Paper Example: Exploring Strain Tensors in 2D and 3D Geometry

Strain or rock deformation as stated by Zoback (2007) can also be described as a second order tensor similar to stress tensor. A 2D geometric strain tensor is described in the equation below:

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i j=12 duidxj+ dujdxi

3D strain tensor is described by Fjaer, et al. as follows:

= xghxyghxzghyxyghyzghzxghzyz

When

= Normal strain

= shear strain

The strain tensor notations defined above will be utilized through the study while the following strain tensor notations will be used in some references (Zoback 2007):

= 111213212223313233

According to Zoback (2007), the potential of leakage of reservoir fluids via induced fractures or faults can be affected by the elevation of the pore pressure; additionally, as pressure increases with depth, it governs the sedimentary basin stress field making the prediction of the pore pressure extremely important. The use of direct measurements such as drilling mud weights, wireline tools, and drill-stem testing tools is required to measure pore pressure in high permeable formations. Techniques for direct pore pressure measurement are difficult to apply for impermeable formations such as shale. Therefore, indirect methods are utilized such as geophysical measurements which employ the concept that an increase in vertical effective stress (Sv - Pp) decreases the porosity of shale (Zoback, 2007). In high permeability formations, loss of porosity is caused due to the occurrence of compaction as the flow of fluid prevents the build-up of pressure as it can discharge from the high permeability formation. Alternatively, low permeability formations do not provide a path for fluid discharge from the pores with the same quantity of lost porosity. Hence, an increase in pore pressure (Zoback, 2007).

To express the association between increase effective vertical stress and reduction in porosity, confined compaction test or hydrostatic compression test can be carried out. In the case of the hydrostatic test, a uniform confining pressure is applied while in the confined compaction test, the sample is subjected to an axial load. Additionally, resistivity measurements, sonic velocity, and seismic data are methods used in determining pore pressure indirectly (Zoback, 2007).

Strain can be described as a measurement of deformation of a body (Harrison and Hudson, 1997). Strain is the ratio of length alterations due to stress to the original length: (Hibbeler, R. C., 2011)

= l / lo.

Strain in rocks exhibiting elastic behavior changes linearly with applied stress. In the absence of the applied stress, the rock is able to reverse the deformation returning to its original state behaving like a spring. In this situation, the Young Modulus (E) represents the proportionality constant and rock stiffness. An example of linear behavior is observed in well cemented sandstone. Alternatively, poorly cemented and weak formation exhibit different characteristics.

On reaching the yield point, some rocks behave plastically while some behave elastically below this point. Elastic behavior on unloading the rocks once again governs the deformation of rocks. The primary factor controlling the required stress rate for specific deformation is the rock's apparent viscosity. Another deformation type belonging to viscous materials is viscoplastic deformation which occurs when they undergo permanent deformation when subjected to stress. Under the elasticity theory, elastic behavior is defined through Hooke's Law which states that strain and stress have a linear relationship. Elasticity is a characteristic of a material to return to its original form once a factor causing the change is removed.

As stated by Schon (2011), rocks mechanical properties can be classified as strength or failure parameters and deformation parameters. Deformation parameters include Young's Modulus, Poisson's ratio, Shear modulus, Bulk modulus and Lame's constant.

Young's Modulus (E) is the elastic deformation strain/stress curve slope. It is the ratio of normal stress to the ensuing normal strain.

E = s

Young Modulus is calculated in megapascals (MPa or N/mm2) or gigapascals (GPa or kN/mm2). N clarifying Hooke's Law, a material is assumed to be cylindrical where a compressive stress is applied in the x-direction. Consequently, the material's shape is changed and becomes smaller in size thus the material has undergone strain or is deformed. As strain is linearly related to stress under Hooke's law, the above equation is useful for stress applied in one direction.

However if the material is subjected to stress in the y or z-direction, in addition to the x-direction, it will undergo strain but less in the x-direction. A strain response in the x-direction in the opposite direction is contributed by a y-direction strain whereby y-direction stress is represented by the Poisson's ratio value (v). Strain in the x-direction is caused by x-direction stress on a material. But at the same time, x-direction strain will be caused by changes of stress applied on the y and z-direction causing an opposing strain caused by the x-direction stress.

Subsequently, Poisson's ratio (v) states that y or z-direction stress in addition to the x-direction stress influences strain in the x-direction. The effect of the y or z-direction stress is equal to the product of the Poisson's ratio and the stress over Young's modulus. It is a constant relating strain in one direction to strain in another direction:

v = - Strain In Direction of Load / Strain at Right Angle of Load

v = - x/ y

Thus in calculating x-direction strain, y and z-directions stresses have to be considered. Also, as shear modulus (G) is the ratio of shear stress over shear strain:

G = txy/ghxy

The relationship between Young's modulus, Poisson's ratio and Shear modulus is:

G = E / [2(1+v)]

Another Elasticity constant is the Bulk modulus (K). on stress application from all surfaces, an object's volume decreases however on removal of the stress it regains its original volume. Bulk modulus or incompressibility it the measurement of an objects ability to withstand compression stresses acting from all sides. It is the ratio of volumetric strain and stress within deformation elastic range and it is measured in Pascal (Pa, N/m2):

K = Volumetric Stress / Volumetric Strain

Bulk modulus is considered for liquids in terms of pressure and volume:

K = - dP / [(dV/V0)]

And in terms of pressure and density

K = dP / [(dr/r0)]

The relationship between Bulk modulus (K), Poisson's ratio (v) and Young's modulus (E) is defined as:

K = E / [3(1-2v)]

In further defining the stress-strain relationship within the elasticity limit, the Lame's constant parameter is also employed. It is described according to Young's modulus, and Poisson's ratio as:

l = {Ev / [(1-2v) (1+v)]}.

Rocks exhibit poroelastic behavior in attendance to oil or water in the pores, but as stated by Fjaer (2008), elastic theory applicable to other solid materials cannot define rocks with permeability and porosity behavior as their stiffness is dictated by the force application rate on the rocks. The major situations of external force application include stress applied in a fast manner. In this case, a portion of the stress is carried in the pores by the fluid resulting in an increase in pore pressure as the fluid drain rate from the rock is less than stress acting on the rock; thus the rock's stiffness increases under this condition.

Alternatively, when stress is applied gradually, the rocks stiffness is unaltered as there is time for the fluid to drain from the rock. Hence it is critical to consider the relationship between rock permeability, loading rate and fluid viscosity. As stated, poroelasticity is applicable to porous material contain fluids as opposed to linear elastic theory. In this case, fluids in the material's pores exert pressure that counteracts the changes made by the applied pressure. In regards to normal and shear pressure, pressure resulting from the fluids can only resist against normal pressure. In the presence of fluid in rock pores, pore pressure is able to resist applied stress, delaying or preventing deformation of the rock. Thus the total stress (sx total), effective stress (sx eff) and pore pressure relationship in the x-directions is as follows:

sx total = sx eff - aP

Rock strength determination from geophysical logs is an alternative lab test utilized in the absence of subsurface cores.

Geophysical logs and rock strength have enabled the development of various relationships in measurable parameters. Based on the relationship from rock strength determination data, several common characteristics have been identified. First, the compressional wave travel time that is equivalent to (t = Vp-1) where (Vp) is P-wave velocity is utilized. Additionally, P-wave velocity and density information determines Young's modulus and the porosity derived from density date is employed in estimating rock strength.

Initially, different rocks possess own compressive strength but when stress applied in compression test surpasses the rocks compressive strength, it fails in compression. Subsequently, rock strength under compression is defined as maximum principal stress value when the rock can no longer withstand applied stress.

In determining rock strength under applied confining pressure, Mohr failure envelope and Mohr circles, and the Drucker-Praeger failure criterion, a general Mohr-Coulomb failure form are used. The major parameters used in these mechanisms are internal friction angle and cohesion. During fluid injection, there is a possibility of fault slip leading to fault activations generating fluid flow paths. As observed by Da Vinci, when the normal to shear ratio goes beyond the material's friction coefficient (m), frictional sliding on the plane occurs. The relationship between friction coefficient (m), effective normal stress (sn) and shear stress (t) is demonstrated by Amontons' law:

t / sn = m

During fluid injection, effective normal stress reduces with an increase in pore pressure causing fault slip. The injection fault stability is assessed using Coulomb failure function (CFF):

CFF = t - m sn

Where a negative CFF value indicate a stable fault with faulting occurring as it approaches zero. Subsequently, data on rock failure under confining pressure is obtained from Mohr envelope. During rock sample failure process, a fault develops under triaxial compression. Through Mohr circle, effective normal stress (sn = Sn -Pp) and shear stress (tf), based on stresses applied (s1, s3), can be evaluated graphically using the equation:

tf = 0.5(s1 - s3) sin2v

sn = 0.5(s1 + s3) + 0.5 (s1 - s3) cos2v

A series of triaxial tests is necessary to describe rock failure in varying confining pressures conditions. In such tests, confining pressures and uniaxial stress are plied simultaneously. Thus the sample is compressed radially and axially under constant confining pressure. Two major stresses are considered, confining pressure (S0) and differential stress (S1 - S0); thus the sample's strength is described by differential stress that it fails in a specific confining pressure.