Shales comprise almost 70% of subsurface sedimentary rocks. They are often the source and seal of a petroleum system. Hence a proper understanding and prediction of the mechanical and elastic properties of shale and its main component, clay is key to the success of petroleum exploration and production.

Deﬁnition

Shale is a ﬁssile sedimentary rock and is composed of clay sized particles. They are complex porous materials consisting of percolating and interpenetrating ﬂuid and solid phases (Hornby, 1995). The solid phase is generally composed of a variety of materials, mainly clay minerals and silt. The shape, orientation, distribution and volume fraction of these minerals control the eﬀective elastic properties of the shales.

Qualitative view of the microstructure of shales at various scales (Sarout & Gueguen, 2008)

Anisotropy in Shales

A transversely isotropic material is a special class of orthotropic material. A transversely isotropic material is one with physical properties that are symmetric about an axis that is normal to a plane of isotropy. This transverse plane has inﬁnite planes of symmetry and thus, within this plane, the material properties are same in all directions. With this type of material symmetry the number of independent constants in the elasticity tensor is reduced to ﬁve. Most shales are microscopically laminated structures composed of clay rich, silt rich and organic matter rich layers. When elastic wavelength is much larger than the thickness of the single layer the ﬁnely layered structure acts as a homogeneous and transversely isotropic medium (Carcione, 2000). There are many possible causes of velocity anisotropy in shales. The widely acceptedexplanation is the preferential orientation of the platy phyllitic clay minerals which form the building blocks of shales (Hornby, 1995; Kaarsberg, 1959; Sayers, 1994). Another possible cause could be the presence of organic matter (kerogen) which may form thin layers leading to the transverse isotropy nature in shales as described by Backus (1962). Yet another explanation that is quite popular is the presence of microcracks oriented parallel to bedding and high aspect ratio pores which might often be caused due to stress eﬀects on the rock.

Frequency dependent behavior of West African Shale. Only the p -wave velocities show dispersion. No eﬀect is seen on the shear wave velocities.(Duranti et al., 2005).

Pore pressure within formations determines the mud weight required to build a balancing ﬂuid pressure downhole. An improper understanding of the subsurface geology and the formation pressures may result in fracturing the formation if the mud weight is too high. Incontrast, if mud weight is too low, then formation ﬂuids can ﬂow into the well, potentially leading to well blowouts if not controlled. Complex geological settings make pore-pressure prediction diﬃcult and often inaccurate due to uncertainty in pressure-generating mechanisms. Estimation of proper pore pressures is necessary for designing stable holes and an optimized casing program. In exploration, knowledge of pore pressures can assist in assessing seal eﬀectiveness and in high-grading reservoir sweet spots. It also provides usefulcalibration information for basin modeling. Hence, it becomes very important to accurately predict pore pressures. Predicting in-situ reservoir and formation pore pressures from seismic velocity, sonic velocity, and resistivity is a general practice within the industry. The relationship between velocity and pore pressure is controlled by eﬀective stress *P**eff* .

*P eff = OB - n.Pp*

where** ****OB** is the overburden or the vertical stress, usually obtained by integrating the density log, *P**p* is the pore pressure, *P**eff* is the eﬀective stress, and n is deﬁned as the eﬀective stress coeﬃcient. Hence, in order to determine subsurface pore pressure from properties like velocity, determination of *n* is a must. The typical method of determining *n* for rocks is through laboratory measurements. Little experimental work has been done on shales, where most instances of overpressure occur. Most models used in velocity-to-pressure transforms are calibrated for sandstone but not for shale. Overpressure is one of the primary concerns of explorationists, and drilling through overpressured shale is still considered a hazard, both in terms of personnel safety and well economics.

As these sediments are buried, mechanical compaction occurs under increasing eﬀective stress, increasing vertical load, and hydrostatically increasing pore pressure. The sediments lose porosity, while sonic velocity and density increase, ultimately approaching a limit beyond which there is no further change in rock properties. This increasingly eﬀective stress path with associated rock properties is called a virgin compaction curve if the pore pressure increases hydrostatically. A similar term used in experimental rockphysics experiments is a loading curve that indicates changing rock properties for compacting sediments under uniformly increasing eﬀective stress conditions. During the compaction process, the rock may see a state where the eﬀective stress is reduced. Compaction is predominantly an inelastic process; however, eﬀective stress reduction results in elastic rebound alone, leading to a diﬀerent unloading curve.

Various mechanisms can cause rocks to be overpressured. The most common is compaction-disequilibrium or undercompaction where the increasing over-burden stress is counteracted by increased pore pressure. Undercompaction itself cannot cause the eﬀective stress to decrease (Bowers, 1995), i.e., the rock does not see a velocity hysteresis and hence does not follow an unloading curve. Instead, its eﬀective stress state can become constant with increasing depth. However, if the cause of overpressure is ﬂuid expansion, then the pore pressure will increase at a faster rate than overburden stress

Vp data on rebound behavior for Cotton Valley shale, showing virgin curve and unloading cases. Vp changes are much more pronounced for the loading case compared to the unloading case. This change in the nature of the velocity path is also termed velocity hysteresis. Note the virgin curve is an estimated curve for normal compaction behavior of shales.(adapted from Bowers (2002))

Schematic showing a case when overpressure is caused by under-compaction (a Gulf of Mexico case). The directions of arrows on the red curves indicate the direction of increasing conﬁning pressure (Sarker, 2010)

Schematic showing a case when cause of overpressure is ﬂuid expansion. Loading curve overestimates eﬀective stress and underestimates pore pressure. The directions of arrows on the red curves indicate the direction of increasing conﬁning pressure (Sarker, 2010)

**Applicability of**** ****n**

Eatons algorithm (Eaton, 1975) is one of the most popular methods for pore-pressure prediction. Ebrom et al. (2003) states that Eatons method may not be the most accurate method for complex geological settings, but it is often used as a standard against which all other pressure prediction models are compared. Although the algorithm claims to be an eﬀective stress approach to computing pore pressure from velocity, it does not actually use eﬀective stress in its true sense. The assumption in Eatons equation is that *n** **= 1* and *P**eff** **=** **P**c** **-** **P**p* . This approach might be true for some Gulf of Mexico wells (Hickers, 1997) for which the equation had actually been derived, but may not hold true for wells from locations where the cause of overpressure generation is not compaction disequilibrium. Eatons exponent *E* is a measure of the sensitivity of the sonic velocities to eﬀective stress (Ebrom et al., 2003). A larger value of *E* indicates insensitivity of the velocities to changes in eﬀective stress. A single value of *E* is often insuﬃcient to predict pore pressures for the entire section, and this number needs to be varied with depth depending on the degree of overpressure in the subsurface. However, if a depth-varying *n* is used instead in the equation, it serves the same purpose and has more physical signiﬁcance. In this way, Eaton’s equation can be extended to areas where there is signiﬁcant overpressure due to other causes.

Sensitivity of E to n. The colored curves indicate the sensitivity of *E* to change in *n*. Note that velocities are most sensitive to pore pressure at*n*=0.7, i.e., when *E* = 1. Study case from North Sea wells (Sarker, 2010)

Another way to see the discrepancies in pressure is having a comparison of the actual mud weight in the well with the pressures predicted using Eatons equation both with the diﬀerential pressure approach and the eﬀective stress approach. In North Sea study chase, An rms error calculated for both cases shows that the eﬀective stress approach reduces error in pressure prediction by 15%

Comparison of surface mud weights with predicted pressure using Eatons equation with *n*=1, *E*=3 and *E*=1, *n*=0.7. The eﬀective stress approach reduces rms error by at least 15% . Study case from North Sea wells (Sarker, 2010)

From the analysis, we understand that the eﬀective stress coeﬃcient is a very important parameter in pore-pressure prediction, and if the coeﬃcient is known, Eatons equation can be used successfully in areas with complex geological settings. More direct laboratory measurements are required for shales, which compose almost 75% of the subsurface sedimentary rocks.

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