Introduction

Petrophysical properties such as porosity and permeability are important indicators of oil or gas reservoir quality, and studying them enhances understanding of the reservoir and potential to predict subsurface characteristics (Morton-Thompson and Woods 1993; Sahin and Saner 2001; Selley 1998; Wilson 1994). Various factors control the properties and distribution of porosity and permeability within clastic reservoir rocks. These include texture, fabric, depositional environment, sorting, packing, amount of matrix, degree of cementation, and other post-depositional diagenetic changes. It is this relatively large number of factors, each of which can vary considerably, that result in the complexity of porosity and permeability evident in sandstone reservoirs (Abdulkadir et al. 2010; Ali et al. 2010; Boggs 2006; Fitch et al. 2015; Gier 2000; Kassab et al. 2015; Ketzer et al. 2002; Sun et al. 2007; Surdam et al. 1989).

This study investigates the Late Ordovician Sarah Formation that crops out in the Qassim region in central Saudi Arabia (Fig. 1). The Sarah Formation in this region is exposed in several glacial paleovalleys in central and northern Saudi Arabia. A stratigraphically equivalent formation to the Sarah Formation is the Late Ordovician Sanamah Formation, which is exposed in the Wajid basin in southwestern Saudi Arabia. The Sarah Formation and its equivalents occupy several sub-basinal areas and represent important subsurface tight gas exploration targets in northern and central Saudi Arabia, and in the Rub’ al Khali basin to the southeast (Abdullatif 2011; Evans et al. 1991; Konert et al. 2001). Subsurface exploration of the Paleozoic succession has revealed a complexity of facies, paleoenvironments, and paleogeography (Al-Mahmoud and Al-Ghamdi 2010; Briner et al. 2010; Khalil 2012). However, very little data are available concerning quantitative assessment of the porosity and permeability properties in the Sarah Formation in these paleovalleys. This study describes and evaluates sedimentological heterogeneity and its impact on porosity and permeability in the Sarah Formation that is exposed in the glacial paleovalleys of the Qassim region of central Saudi Arabia.

Fig. 1
figure 1

Geological map showing the Late Ordovician glacio-fluvial paleovalleys (denoted by P.V.) of the Sarah Formation, central Saudi Arabia. Note the positions of the Bukayriyah, Hanadir, Sarah, and Khanasir Sarah paleovalleys (Senalp and Al-Laboun 2000)

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Geological background

Late Ordovician glaciation resulted in the formation of unconformities and overlying glacial, glacio-marine, and glacio-fluvial deposits laid down as influxes into deeply incised paleovalleys (McClure 1978; Senalp and Al-Laboun 2000). The Sarah Formation is characterized by low porosity and very low permeability, and considered an important tight gas reservoir target in northwestern Saudi Arabia and the Rub’ al Khali basin (McGillivray and Husseini 1992).

Vaslet (1987, 1989, 1990) carried out mapping projects in the Sarah paleovalleys, while Senalp and Al-Laboun (2000) studied the sedimentological significance of the Late Ordovician Zarqa and Sarah Formations in the Qassim and Hail regions of Saudi Arabia. Paleovalleys of the Sarah Formation were thoroughly investigated by Clark-Lowes (2005) with extensive image-based and sedimentological work. In addition, Moscariello et al. (2009) studied the sedimentology and paleogeography of the Sarah Formation based on well data and outcrop observations. Several recent studies of the Sarah Formation have dealt with the facies, paleoenvironments, and sandstone composition in the Wajid basin of southwestern Saudi Arabia (Abdullatif 2011, Abdullatif et al. 2013), while Al-Zayer et al. (2013) carried out well testing analysis on Sarah Formation reservoirs in the subsurface. Other works have studied reservoir heterogeneity and diagenesis (El-Deek and Abdullatif 2013; El-Deek et al. 2013; El-Deek et al. 2014a), integration of sedimentology, petrophysics, heterogeneity and statistics (El-Deek et al. 2014b), and sedimentological and petrophysical heterogeneity of a glacial paleovalley of the Sarah Formation in the Hail region of central Saudi Arabia (Razzaq et al. 2014).

Heterogeneity

Heterogeneity in a sandstone reservoir depends on various controlling factors, such as geometry, reservoir structural parameters, facies distribution, sedimentary structures, lamination, and bedding, as well as the effects of diagenesis on both porosity and permeability (Ahmed 2010; Higley et al. 1997; Milliken 2001; Morad et al. 2010). The diagenetic factors, such as compaction, cementation, leaching, and dissolution, may affect the petrophysical parameters either by preservation, destruction, or enhancement. Thus, heterogeneity occurs at different scales: (1) a megascopic scale (basin-scale paleogeography), (2) a macroscopic scale (formation-scale depositional environments and relationships with other Paleozoic formations), (3) a mesoscopic scale (the lithofacies variability), and (4) a microscopic scale (the factors controlling the variation of petrophysical properties, e.g., textural, compositional, depositional, and diagenetic factors) (Adams et al. 2011; Jennings Jr and Ward 2000; Pranter et al. 2006; Sahoo et al. 2016; Tiab and Donaldson 2004).

Heterogeneity can be measured as a geostatistical characteristic that provides numerical values for studying the distribution thereof, and these values can be used for comparing heterogeneity with other distributions or in other reservoirs (Jensen 2000; Lake and Jensen 1991). The heterogeneity of the sandstone reservoir of the Sarah Formation is discussed in this study on the basis of the three most common statistical measures: coefficient of variation, Dykstra-Parsons coefficient, and Lorenz coefficient.

Methods and data set

This study is based on field and laboratory investigations. The field investigation involves lithofacies descriptions and analysis of the four paleovalleys (Bukayriyah, Hanadir, Sarah, and Kanasir Sarah) of the Sarah Formation in the Qassim region. The outcropping sections were studied in the field, and rock samples were collected.

The laboratory investigations included thin section petrography, X-ray powder diffraction (XRD), and scanning electron microscopy (SEM) analysis. Porosity and permeability measurements were also performed on the sandstone lithofacies.

Lithofacies description and analysis

Lithofacies descriptions, analysis, and classification were carried out on vertical and lateral outcrop sections in each of the paleovalleys. The lithofacies identified in these sections include trough cross-stratified sandstone, horizontally and low-angle stratified sandstone, planar cross-stratified sandstone, massive sandstone, shallow scour sandstone, finely laminated rippled siltstone, massive siltstone, and stratified glacial diamictite (Fig. 2). With the exception of the stratified glacial diamictite, all facies represent periods of ice retreat that is reflected in the glacial braided plain outwash facies within channels and overbank sub-environments. The glacial diamictite represents periods of ice advance. The lithofacies characteristics within the Sarah Formation paleochannel are summarized in Table 1.

Fig. 2
figure 2

Representative lithofacies sections of the glacio-fluvial paleovalleys of the Sarah Formation. a Interbedded trough cross-bedded sandstone. b Laminated siltstone overlain by pebbly cross-bedded sandstone. c Stacked horizontally bedded sandstone and laminated siltstone. d Horizontally bedded sandstone interbedded with minor trough cross-bedded sandstone and diagenetic hematite (H) and limonite (L) concretions (millimeters in size)

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Table 1 Facies types recognized in the Sarah Formation paleovalleys; facies codes modified after Miall (1996)
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Grain sizes vary from silt to very fine sand and ultimately very coarse sand. Pebbles also occur locally. The measured porosity is in the 17–56.7% range, with an average value of 28%. Permeability values are in the 0.1 mD to 2.27 Darcy range, with an average value of 0.25 Darcy. Porosity and permeability values for the Sarah Formation show variation at the outcrop scale (Fig. 3). Lithofacies within the Sarah Formation and their abundances are shown in Table 2 and Fig. 4. The sandstones are classified as quartz arenites, with angular to sub-rounded clasts that are poorly to moderately sorted, or well sorted (Fig. 5).

Fig. 3
figure 3

Photographs showing the lithofacies sections of the glacio-fluvial paleovalleys of the Sarah Formation. a Horizontally stratified sandstone (Sh). b Low-angle cross-stratified sandstone (Sl). c Planar cross-stratified sandstone (Sp). d Trough cross-stratified sandstone (St), planar cross-stratified sandstone (Sp), and massive rootlet siltstone (Fr). e Glacial diamictite (Dm). f Finely laminated (Fl) and rippled (Fr) siltstone facies

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Table 2 The lithofacies within the Sarah Formation paleovalleys with their codes and percentage abundances
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Fig. 4
figure 4

Pie chart showing the lithofacies abundances in the Sarah Formation outcrops which indicates the predominance of stratified sandstone (Sh), cross-stratified sandstone (St), and massive sandstone (Sm). Fl finely laminated siltstone, Dm glacial diamictite, Gh horizontally imbricated gravels

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Fig. 5
figure 5

Microphotographs showing a thin section image (in plane polarized light) of moderately sorted, sub-angular to sub-rounded clasts in quartz arenite (φ = 22.8%, K = 9.45 mD); b thin section image (in plane polarized light) of grain dissolution and presence of ferruginous cement between the quartz grains (φ = 30.21%, K = 26.81 mD); c SEM image of quartz arenite with primarily rounded to sub-rounded quartz grains (φ = 32.24%, K = 1056.5 mD); and d SEM image of quartz arenite with kaolinite cement blocking pore spaces between quartz grains (φ = 27.69%, and K = 1.87 mD)

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Statistical measures of heterogeneity

Coefficient of variation

The coefficient of variation (CV) is used as a measure of heterogeneity and is calculated by dividing the standard deviation of a set of samples by the mean of those samples, provided that the mean does not equal zero.

$$ \mathrm{CV}=\frac{\sigma }{\mu } $$
(1)

where, CV is the coefficient of variation, σ is the standard deviation, and μ is the mean.

The CV of permeability values is < 0.5 in a homogenous medium, between 0.5 and 1.0 in a heterogeneous medium, and > 1.0 in a medium that is very heterogeneous (Lake and Jensen 1991).

Permeability distributions in the Bukayriyah, Hanadir, Sarah, and Khanasir Sarah paleovalleys have CV values of 0.62, 1.24, 1.94, and 1.17, respectively. These values indicate heterogeneous to very heterogeneous permeability distributions.

Dykstra-Parsons coefficient

The Dykstra-Parsons coefficient (Dykstra and Parsons 1950) is also known as the “coefficient of permeability variation” or the “Reservoir Heterogeneity Index”. It is the most common measure of heterogeneity. Tiab and Donaldson (2004) described it as an excellent tool for characterizing the heterogeneity of a reservoir. It is based mainly on permeability variations, and is calculated as follows:

$$ {V}_{\mathrm{DP}}=\frac{k_{0.50}-{k}_{84.1}}{k_{0.50}} $$
(2)

where, k 0.50 is the median permeability and k 84.1 is the permeability value at one standard deviation above k 0.50 on a log-permeability probability plot. The value of V DP is equal to zero for an ideally homogeneous reservoir, 0–0.25 for a slightly heterogeneous reservoir, 0.25–0.5 for a heterogeneous reservoir, 0.5–0.75 for a very heterogeneous reservoir, 0.75–1 in an extremely heterogeneous reservoir, and > 1 for a perfectly heterogeneous reservoir (Tiab and Donaldson 2004).

The Dykstra-Parsons coefficient has been calculated from a set of permeability values from the Sarah Formation paleovalleys. The permeability measurements from each outcrop are sorted in descending order, followed by computation of the percent of permeability values greater than each permeability value in the distribution (Tables 4, 5, 6, and 7). Finally, the results are plotted on a log-probability curve (Fig. 6). The Dykstra-Parsons coefficient of permeability variation is then calculated for each of the four paleovalleys, as described below:

Fig. 6
figure 6

Dykstra-Parsons plot of permeability for Bukayriyah paleovalley (k 0.50 = 7.2, k 84.1 = 3.2) (a); Hanadir paleovalley (k 0.50 = 95, k 84.1 = 11) (b); Sarah paleovalley (k 0.50 = 34.05, k 84.1 = 9.56) (c), and Khanasir Sarah paleovalley (k 0.50 = 147.47, k 84.1 = 31.72) (d)

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For Bukayriyah paleovalley:

$$ {k}_{0.50}=7.2,{k}_{84.1}=3.2;{V}_{\mathrm{DP}}=\left({k}_{50}-{k}_{84.1}\right)/{k}_{50}=\left(7.2\hbox{--} 3.2\right)/7.2=0.56 $$
(3)

For Hanadir paleovalley:

$$ {k}_{0.50}=95,\kern0.75em {k}_{84.1}=11;{V}_{\mathrm{DP}}=\left({k}_{50}-{k}_{84.1}\right)/{k}_{50}=\left(95\hbox{--} 11\right)/95=0.88 $$
(4)

For Sarah paleovalley:

$$ {k}_{0.50}=34.05,\kern0.5em {k}_{84.1}=9.56;{V}_{\mathrm{DP}}=\left({k}_{50}-{k}_{84.1}\right)/{k}_{50}=\left(34.05\hbox{--} 9.56\right)/7.2=0.72 $$
(5)

For Khanasir Sarah paleovalley:

$$ {k}_{0.50}=147.47,\kern0.5em {k}_{84.1}=31.72;{V}_{\mathrm{DP}}=\left({k}_{50}-{k}_{84.1}\right)/{k}_{50}=\left(147.47\hbox{--} 31.72\right)/7.2=0.78 $$
(6)

The range of Dykstra-Parsons coefficients indicates that the Sarah Formation reservoir is very heterogeneous to extremely heterogeneous (0.56–0.88). The Bukayriyah and Sarah paleovalleys represent very heterogeneous reservoirs, whereas the Hanadir and Khanasir Sarah paleovalleys are extremely heterogeneous.

Lorenz coefficient

Lorenz (1905) developed the Lorenz coefficient (Lc) as a measure of the concentration of wealth. It was later introduced to geostatistics as an important parameter for characterizing the heterogeneity of porosity and permeability within a reservoir (Schmalz and Rahme 1950).

Generally, the Lc is obtained by plotting cumulative flow capacity (F m) against cumulative thickness (H m). The values of F m and H m are calculated from the following equations (Ahmed 2010; Lake and Jensen 1991; Peters 2012; Schmalz and Rahme 1950; Tiab and Donaldson 2004):

$$ {F}_{\mathrm{m}}=\frac{\sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{m}}{\mathrm{k}}_{\mathrm{i}}{\mathrm{h}}_{\mathrm{i}}}{\sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{n}}{\mathrm{k}}_{\mathrm{i}}{\mathrm{h}}_{\mathrm{i}}} $$
(7)
$$ {H}_{\mathrm{m}}=\frac{\sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{m}}{\mathrm{h}}_{\mathrm{i}}}{\sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{n}}{\mathrm{h}}_{\mathrm{i}}} $$
(8)

However, Fitch et al. (2013) developed an alternate version of the Lc in the event that there are pairs of variables that are all positive, and every pair is measured at the same location and the thickness (h) is not known. The alternate version takes the cumulative sums of the first variable “A”, sorts them in descending order, and then these values are plotted against the cumulative sums of the other variable “B” (Fig. 7). Thus, the Lc can be calculated directly from porosity and permeability.

Fig. 7
figure 7

Schematic illustration of a Lorenz plot showing the Lorenz curve, and line of perfect equality where Lc = 0 (after Fitch et al. 2013)

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The Lc, which is twice the area between the Lorenz curve and the diagonal line in Fig. 7, ranges from zero in a homogeneous reservoir to one in a heterogeneous reservoir.

The total area below the Lorenz curve is calculated as follows:

$$ \mathrm{Area}\ \mathrm{below}\ \mathrm{Lorenz}\ \mathrm{curve}=\sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{n}}\left({\varnothing}_{\mathrm{cum},\kern0.75em \mathrm{i}+1}-{\varnothing}_{\mathrm{cum},\kern0.75em \mathrm{i}}\right)\times \frac{K_{\mathrm{cum},\kern0.75em \mathrm{i}}+{K}_{\mathrm{cum},\kern0.75em \mathrm{i}+1}}{2} $$
(9)

The area of the plot below the line of perfect equality in Fig. 7 is equal to half the total area of the plot (a square), and so the area between the Lorenz curve and the line of perfect equality is equal to the total area below the Lorenz curve minus the area of the plot below the line of perfect equality.

In this study, the values of Lc, Ø, and K were calculated for each paleovalley of the Sarah Formation and arranged in descending order. Then, the cumulative sums were calculated and normalized to 1 Tables 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18), after which the normalized porosity values (∑Ø) were plotted as a function of the normalized cumulative permeability values (∑K) (Fig. 8).

Fig. 8
figure 8

Lorenz plot showing normalized porosity values (∑Ø) plotted as a function of the normalized cumulative permeability values (∑K) for a Bukayriyah, b Hanadir, c Sarah, and d Khanasir Sarah paleovalleys

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The Lc values for the Bukayriyah, Hanadir, Sarah, and Khanasir Sarah paleovalleys are 0.30, 0.59, 0.58, and 0.54, respectively, indicating high variability in the distribution of porosity and permeability.

Summary of the heterogeneity parameters

Table 3 provides a comparison of the heterogeneity parameters for the Sarah Formation paleovalleys. The coefficient of variation values for permeability distributions is in the 0.63–1.93 range, indicating extremely heterogeneous distributions. The Dykstra-Parsons coefficient values are in the 0.56–0.88 range, suggesting very heterogeneous to extremely heterogeneous reservoirs (Tables 4, 5, 6, and 7). The Lorenz coefficients correlate well with the Dykstra-Parsons coefficients (Fig. 9 and Tables 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18).

Table 3 Heterogeneity parameters for the Sarah Formation paleovalleys
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Fig. 9
figure 9

Graphical illustration of the correlation between the Lorenz and Dykstra-Parsons coefficients for the Sarah Formation paleovalleys (R 2 = 0.7831)

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Overall, the parameters indicate that the Sarah Formation paleovalleys represent heterogeneous to very heterogeneous reservoirs, and variations in porosity and permeability can be attributed to textural variation, compaction, cementation, dissolution, replacement, quartz overgrowth, and pore size distribution (El-Deek et al. 2014b).

Interpretation and discussion

Analysis of rocks exposed in paleovalleys of the Sarah Formation reveals approximately six lithofacies. The lithologies range from sandstone to siltstone to diamictite, some with a pebbly component. Horizontally laminated sandstone is the most common lithofacies, occupying approximately 60% of the total composite section. Petrographic analysis indicates that the sandstones are predominantly fine- to medium-grained, moderately to well-sorted quartz arenites. Alkali feldspars and plagioclase are the main detrital components of the samples. The grains are mainly sub-angular to sub-rounded. Mud is the predominant matrix type, but constitutes < 2% of the total rock volume. The cementing materials (5–10%) include clay and iron oxides, while SEM and XRD analyses show kaolinite to be the predominant cementing material. The abundance of kaolinite and ferruginous cement is responsible for the critical reduction in permeability, while the coating of quartz grains by kaolinite retards compaction, inhibits quartz overgrowth, and hence plays a significant role in preserving the initial porosity and permeability. Such preservation of porosity allows the dissolution of feldspar grains by pore fluids (e.g., meteoric water), which creates secondary porosity and subsequently enhances the permeability.

The heterogeneity of the Sarah Formation reflects the complexity of facies, paleoenvironments, and paleogeography of the Late Ordovician deposits, which makes the prediction of reservoir quality difficult, especially relying on insufficient borehole data. While the borehole data cannot resolve the relationship of such multiple closely spaced rock bodies, the outcrop analog of the Sarah Formation helps to investigate such complexity, as it provides information about rock body dimension, size, and orientation, thus revealing details on the reservoir continuity and connectivity within the glacio-fluvial environments and their vertical and lateral relationships with other Paleozoic formations at a resolution unavailable from the subsurface (Fig. 10). Such information is useful as it fills the gap in knowledge within the interwell spacing (Abdullatif and Makkawi 2010; Al-Ajmi et al. 2011; Grammer et al. 2004; North and Prosser 1993; Thurmond et al. 2005; Tye 2004).

Fig. 10
figure 10

Schematic model illustrating the heterogeneity of the Sarah Formation in terms of vertical and lateral facies variability and relationships with other Paleozoic formations

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Based on field observations and laboratory analyses, the heterogeneity of the Sarah Formation paleovalleys occurs at different scales. At the megascopic and macroscopic scales, the Late Ordovician glaciation resulted in the formation of unconformities and overlying glacial, glacio-marine, and glacio-fluvial deposits laid down as influxes into deeply incised paleovalleys cutting through various depositional environments, e.g., the storm-dominated, shallow marine depositional system of Hanadir and Kahfah, and the upper progradational sequence of the offshore marine Ra’an shale and the tide-dominated Quwarah sandstone. In addition, the Sarah Formation cuts further into the fluvial and shallow marine sandstones of the Saq Formation in Khanasir Sarah paleovalley (Clark-Lowes 2005). This indicates the basin-scale heterogeneity in terms of depositional environments that may be detected in a single borehole. At the mesoscopic scale, the depositional environments have impacted the lithofacies variability within the glacio-fluvial paleovalleys where several lithofacies were identified (e.g., Figs. 2, 3, and 4, Table 2). At the microscopic scale, these lithofacies vary in terms of texture (grain size, shape, roundness), composition of detrital mineralogy, and all superimposed diagenetic changes. This might eventually reflect on porosity and permeability variation within the paleovalleys.

Therefore, all these aspects discussed earlier collectively might be responsible for the heterogeneities as revealed from the statistical parameters (Table 3). Understanding the different scales of heterogeneity that occur in the Sarah Formation, from pore-scale to basin-scale, might help in better evaluating and predicting the reservoir quality and architecture in the subsurface.

Conclusions

The heterogeneity of the Sarah Formation reservoirs may be attributed to complex depositional and diagenetic variations that have affected the porosity and permeability distributions. Understanding the geologic controls on the variation of reservoir quality increases the understanding of reservoir heterogeneity in the subsurface and is a key factor for predicting the distribution of the potential reservoir facies, and their petrophysical properties.

The reservoir heterogeneity analysis conducted using the three statistical measures, coefficient of variation, Dykstra-Parsons coefficient, and Lorenz coefficient, shows that the Sarah Formation represents a heterogeneous to very heterogeneous reservoir. The integration of outcrop analog data with subsurface information may provide a better understanding of the glacio-fluvial reservoir quality and architecture.