Log analysis in shaly formations

Shales are one of the more important common constituents of rocks in log analysis. Aside from their effects on porosity and permeability, this importance stems from their electrical properties, which have a great influence on the determination of fluid saturations.

Differences in shale formations

Shales are loose, plastic, fine-grained mixtures of clay-sized particles or colloidal-sized particles and often contain a high proportion of clay minerals. Most clay minerals are structured in sheets of alumina-octahedron and silica-tetrahedron lattices. There is usually an excess of negative electrical charges within the clay sheets. The substitution of Al⁺⁺⁺ by ions of lower valence is the most common cause of this excess; the structure of the crystal remains the same. This local electrical imbalance must be compensated to maintain the electrical neutrality of the clay particle. The compensating agents are positive ions which cling to the surface of the clay sheets in a hypothetical dry state:

• Cations
• Counterions

The positive surface charge is usually measured in terms of milli-ions equivalents per 100 grams of dry clay minerals and is called the cation exchange capacity (CEC). When the clay particles are immersed in water, the Coulomb forces holding the positive surface ions are reduced by the dielectric properties of water. The counterions leave the clay surface and move relatively freely in a layer of water close to the surface (the electrical balance must be maintained so that the counterions remain close to the clay water interface) and contribute to the conductivity of the rock.

The Archie water saturation equation, which relates rock resistivity to water saturation, assumes that the formation water is the only electrically conductive material in the formation. The presence of another conductive material (i.e., shale) requires either that the Archie equation be modified to accommodate the existence of another conductive material, or that a new model be developed to relate rock resistivity to water saturation in shaly formations. The presence of clay also complicates the definition or concept of rock porosity. The layer of closely bound surface water on the clay particle can represent a very significant amount of porosity. However, this porosity is not available as a potential reservoir for hydrocarbons. Thus, a shale or shaly formation may exhibit a high total porosity, yet a low effective porosity as a potential hydrocarbon reservoir.

The way shaliness affects a log reading depends on the amount of shale and its physical properties. It may also depend on the way the shale is distributed in the formation. Shaly material can be distributed in the formation in three ways:

• Shale can exist in the form of laminae between which are layers of sand. The laminar shale does not affect the porosity or permeability of the sand streaks themselves. However, when the amount of laminar shale is increased and the amount of porous medium is correspondingly decreased, overall average effective porosity is reduced in proportion.
• Shale can exist as grains or nodules in the formation matrix. This matrix shale is termed structural shale; it is usually considered to have properties similar to those of laminar shale and nearby massive shales.
• The shaly material can be dispersed throughout the sand, partially filling the intergranular interstices. The dispersed shale may be in accumulations adhering to or coating the sand grains, or it may partially fill the smaller pore channels. Dispersed shale in the pores markedly reduces the permeability of the formation.

All these forms of shale can, of course, occur simultaneously in the same formation.

Relating resistivity and saturation in shales

Over the years, a large number of models relating resistivity and fluid saturations have been proposed. Many have been developed assuming the shale exists in one of the three specific geometric forms. All these models are composed of a clean sand term, described by the Archie water saturation equation, plus a shale term. The shale term may be fairly simple or quite complex; the shale term may be relatively independent of, or it may interact with, the clean sand term. All the models reduce to the Archie water saturation equation when the fraction of shale is zero; for relatively small amounts of shaliness, most models and methods yield quite similar results.

Only a very few of these models will be reviewed here to provide some flavor and understanding for the evolution of shaly-sand interpretation logic.

Laminated sand/shale simplified model

In this laminar shale model, R_t, the resistivity in the direction of the bedding planes, is related to R_sh (the resistivity of the shale laminae) and to R_sd (the resistivity of the clean sand laminae) by a parallel resistivity relationship,

....................(1)

where V_lam is the bulk-volume fraction of the shale, distributed in laminae, each of more-or-less uniform thickness.

For clean-sand laminae, , where F_sd is the formation resistivity factor of the clean sand. Because (where ϕ_sd is the sand-streak porosity) and f = (1– V_lam )ϕ_sd (where ϕ is the bulk-formation porosity), then

....................(2)

To evaluate S_w by the laminated model, R_t, R_w, ϕ, V_lam, and R_sh must be determined.

For the determination of R_t, the problem is the same as for clean formations. If R_w is not known, its determination usually involves looking at a nearby clean sand and solving for R_w using the SP measurement. If the formation is water-bearing, the resistivity and porosity measurements can be used.

For the determination of ϕ and V_lam, a combination of porosity logs can be used. For example, as illustrated in Fig. 1, a crossplot of ϕ_N and ϕ_B from a density log is effective. The triangle of the figure is defined by the matrix point, water point, and shale point. In this example, the matrix point is at ϕ_N = 0 (the neutron log was scaled in apparent sandstone porosity) and ϕ_ma = 2.65 g/cm³ (quartz matrix). The shale point is at ϕ_N = 50 p.u. and ϕ_sh = 2.45 g/cm³. These values were taken in a nearby thick shale bed; it is assumed that shale laminae in the shaly sand under investigation are similar to the nearby massive shale beds. The water point is, of course, located at ϕ_N = 100 p.u. and ϕ_B = 1 g/cm³. The matrix-water line and shale-water line are linearly divided into porosity; the matrix-shale line and water-shale line are linearly divided into shale percentages.

• 

  Fig. 1 – Neutron-density crossplot showing matrix, water, and shake points, scaled for determination of V_shale and porosity.

Point A, plotted as an example, corresponds to log readings of ϕ_B = 2.2 g/cm³ and ϕ_N = 33 p.u. Interpretation by the lines on the plot yields 23% and V_sh (or V_lam) = 16 %.

Direct use of this crossplot assumes 100% water saturation in the zone investigated by the tools. Because oil has a density and hydrogen content normally not greatly different from water, this crossplot technique can be used with acceptable accuracy in oil-bearing formations. The presence of gas or light hydrocarbon, however, decreases ϕ_N and decreases ϕ_B. This would cause the point to shift in a northwesterly direction. When gas or light hydrocarbons are present, an additional shaliness indicator, such as GR or SP data, is needed to evaluate the amount of the shift.

Using the laminated model, an equation for R_xo analogous to Eq. 2 could be written. S_xo would replace S_w, and R_mf would replace R_w. The other terms (ϕ, V_lam, and R_sh) remain the same in the two equations. Assuming S xo = S w 1/5 (as in the flushed-zone ratio method) and the ratio of the PSP (SP deflection in the shaly sand) to the SSP (SP deflection in a nearby clean sand of similar formation water) is a measure of shaliness, V_lam, water saturation could be calculated from R_xo/R_t and PSP in the shaly sand and SSP (or R_mf/R_w) in a nearby clean sand.

Dispersed clay

In this model, the formation conducts electrical current through a network composed of the pore water and dispersed clay. As suggested by de Witte, it seems acceptable to consider that the water and the dispersed shale conduct an electrical current like a mixture of electrolytes. Development of this assumption yields

....................(3)

where ϕ_im = intermatrix porosity, which includes all the space occupied by fluids and dispersed shale; S_im = the fraction of the intermatrix porosity occupied by the formation-water, dispersed-shale mixture; q = the fraction of the intermatrix porosity occupied by the dispersed shale; and R_shd = the resistivity of the dispersed shale. Also, it can be shown that S_w = (S_im − q)/(1 − q), where S_w is the water saturation in the fraction of true effective formation porosity.

Combining these relations and solving for S_w yields

....................(4)

Usually, ϕ_im can be obtained directly from a sonic log because dispersed clay in the rock pores is seen as water by the sonic measurement. The value of q can be obtained from a comparison of a sonic and density log. Indeed, if ρ_shd ≅ ρ_ma, then qsv ϕ (ϕ_SV − ϕ_D)/ ϕ_SV, where ϕ_SV and ϕ_D are the sonic and density derived porosities, respectively. In this case, ϕ_D approximates ϕ, the effective porosity available for fluid saturation.

The value of R_sh is more difficult to evaluate. It is usually taken as equal to R_sh in nearby shale beds. Fortunately, its value is not too critical if it is at least several times greater than R_w. In fact, when R_w is small compared to R_sh and the sand is not too shaly, Eq. 4 can be simplified to a form independent of R_sh:

....................(5)

Total shale relationship

Based upon the previously described ideas, laboratory investigations, and field experience, it has been found that a simple relationship of the following form works well for many shaly formations independent of the distribution of the shale and over the range of S_w values encountered in practice:

....................(6)

In using this equation, R_sh is taken equal to the resistivity of the adjacent shale beds, and V_sh is the shale fraction as determined from a total shale indicator.

Before the Waxman-Smits formulation, equations of the form of Eq. 3 and 6 gained wide acceptance in the evaluation of shaly sands. These equations have a general form of

....................(7)

where α denotes a predominant sand term that is dependent on the amount of sand, its porosity, and the resistivity of the saturating water. The sand term always reduces to Archie’s water saturation equation when the shale fraction is zero. γ denotes a predominant shale term that depends on the amount and resistivity of the shale.

Dual water model

In 1968, Waxman and Smits proposed, based on extensive laboratory work and theoretical study, a saturation-resistivity relationship for shaly formations that related the resistivity contribution of the shale (to the overall resistivity of the formation) to the CEC of the shale. The Waxman-Smits relationship is

....................(8)

where F * is the formation factor of the interconnected porosity, S_w also relates to the interconnected pores, B is the equivalent conductance of the sodium clay-exchange cations as a function of the formation water conductivity, and Q_v is the CEC of the rock per unit pore volume. Unfortunately, a continuous in-situ measurement of rock CEC was not available when this study was presented. As a result, the dual water model was developed as a practical solution. The dual water method is based on three premises:

• The conductivity of clay is because of its CEC.
• The CEC of pure clays is proportional to the specific surface area of the clay.
• In saline solutions, the anions are excluded from a layer of water around the surface of the grain. The thickness of this layer expands as the salinity of the solution (below a certain limit) decreases, and the thickness is a function of salinity and temperature.

Therefore, because CEC is proportional to specific area (area per unit weight) and to the volume of water in the counter-ion exclusion layer per unit weight of clay. Consequently, the conductivity of clay is proportional to the volume of the counter-ion exclusion layer, this layer being "bound" to the surface of the clay grains. For clays, this very thin sheet of bound water is important because of the large surface areas of clays relative to sand grains (several magnitudes greater). Therefore, in the dual water model, a clay is modeled as consisting of two components: bound water and clay minerals.

The clay minerals are modeled as being electrically inert; the clay electrical conductivity is modeled as being derived from the conductivity of the bound water, C_wb. C_wb is assumed to be independent of clay type (from the second postulate described previously). The amount of bound water varies according to clay type, being higher for the finer clays (with higher surface areas), such as montmorillonite, and lower for coarser clays, such as kaolinite. Salinity also has an effect; in low-salinity waters (roughly < 20,000 ppm NaCl), the diffuse layer expands.

The bound water is immovable under normal conditions; therefore, the volume it occupies cannot be displaced by hydrocarbons. Because the clay minerals (dry colloids) are considered electrically inert, they may be treated just as other minerals. Schematically, shaly formations are modeled with the dual water model, as illustrated in Table 1.

• 

  Table 1

For most rocks (except for conductive minerals such as pyrite, which cannot be treated in this way) only the porous part needs to be considered when discussing electrical properties, and it is treated according to the Archie water-saturation equation. The equation becomes

....................(9)

where a, m, and n have the usual Archie connotations. σ t is the conductivity of the noninvaded, virgin formation (1/R_t), and σ_we is the equivalent conductivity of the waters in the pore space.

Note that ϕ_t and S_wt refer to total pore volume; this includes the pore volumes saturated with the bound water and the formation connate water (sometimes called the "free" water). The equivalent water conductivity, σ_we, is

....................(10)

where V_w and V_wb are the bulk volumes of formation water and bound water, respectively, and σ_w and σ_wb are their conductivities.

In terms of saturation, Eq. 10 becomes

....................(11)

or

....................(12)

or ....................(13)

where S_wb is the bound water saturation (i.e., the fraction of the total pore volume occupied by the bound water).

Eq. 13 describes the equivalent-water conductivity as a function of the formation water conductivity plus the bound-water conductivity. The saturation equation (Eq. 9) becomes

....................(14)

The porosity and water saturation of the sand (clean formation) phase (that is, the nonclay phase) of the formation is obtained by subtracting the bulk-volume fraction of bound water (ϕ_t S_wb). Therefore, the effective porosity is

....................(15)

and the water saturation is

....................(16)

To evaluate a shaly formation using the dual water model, four parameters must be determined. They are σ_w (or R_w), σ_wb (or R_wb), ϕ_t, and S_wb. A neutron-density crossplot provides a good value of ϕ_t. S_wb is obtainable from a variety of shale-sensitive measurements (SP, GR, ϕ_N, R_t, ϕ_N – ρ_B, t – ρ_B, etc.). R_wb and R_w are usually determined by the log analyst and entered as input parameters.

Nomenclature

Rt	=	resistivity of the uninvaded formation
Rw	=	resistivity of the formation connate water (ohm•m)
Sw	=	water saturation in the uninvaded zone
Vsh	=	fraction of the total formation volume that is shale
F	=	formation factor relating resistivity to porosity
σ	=	conductivity, mS/m
ϕ	=	porosity
ϕim	=	intermatrix porosity
Sim	=	the fraction of the intermatrix porosity occupied by the formation-water
Sw	=	(Sim − q)/(1 − q)
Rshd	=	the resistivity of the dispersed shale
q	=	the fraction of the intermatrix porosity occupied by the dispersed shale
Swb	=	bound water saturation
Vw	=	the bulk volume of formation water
Vwb	=	the bulk volume of bound water
σt	=	the conductivity of the noninvaded, virgin formation
σwe	=	the equivalent conductivity of the waters in the pore space

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See also

Water saturation determination

Well log interpretation

Resistivity and spontaneous (SP) logging

PEH:Resistivity_and_SP_Logging