Rock Mechanics as a Multidisciplinary Science, Roegiers (ed.) ¸ 1991 Balkema, Rotterdam ISBN906191 194X

Critical state shale mechanics

Ronald P. Steiger & Peter K. Leung Exxon Production Research, Houston, Tex.

ABSTRACT: Previously, observed similarities in the mechanical behavior of soils and shales indicated that critical state mechanics models

could possibly be developed to better describe the complex mechanical behavior of shales. Since then, new equipment and techniques have been discovered to measure quantitatively the consolidation properties of low permeability shales. The measurements require a heavy duty triaxial test load frame and a newly developed, precisely controlled, pore-water drainage system (patents pending) to obtain accurate volumetric measurements of pore volume compressibility during the tests. Data from shale consolidation tests and undrained strength tests have been used to develop parameters to describe shale stress- strain behavior over a wide range of loads using the critical state shale mechanics approach.

I INTRODUCTION

Critical state soil mechanics models have been used widely by geo- technical engineers to predict the complex mechanical behavior of soils and to design geotechnical structures such as foundations for offshore structures, tunnels, etc. A critical state soil mechanics model provides an important framework for generally describing, over a wide range of loads, the stress-strain behavior of soils with signifi- cantly different stress histories.

In an earlier paper (Steiger & Leung, 1988) on the strength measure- ments of shales, we indicated that we had observed similarities in the mechanical behavior of soils and shales and that critical state

mechanics models could possibly be used to better describe the complex mechanical behavior of shales. Since then, we have developed new equipment and techniques (patents pending) to measure quantitatively the consolidation properties of low permeability shales. Data from consolidation and triaxial tests have been used to develop parameters to model shale behavior using the critical state mechanics approach.

This paper describes a research test program to accurately determine the consolidation (drainage) behavior of low permeability shales. Well preserved samples were cored from large, undisturbed block samples. The tests require a heavy duty triaxial test load frame and a newly developed, precisely controlled, pore-water drainage system to obtain accurate volumetric measurements of pore volume compressibility

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during the tests. The tests can be conducted under either K o or hydrostatic conditions.

Critical state mechanics constitutive models for shales in conjunc- tion with nonlinear finite element analyses can provide significantly improved, more descriptive and less conservative, wellbore stability predictions that are needed for highly deviated and extended-reach oil and gas wells. Furthermore, critical state shale mechanics can be used in several other geomechanical applications, such as drilling rate optimization, predicting source rock and reservoir seal behavior, abnormal pore pressure prediction, subsidence, tunneling, mining, etc.

2 SHALES

Shales are fine grained, extremely low permeability sedimentary rocks that contain significant amounts of clay minerals. We use the term "shale" to include all the low permeability clay-bearing rocks that range from illitic siltstones to smectitic mudstones. These rocks are extremely difficult to characterize and test. Hydratable clays make them water sensitive. Uncontrolled hydration and swelling or drying and shrinking cause rapid deterioration of the rock structure and destruction of the integrity of a representative test sample. Addi- tionally, shales have complex, ill defined compositions and extremely low permeabilities in the microdarcy to nanodarcy range. Only a few years ago, they were referred to as impermeable rocks and were pre- cluded from direct pore pressure measurements during triaxial testing.

Recent research (Steiger & Leung 1988) demonstrated that with new techniques pore pressures of low permeability shales can be measured during triaxial strength tests. As a consequence, we showed directly that the strengths of shales vary as a function of mean effective stress, like other higher permeability geological materials, such as sandstones (Jaeger & Cook 1979), clays and soils (Terzaghi 1943). The effective stress represents the intergranular stress between solids of a soil or rock or the difference between the total confining stress and the pore pressure. According to the effective stress concept, all the measurable effects of a change in stress, such as compression and change in shearing resistance are due to changes in the effective stresses. The significance of the effective stress concept is that macroscopic deformation behavior is governed by effective rather than total stresses. Recognition of the effective stress principle has provided a valuable tool to explain soil and rock behavior. It has led to the development of many useful failure criteria and valuable soil constitutive relationships that present more realistic represen- tations of soil behavior by a concept called critical state soil mechanics (Schofield & Wroth 1968).

3 CRITICAL STATE MECHANICS

Critical state mechanics theory describes soil behavior in terms of sophisticated relationships between mean effective stress, stress difference (a function of shear stress) and specific volume (a func- tion of porosity or void ratio). It combines several aspects of soil stress-strain behavior, such as concepts of consolidation (compac- tion), yielding and failure, within a single theoretical framework to produce a more complete constitutive model of soil mechanical behavior (Atkinson & Bransby 1978). This theory is used to predict how a soil

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tested under one set of conditions (e.g. undrained triaxial tests) will behave under another set of conditions. This approach has allowed engineers to generalize soil behavior under different complex conditions based on consolidation and triaxial tests on soils.

The recent approaches to shale problems in drilling have relied upon assumptions of simple linear elastic theory to describe rock behavior. Those assumptions are adequate to describe stress-strain behavior for small strains but not for large strains near failure. Critical state mechanics combines the fundamentals of continuum mechanics and the

theories of elasticity and plasticity to produce a complete elastoplastic constitutive model. Figure 1 illustrates the relation- ship between stress difference (shear stress or deviatoric stress), effective stress and void ratio that produces a complete state bounda- ry surface for a soil during shearing under most conditions. For example, when the soil is loaded under any stress path condition, the stress will approach a line on the surface called the critical state line. The critical state line is the locus of failure points at which the soil will continue to yield at constant volume under constant effective stresses, thus defining the strength of the soil under any loading condition regardless of the loading path or sequence. An undrained sample with void ratio A will fail at the stress point on the critical state line at the intersection of the critical state line and the curve AB. These relationships in conjunction with finite element analysis methods provide a means to produce a more complete elastoplastic constitutive law for soil mechanical behavior.

Figure 1. Critical state mechanics stress difference - effective stress -

void ratio relationship produces a complete state boundary surface.

Critical State Effective Stress, p

In order to model shale mechanical behavior using critical state mechanics, special tests have to be conducted to quantify the critical state and the normal consolidation lines. The projection of the critical state line onto the effective stress - stress difference

(p-q) plane can be described by: q = M x p. M is the slope of the critical state line on the p-q plane. The M parameter can be obtained by a series of consolidated undrained triaxial tests with different consolidation pressures. It can also be obtained by several censoli- dated drained triaxial tests.

The normal consolidation line defines the 'base' of the mean effec-

tive stress-stress difference-void ratio space. It requires two parameters, x and el, to describe it. The x parameter represents the shape of the'normal consolidation line and e 1 represents the void ratio of a normally consolidated shale with effective stress, p : 1

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kPa. These parameters can be determined by a consolidation test using the specially designed test equipment and procedures.

4 SHALE RESEARCH PROGRAM

In 1984, Exxon initiated a comprehensive research program to measure mechanical properties of shales. Techniques and equipment necessary to prepare samples and obtain accurate, direct pore pressure measure- ments of low permeability shales were developed. As a result, shale strength data on nanodarcy permeability shales can now be measured on a routine basis. In the following several years, a shale strength data base was developed by testing several types of shales, ranging from an illitic siltstone to a smectitic mudstone.

In 1988, Steiger and Leung reported the development of the testing techniques and showed data to prove directly that low permeability shales follow the effective stress principle. They {1990} presented an overview of the advances in shale technology and results that show relationships between cuttings index properties and strengths measured by triaxial tests. The advances in the shale technology, including measuring techniques and the data base, provide the key to practical wellbore stability predictions for deep oil and gas wells (Steiger & Leung 1991). Reliable wellbore stability predictions could signifi- cantly impact drilling risks and costs, particularly in expensive wells under high-angle, extended-reach and severe tectonic conditions.

Encouraged by the success of the shale strength measurement program, Exxon in 1988 initiated another shale test program to develop more shale data from other advanced tests to define the stress-history dependent constitutive relationships. Predictions based on elasticity theory provide limited descriptions of wellbore behavior and often yield results that are too conservative. An advanced model based on critical state shale mechanics in conjunction with numerical analyses could lead to greatly improved wellbore stability predictive capabili- ties. The thrust of the program was to develop advanced techniques and equipment required to accurately drain pore fluid from or inject pore fluid into low permeability shales on a volumetric basis during triaxial tests. During the last 3 years, we had developed and refined several new experimental procedures for shales. Highly accurate, void ratio versus effective stress data and critical state mechanics parameters can now be measured on a routine basis.

5 EXPERIMENTAL METHODS

5.1 Testing program

A special test program was initiated to obtain the parameters needed to describe the mechanical behavior of shales using the critical state approach. The first phase of the test program was to design and fabricate end caps and a pore fluid accumulation system. Several versions of fluid extraction systems, such as conventional high pressure generators, were tested before useful data were collected using the specially designed microaccumulator. Scoping tests were conducted to determine the time required to achieve full consolidation for each loading step. It was found that the consolidation time varies from about 1 to 4 days, depending on the state of consolidation and the types and quantities of clays in the sample.

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The test program included 2 or 3 consolidation tests with different cycles of increasing and decreasing confining pressures. A series of consolidated undrained triaxial tests at different consolidation

pressures were conducted to obtain undrained strength measurements. Consolidation pressures were varied by draining water from the sample located inside the test cell prior to the triaxial test. Triaxial drained tests were conducted also.

5.2 Shale samples

Shale test samples were cut from undisturbed block samples or large field cores under controlled conditions with an inert coring fluid to prevent damage. Both block and test samples were handled with extreme care to minimize disturbance and loss of moisture. The block samples and large field cores were wrapped by multilayers of plastic wrap, aluminum foil and plastic sealant. The small core samples were stored in tightly sealed Teflon containers. All block samples, field cores and small core samples were stored in a humid controlled environment.

The shale test samples were in the shape of a right circular cylin- drical core. The sample size was typically smaller than the sample size used for testing high permeability rocks and soils, such as sand stone and clay soil. The small sample size was chosen to allow pore pressure equilibrium throughout the sample and pore fluid drainage from the sample to occur in a reasonably short time period.

5.3 Special test equipment and procedures

Both consolidation and triaxial tests were conducted using a heavy duty, high load capacity (about 1,100,000 N) triaxial load frame with a high pressure (about 200 MPa) confinement vessel. The test system was equipped with a computer control and data acquisition system. A computer program was written to provide very accurate data acquisition and control of the triaxial test equipment during a test.

Figure 2. Triaxial test apparatus to measure pore volume changes in the shale sample.

I Load Piston

I1,, .)! oace,, m Pis•n

• ........ -.• Microa•umulaor •ntml Va•e • Pre•ure

••m•ment x Fluid Re•oir

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As shown in Figure 2, for each test a sample core was mounted between the end caps {platens} and sealed in an impermeable, flexible jacket. The sample core was instrumented with four strain-gauged, cantilever-beam transducers to measure orthogonal transverse strains and four linear variable differential transformers {LVDT} parallel to the core axis to measure axial strain during the test. Axial load was determined by a load cell located at the bottom of the sample/end cap stack on the inside of the confinement vessel. Confining pressure was measured with a strain-gauged pressure transducer in the test cell.

The pore pressure during testing was measured by a miniaturized pore pressure transducer in the top end cap or the bottom end cap or both. The design shown in Figure 2 allows pore pressure measurement to be obtained from the top end cap. The pore pressure system consists of a small-volume channel and an accurate pressure transducer within a pore pressure chamber or port. The voids in the pore pressure port and channel were filled with an inert fluid to provide direct pressure transmission.

The fluid drainage system consists of a fluid channel in the bottom end cap, a very stiff heavy wall conduit connected to the base of the bottom end cap to extend the fluid channel to outside the pressure vessel, and a microaccumulator to provide a reservoir to retain the fluid ejected from the sample. The microaccumulator is a specially designed low volume, high pressure {100 MPa} vessel. The cavity inside the microaccumulator is divided into two compartments by a low friction piston. The piston was specially designed so it can seal off the two compartments and yet slide freely inside the cavity if the pressures of the two compartments are slightly unequal. One of the compartments connected to the fluid channel in the bottom end cap by a heavy wall conduit, is a reservoir for the pore fluid to be drained out from the sample. The fluid channel in the end cap and conduit and the fluid reservoir in the microaccumulator are filled with an inert fluid to provide direct pressure transmission. The other compartment in the microaccumulator is connected to a pressure source which provides a constant back pressure to the sample. In a typical test, a back pressure of about 7 MPa is maintained by applying gas pressure to the back pressure compartment.

The microaccumulator used in the tests has 2.5 cm 3 retaining capaci- ty and can accurately measure volume changes as low as 0.2 microliters. The high sensitivity of the microaccumulator is essen- tial for pore volume compressibility measurements since shales can have low porosities.

5.4 Consolidation tests

Consolidation tests can be conducted under isostatic or K o conditions. During an isostatic consolidation test, a shale sample is placed in the triaxial cell. Then, the sample is consolidated under different stages of equal all-around confining pressures. The control valve which controls the communication between the microaccumulator and the bottom end cap is opened to allow withdrawal of fluid from the sample to the fluid reservoir in the microaccumulator. A constant back pressure of about 7 MPa is applied to saturate the sample and to dissolve any air inside the sample core space. Confining pressure, axial and radial strains, pore pressure, and fluid volume drained from

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the sample are carefully monitored during each isostatic loading stage. After the fluid volume change becomes nearly constant, another level of confining pressure is applied to the sample. The test requires several cycles of loading and unloading to define the consol- idation behavior of a shale.

A K o consolidation test allows no lateral sample deformation during the test. During the test, an axial load is applied to a sample inside the triaxial test cell. The confining pressure (lateral pressure} is adjusted by computer control to prevent radial strain. The microaccumulator control valve is opened to allow fluid to move in (or out) of the sample to (or from) the microaccumulator reservoir. The test is conducted in multiple cycles of loading and unloading.

5.5 Triaxial tests

Consolidated undrained (CU) triaxial tests were run for most of the strength measurements. During a test, a shale sample core is placed in the triaxial test cell and consolidated under a constant confining pressure. The microaccumulator control valve is opened to allow fluid drainage from the sample. After the sample consolidation stage is complete, the microaccumulator control valve is closed to prevent fluid drainage from the sample and an axial load is applied until the sample reaches a post-failure, residual-stress condition. During the axial loading stage, a constant confining pressure is maintained.

6 RESULTS AND DISCUSSIONS

6.1 Consolidation test data

Figure 3 shows a typical void ratio versus mean effective stress response for one of the consolidation tests. Mean effective stress is presented in the natural log form. The sample was loaded isostatically to approximately 25 MPa in four load increments. Then, it was unload- ed to 7 MPa and reloaded to 45 MPa. During the final stage, the sample was unloaded to 19 MPa. Because of the extremely low permeability, the test took over one month to complete. At the peak load, the sample expelled about 1.0 cm 3 of pore fluid to the microaccumulator fluid reservoir. The fluid drained from the sample represents approximately 25 % of the total pore fluid in the sample.

0.55

0.5

Figure 3. Isostatic consol idation test to obtain critical state mechanics para- meters.

0.4

0.35 1

Unloading/reloading

'•ding Unloading•

1.5 2 2.5 3 3.5

In (mean effective stress), MPa

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The results of the consolidation test were used to develop the critical state mechanics parameters. Using the test shown on Figure 3, the x parameter that describes the normal consolidation line {loading) equals 0.164. The • parameter that describes the reconsolidation {unloading - reloading) has an average value of 0.024. The • calcu- lated from the first unloading-reloading was equal to .023 and the second unloading cycles are in good agreement with a value of .025.

The maximum past pressure that the rock experienced while in the earth can be estimated from the consolidation data. Using the method {Lambe & Whitman 1969} to determine maximum compression pressures of soils, the maximum past pressure {MPP) of the shale sample is about 24 MPa. Observations from the triaxial strength test data indicated that the MPP estimate from the test is reasonably accurate. Triaxial tests conducted on heavily overconsolidated samples {i.e. effective consoli- dation pressure << 24 MPa) displayed dilational, residual-strength behavior while triaxial tests conducted on normally consolidated {i.e. effective consolidation pressure > 24 MPa)or slightly overconsolidated samples displayed non-dilational, peak-strength behavior.

6.2 Triaxial test data

Figure 4 shows the results of several triaxial tests. The stress difference at failure increases linearly with an increase in mean effective stress. The critical state line parameter, M, which is the slope of the stress difference versus mean effective stress line is equal to 0.53.

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Figure 4. Critical state line from undrained triaxial tests.

Critical State Line

10 20 30

Mean Effective Stress, MPa

Several undrained triaxial tests were run to determine responses of samples with different consolidation ratios (OCR). OCR is defined as the ratio between the maximum past pressure and the present effective stress. Samples for these tests were isostatically consolidated first. Then they were unloaded to different stresses to achieve different OCRs. Figure 5 shows the normalized stress paths {shear stress versus mean effective stress} of these set of tests. Both the shear stress and mean effective stress were normalized by the mean effective stress measured after the consolidation stage and at the start of the triaxial loading.

The state boundary surface that provides the critical state descrip- tions {Figure 1) of the shale can be obtained from the triaxial test

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Figure 5. Stress paths of CU triaxial tests

samples with OCR's from 1.0 to 2.5.

0.4

= o.2r I %øø % • I • i •o o

o ' I OCR 2.5• 1.3^ol.2 • 1.0

Zol , , 0 0.2 0.4 0.6 0.8 1

Normali•d Mean Ef•cti• 8trees

1.2

Figure 6. Using the modified Cam clay model to define the Roscoe surface.

0.4

•o.e

0•0.2

.õ

z

o o

critical State Line Impossible State m/Modified Cam ClayJ

0.2 0.4 0.6 0.8 I 1.2

Normalized Mean Effective Stress

data shown in Figure 5. For clarity, Figure 6 simplifies Figure 5 to show features of the state boundary surface.

The total state boundary surface shown in Figure 6 consists of the Roscoe and Hvorslev surfaces. The Roscoe surface (path B-C) repre- sents the stress paths of normally consolidated samples prior to failure. The Hvorslev surface (path A-B) represents the stress paths of heavily overconsolidated sample prior to failure. Both surfaces meet at the critical state line (point B). Boundary surfaces of all constant void ratio cross sections have the same shape but are of different sizes. Higher void ratio results in a smaller yield sur- face. The sections may be scaled to a single normalized curve as shown in Figure 6. Figure 6 also shows that the Roscoe surface fits within the framework of the modified. Cam-clay model. The modified Cam-clay model is a variation of critical state mechanics models that assumes the Roscoe surface has an elliptical shape (Roscoe & Burland 1968). The comparison between the test data and the modified Cam-clay mathematical function shows that the state boundary can be modeled by a mathematical relationship, which can be defined by a few measurable engineering parameters.

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7 CONCLUSIONS

New measurement techniques have been developed to quantify consolida- tion behavior of low permeability shales. Highly accurate volumetric measurements of pore volume compressibility of shales can be obtained on a routine basis. As a result, parameters for critical state shale mechanics models can measured accurately. However, additional tests on different shale types are needed to expand the data base.

Critical state mechanics constitutive models for shales in conjunc- tion with nonlinear finite element analysis can provide significantly improved, more descriptive and less conservative, wellbore stability predictions that are needed for highly deviated and extended-reach oil and gas wells. Furthermore, critical state shale mechanics can be used in several other geomechanical applications, such as drilling rate optimization, predicting source rock and reservoir seal behavior, abnormal pore pressure prediction, subsidence, tunneling, mining, etc.

ACKNOWLEDGEMENTS

The authors thank Exxon Production Research Company for support of this project and permission to publish this paper. Special thanks are given to W. E. Kline for his management support and encouragement, to Rudy Stankovich for his diligent laboratory work and to David Kent for his assistance in data reduction and preparing this manuscript. We also thank Terra Tek, Inc. and other project team members.

REFERENCES

Atkinson, J.H. & P.L. Bransby 1978. The Mechanics of Soils. London: McGraw-Hill.

Lambe, T.W. & R.V. Whitman 1969. Soil Mechanics. New York: John Wiley and Sons.

Roscoe, K.H. & J.B. Burland 1968. On the generalized stress-strain behavior of "wet" clay. Engineering Plasticity. Cambridge: Cambridge University Press.

Schofield, A. & P. Wroth 1968. Critical State Soil Mechanics. New York: McGraw-Hill.

Steiger R.P. & P.K. Leung 1988. Quantitative Determination of The Mechanical Properties of Shales. SPE paper 18024.

Steiger R.P. & P.K. Leung 1990. Lecture: Predictions of Wellbore Stability and Shale Formations at Great Depth. Proc. ISRM-SPE Int. Symp. Rock at Great Depth proceedings, vol 3., p.1209-1218. Rotter- dam:Balkema.

Steiger R.P. & P.K. Leung 1991. Advances in Shale Mechanics - Key to Wellbore Stability Predictions. Comprehensive Rock Engineering, Vol. 6. To be published.

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