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Understanding geopressure mechanisms using micro-mechanicsYaping Zhu*, Statoil

SummaryGeopressure (or overpressure) presents significant risks todrilling if not predicted accurately. To better understand

mechanisms of abnormal fluid pressure, I choose aninclusion-based micromechanical model by using the

concept of eigenstrain and characterize the microstructureof rocks as a product of the compaction processes. Analysisand numerical tests demonstrate that fluid pressure andeffective velocities are functions of the microstructure (e.g.,

shape of inclusions) and the stress state of the rock, whichsuggests the possibility of linking pressure and velocities togeological parameters.

Introduction

Abnormal pore pressure has become an increasinglyimportant issue in exploration in the Gulf of Mexico andoften results in drilling hazards such as kicks, mud

volcanoes, wellbore instability, and lost circulation (Sayers,2006). Commonly used pressure prediction approachessuch as Eatons (1975) and Bowers (1995) methodsgenerally focus on building empirical relationship between

observed geophysical properties (e.g., velocity) and porepressure, which, although work well for many applications,may also cause large errors in the prediction due to highuncertainty in the estimate of velocities and the lack of

understanding of the relationship between geophysicalproperties and microstructural features of the rocks.

In this study, I use a micromechanical pore pressure

prediction model by representing rocks as matrix withspheroidal inclusions (pores and cracks). By introducingthe concept of eigenstrain, I represent fluid pressure

through confining stress and residue pressure, properties ofrock components (minerals and fluids), as well as thecharacteristics of the microstructure of the rock (e.g., theshape and alignment of inclusions). Effective physical

properties such as velocity are then calculated usingdifferential effective medium theory.

The proposed method allows a study of the behavior ofeffective properties in response to different in-situ rock

properties such as fluid pressure, porosity, and crack

density. It also allows an understanding of variousmechanisms related to geopressure (e.g., Zoback, 2007).

For example, during normal compaction process, rocksbehave as a drained (a.k.a. open) system where fluid isexpelled at the same time as formation porosity is reduced

and cracks are closed. This results in an increase of velocityas a function of effective stress. When the formation

permeability becomes too low to maintain the free fluidflow into and out of the system, compaction disequilibrium

starts to develop. This causes the formation to become an

undrained (a.k.a. closed) system and fluid pressure starts tobuild up above the hydrostatic pressure. During the thermalexpansion or maturation process, microcracks may begenerated in response to elevated fluid pressures. The

presence of microcracks can have pronounced effects onvelocity and anisotropy, thus providing potentially usefulattributes for estimating abnormal pressure zones.

Hereafter the discussion is limited to spheroidal inclusions,whose shape can be characterized with the aspect ratio ,ratio of the semi minor axis to the semi major axis.

Theory of micro-mechanical model

From a micromechanical point of view, pores and cracks

that contain fluids can be considered as inclusionsimbedded in the solid background of rocks. An inclusion

differs from its background in that 1) fluid and solid havedifferent physical properties (i.e., heterogeneity), and 2)extra fluid pressure (e.g., due to thermal expansion of fluidsor hydrocarbon generation) may develop in inclusions.

According to Eshelby (1957), the extra pressure (denotedas residue pressure), or its general form if inclusionmaterial is a solid (denoted as residue stress), can berepresented through the eigenstrain, which usually refers to

strain remaining in a body when the body is self-equilibrated. Indeed, heterogeneity problem can also berepresented as an equivalent eigenstrain problem (e.g., Xu,1998). Here I follow the equivalent inclusion method

introduced by Mura (1987) to treat both heterogeneity andresidue fluid pressure as an eigenstrain problem andcalculate stress and strain of an inclusion in an extended

background with applied stress at infinity (Figure 1).

Figure 1. Illustration of equivalent inclusion method.

Consider a background material with elastic stiffness (itscompliance being), containing an infinitesimal spheroidalinclusion with different stiffness (its compliance being

) and residue stress . For fluids, the term residue stressis interchangeable with residue pressure and can be

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Understanding geopressure mechanisms using micro-mechanics

expressed as , where is the magnitude ofresidue pressure and is a unit tensor, and is linked toresidue strain as

. (1)Due to the presence of heterogeneity and residue stress, the

stress field of the inclusion deviates from the backgroundstress and can be expressed as (Mura, 1987):

( ) ( ),(2)

where and are, respectively, the perturbation of stressand strain from the background (or applied) stress andstrain , and the applied stress has . Parameteris the eigenstrain that is related to the strain perturbationthrough the Eshelby tensor (Eshelby, 1957) as:

( ) , (3)where a new eigenstrain is introduced as .Substituting equation 3 into equation 2 yields

[ ] . (4)

Fluid pressure

Consider a closed inclusion where fluid is trapped, thestrain deformation of the inclusion can be given as

, (5)where is the stress of the inclusion material, which isalso called fluid pressure if the inclusion is filled with fluid.

In such a case, is given as ,where denotes the magnitude of the total fluid pressurein the inclusion. Solving equations 4-5 yields

, (6)where , , and [ ]. Note the difference between the stressof inclusion material and residue pressure . Here has the contribution from both applied stress and residue

pressure. Also note that tensors , ,, and depend onthe strength of the background material and geometry ofthe inclusions such as the shape of pores and cracks.

Tensors and also depend on fluid properties.

Effective elasticity

After calculating the average stress and strain of the

system, I write the effective compliance as a function ofthe applied stress and residue pressure:

, (7)where and denotes, respectively, the compliance ofthe background material and the fluid. Parameter denotesthe porosity and is the differential stress (or

commonly referred to as effective stress). It is clear that if . Note equation 7 was derived for diluteconcentration of inclusions where the interaction amonginclusions is ignored. To account for higher volumeconcentration of pores and cracks, modeling schemes suchas differential effective medium (DEM) can be used.

Equation 7 suggests that effective velocity can be expressed

through the stress state of the rock and residue pressure

generated in the inclusions. It also suggests that the verticaleffective stress alone (e.g., Bowers, 1995) may not be

sufficient to determine the effective elastic properties of

rocks for regions where tectonic forces become prominent.

In a special case where is hydrostatic confining pressure(i.e., where is the magnitude ofthe confining stress), equation 7 can be simplified as

, (8)where is the normalized differential stressand is pore fluid pressure. Quantity is theresidue pressure normalized by confining pressure, and

is the combination of

several components of tensor . As shown in equation 8,the effective compliance reduces with the increase of thenormalized differential stress. Hence, as the confining

pressure increases, the effective velocities increase.

Interpretation of fluid pressure

Equation 6 suggests that fluid pressure is a function of themicrostructure and medium properties of rocks, as well asapplied stress and residue pressure. If residue pressure is

absent ( ), it reduces to , which isconsistent with pore pressure obtained by Xu (1998).According to equation 6, the differential stress can bewritten as

, (9)where the first term, , is caused by theheterogeneity of the medium, i.e., due to the difference

between and . For example, if the inclusion materialhas the same properties as the background ( ), this

gives and hence . If the inclusionmaterial becomes extremely compliant (), tensor reduces to 0 and approaches . Since in such cases theinclusion material is too compliant to support any residue

stress (i.e., ), the stress of inclusion material, ,reduces to 0, making the inclusion behaving like a vacuum.

If fluid pressure deviates from hydrostatic pressure, then

abnormal pressure appears, which is also termedoverpressure when fluid pressure exceeds hydrostatic

pressure. As shown in Figure 2, hydrostatic pressure can be

given as

, where is brine density, is

gravitational acceleration, and is the reference depthwhere the integration starts from, e.g., the mean seal level

in an offshore exploration case. In a 1D case where thelithostatic stress can be obtained using vertical integration

of bulk density () of rocks, we have

.

For regions where tectonic forces become significant suchas areas near salt that has irregular geometry, 1Dassumption of overburden stress can break down and hencemore sophisticated modeling of the stress field is needed.

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Figure 2 illustrates the relation between different types of

pressures. Here the abnormal pressure emerges at depth and residue pressure starts to develop at depth . Hence,

starting from the reference hydrostatic pressure

, the abnormal pressure is given as

, where stands for the traceof the corresponding matrix.

Figure 2. Schematic plot showing abnormal pressure is influencedby heterogeneity and residue pressure, and varies with depth.

Figure 3 shows the dependence of pore fluid pressure on

the aspect ratio of a horizontally alignedspheroidal pore in a constant solid background. A flat,

penny-shaped pore ( ) is generally more compliantthan a round pore ( ), thus making the pressure in aflat pore higher than in a round pore. If , fluid

pressure approaches the confining pressure or theoverburden stress, causing fluid pressure to be abnormally

high and drilling through this area more risky. If thepressure buildup in a pore is higher than a critical strength

of the rock (e.g., leak-off point or fracture propagationpressure), one would expect that new cracks or fracturesmay initiate and even propagate.

Figure 3. Normalized pore pressure as a function of pore aspect

ratio for a) confining stress psi, and b) uniaxial(vertical) stress psi. Model parameters: background km/s, km/s, g/cc; pore fluid km/s, g/cc. Here the aspect ratio is defined as /so that it varies over a wide range of 0.001~100.

In a hydrostatic stress state where all three principle stress

components are identical (Figure 3a), a spherical pore( ) becomes stiffest as the fluid pressure in the pore

becomes minimum. In a uniaxial (vertical) loading process

(Figure 3b), however, a spherical pore is not necessarilystiffest since no lateral constraints are applied to the rock.

In such a case, the fluid pressure keeps dropping for ,albeit gradually, as the aspect ratio increases.

Although not shown here, pore fluid pressure is alsoinfluenced by other factors such as pore orientation and the

connectivity of the pore system. For example, in a closedpore, fluid cannot escape, thus causing pressure buildup.

Micromechanical analysis of pressure mechanisms

Here I attempt to interpret a few mechanisms using theproposed micromechanical model. During the early stage ofthe compaction, sedimentary rocks usually contain pores

with high porosity. As sedimentation continues, increasingoverburden stress causes fluids being expelled out of the

pores and porosity reduces. If the sedimentation rate isrelatively low, or if the formation permeability is high

enough to allow free fluid flow out of the inclusion space,the equilibrium between the pore fluid pressure andexternal fluid pressure such as hydrostatic pressure ismaintained (Zhang, 2011). In such a case, the rock

undergoes normal compaction and can be treated as anopen pore system with drained boundary condition for thefluids. The effective compliance can then be expressed as

[ ]. (10)If the sedimentation rate is relatively high or if formation

permeability becomes too low to allow free fluid flow, the

pore system starts to close and this marks the emergence of

disequilibrium compaction. For such a system containingclosed pores, the effective compliance can be written as

[ ( )

], (11)where denotes the stiffness of the fluid in a tensorialform. Equation 11 can also be rewritten in terms of applied

stress, as shown in equation 7 (with set to 0).

A transition zone usually occurs where normal compactionis giving place to disequilibrium compaction. In theserocks, both open and closed inclusions coexist. Effectivecompliance is then a combination of equations 10 and 11.

Under favorable geological conditions, hydrocarbons cangenerate from thermal maturation of kerogen in source

rocks. This can cause a significant increase of volume ofpore fluids and results in overpressure. Alternatively,thermal expansion of fluids can cause pressure to increasein a closed pore. Effective compliance due to this

mechanism can be described using equation 7.

Note that the microstructure of rocks evolves with thegeological processes. For example, an exponential



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relationship between porosity and effective stress was used

by Hart et al. (1995) to characterize the reduction ofporosity at geologic depths for normal compaction process.

Example

The method can be used to simulate several compactionprocesses such as normal compaction, disequilibriumcompaction, and thermal fluid expansion. In the followingexample, I use Monte-Carlo simulation to calculate

effective velocity and fluid pressure. I assume that therocks contain pores and cracks characterized as spheroidswith different aspect ratios. To account for the reduction ofinclusion space, I assume that porosity and crack density

generally follow exponential trends over the depth range of5000-19000 ft. Figures 4b-d show modeling results ofeffective velocity and pressure profiles. Also shown aremodeling parameters such as volume concentration (Figure

4a) and alignment (Figure 5) of pores and cracks.

Figure 4. Simulated depth-varying physical properties: a) Porosity(blue) and crack density (red), b) effective vertical P-wavevelocity, c) hydrostatic- (blue), fluid- (red), and overburden-(black) pressures, and d) hydrostatic- (blue), fluid- (red), andoverburden- (black) pressure gradients.

Figure 5. Pore and crack alignment at various depth.

Normal compaction starts from the mudline at 5000 ftwhere fluids are free to escape out of pores and cracks. As

sedimentation continues, crack density drops more quicklywith burial depth, which reflects the closure of some cracksdue to increased overburden pressure. As cracks close and

porosity reduces, velocity increases (Figure 4b). A

transition zone appears at 9000-11000 ft where velocityincreases significantly since rocks become stiffer as the

pore system changes from drained to undrained conditions.

Below the depth of 11000 ft, rocks experience

disequilibrium compaction where porosity reduces slowlyand velocity varies more gradually. Note that

disequilibrium compaction results in elevated fluid

pressure, indicating that overpressure is observed and fluidpressure gradient is above hydrostatic pressure gradient(Figure 4c-d). Thermal fluid expansion kicks in below

19000 ft, causing the development of residue pressure.When fluid pressure is sufficiently high, new cracks mayinitiate and propagate, resulting in significant drop ofvelocity. Figure 6 shows velocity as functions of effective

stress and density for the simulated geologic processes. Forexample, velocity increases with effective stress duringdisequilibrium compaction (loading process) and decreasesduring fluid expansion (unloading process). Numerical

simulation also suggests that the change of density duringfluid expansion is negligible, which is similar to the trend B(unloading) discussed by Swarbrick (2012).

Figure 6. Effective vertical P-wave velocity as a function ofeffective stress and density.

Conclusions

In this study, an inclusion-based micromechanical modelwas used to characterize rock properties (effective velocityand fluid pressure). The proposed method extends Xus

(1998) poroelastic model for the undrained fluid system byaccounting for residue pressures due to, e.g., thermal fluidexpansion and hydrocarbon generation. Heterogeneity dueto the presence of inclusions and residue pressure in

inclusions are treated consistently as an eigenstrainproblem, and effective elasticity of the medium isexpressed through applied stress and residue pressure.

I analyzed the microstructure of rocks under variousgeological conditions and used the model to characterizevelocity and fluid pressure for various mechanisms that

have been proposed for pressure effects, e.g., normalcompaction, disequilibrium compaction, and thermal fluidexpansion. Other possible mechanisms such as clay

diagenesis are out of the scope of this work.

Acknowledgments

The author would like to thank Statoil for the permission to

publish this work. Thanks also go to (but not limited to)Ivar Brevik, Chez Uzoh, Hege M. Nordgrd Bols, TomSun, Dan Ebrom, and Cem Ozan.



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EDITED REFERENCES

Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2014

SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for

each paper will achieve a high degree of linking to cited sources that appear on the Web.

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Hart, B. S., P. B. Flemings, and A. Deshpande, 1995, Porosity and pressure: Role of compactiondisequilibrium in the development of geopressures in a Gulf Coast Pleistocene basin: Geology, 23,no. 1, 4548, http://dx.doi.org/10.1130/0091-7613(1995)0232.3.CO;2.

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