## Abstract

Dimensional analysis has been used as a tool for developing a non-linear empirical model giving the flow zone indicator (FZI) as a function of open-hole log measurements. Dimensional analysis confirms that interval transit time (Δ*t*), true resistivity (*R*_{t}), bulk density (*ρ*_{b}), the apparent water resistivity (*R*_{wa}) and the photo-electric absorption (*P*_{e}) are essential well-log measurements for estimating the FZI in sandstone formations.

A unique power-law relationship has emerged between a dimensionless FZI group (*λ*) and a dimensionless resistivity group for distinct hydraulic flow units. Petrotyping, using either the discrete rock type (DRT) approach or the global hydraulic elements (GHE) approach, appears to provide a credible framework for comparative hydraulic flow unit description. Dimensionless groups permitted the analysis of the effect of several operational variables. The FZI increases with increasing bulk density, and with increasing interval transit time. On the other hand, the FZI decreases with increasing true resistivity (*R*_{t}). The relationship between the FZI and the photo-electric absorption is more intricate, though, since it depends on the value of the power-law exponent.

Conventional log data from an oil well (well B), penetrating two distinct sandstone oil reservoirs in an onshore sandstone oilfield in the Middle East, were used to validate the dimensionless groups. For the field case presented, the FZI empirical model prediction capability does not appear to be diminished by the existence of a capillary transition zone. The FZI-based model, used for estimating permeability, represents an improvement in the prediction of permeability provided that reliable estimates of the FZI are available. Dimensional analysis proved to be a powerful modelling tool capable of revealing fine relationships that are invisible to exhaustive data mining techniques.

Evaluation of a net pay production schedule usually requires estimates of rock permeability. Precise prediction of reservoir rock permeability from open-hole wireline logs in uncored intervals is of fundamental importance. There is no single routine open-hole wireline log instrument that measures permeability directly. As a consequence, the current state of knowledge relies upon empirical relationships that use either porosity or resistivity data. Mohaghegh *et al.* (1997) provided a summary of empirical models based on the correlation between permeability, porosity, resistivity and the irreducible water saturation. A limited number of researchers have published empirical models that estimate permeability using open-hole log measurements (Saner *et al.* 1997; Xue *et al.* 1997; Yao & Holditch 1993). The log measurements used for permeability estimation included gamma ray (GR), induction deep resistivity, sonic and bulk density. These models show, in general, good correlations when applied, locally, to clean sandstone rocks with interparticle porosity. The applicability of these models to highly heterogeneous shaly sandstones, however, does not usually give plausible results (Perez *et al.* 2005). Some of these empirical models are often diffuse when used at water saturations that are not irreducible (Coates & Dumanoir 1973; Mohaghegh *et al.* 1997).

Nuclear magnetic resonance (NMR) log permeability models are usually constructed based on the relationship between the transversal relaxation time (*T*_{2}) distribution and the pore-size distribution (Di & Jensen 2016). For well-sorted clean clastic rocks, this relationship is fairly consistent, and the prediction of permeability usually gives reasonable results (Kenyon *et al.* 1988). But in heterogeneous sandstones and in tight rocks, the prediction capability of permeability models using the *T*_{2} distribution poses a great challenge, and is sensitive to different rock lithofacies (Davis *et al.* 2006; Di & Jensen 2016).

The general trend of increasing permeability with increasing porosity is violated, especially for fractured reservoirs and for low-porosity formations with poor sorting, or with authigenic clay and mineral growth (Beard & Weyl 1973; Ehrenberg *et al.* 2008). Porosity depends on the size and number of pore bodies. On the other hand, permeability depends on the coordination number, which is the number of pore throats that intersect a pore body (Sharma *et al.* 1991) or the number of contacts per grain. In essence, the coordination number distribution has a significant impact on the pore network topology and the rock flow properties. Permeability depends also on the aspect ratio, which is the ratio of pore body diameter to the pore throat diameter (Garrouch 1999). The pore network structure of hydrocarbon reservoirs is the result of depositional and diagenetic processes. The depositional and diagenetic processes are the prominent geological factors controlling rock type, and hence the reservoir quality (Saberi *et al.* 2009; Hollis 2011; Correia & Schiozer 2016). The depositional process includes the mineral composition and the rock texture. Parameters influencing the rock texture consist of the grain size, sorting, grain shape, grain packing and grain roundness (Brayshaw *et al.* 1996). On the other hand, the diagenetic processes include compaction cycles, cementation, leaching and fracturing (Glennie *et al.* 1978; Ehrenberg & Nadeau 2005; Fitch *et al.* 2013). Reservoir permeability is affected by all depositional and diagenetic processes (Brayshaw *et al.* 1996). Grain size and sorting have a great impact on permeability of porous rocks; porosity, on the other hand, is not affected by the grain size (Thomas & Steiber 1975; Brayshaw *et al.* 1996). Thus, one should not expect a general relationship between porosity alone and permeability. Coates & Dumanoir (1973) reported an empirical model for estimating permeability using porosity and resistivity data, as follows:
(1)
(2)
(3)

In the above equations, *k* is permeability (in mD), *ϕ* is porosity (fraction), *R*_{tirr} is deep resistivity from a zone at *S*_{wirr} (in ohm m), *R*_{w} is formation water resistivity at the formation temperature (in ohm m) and *ρ*_{h} is the hydrocarbon density (g cm^{−3}).

It is clear from equations (1)–(3) that the Coates & Dumanoir (1973) model is limited to reservoir formations at irreducible water saturations.

Rock typing, a methodology for classification of rocks based on sensible geological parameters and the physics of flow at the pore scale, has improved the prediction of permeability data (Amaefule *et al.* 1993; Salazar *et al*. 2005; Al-Tooqi *et al.* 2014; Griffiths 2015; Eltom *et al.* 2017). A rock type, or a hydraulic flow unit, is defined as a distinct interval of rocks with constrained petrophysical properties leading to similar fluid flow attributes. The concept of using rock types as an aid in petrophysical evaluations has been used by a number of researchers (Myers 1991; Davies *et al.* 1999; Tricoranto 2002; Perez *et al.* 2005; Garrouch *et al.* 2009; Knackstedt *et al.* 2010; Al-Tooqi *et al.* 2014; Griffiths 2015; Gupta *et al.* 2017). A number of researchers have used the hydraulic flow zone indicator to group data into rock types (Amaefule *et al.* 1993; Guo *et al.* 2007; Corbett & Mousa 2010; Xu & Torres-Verdin 2012). A hydraulic flow zone indicator is defined as the ratio of rock quality index (RQI) to the normalized porosity index (*ϕ*_{z}) as follows:
(4)RQI is given by the following equation (Amaefule *et al.* 1993):
(5)In the above notation, permeability is in millidarcies (mD) and RQI is in micrometres (µm). The normalized porosity index (*ϕ*_{z}) is given as a function of the effective porosity (*ϕ*_{e}) as follows:
(6)It has been confirmed that rock samples lying on the same straight line, with a unit slope, on a log–log plot of RQI v. *ϕ*_{z} belong to the same hydraulic flow unit (Amaefule *et al.* 1993; Gunter *et al.* 1997; Guo *et al.* 2007; Fitch *et al.* 2013). A particular hydraulic flow unit has a flow zone indicator (FZI) distribution, characterized by an average FZI value equal to the intercept of RQI v. *ϕ*_{z} log–log plot at a *ϕ*_{z} value equal to unity (Amaefule *et al.* 1993). That unique FZI intercept is a key characteristic of the hydraulic flow unit. The FZI distribution for a particular rock type stems from two factors (Abbaszadeh *et al.* 1996):

measurement errors associated with core analysis;

depositional and diagenetic controls causing minor textural and geometrical pore structure variations within the same hydraulic flow unit.

Rock types are segregated by calculating the discrete rock type value (DRT) as follows:
(7)In equation (7), the *Round* function calculates the nearest integer of the argument between brackets. Core samples of similar DRT values are believed to belong to the same rock type (Gunter *et al.* 1997; Guo *et al.* 2007). Recent approaches for estimating permeability, based on rock typing, estimate the FZI from either simple linear regression of log responses (Guo *et al.* 2007) or by using probabilistic and neural network models (Abbaszadeh *et al.* 1996; Soto *et al.* 2001; Tricoranto 2002). Permeability is then estimated from the FZI and effective porosity data using Amaefule *et al.* (1993) empirical model, given as follows:
(8)The above model represents an improvement in the quantitative prediction of permeability in uncored intervals, provided that precise estimates of the FZI and effective porosity values are accessible. Empirical inferences for estimating the FZI (Guo *et al.* 2007) are often diffuse and are plagued by large scatter, especially in highly heterogeneous reservoirs. These models generate considerable spread and are fraught by great uncertainty when applied to other locations with different depositional characteristics to the original location pertinent to the model data (Tricoranto 2002). This is possibly due to the lack of scientific foundation for selecting appropriate log measurements when coming up with FZI empirical and neural network models. A haphazard procedure is usually followed for selecting the input log measurements instead (Guo *et al.* 2007).

The objective of this study is to use dimensional analysis in order to explore the link between the FZI and key open-hole wireline log measurements. The following section illustrates how dimensional analysis was used to formulate empirical models for predicting the flow zone indicator and rock permeability.

## Dimensional analysis

Dimensional analysis is a robust tool applied in the general areas of physics and engineering for analysing relationships between dependent and independent variables (Munson *et al.* 2010). Dimensional analysis also helps in finding similitudes and scale factors in model studies by formulating non-dimensional groups. Derivation of non-dimensional groups is founded on the correct identification of the relevant independent variables. Empirical models may be obtained between the non-dimensional groups by plotting one group v. another using relevant experimental data (Garrouch 2018). When the dimensionless groups include dimensionless variables, additional effort is required in defining the relationship between the groups by arbitrarily adjusting the exponents of the dimensionless variables (Zendehboudi *et al.* 2011). Dimensional analysis is effective in finding intricate relationships that mimic the physical phenomena being studied, without going through an exhaustive trial-and-error process inherent with non-linear regression analysis. In this section, dimensional analysis is performed to seek a functional relationship that captures petrophysical controls of the FZI. It is postulated here that the FZI may be a function of the wireline log responses, as follows:
(9)In the above notation, Δ*t* is the compressional sonic wave travel time (in µs/ft), *ρ*_{b} is the bulk density (in g cm^{–3}), *P*_{e} is the photoelectric absorption (in barns/electron), *R*_{t} is the true resistivity (ohm m) and *R*_{wa} is the apparent water resistivity (ohm m).

The apparent water resistivity, *R*_{wa}, is given by the following equation (Bassiouni 1994):
(10)

In the above notation, *ϕ* is the effective porosity and m is the cementation exponent. The apparent water resistivity (*R*_{wa}) is equal to the water resistivity (*R*_{w}) only in a water zone. Porosity, or the resistivity formation factor (*F*), might have been used in equation (9) instead of *R*_{wa}. Nevertheless, the dimensional analysis becomes erratic unit-wise. As a rule of thumb, validation of the dimensionless groups, using the relevant experimental data, requires less effort when all independent variables of the generic equation (9) have dimensions (Zendehboudi *et al.* 2011).

Dimensional analysis utility derives from its ability to study similar systems for which the defining equations and boundary conditions are not completely articulated. An *MLTQ* dimensional analysis for the dependence of the FZI on open-hole well-log measurements is performed, in an attempt to identify dimensionless groups that are crucial for estimating the FZI. *MLTQ* stands for the fundamental dimensions of mass (*M*), length (*L*), time (*T*) and charge (*Q*). Table 1 lists all variables used in the dimensional analysis, their units and their fundamental dimensions. With the four fundamental dimensions (*MLTQ*) used in the analysis, and the six total variables expressed in equation (9), the Buckingham's pi theorem (Munson *et al.* 2010) depicts two dimensionless groups (*λ* and ). When applied to any physical system, the Buckingham's pi theorem states that a dimensionally homogeneous equation, involving *n* variables, can be reduced to an equation with (*n*−*k*) dimensionless groups, with *k* being the number of independent reference, or fundamental, dimensions (Buckingham 1914).

Taking the FZI and *R*_{t} as the non-repeating variables, and the parameters Δ*t*, *ρ*_{b}, *P*_{e} and *R*_{wa} as the repeating variables, allows the following two dimensionless groups to be formulated adequately:
(11)
(12)The selection of the repeating and the non-repeating variables is constrained by the following three rules (Munson *et al.* 2010):

The dependent variable (the FZI) must appear as a non-repeating variable in only one of the dimensionless groups.

The repeating variables have to be dimensionally independent of each other. In addition, the dimensions of any of the repeating variable must not be reproduced by some combination of products of powers of the remaining repeating variables dimensions.

All of the fundamental dimensions used in the analysis must be represented within a particular set of the repeating variables.

Rearranging the variables of equations (11) and (12) in their *MLTQ* fundamental dimensions, knowing that *λ* and are dimensionless, and solving for the exponents *α, β, γ, θ, ε, χ, τ* and *δ*, the following two dimensionless groups are obtained:
(13)
(14)The dimensionless group *λ* is denoted as the FZI number, whereas is denoted as the resistivity number. The contraction of the physical relationship between the flow zone indicator and the log-derived variables into a more succinct functional form given by equations (13) and (14) illustrates the main utility of dimensional analysis. For carefully defined variables contributing to the FZI variance (equation 9), dimensional analysis is likely to lead to a general relationship between the dimensional groups that may be applied for similar systems. This particular inference can only be confirmed with log data.

Log and core data are obtained for well B from an outstanding oilfield, in the Arabian Peninsula, penetrating two distinct terrigenous sandstone oil reservoirs (R_{1} and R_{2}) with excellent quality and abundant oil production (Al-Sultan 2017). The log measurements consist of GR, caliper, neutron porosity, bulk density, compressional wave travel time (sonic log), laterolog deep resistivity and photoelectric absorption (Figs 1 and 2). All logs were depth shifted, and environmentally corrected, and corrected for washout effects (Al-Sultan 2017). Lithology in both reservoirs is composed mainly of sand and shale layer sequences, with a minor presence of siderite nodules, glauconite, and fragments of marine fossils and coal beds. Both reservoirs lie in an anticlinal structure composed of relatively complex geological features such as lateral lithology changes, and the existence of normal faults in some sections near the crest. The anticline structure has an elongated nose, with the crest of the structure appearing to be broad and relatively undeformed (Al-Sultan 2017). Sandstone layers of reservoirs R_{1} and R_{2} appear to belong to a sedimentary depositional environment that may be generally interpreted to be fluvial channels on a delta plane close to a river mouth (Al-Sultan 2017). Reservoir units consist of sandstone formations with different grain sizes: fine to medium grained, coarse grained and even conglomerates. Descriptive statistics of the data used to validate the dimensional analysis groups are given in Table 2. Core data of helium porosity and air permeability have been obtained for 335 depth intervals from the same well. As shown in Figures 1 and 2, core porosity measurements match reasonably well with the porosity values obtained from neutron and density logs (PHIE ND curves in Figs 1 and 2). More details about the data description and analysis are given in Appendix A.

A plot of the FZI number (*λ*) v. the resistivity number , in a log–log paper using well B data, yields a straight-line trend which suggests a power-law relationship between the two dimensionless groups (Fig. 3). As shown in Figure 3, *λ* decreases as increases. An increase in flow resistance causes a decrease in permeability, and a decrease in the FZI, as a consequence. The power-law trend is fraught by some scatter, though, with a correlation coefficient of *c.* 0.88. The premise of this dimensional analysis is that the dimensionless groups obtained (equations 13 and 14) appear to account for sufficient wireline log responses that may reflect the FZI of siliclastic rocks.

The power-law relation between *λ* and is scrutinized further for specific rock types. As shown in Table 3, eight discrete rock types are identified using equation (7). Three rock samples with DRT = 18 and two rock samples with DRT = 8 have been eliminated since no meaningful statistical analysis can be based on a very small number of observations. Figure 4 illustrates the excellent fit between *λ* and , when the data are partitioned into unique rock types. Table 4 illustrates the intercepts and exponents of the power-law relationships, between *λ* and for various DRT values, with corresponding coefficients of determination. The fact that the data for 330 logged and cored depths (of well B) appear to yield unique power-law relationships, with an excellent fit between *λ* and (Table 4), validates the dimensional analysis. The general power-law relationships between the dimensionless groups *λ* and may be expressed as follows:
(15)where *η* and consist of the power-law intercept and exponent, respectively. The premise of the general power-law relationship, given by equation (15), is that it remains valid for sandstone reservoirs of other oil and gas fields. Substituting for the expressions of *λ* and , given by equations (13) and (14) in equation (15), an empirical model is obtained for rock permeability as a function of routine log measurements:
(16)

Equation (16) is analogous to Amaefule *et al.* (1993) model for permeability, given by equation (8). This analogy leads to a functional relationship between the FZI and conventional log responses as follows:
(17)It is clear from equation (17) that the relationship between the FZI and the wireline log measurements is highly non-linear, contrary to what has been postulated by the Guo *et al.* (2007) model which expresses the FZI in terms of normalized log values as follows:
(18)In the above notation, NXRD is the normalized resistivity, NXRHO is the normalized bulk density, NXGR is the normalized gamma-ray measurement, NXSP is the normalized spontaneous potential, NXDT is the normalized compression travel time from the sonic log and NXNPH is the normalized apparent neutron log porosity.

The above normalized values are obtained as follows (Guo *et al.* 2007):
(19)where is any of the selected log's values at a particular depth, is the minimum observed value of the log and is the maximum observed value of the log.

Figure 5 illustrates a comparison between the FZI values calculated using the Guo *et al.* (2007) and measured FZI values, for reservoirs R_{1} and R_{2} of well B. The FZI estimates appear to be fraught with considerable uncertainty. For relatively low FZI values (less than 2 µm), the Guo *et al.* (2007) model appears to overestimate the predicted FZI. On the other hand, for high FZI values (greater than 2 µm), the Guo *et al.* (2007) model appears to underestimate the predicted FZI. Indeed, a linear regression analysis for estimating the FZI from wireline log measurements appears to be inadequate. Figure 6 illustrates a comparison between FZI values calculated using the FZI empirical model introduced in this study (equation 17) and measured FZI values, also using the same dataset applied in generating Figure 5. A satisfactory agreement is obtained between predicted and measured FZI values.

Equation (17) infers the link between rock FZI and conventional well-log measurements. In retrospect, texture stands as a prominent reservoir quality control affecting the FZI (Correia & Schiozer 2016). Among all the open-hole routine logs, resistivity and sonic logs are the most texture-sensitive. Rock texture is influenced by parameters like the grain size, sorting, grain shape, grain packing and grain roundness. The sonic log is a matrix response log. The sonic travel time is, therefore, greatly affected by the rock compaction. The density log is affected mainly by the pore-filled fluids and by the grain density. The density log represents the dependence of the FZI on porosity, and on rock grain density. On the other hand, the photoelectric absorption (*P*_{e}) response is mainly affected by rock mineralogy and texture. *P*_{e} in equation (17) represents the dependence of the FZI on lithology. The GR is implicitly accounted for in the independent variables used to estimate the FZI. GR log measurements are used to estimate the shale volume fraction (*V*_{sh}), which is implicitly accounted for in the sonic travel time and in the bulk density response. As a matter of fact, the sonic travel time may be expanded in terms of individual contributions of the fluid in the pore space, the clean matrix and the shale content as follows (Bassiouni 1994):
(20)Likewise, the bulk density may be expanded in terms of individual contributions of the fluid in the pore space, the clean matrix and the shale content as follows (Bassiouni 1994):
(21)Therefore, including GR response, as an independent variable, for estimating the FZI becomes redundant. Neutron log porosity is also used to calculate the total porosity (see equation A4). The total porosity is used in return to calculate *R*_{wa} (equation 10). Thus, the neutron log porosity is implicitly accounted for in estimating the FZI.

The dependence of the FZI on rock texture, fluid-filled porosity and on lithology, and ultimately on rock type, is, therefore, satisfied by the explicit use of resistivity, sonic log, bulk density and the photoelectric absorption measurements. Equation (17) depicts an increase in the FZI as *ρ*_{b} increases. FZI also increases with increasing Δ*t*. However, the rate of change of the FZI with *ρ*_{b} is higher than that with Δ*t*. For the range of values observed (Table 4), the FZI appears to decrease with increasing *R*_{t}. The behaviour of the FZI with respect to *P*_{e}, on the other hand, depends on the value of . For , the FZI increases with increasing *P*_{e}. For , the FZI decreases as *P*_{e} increases.

The empirical model of permeability as a function of routine log measurements, given by equation (16), has been used to estimate permeability for the various rock types of reservoirs R_{1} and R_{2} of well B. As shown in Figure 7, the estimated permeability values using this approach were in satisfactory agreement with measured core data values. In general, estimated permeability values (Fig. 7), using the general empirical model given by equation (16), were much closer to the permeability values (Fig. 8) estimated using the Coates & Dumanoir (1973). Permeability estimates obtained from the Coates & Dumanoir (1973) appear to be plagued by a considerable spread and seem to be fraught with significant uncertainty.

The uncertainty associated with the Coates & Dumanoir (1973) permeability predictions for this field case might be caused by a violation of the irreducible water-saturation condition in reservoir R_{1}, and by a violation of the stationarity condition. An irreducible water-saturation condition, in the reservoir, would be confirmed by a constant bulk volume water (BVW) profile as a function of depth (Asquith & Krygowski 2004). It is the product of porosity by water saturation at any logged depth. Figure 9 indicates a variable BVW for all depth intervals above the water zone identified by the lowest resistivity (*R*_{t}) profile shown in Figure 1. Indeed, the resistivity profile of the middle permeable zone indicates an oil column above a water aquifer. The gradual increase in resistivity above the water aquifer indicates a capillary transition zone in reservoir R_{1}. This fact is also confirmed by the production data of this reservoir. As indicated by the resistivity (*R*_{t}) profile of Figure 2, none of the permeable zones, of reservoir R_{2}, containing hydrocarbon is in communication with a water aquifer. Figure 10 indicates that the BVW is relatively constant with depth for all the hydrocarbon zones of reservoir R_{2}. Therefore, reservoir R_{2} does not have any capillary transition zone. Indeed, all of the permeable zones of reservoir R_{2} are at an irreducible water-saturation condition. This is also confirmed by the production data of this reservoir. As shown in Figure 7, the satisfactory prediction capability of permeability using the dimensional analysis model (equation 16) does not appear to be affected by the presence of a capillary transition zone in reservoir R_{1}.

Perhaps, an adequate variable that may be used to plan a production schedule is the volumetric flow rate (*q*). An error analysis applied on the volumetric flow rate gives a preliminary assessment of the improvement in the production schedule due to the use of the introduced permeability model (equation 16). The volumetric flow rate for radial steady-state flow of an incompressible liquid in horizontal layers is given by Darcy's law (Tiab & Donaldson 2012) as follows:
(22)In the above notation, *H* is the net pay thickness, *µ* is the fluid viscosity, Δ*P* is the pressure drop, and *r*_{e} and *r*_{w} are the reservoir drainage radius and the well bore radius, respectively. All variables of equation (22) are expressed in SI units. Application of the Chain Rule (Kreyszig 2006) in equation (22) gives an approximation of the error in the volumetric flow rate (Δ*q*), as a function of the errors in the independent variables of equation (22) as follows:
(23)where is the error in permeability, is the error in the net pay thickness, is the error in the pressure drop, is the error in viscosity, is the error in the drainage radius and is the error in the wellbore radius.

The error in the volumetric flow rate caused by the uncertainty in permeability is obtained by cancelling all error terms in equation (23), except that of permeability. By taking the derivative of equation (22) with respect to permeability and substituting the resulting term in equation (23), an expression for the relative error in the volumetric flow rate caused by error in permeability is obtained as follows:
(24)The average relative error obtained for estimating permeability, by applying the dimensional analysis model (equation 16) on reservoir R_{1} and R_{2} data, is *c.* 36% with a standard deviation of about 32% (Fig. 7). This relative error implies that the predicted permeability values are within less than an order of magnitude variation of the measured permeability values. As indicated from equation (24), 36% error in permeability also leads to an average relative error in the production schedule of about 36%. On the other hand, estimates of permeability using the Coates & Dumanoir (1973) model indicate, on average, about a three orders of magnitude variation from measured values (Fig. 8). Likewise, this error in permeability leads to estimates of the production schedule within a three orders of magnitude variation from the actual production schedule. The application of the dimensional analysis model for estimating permeability (equation 16) appears to improve the evaluation of the net pay production schedule considerably for the field case investigated.

## Validation of rock type classification

As shown in Figure 11, a plot of rock quality index (RQI) v. normalized porosity index (*ϕ*_{z}) shows no clear delineation of rock types since for a finite number of straight lines a given data point could belong to more than one straight line (or rock type) because of its proximity to either lines defining the rock types. On the other hand, calculating the discrete rock type (DRT) using equation (7) leads to eight distinct rock types with DRT values ranging from 9 to 16 (Table 4). As shown in Figure 12, RQI v. *ϕ*_{z} plots for all rock types display non-intersecting trend lines with a unity slope; all of the trend lines seem to be reasonably correlated except for the DRT = 9 line. Representative FZI values for each DRT have been estimated using the following two approaches:

Extrapolating the log–log plot of RQI v.

*ϕ*_{z}for each DRT (Fig. 12) and then reading the value of RQI at*ϕ*_{z}= 1. The extrapolated RQI value corresponds to the rock type characteristic FZI value. This value is denoted as in Table 5.Calculating the FZI for each rock sample using equation (4), and then estimating an arithmetic average of the FZI values for a given DRT. This value is denoted as in Table 5.

As shown in Table 5, there is a reasonable match between and , except for DRT = 9. The small number of observations with DRT = 9 makes it hard to make a robust statistical inference. These results confirm the strength of the DRT for delineating unique hydraulic flow units. In general, a no match between and undermines the use of DRT for making a distinction between various rock types.

To confirm the robustness of the DRT calculations in the delineation of various rock types, a linear multivariate discriminant analysis was run using variables that are thought to influence rock classification. Seven variables were selected as being pertinent, either by themselves or in conjunction with other variables, namely FZI, Δ*t*, *R*_{t}/*R*_{w}, *F*, *ρ*_{b}, *P*_{e} and *ϕ*_{e}. The discriminant analysis is based on a *a priori* assumption that an individual rock sample belongs to one of the eight rock types: π_{1}, …, π_{8}. For each rock type characterized by a specific probability density function with a known prior probability (π_{c}) of rock type c, Bayes’ theorem (Lee & Datta-Gupta 1999) gives an estimate of the posterior distribution of the rock types for a particular observation **x** as follows:
(25)

In the above notation, *c* is the rock type, *p* is the posterior distribution, *x* is the data observation. The Bayesian maximum likelihood rule is used to evaluate the membership of a given observation **x** to the rock type c for which the log likelihood function of equation (25) is largest (Lee & Datta-Gupta 1999), corresponding to the minimum of *Q*_{c} function given by:
(26)When the rock types have a common covariance, , the differences in *Q*_{c} for the rock types are linear functions of **x**, creating linear decision boundaries between rock types. In this case, discriminant analysis computes a set of linear functions (*F** _{i}*, for ) that are used to specify the membership of a particular rock into one of the eight rock types, as follows:
(27)Table 6 displays the matrix of the classification coefficients

*a*,

_{i}*b*,

_{i}*c*,

_{i}*d*,

_{i}*e*,

_{i}*f*,

_{i}*g*and

_{i}*C*for the eight classification functions of equation (27). For a given set of variables of vector

_{i}**x**, the classification functions of equation (27) calculate eight scores. The rock type of vector

**x**is equal to the particular function rank (

*i*) for which

*F*gives the maximum score. As shown in Table 6, the coefficient

_{i}*e*is systematically equal to zero for all classification functions. This implies that the variable

_{i}*R*

_{t}/

*R*

_{w}appears to have no influence on segregating rock types. As a consequence, equation (27) may be simplified as follows: (28)

Figure 13 compares the rock types predicted from discrete rock type values (equation 7) with rock types predicted based on the classification functions scores (equation 28). The exact matching of rock types occurs for 288 samples out of 330 samples tested (Fig. 13), corresponding to a success rate of *c.* 87% (Table 7). The observations of rock type 1 falsely identified, are all identified as rock type 2. The falsely identified rock type 2 is almost equally identified as either rock type 1 or type 3. Similarly, nine out of 11 falsely identified rock type 3 are identified as rock type 2. Indeed, as indicated in Table 3, rock types 1, 2 and 3 are characterized by a low range of permeability and by a great overlap in the porosity range. This fact mitigates the effects of false identification of rock type on the estimation of permeability. Nevertheless, it seems that the DRT approach for rock type delineation is more robust for rock types in the high-permeability range than for rock types in the low-permeability range. The reasonable matching frequency between the discriminant functions rock type output and the rock types obtained using the DRT analysis asserts the sturdiness of the rock type identification using the DRT alone (equation 7). Indeed, the delineation of rock types using the DRT analysis yields reasonable correlations for permeability as a function of the effective porosity (Fig. 14). This fact confirms that the resulting rock types mimic the geological controls of rock texture and diagenetic processes affecting the structure of the hydraulic flow units.

The global hydraulic elements (GHE) approach for rock typing (Corbett & Potter 2004; Corbett & Mousa 2010) has been applied to reservoirs R_{1} and R_{2} data of well B, in order to study the effect of reducing the number of rock types on the dimensional analysis validation and on the prediction of permeability. The GHE approach is based on a series of 10 arbitrarily chosen values of the FZI, varying from 0.0938 to 48 µm (Corbett & Mousa 2010). The *i*-th ranked FZI value is computed as a function of the preceding FZI value as follows:
(29)The application of the GHE approach identified six global hydraulic flow units (GHE-1–GHE-6). Originally, eight hydraulic flow units were identified using the DRT analysis. A distinct trend in texture contrast appears to be associated with the global hydraulic elements. GHE-3 and GHE-4 are associated with fine and poorly sorted sands with clay- and silt-rich shales, respectively. These two global hydraulic elements have the worst reservoir rock quality (Tables 3 and 8). On the other hand, the coarse-grained well-sorted sands and the clean conglomerates are associated with GHE-8, which has the best reservoir rock quality (Tables 3 and 8).

As indicated in Figure 15, the relationship between the *λ* and groups does not appear to be distorted as a consequence of this reduction in rock types suggested by the GHE approach. Table 8 illustrates the power-law relationships for the six identified global hydraulic elements, with the associated coefficients of determination. Likewise, the GHE approach leads to power-law relationships between the *λ* and groups that are highly correlated. Moreover, the prediction of permeability using the GHE petrotyping approach (Fig. 16) appears to yield comparable results to permeability prediction using the DRT approach (Fig. 8). The average relative error in estimated permeability using the DRT approach is about 36% with a standard deviation of 32%. The average relative error in permeability using the GHE petrotyping approach is about 35% with a standard deviation of 41%. Both petrotyping approaches yield, on average, satisfactory permeability estimates within less than an order of magnitude variation from the measured permeability.

## Conclusions

This treatise applies the principles of dimensional analysis for identifying the open-hole log controls reflecting the rock flow zone indicator (FZI). Dimensional analysis reveals that log measurements such as the bulk density (*ρ*_{b}), photoelectric adsorption (*P*_{e}), the sonic interval transit time (Δ*t*), the apparent water resistivity (*R*_{wa}) and the deep resistivity (*R*_{t}) provide a satisfactory inference for the reservoir rock FZI. The analysis reveals the existence of two dimensionless groups namely, the dimensionless FZI (*λ*) group and the dimensionless resistivity group. An excellent fit between *λ* and is obtained when the data are partitioned into unique hydraulic flow units. The dimensionless groups have been validated with sandstone reservoir data from a prominent onshore oilfield from the Middle East. A non-linear empirical model for predicting the rock FZI using open-hole log measurements has emerged as a consequence. Dimensional analysis reveals that the FZI is an intricate non-linear function of open-hole log responses. Therefore, the use of linear regression analysis models for estimating the rock FZI as a function of open-hole log measurements may be inadequate. The dimensional analysis approach introduced does not appear to require reservoir layers to be at an irreducible water-saturation condition.

The field case presented shows that it is possible to generate reliable estimates of FZI and rock permeability from wireline open-hole log measurements when rocks are properly classified into distinct hydraulic flow units. The systematic classification of hydraulic flow units using a standard reference approach, such as the discrete rock type (DRT) or the global hydraulic elements (GHE), appears to identify clear rock-quality progression trends with increasing DRT and GHE values.

## Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

## Appendix A

#### Data description

The data (Figs 1 and 2) consist of gamma ray (GR), caliper, neutron porosity, bulk density, compressional wave sonic interval transit time, laterolog deep resistivity and photoelectric absorption. The data belong to two sandstone reservoirs, R_{1} and R_{2}, from well B of a prominent sandstone oilfield in the Middle East.

The well is cored at 335 depth intervals. Conventional core analysis was performed on all collected samples. The permeability of the cored intervals is measured using an air permeameter at a pressure of 1000 psi. The coupled interaction between the deposition and the diagenesis processes has caused the porosity and permeability to be highly variable. Permeability values for this well vary over roughly six orders of magnitude from 0.04 mD in low-quality zones to 10 000 mD in high-quality zones. As shown in Figure A1, permeability appears to have a bimodal distribution with a positively skewed left mode and a negatively skewed right mode. Porosity varies from 5.7 to *c.* 35.5%, and appears to have a negatively skewed distribution (Fig. A2). As shown in Figure A3, the flow zone indicator (FZI) values calculated from the core data appear to have a bimodal distribution, with the central mode pseudo-normal. As shown in Figure A4, a plot of permeability v. porosity displays a general trend of increasing permeability with increasing effective porosity, with a sizeable amount of data scatter. As shown in Figure A4, for a fixed effective porosity of 15 or 20%, permeability varies over approximately three orders of magnitude.

In general, the minimum of the following two shale volume indicators was found most suitable for estimating the shale volume fraction:
(A1)
(A2)where GR_{shale} is the GR log reading in a shale zone and GR_{clean} is the GR log reading in a shale-free zone. Log shale indicators were calibrated to actual shale weight percentages derived from X-ray diffraction (XRD) performed on a limited number of core samples (28 samples from reservoir R_{1} and 32 samples from reservoir R_{2}). XRD is applied in order to identify clay types and to quantify their relative abundance (Al-Sultan 2017). Identifying the presence of minerals, such as orthoclase feldspar and mica, and metamorphic rock fragments is essential for estimating credible values of shale volume fraction. These minerals can affect the GR response, and their presence may result in an overestimation of the shale volume fraction using equation (A2). In a pure shale zone, the neutron porosity is usually greater than the density porosity because of the high hydrogen index in shale caused by the presence of bound water between the clay layers. The maximum disparity between the neutron and density porosities normally takes place in shale (Bassiouni 1994). Therefore, equation (A1) is a sort of linear interpolation for the shale volume fraction in a shaly sand based on the maximum disparity between neutron and density porosities in a shale zone. This method for evaluating the shale volume fraction (equation A1) is likely to work well provided that there is no gas in the formation, since gas has a low hydrogen index and decreases the apparent neutron porosity. Similarly, the density tool response is affected by heavy minerals, such as siderite and pyrite. Equation (A1) is likely to account for these structural heavy minerals as shale, and therefore overestimate the shale volume fraction. In these cases, equation (A2) becomes more reliable for estimating the shale volume fraction. The effective porosity (*ϕ*_{e}) is calculated using the following equation:
(A3)The total porosity (*ϕ*_{t}) in the above equation is given by
(A4)where *ϕ*_{d} is the apparent density log porosity and *ϕ*_{n} is the apparent neutron log porosity. As indicated by Thomas & Steiber (1975), the approximation of the effective porosity using equation (A3) is appropriate for a pore-lining dispersed shale. Otherwise, an evaluation of the clay distribution is fundamental for an accurate estimation of the effective porosity. For instance, equation (A3) does not adequately approximate the effective porosity for a structural or laminated shale.

Isolated pores in sedimentary rocks are rare, unless some of the detrital material of the rocks originates from volcanic deposits that are characterized by vesicular pores (Thomas & Ausburn 1979). Thus, in general, all pores of sedimentary rocks are interconnected, and so the measured porosity using any of the log instruments, or any of the laboratory techniques, is the total porosity (Swanson & Thomas 1979; Thomas & Ausburn 1979). The effective porosity constitutes part of the total porosity saturated by fluids that contribute to fluid flow (e.g. hydrocarbons and moveable water). For instance, water trapped in the smallest pores and capillary-bound water, as well as clay-bound water, are factored out from the total porosity when calculating the effective porosity. In clean zones that have no shale, the effective porosity is approximated as follows:
(A5)In the above notation, *S*_{wirr} is the irreducible water saturation obtained from capillary pressure measurements.

- © 2018 The Author(s). Published by The Geological Society of London for GSL and EAGE. All rights reserved