, unconfined samples can fail in a variety of ways that do not provide a good measure of Co for use with a Mohr-Coulomb model. Furthermore, it is impossible to measure μi using one test unless a clearly defined failure surface is produced, the angle of which with respect to the loading axis can be measured. For these reasons, a series of triaxial tests is preferred.

An alternative method that does require testing in a triaxial cell is to carry out a series of tests on a single sample. The process proceeds by establishing a low-confining pressure and then increasing the axial stress until the sample just begins to yield. At that point, the test is stopped, the confining pressure is increased, and again the axial stress is increased until yielding occurs. In comparisons of this method against multiple triaxial tests, it is often the case that the yield stress derived from the multistage test is systematically lower, and the internal friction is also systematically lower, than the stress at failure and the internal friction derived from the triaxial tests. This is because, once the initial yielding has begun, the sample is already damaged and thus is weaker than it would be had this not occurred. However, by using this method, it may be possible to characterize the yield envelope of a plastic rock.

Scratch and penetrometer testing

A number of techniques have been developed to replace or augment triaxial tests to measure the strength properties of rocks. One such technique, which has a demonstrated ability to provide continuous, fine-scale measurements of both elastic and strength properties, is the scratch test.

Scratch testing

The scratch test involves driving a sharp cutter across a rock surface. By monitoring the vertical and lateral forces required to maintain a certain depth of cut, it is possible to determine the uniaxial compressive strength, Co. The Young’s modulus, E, can also be estimated in some cases. Fig. 5 shows a comparison of Co derived by scratch testing to laboratory core measurements and log-derived Co. The results are quite similar.

The advantage of scratch testing is that no special core preparation is required. This is in contrast to the extensive preparations required prior to triaxial testing. The test can be conducted either in the lab or, in principle, on the rig, almost immediately after recovery of core material. No significant damage occurs to the core, which makes this a very attractive substitute for triaxial testing when little material is available. In fact, research is now under way to evaluate the feasibility of designing a downhole tool to carry out this analysis.

Penetrometer testing

In a penetrometer test, a blunt probe is pressed against the surface of a rock sample using continuously increasing pressure. The unconfined compressive strength is then computed from the pressure required to fracture the sample. As in the case of scratch testing, no special sample preparation is required. In fact, any sample shape can be used for a penetrometer test, and even irregular rock fragments such as those recovered from intervals of wellbore enlargement because of compressive shear failure can be tested. Recently, methods have been developed to apply penetrometer tests to drill cuttings. Although these have not been widely used, they show considerable promise, and in the future they may become an important component of the measurement suite required to carry out wellbore stability analysis in real time.

Estimating strength parameters from other data

It is relatively straightforward to estimate Co using measurements that can be obtained at the rig site. Log or logging while drilling (LWD) measurements of porosity, elastic modulus, velocity, and even gamma ray activity (GR) have all been used to estimate strength. For example, Fig. 6[3] shows a lot of Co computed from P-wave modulus (ρbVp2) for Hemlock sands (Cook Inlet, Alaska).

It is possible to develop an empirical relationship between any log parameter (even GR—see Fig. 7) and Co or internal friction μi. Measurements of cation exchange capacity (CEC) and P-wave velocity have both been used for this purpose.

Because velocity, porosity, and GR can be acquired either using LWD or by measurements on cuttings carried out at the drilling rig from which CEC can also be derived, it is now possible to produce a strength log almost in real time. It is important, however, to recognize that different rock types will have very different log-strength relationships, based on their:

  • Sand/Shale
  • Limestone
  • Dolomite
  • Age
  • History
  • Consolidation state

Therefore, it is important to be careful to avoid applying to one rock type a relationship calibrated for another.

Nomenclature

Co = unconfined compressive strength, MPa, psi
G = acceleration of gravity, m/s2
k = Bulk modulus, GPa
ll = length in the l direction
Mijkl = Component of the modulus tensor that relates the ij component of the stress tensor to the kl component of the strain tensor, MPa, psi
S1 = greatest principal stress, MPa, psi
S3 = least principal stress, MPa, psi
SHmin = least horizontal stress, MPa, psi, lbm/gal
SHmax = greatest horizontal stress, MPa, psi, lbm/gal
Sv = vertical stress, MPa, psi
Vp = compressional-wave velocity, km/s
Vs = shear-wave velocity, km/s
ε kl = component of strain acting in the l direction per unit length in the k direction
μ = coefficient of sliding friction on a pre-existing weak plane, where μ = tanΦ
ρ = density, gm/cm3
ρb = bulk density, gm/cm3
σ = Terzaghi effective stress, MPa, psi
σij = effective stress acting in the i direction on a plane perpendicular to the j direction, MPa, psi
σn = effective stress acting normal to a plane, MPa, psi

Subscripts

i = index
j = index

References

  1. 1.0 1.1 Moos, D. and Chang, C. 1998. Relationships between Porosity, Pressure, and Velocities in Unconsolidated Sands. Proc., Overpressure in Petroleum Exploration Workshop, Pau, France.
  2. Schutjens, P.M.T.M., Hanssen, T.H., Hettema, M.H.H. et al. 2001. Compaction-induced porosity/permeability reduction in sandstone reservoirs. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, 30 September-3 October. SPE-71337-MS. http://dx.doi.org/10.2118/71337-MS.
  3. 3.0 3.1 Moos, D., Zoback, M.D., and Bailey, L. 2001. Feasibility Study of the Stability of Openhole Multilaterals, Cook Inlet, Alaska. SPE Drill & Compl 16 (3): 140-145. SPE-73192-PA. http://dx.doi.org/10.2118/73192-PA.

Noteworthy papers in OnePetro

Sharma, Bijon. 1992. A New Technique To Estimate Vertical Variability of Rock Properties for Reservoir Rock Evaluation, SPE Annual Technical Conference and Exhibition, 4-7 October. 24724-MS. http://dx.doi.org/10.2118/24724-MS.

Mohammed Y. Al-Qahtani and Zillur Rahim, 2001. A Mathematical Algorithm for Modeling Geomechanical Rock Properties of the Khuff and Pre-Khuff Reservoirs in Ghawar Field, SPE Middle East Oil Show , 17-20 March. 68194-MS. http://dx.doi.org/10.2118/68194-MS.

External links

See also

Page champions

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

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