## Abstract

Distal intervals of interbedded sandstones and mudstones in shallow-marine, wave-dominated shoreface and deltaic reservoirs may contain significant hydrocarbon resources, but their reservoir properties are difficult to predict. Relatively small-scale (200 × 100 × 20 m) three-dimensional object-based reservoir models, conditioned to outcrop analogue data, have been used to investigate the controls on the proportion of sandstone, the proportion of sandstone beds that are connected by sandstone-filled erosional scours and the effective vertical-to-horizontal permeability ratio (*k*_{v}/*k*_{h}) of such intervals. The proportion of sandstone is controlled by sandstone-bed and mudstone-interbed thickness, and by parameters that describe the geometry, dimensions and lateral-stacking density of sandstone-filled scours. Sandstone-bed connectivity is controlled by the interplay between the thickness of mudstone interbeds and sandstone-filled erosional scours. Effective *k*_{v}/*k*_{h} is controlled by the proportion of sandstone, which represents the effects of variable distributions and dimensions of mudstones produced by scour erosion, provided that scour thickness is greater than mudstone-interbed thickness. These modelling results provide a means of estimating the effective *k*_{v}/*k*_{h} at the scale of typical reservoir-model grid cells using values of mudstone-interbed thickness and the proportion of sandstone that can potentially be provided by core data.

Sandstones deposited in shallow-marine, wave-dominated shoreface and deltaic environments form an important reservoir type for the global petroleum industry. Examples include the Jurassic Brent Group reservoirs of the North Sea, offshore UK and Norway (e.g. Husmo *et al.* 2003); the Eocene Jackson Group and Oligocene Frio Formation reservoirs of Texas, onshore USA (Fisher *et al.* 1970; Galloway & Morton 1989); and the Tertiary reservoirs in the Niger Delta province, offshore Nigeria (Larue & Legarre 2004), the Baram Delta province, offshore Brunei (Atkinson *et al.* 1986) and the Columbus Basin, offshore Trinidad and Tobago (Sydow *et al.* 2003). These reservoirs are heterogeneous over lengthscales ranging from parasequences (1 km to tens of kilometres laterally, tens of metres vertically) to pores (micrometres) (Fig. 1) (e.g. Weber 1982; Kjønsvik *et al.* 1994; Larue & Legarre 2004; Hampson *et al.* 2008; Sech *et al.* 2009). Typically, only large-scale heterogeneities (Fig. 1a) can be observed using seismic data, while smaller-scale heterogeneities (e.g. Fig. 1b, c) require interpretation of wireline-log, core and production data using geological concepts that describe the architecture of interwell volumes and which may be derived from outcrop analogues (e.g. Larue & Legarre 2004; Hampson *et al.* 2008). Distal deposits in wave-dominated shoreface and deltaic environments consist of interbedded sandstones and mudstones (Fig. 1c). The marked permeability contrast between sandstone beds of good reservoir quality and poor-quality mudstone in distal lower-shoreface and wave-dominated delta-front deposits has the potential to create low effective vertical permeability. As a result, it may be difficult to produce hydrocarbons hosted in these intervals. Previous work has demonstrated that, in the absence of significant modification by bioturbation (e.g. Pemberton & Gingras 2005), hydrocarbon recovery from heterolithic intervals of interbedded sandstone and mudstone is controlled by the sandstone to mudstone ratio (cf. the proportion of sandstone), the connectivity of sandstone beds, and the lateral extent and continuity of mudstone interbeds (e.g. Kjønsvik *et al.* 1994). Outcrop analogues indicate that heterolithic distal lower-shoreface and wave-dominated delta-front deposits can contain sandstone-filled erosional scours with the potential to erode through mudstone interbeds, and thereby enhance vertical connectivity between sandstone beds and reduce mudstone continuity (e.g. Eide *et al.* 2015; Collins *et al.* 2017; Onyenanu *et al.* 2018). Such scours are commonly inferred to have a near-linear channelized geometry (gutter cast) or a semi-ovoid geometry (pot cast) in three dimensions (e.g. Whitaker 1973; Myrow 1992). Scours also vary in their distribution, being more densely spaced in locations offshore of coeval river mouths, where a combination of storm-generated currents and riverine sediment influx increased erosional potential during storm events (Eide *et al.* 2015; Collins *et al.* 2017; Onyenanu *et al.* 2018).

The aims of this article are three-fold: (1) to identify the key sedimentological heterogeneities that influence sand-body connectivity and the proportion of sandstone within distal lower-shoreface and wave-dominated delta-front deposits; (2) to characterize the associated extent and distribution of mudstone barriers that are remnants of erosional scouring in these deposits; and (3) to determine the impact of scour-related heterogeneities on effective vertical permeability. These aims are addressed via a suite of reservoir models that are calibrated to data derived from a well-exposed outcrop analogue (Onyenanu *et al.* 2018).

## Outcrop analogue dataset

The geometry and distribution of scours, and their facies architectural context have been documented in distal lower-shoreface and wave-dominated delta-front deposits of an outcrop analogue: the ‘G2’ parasequence of the Grassy Member, Blackhawk Formation (Late Cretaceous), exposed in the Book Cliffs of east-central Utah, USA (see Onyenanu *et al.* 2018 for details). These deposits are exposed in nearly continuous cliff faces along the walls of narrow, sinuous canyons that provide some three-dimensional control in a studied volume of *c.* 260 × 250 × 15 m, and have been characterized using conventional measured sections combined with a high-resolution (20 000 points/m^{2}) digital outcrop model constructed using photogrammetry (Onyenanu *et al.* 2018).

The studied strata contain two upwards-thickening successions of sandstone beds (‘bedsets’ of O'Byrne & Flint 1995) (Figs 2a and 3a). Individual and conjoined (amalgamated) sandstone beds range from 0.125 to 2.0 m thick in the upwards- thickening successions (Figs 2a and 3a). At their bases, many beds have steep-sided (up to 35°) erosional scours that are up to 1.1 m in thickness and 15.1 m in apparent width (Figs 2b and 4). The mean lateral spacing of scours along the base of each bed (scour density) is variable, from 0 to 0.06 m^{−1} in different beds, with higher values typically associated with conjoined sandstone beds over 0.5 m thick. The erosional scours do not show systematic variations in their cross-sectional geometry with local cliff-face orientation, implying a pot-cast geometry in three dimensions. The erosional scours have a mean orientation approximately perpendicular to the palaeocoastline (N099–N279 with a standard deviation of ±36^{°}).

## Methodology

### Design of modelling experiments

We use the outcrop analogue dataset described above as a basis to identify key heterogeneities to be investigated in reservoir modelling experiments (Table 1). We are particularly interested in capturing heterogeneities at the bed scale (Figs 2 and 3a) in order to evaluate the impact of sandstone-filled erosional scours on sand-body connectivity, and on mudstone extent and distribution. Using qualitative observations and quantitative measurements of the outcrop data as inputs, we first conducted a screening study to identify the heterogeneities with the largest impact on sand-body connectivity and the proportion of sandstone (as a proxy for net-to-gross ratio, NTG). In this screening study, experimental design techniques (Box *et al.* 1978; Damsleth *et al.* 1992; Jones *et al.* 1995; White & Royer 2003) and analysis of variance (Box *et al.* 1978) were used to explore the parameter space defined by the six heterogeneities described below. A two-level fractional factorial design was applied in the screening study, in which each factor (heterogeneity) can take one of two settings (i.e. high and low values of parameters that describe a particular heterogeneity) (Table 1). The experimental design allows us to quantify efficiently the effect of varying each factor from setting 1 to setting 2. The results of this screening study were used to design sensitivity studies that characterize the response of sand-body connectivity, the proportion of sandstone, and mudstone extent and distribution to variations in the parameters that describe key heterogeneities (Tables 2 and 3).

### Modelled heterogeneities and settings

Six sedimentological heterogeneities were investigated (Table 1).

#### Scour dimensions

Erosional scours in the outcrop dataset show variability in their apparent cross-sectional geometry and dimensions (Fig. 4). The range of cross-sectional scour dimensions used in our modelling experiments was selected to represent the distribution of observed data, from minimum values of 0.125 m (thickness) × 1.0 m (width) to maximum values of 0.50 m (thickness) × 12.0 m (width). Scour dimensions can have a significant impact on sand-body connectivity (e.g. in low-NTG channelized reservoirs: Jones *et al.* 1995; Larue & Hovadik 2006) and may influence the continuity of mudstone barriers between sand bodies.

#### Orientation

We modelled scenarios in which all scours were orientated with the observed mean orientation (N099) or had a range of orientations defined by the observed mean and standard deviation (N099 ± 36°). The range of scour orientations may influence sand-body connectivity (e.g. in low-NTG channelized reservoirs: Larue & Hovadik 2006).

#### Scour density

Scour density, defined as the mean number of scours along the base of each sandstone bed along a planar cross-sectio, such as a cliff-face exposure, varies from 0 to 0.06 m^{−1} in the outcrop analogue dataset (fig. 11B in Onyenanu *et al.* 2018). A range of scour densities of 0 to 0.04 m^{−1} was investigated in our models. Scour density controls the number of scours in the models, and thus is expected to influence the proportion of sandstone, sand-body connectivity, and the extent and continuity of mudstone barriers.

#### Scour geometry

Two geometrical templates were investigated: pot casts and gutter casts. These geometrical templates differ in their planform. Pot casts are semi-ovoid scours with a steep-sided bulbous nose facing upflow, and they become progressively narrower and shallower downflow (Knaust & Langbein 1992; Myrow 1992), whereas gutter casts are represented as straight channels (cf. Whitaker 1973; Myrow 1992). An individual gutter cast has a greater volume than an individual pot cast; thus, it is expected that the choice of geometrical template will influence the proportion of sandstone.

#### Lateral trends in the distribution of scours

We hypothesize that lateral trends in the distribution of erosional scours may influence the distribution of mudstone barriers. Although no apparent lateral trend in scour density is observed in the outcrop analogue dataset (fig. 12 in Onyenanu *et al.* 2018), we modelled scenarios with and without a directional lateral trend in scour density.

#### Bed thickness

In the outcrop analogue, sandstone bed thickness increases upwards, and the thickness of interbedded mudstones correspondingly decreases upwards in upwards-coarsening successions (‘bedsets’ of O'Byrne & Flint 1995) (Figs 2 and 3a). Individual sandstone beds and mudstone interbeds vary in thickness from 0.1 to 2.0 m (fig. 11A in Onyenanu *et al.* 2018). We choose not to model vertical trends in bed thickness but, instead, model scenarios in which the thickness of sandstone beds and mudstone interbeds is uniform. Consequently, different parts of a single upwards-coarsening succession (‘bedset’) correspond to different modelled scenarios. The minimum and maximum values of sandstone-bed and mudstone-interbed thickness in the various models are 0.125 and 1.0 m, respectively.

### Model dimensions, grids and construction

All of the reservoir models described herein have a relatively small volume of 200 m (length) × 100 m (width) × 20 m (height), which is consistent with the dimensions of the outcrop analogue (Fig. 3b) and with scales represented by several tens of full-field model grid cells or a sector model in many reservoirs. All models were constructed using a conventional object-based modelling approach (Caers 2005) in industry-standard commercial software. The models contain 12 800 000 rectangular grid cells that each measure 0.5 m (length) × 0.5 m (width) × 0.125 m (height) (Fig. 5). This grid resolution is sufficiently fine to allow the smallest erosional scours to be represented in the models. The model grid is orientated parallel to the mean orientation of the outcrop face (N031), such that the areal geometries of scours were subject to minimal distortion. Sandstone beds and mudstone interbeds were modelled deterministically as sheets of uniform thickness in each model. Erosional scour-fill sandstones were modelled as discrete objects with tops that coincide with the bases of sandstone beds, to mimic erosion into the underlying mudstone interbeds. In plan view, the distribution of erosional scour-fill objects is stochastic, and conditioned to a probability map for object insertion that is either uniform or contains a simple directional trend. Five realizations of each reservoir model were generated, in order to assess the stochastic variability inherent to each modelled scenario.

### Comparison of models

Models are compared using three measures of reservoir architecture. First, the proportion of sandstone is reported as a proxy for NTG. Secondly, the fraction of sandstone in the largest connected sand body is reported as a static measure of sand-body connectivity (cf. Larue & Hovadik 2006). Thirdly, the areal extent of preserved mudstone bodies is assessed using the volumetric fraction of mudstone present in each body. We use values of lacunarity measured from plan-view slices through the remnant mudstone interbeds and associated erosional scours to assess mudstone body size, shape and lateral distribution. Lacunarity is a pixel-based method that describes patterns of spatial dispersion over multiple lengthscales (Plotnick *et al.* 1996), and has been used previously in several geological studies (e.g. Plotnick *et al.* 1996; Roy *et al.* 2010; Flood & Hampson 2015). In this study, lacunarity is calculated using a sliding-box algorithm that is applied to a binary image of a plan-view section through a model in which mudstone bodies in the ‘foreground’ are distinguished from sandstones in the ‘background’. The algorithm uses boxes of different sizes to sample the binary image (Allain & Cloitre 1991; Plotnick *et al.* 1996). The box slides incrementally across the binary image, and the number of foreground pixels in the box are counted after each increment of slide, until the entire image has been scanned. Nine grid box sizes were applied to each binary image, and boxes range in size between 0.25 and 25% of the image area. This range of box size was chosen so that small boxes are larger than individual grid cells, and large boxes are sufficiently small to avoid point statistical errors (Karperien 1999–2013). Lacunarity has been used to compare spatial heterogeneity in mudstone distributions at different lengthscales, by comparing the results for different box sizes (e.g. Plotnick *et al.* 1996; Roy *et al.* 2010). Low values of lacunarity are obtained from spatial patterns with evenly distributed, translationally invariant gaps between ‘foreground’ pixels (mudstones) of similar size (e.g. regular patterns), whereas complex patterns that display translational variance and contain unevenly distributed gaps of heterogeneous size (e.g. clustered patterns) are characterized by high values of lacunarity.

## Results

### Screening study: ranking of modelled heterogeneities by impact on static measures of reservoir architecture

The screening study results indicate that scour cross-sectional dimensions, scour density, scour geometry, and the thickness of sandstone beds and mudstone interbeds have a significant impact on the proportion of sandstone (Fig. 6a), whereas scour cross-sectional dimensions and bed thickness control sand-body connectivity (Fig. 6b). Scour orientation and scour distribution have a much smaller impact on the proportion of sandstone and sand-body connectivity, with the average effect of varying these factors over the modelled parameter settings being close to or below the level of estimated uncertainty in the results (Fig. 6). Models with relatively thin sandstone beds and mudstone interbeds (0.25 m) have an average proportion of sandstone of 0.76 and a maximum fraction of sandstone in the largest connected sand body of 100%. In contrast, models with relatively thick sandstone beds and mudstone interbeds (1.0 m) have an average proportion of sandstone of 0.61 and a maximum fraction of sandstone in the largest connected sand body of 10%. These results are investigated further in two sensitivity studies, described below.

### Sensitivity study 1: impact of bed thickness and scour cross-sectional dimensions on sand-body connectivity

The impact of sandstone bed and mudstone interbed thickness on sand-body connectivity was investigated by constructing different models using the range of bed thicknesses measured from the outcrop analogue, from 0.125 to 1.0 m (Table 2), and erosional scours of uniform geometry (pot casts), cross-sectional dimensions (12 m wide × 0.5 m thick), orientation (N099 ± 36°) and density (0.4 m^{−1}). In models in which mudstone-interbed thickness is smaller than or equal to scour thickness, all sandstone is connected to form a single connected sand body (Figs 7a and 8a, b). Where mudstone-interbed thickness is greater than scour thickness, sand-body connectivity is much lower, and the largest connected sand body constitutes an individual sandstone bed and the erosional scour-fill sandstones at its base (Figs 7a and 8c). Similarly, we investigated the impact of scour cross-sectional dimensions on sand-body connectivity using models in which the dimensions of pot-cast scours were varied (from 0.9 m wide × 0.125 m thick to 12 m wide × 0.5 m thick) and bed thickness was held constant (0.25 m) (Table 2). In models in which scour thickness is greater than or equal to mudstone-interbed thickness, all sandstone beds are connected (Figs 7b and 8e, f). Where scour thickness is smaller than mudstone-interbed thickness, sand-body connectivity is low because sandstone beds are isolated from each other (Figs 7b and 8d). These results indicate that the ratio between mudstone-interbed thickness and scour thickness defines a threshold for sand-body connectivity (Fig. 7). Although this result appears self-evident, it is worth emphasizing that the thickness of laterally impersistent sandstone-filled scours is not a parameter that can be measured robustly from well data, unlike the thickness of laterally extensive mudstone-interbed sheets. Instead, values of scour thickness must be taken from appropriate analogue data (e.g. Fig. 4).

### Sensitivity study 2: impact of scour geometry and scour density on proportion of sandstone and mudstone distribution

We investigated the impact of gutter-cast and pot-cast scour geometries on the proportion of sandstone and mudstone distribution by constructing different models using the range of scour densities measured from the outcrop analogue, from 0.01 to 0.04 m^{−1}, in models containing sandstone beds and mudstone interbeds of uniform thickness (0.25 m) (Table 3). Scour cross-sectional dimensions, scour orientations and scour distributions are held constant in all models. The proportion of sandstone increases as scour density increases, and is also higher for models with gutter-cast scour geometry (Fig. 9). An increase in scour density results in the addition of sandstone-filled scours to the model, which thus raises the proportion of sandstone. Although the gutter-cast and pot-cast templates of scour geometry appear similar in their cross-sectional (axis perpendicular) dimensions, they differ in their along-axis continuity, such that an individual gutter cast has a greater volume than an individual pot cast. As a result, a model containing gutter casts has a higher proportion of sandstone than a model of the same scour density that contains pot casts (Figs 9 and 10). Over the range of modelled scour density values, the proportion of sandstone increases almost linearly with scour density in models containing pot casts, whereas the rate of increase in the proportion of sandstone progressively decreases with increasing scour density in models containing gutter casts (Fig. 10). This effect results from the differences in areal extent and volume between gutter casts and pot casts. Over the range of modelled scour density values, pot casts are largely non-amalgamated in plan view (e.g. Fig. 10a, b), such that increasing scour density results in the removal of predominantly mudstone-interbed volumes by pot casts. In contrast, gutter casts are amalgamated in plan view even for low values of scour density (e.g. Fig. 10c, d), such that increasing scour density results in the removal of parts of existing scour-fill sandstone volumes, as well as remnant patches of mudstone interbeds by gutter casts. These results imply that scour geometry may potentially have a significant impact on the dimensions, geometries and distributions of mudstones in the models.

The areal dimensions and extent of remnant mudstone bodies that result from the erosion of mudstone interbeds by pot casts and gutter casts (e.g. Fig. 10b, d) is first assessed using the frequency distribution of mudstone-body sizes in the sensitivity-study models (Fig. 9). Mudstone-body size is defined as the relative proportion of the mudstone fraction in a particular model. The frequency of smaller mudstone bodies increases as scour density increases (Fig. 11). Small mudstone bodies are also more abundant for a given scour density in models containing gutter casts (Fig. 11a, c, e & g) than in those containing pot casts (Fig. 11b, d, f & h). These quantitative results (Fig. 11a–h) are consistent with qualitative observations of mudstone-body continuity and extent in representative plan-view sections from the sensitivity-study models (Fig. 12a–h). Models containing gutter casts are characterized by many small mudstone bodies of limited lateral continuity for all of the modelled values of scour density (Fig. 12a, c, e & g). In contrast, models containing pot casts of low scour density are characterized by laterally extensive mudstone bodies with irregularly distributed holes (‘punctured sheets’) (Fig. 12b, d). As scour density increases, the ‘punctured sheets’ of mudstone are gradually replaced by discrete mudstone bodies of limited lateral continuity (Fig. 12f, h); this transition is represented by a bimodal distribution of mudstone-body sizes in models containing pot casts (Fig. 11b, d, f & h). Mudstone-body distributions in representative plan-view sections from the sensitivity-study models have also been characterized by lacunarity analysis (Fig. 12i, j). In all models, lacunarity generally increases, indicating more heterogeneous mudstone distributions, as scour density increases and as the scale of observation (i.e. box width) decreases (Fig. 12i, j). Lacunarity values are also higher in plan-view sections from models containing gutter casts (Fig. 12i) than in those from models containing pot casts (Fig. 12j), indicating less heterogeneous mudstone distributions in the latter.

## Discussion

### Impact on effective permeability

The modelling experiments documented herein provide quantitative three-dimensional descriptions of mudstone-body sizes and distributions that are extrapolated from outcrop analogue data. These model-derived descriptions of mudstone parameters are used to estimate the effective vertical to horizontal permeability ratio (*k*_{v}/*k*_{h}) of heterolithic, distal intervals of lower-shoreface and wave-dominated delta-front reservoirs that contain erosional scours using two different methods that are based on simple averaging and analytical approaches (Figs 13 and 14).

The streamline-based statistical method of Begg & King (1985) has been used to estimate the mean effective *k*_{v}/*k*_{h} of the modelled scenarios (Table 3) at the scale of the model volume, assuming that permeability in the sandstone beds is isotropic and that the mudstones are impermeable:
(1)where *Fs* is the proportion of mudstone, *S* is the density of mudstones in vertical section and *l* is the mean mudstone length. *Fs* is measured directly in each model, as the complement to the proportion of sandstone. *S* is calculated by multiplying the density of mudstone interbeds in the absence of erosional scours by the preserved fraction of mudstone interbeds in a particular model. *l* is estimated in each model using the three-step procedure outlined below: (1) the mean volume of a single mudstone body is calculated from the frequency distribution of mudstone body sizes (e.g. Fig. 11), the proportion of sandstone in the model and the model volume; (2) the mean volume of a single mudstone body is divided by the mudstone-interbed thickness to give the mean area of a single mudstone body; and (3) we assume a disc geometry for the mudstone bodies to calculate the mean mudstone-body length (i.e. disc diameter) from the mean area of a single mudstone body. Values of mean mudstone-body length estimated by this method from models containing pot casts varies from 58 to 146 m for scour densities of 0.04 to 0.01 m^{−1}, and from 8.5 to 25 m from models containing gutter casts for scour densities of 0.04 to 0.01 m^{−1}. These values are consistent with the mean length of 20 m measured by Arnot (2001) for approximately elliptical mudstone bodies in outcrops of the ‘G1’ and ‘G2’ parasequences of the Grassy Member in a region that encompasses the area and stratigraphic interval studied by Onyenanu *et al.* (2018), and also with the mean mudstone-body length of 82 m measured by Zeito (1965) in a collection of 12 unspecified marine sandstones at outcrop. Our estimated values of effective *k*_{v}/*k*_{h} are plotted in Figures 13 and 14 (red lines) as a function of scour geometry, scour density and the proportion of sandstone. Estimated *k*_{v}/*k*_{h} values are higher for models containing gutter casts (0.02–0.8 for a scour density of 0.01–0.04 m^{−1}) than for those containing pot casts (0.00008–0.002 for a scour density of 0.01–0.04 m^{−1}), and increase with scour density and the proportion of sandstone.

We also applied a conventional upscaled averaging method to estimate effective *k*_{v}/*k*_{h}. Isotropic permeability values of 20 and 0.01 mD were assigned to sandstone and mudstone lithologies, respectively. These permeability values are consistent with petrophysical properties documented in heterolithic, distal lower-shoreface and wave-dominated delta-front deposits of Jurassic Brent Group reservoirs, North Sea (e.g. Husmo *et al.* 2003) and Tertiary deltaic reservoirs, offshore Trinidad (e.g. Sydow *et al.* 2003), and are similar to those used in previous modelling studies of distal lower-shoreface and wave-dominated delta-front deposits (e.g. Kjønsvik *et al.* 1994; Jackson *et al.* 2009; Sech *et al.* 2009). Effective horizontal (*k*_{h}) and effective vertical permeabilities (*k*_{v}) were then calculated for each model volume using the arithmetic average and harmonic average, respectively (cf. Weber & van Geuns 1990). The harmonic average makes the assumption that the reservoir consists of continuous layers that have a constant permeability value. We recognize that this simple upscaled averaging method therefore gives a theoretical lower limit of effective *k*_{v}/*k*_{h}, assuming that the values of permeability assigned to the mudstone and sandstone lithologies are representative, but does not fully capture the vertical connectivity of sandstone beds. It is used here as a simple descriptor for comparison to the streamline-based statistical method. The resulting *k*_{v}/*k*_{h} estimates are higher for models containing gutter casts (0.004–0.2 for a scour density of 0.01–0.04 m^{−1}) than for those containing pot casts (0.003–0.02 for a scour density of 0.01–0.04 m^{−1}), and increase with scour density and the proportion of sandstone (blue lines in Figs 13 and 14).

Although the estimates of effective *k*_{v}/*k*_{h} derived from the streamline-based statistical method and upscaled averaging method (red and blue lines, respectively, in Figs 13 and 14) both increase with increasing scour density and proportion of sandstone, there are significant differences between the two sets of estimates. In particular, the streamline-based statistical method gives higher estimates of effective *k*_{v}/*k*_{h} for the models containing gutter casts, and lower estimates for the models containing pot casts. These differences are attributed principally to the upscaling approach overestimating effective *k*_{v}/*k*_{h} in models containing pot casts and underestimating effective *k*_{v}/*k*_{h} in models containing gutter casts (Figs 13 and 14), because the underlying assumption that each layer is laterally continuous does not take the geometry of the scours into account. The permeability calculated for each layer of laterally discontinuous mudstones and sandstone-filled scours in the models (e.g. Fig. 10b, d) is the arithmetic average resulting from the proportion of mudstone and sandstone in that layer. These calculated layer-average permeabilities over-represent the impact of sandstone-filled pot casts in providing vertical connectivity when they are subsequently harmonically averaged to calculate effective *k*_{v}, particularly in models of low scour density that contain laterally isolated pot casts (Figs 10b and 13). On the other hand, the calculated layer-average permeabilities over-represent the impact of mudstones in models containing gutter casts when they are subsequently harmonically averaged to calculate effective *k*_{v}, particularly in models of low scour density that contain only small, isolated mudstone patches (Fig. 10c and 13). Four additional reasons may also contribute to the differences in effective *k*_{v}/*k*_{h} estimated by the streamline-based statistical method and upscaled averaging method. (1) The mudstone bodies in the streamline-based statistical approach are assumed to be impermeable, while they are assigned a low but non-zero permeability (0.01 mD) in the upscaled averaging method. Assuming a lower permeability in the latter method would result in lower estimates of *k*_{v}/*k*_{h}. (2) The disc shape assumed for calculating the mean mudstone body length in the streamline-based statistical method maximizes the estimated effective *k*_{v}/*k*_{h} (Begg & King 1985). Assuming a more irregular geometry would result in lower estimates of *k*_{v}/*k*_{h} using this method. (3) The streamline-based statistical approach also overestimates the length of low-tortuosity streamlines for low values of shale density (Begg & King 1985), which may result in overestimation of *k*_{v}/*k*_{h} in models containing gutter casts with scour densities of 0.02–0.04 m^{−1}. (4) The streamline-based statistical approach assumes that mudstone barriers are smaller in area than the model, with mudstone-body lengths smaller than two-thirds of the model length or width (Begg & King 1985). This assumption does not hold true for models containing pot casts with scour densities of 0.01–0.03 m^{−1}, such that *k*_{v}/*k*_{h} may be overestimated in these models.

The limitations of both methods imply that the resulting estimates of effective *k*_{v}/*k*_{h} are indicative. However, it is noteworthy that estimates of effective *k*_{v}/*k*_{h} for either method appear to form a single consistent relationship when plotted against the proportion of sandstone for models containing gutter casts and models containing pot casts (Fig. 14). This implies that the combined effect of scour geometry and scour density controls effective *k*_{v}/*k*_{h}, provided that scour thickness is greater than the mudstone-interbed thickness. If this condition is valid, then the proportion of sandstone is sufficient to estimate effective *k*_{v}/*k*_{h} using either method. Future work could refine the results presented above by applying more sophisticated computational methods such as flow-based upscaling (e.g. Pickup *et al.* 1994, 1995; Renard & de Marsily 1997) or flow simulation of more geometrically complex surface-based models (e.g. Massart *et al*. 2016*a*, *b*).

### Application to reservoir characterization studies

Distal intervals of interbedded sandstones and mudstones in wave-dominated shoreface and deltaic environments that are associated with erosional scours are interpreted to occur in regions directly offshore of river mouths (Eide *et al.* 2015; Collins *et al.* 2017; Onyenanu *et al.* 2018). In these settings, offshore-directed downwelling storm flows coincided with peak riverine sediment delivery due to storm-driven precipitation (‘storm floods’ *sensu* Collins *et al.* 2017), resulting in deep, localized erosional scour at the base of storm-event sandstone beds. Reservoirs in such settings may be characterized by pronounced lateral variations between wells in the number and thickness of storm-event sandstone beds that are separated by mudstone interbeds (e.g. Atkinson *et al.* 1986), and by production data which indicate that such sandstone beds are connected, at least to some extent (Baillie & James-Romano 2010). The modelling results presented above can be applied to the characterization of such reservoirs.

Sandstone beds in distal wave-dominated shoreface and deltaic successions generally exhibit an upwards increase in thickness and scour density, and an upwards decrease in the thickness of mudstone interbeds (e.g. Figs 2a and 3a) (Eide *et al.* 2015; Onyenanu *et al.* 2018). In the outcrop analogue on which this modelling study is based, sandstone beds thicken upwards from 0.1–0.2 m at the base to 1.0–2.0 m at the top of each upwards-coarsening succession (Onyenanu *et al.* 2018). Scour density increases upwards from 0 to 0.02–0.06 m^{−1} in each upwards-coarsening succession (Onyenanu *et al.* 2018). The different models analysed in the two sensitivity studies (Table 2 and 3) are applicable to different intervals of an upwards-coarsening succession. In the lower part of such a succession, where mudstone interbeds are thicker than scour-fill sandstones and/or scours are absent, the sandstone beds are isolated (cf. Fig. 8d) and effective *k*_{v}/*k*_{h} is zero. In the central part of the succession, sandstone beds are connected vertically by scour-fill sandstones of low scour density (cf. Fig. 12b) and effective *k*_{v}/*k*_{h} is low but non-zero (e.g. 0.0001–0.002 for pot casts with scour densities of 0.01 m^{−1}: Fig. 13). In the upper part of the succession, sandstone beds are connected by densely spaced scour-fill sandstones (cf. Fig. 12h), resulting in a relatively high effective *k*_{v}/*k*_{h} (e.g. 0.001–0.002 for pot casts with scour densities of 0.04 m^{−1}: Fig. 13). Our results indicate that the proportion of sandstone provides a sound basis for approximating the range of effective *k*_{v}/*k*_{h} (Fig. 14), provided that sandstone scour-fill thickness is greater than mudstone-interbed thickness. The proportion of sandstone can readily be measured in core, and potentially indirectly deduced in wireline-log data using careful petrophysical analysis (e.g. Henderson *et al.* 2010). However, the thickness of scour-fill sandstones is unlikely to be a parameter that can be robustly measured in core and wireline-log data (e.g. Onyenanu *et al.* 2018), and must be constrained instead from appropriate analogue data (e.g. Fig. 4).

Although the models presented herein are constructed specifically for heterolithic, distal lower-shoreface sandstones, the results are applicable to heterolithic deposits of similar geometrical configuration and scale in other depositional environments. Intervals of interbedded, tabular mudstone interbeds and sandstone beds with basal erosional scours are characteristic of fluvial crevasse-splay, fluvial-dominated delta-front, and deep-water levee and lobe deposits (e.g. Scholle & Spearing 1982). Effective *k*_{v}/*k*_{h} can thus be estimated in such intervals using the relationships shown in Figure 14, in combination with measurements of sandstone proportion, sandstone scour-fill thickness and mudstone-interbed thickness.

## Conclusions

Distal intervals of wave-dominated shoreface and deltaic sandstone reservoirs comprise sandstone beds and mudstone interbeds, in which sandstone beds may be vertically connected by sandstone-filled scours of variable geometry. A suite of relatively small-scale (200 × 100 × 20 m) three-dimensional reservoir models based on data from an outcrop analogue (‘G2’ parasequence, Grassy Member, Blackhawk Formation exposed in Book Cliffs, east-central Utah, USA) have been analysed to establish the controls on the proportion of sandstone, sand-body connectivity and effective vertical to horizontal permeability ratio (*k*_{v}/*k*_{h}). These models bridge the gap in scale between well data, which sparsely sample a reservoir at high resolution, and typical reservoir-model grid cells.

The proportion of sandstone is controlled by the thickness of sandstone beds and mudstone interbeds, and by parameters that describe the geometry, dimensions and lateral-stacking density of sandstone-filled scours. Scours may either have a channelized geometry (gutter casts) or a semi-ovoid geometry (pot casts); each gutter cast has a larger volume than a pot cast of the same cross-sectional geometry. Successions containing abundant, wide and deep gutter casts have a higher proportion of sandstone than successions of similar bed thickness that contain sparse, narrow and shallow pot casts.

Sand-body connectivity is controlled by mudstone-interbed thickness and sandstone scour-fill thickness. The ratio between mudstone-interbed thickness and sandstone scour-fill thickness defines a threshold for sand-body connectivity. Erosion of the mudstone interbeds by scours of different geometries and lateral-stacking densities results in distinctly different spatial distributions and dimensions of mudstone-interbed remnants.

Effective *k*_{v}/*k*_{h} in the models is estimated using a streamline-based statistical method and by upscaled averaging. Although estimates of effective *k*_{v}/*k*_{h} derived from the two methods differ from each other by up to 1.5 orders of magnitude, they increase consistently as the proportion of sandstone is increased for both methods over the range of investigated parameters. Thus, for the purpose of effective *k*_{v}/*k*_{h} estimation, the proportion of sandstone serves as a robust proxy for the spatial distributions and dimensions of mudstone-interbed remnants that emerge from the combined effects of scour geometry and scour density, provided that scour thickness is greater than mudstone-interbed thickness.

These results indicate that sand-body connectivity and effective *k*_{v}/*k*_{h} can be estimated in distal intervals of interbedded sandstones and mudstones in wave-dominated shoreface and deltaic sandstone reservoirs, provided that mudstone-interbed thickness, the thickness of sandstone scour-fills and the proportion of sandstone are known. The thickness of sandstone scour-fills can be derived from appropriate outcrop analogue data, and mudstone-interbed thickness and the proportion of sandstone can be measured in core and wireline-log data. Outcrop analogues show that mudstone-interbed thickness decreases upwards and the proportion of sandstone increases upwards within upwards-coarsening successions in distal wave-dominated shoreface and deltaic deposits, while thick sandstone-filled scours preferentially occur offshore of coeval river mouths.

## Acknowledgements

The authors thank Schlumberger for providing access to the Petrel software package via an academic software donation. Lacunarity analysis was carried out using ImageJ and FracLac software, developed by Wayne Rasband (Research Services Branch, National Institute of Mental Health, Washington D.C., USA) and Audrey Karperien (Charles Sturt University, Sydney, Australia), respectively; we are grateful to the authors of these software packages for making them freely available. Finally, we are indebted to two anonymous reviewers and editor Sebastian Geiger for their constructive criticism and suggestions.

## Funding

Financial support for this work is gratefully acknowledged from the Petroleum Technology Development Fund (PTDF) of Nigeria (grant No. PTDF/E/OSS/PHD/UEU/549 to G. I. Onyenanu) and the Global Student Relief Fund of Imperial College London.

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