# Stochastic modelling of spatial variability of petrophysical properties in parts of the Niger Delta Basin, southern Nigeria

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## Abstract

Three-dimensional models of petrophysical properties were constructed using stochastic methods to reduce ambiguities associated with estimates for which data is limited to well locations alone. The aim of this study is to define accurate and efficient petrophysical property models that best characterize reservoirs in the Niger Delta Basin at well locations and predicting their spatial continuities elsewhere within the field. Seismic data and well log data were employed in this study. Petrophysical properties estimated for both reservoirs range between 0.15 and 0.35 for porosity, 0.27 and 0.30 for water saturation, and 0.10 and 0.25 for shale volume. Variogram modelling and calculations were performed to guide the distribution of petrophysical properties outside wells, hence, extending their spatial variability in all directions. Transformation of pillar grids of reservoir properties using sequential Gaussian simulation with collocated cokriging algorithm yielded equiprobable petrophysical models. Uncertainties in petrophysical property predictions were performed and visualized based on three realizations generated for each property. The results obtained show reliable approximations of the geological continuity of petrophysical property estimates over the entire geospace.

## Keywords

Petrophysical property Stochastic techniques Geostatistics Horizon picking Niger Delta## Introduction

Evaluation of petrophysical properties is a vital component in reservoir characterization in the oil and gas industries. The conventional petrophysical properties needed for hydrocarbon exploration, characterization, field development, evaluation and reservoir management are lithology, porosity, permeability, net-to-gross and fluid saturation (Emujakporue 2017). Detailed petrophysical property characterization is critical during the oil and gas exploration, production, management and surveillance stages. Parameters such as porosity, permeability and volume of sand and shale that are of interest to geoscientists and engineers are products of complex chemical and physical processes. It is significant to understand their spatial orientations and scales as well as the uncertainties that characterize these variables in space from one well to another in a particular field, for optimal hydrocarbon production. Permeability which is a dynamic property of the reservoir is difficult to characterize due to the irregular nature of pore structures occasioned by depositional and diagenetic alterations. The spatial distribution of rock permeability in heterogeneous reservoir systems is problematic since its determination has no direct solution (Fegh et al. 2013). Generally, two direct and reliable ways of determining permeability are the laboratory measurements and well test methods (Noah and Shazly 2014), which are expensive, limited to either the well bore or certain sections of the well bore and cannot be applied to all the wells drilled in the field. Alternatively, estimation of permeability from porosity–permeability cross-plot generated through linear correlation of porosity and permeability data is being used and has gained wide patronage in the oil and gas industries (Jennings Jr and Lucia 2001; Rezaee et al. 2006). However, this method can provide adequate result in sandstone reservoirs with little or no complexities and may not be relied upon when considering reservoirs with higher degree of heterogeneities due to episodes of diagenetic alterations (Edigbue et al. 2015; Ebong et al. 2019).

The application of indirect method, i.e. geostatistical methods in complex reservoir situations can provide information such as rock types, fluid contents and other physical processes within the formation. It involves the use of seismic sections, wireline logs and measurement while drilling and yields reliable results in a less expensive and faster way and can be performed over a large area (Dowell et al. 2006). Results generated from direct methods which are localized are used to estimate petrophysical properties elsewhere beyond wells via stochastic and fuzzy logic processes (Esmaeilzadeh et al. 2013). Geostatistical assisted modelling which is one of the many methods often applied to reservoir characterization (Bueno et al. 2011) has the ability to integrate several groups of information in the generation of suitable reservoir models that fit any given subsurface geologic condition of interest (Caers and Zhang 2002; Liu et al. 2004). Geostatistical principles also ensure that the geologic realities of the reservoirs are not lost in the course of the model building (Chen et al. 2011). The geologic model of the training image includes lithofacies assemblages from seismic sections, well log and production data as constraints and the petrophysical model which consist of the parameters of each facie (Caers 2002; Gonzalez and Reeves 2007). Akin to the deterministic approach, hard data points are usually conserved where it exists and have been interpreted and soft data where they are useful (Zhou et al. 2014; Wilson et al. 2011). In contrast to the deterministic approach, geostatistics provides several plausible outcomes (Zarei et al. 2011). When performed over a large area of the reservoir, results from such spatial investigations permit better characterization of the reservoir and effective recoverable reserves needed for adequate reservoir management can be estimated (Soleimani et al. 2017). In this study, the stochastic technique which is capable of generating more reliable numerical petrophysical models was used, due to the difficulty of predicting the spatial distribution of these properties deterministically (Ma et al. 2008).

This study which involves the estimation of petrophysical properties is aimed at estimating reservoir properties at well locations and predicting the spatial variability of these properties elsewhere within the field.

## Stochastic reservoir modelling in brief

Petrophysical properties such as porosity, permeability and fluid saturation are the requirements for building 3D grid during flow simulation. These properties are limited to well locations and can pose a great deal of uncertainty in grid block property assignment. Building alternative numerical models or images of these petrophysical properties, taking into consideration the unknown aspects of the spatial distribution is generally referred to as stochastic reservoir modelling (Deutsch 1992). The traditional geostatistical approach to reservoir property modelling is through the sequential simulation of facies and/or petrophysical properties (Arpat 2005). The sequential simulations usually performed for stochastic reservoir modelling are the sequential Gaussian simulation which is parametric and the sequential indicator simulation, i.e. non-Gaussian and nonparametric approach (Xu et al. 1992). These two are performed using computer codes (Deutsch and Journel 1998). The sequential simulation path is such that it visits each node of the model and simulated values are drawn from conditional distribution of the values at the node given the neighbouring subsurface data and previously simulated values (Arpat 2005). Simulation techniques generate multiple realizations of unknown spatial distribution of observations, without losing the originality of the observed data. The differences between realizations provide quantitative measure of uncertainty (Vasquez 2014).

Since sequential simulation implies the reproduction of desired spatial properties through sequential use of conditional distributions, we can represent the primary variable of interest distributed over a field * A* as, \(z_{1} (\varvec{u})\) and the coordinate vector, \(\varvec{u} \in \varvec{A}\). The primary variable in this work is the well log data from which porosity, permeability, fluid saturation, \(\varvec{u},\) were derived on a three-dimensional scale. The integrals of \(z_{1} (\varvec{u})\) are associated with total pore volume. Randomly distributed values \(z_{1} (\varvec{u}_{\alpha } ),\alpha = 1, \ldots , n_{1}\) are available at well locations \(\varvec{u}_{\alpha } \in A\), and the seismic data \(z_{2} (\varvec{u}'_{\alpha } ),\alpha = 1, \ldots , n_{2}\) of related nature are available at much larger number \(n_{2}\) locations, \(\varvec{u}'_{\alpha } \in \varvec{A}\). The purpose of this coupled approach is to produce several highly resolved maps of the distribution \(z_{1} (\varvec{u})\) over the field,

*A*, making the most of both hard \(\{ z_{1} (\varvec{u}_{\alpha } ),\alpha = 1, \ldots , n_{1} \}\) and soft \(\{ z_{2} (\varvec{u}'_{\alpha } ),\alpha = 1, \ldots , n_{2} \}\) data. The term “hard data” is used to emphasize the fact that the modelling method should exactly reproduce the point data obtained from wells at its location. On the other hand, “soft data” which is the seismic section is used as constraint to guide the process and may not be reproduced exactly.

Data conditioning is significant especially when seismic data is utilized in estimating petrophysical properties. Hence, both hard and soft data must be conditioned before usage. Hard data conditioning may not only mean an exact reproduction of the original data but also requires the generation of adequate continuity around the defined area as dictated by the geological continuity model (Arpat 2005). For instance, if the geology of the reservoir shows evidence of thin, continuous horizontal layer of shale, a well data which indicates the shale value at a particular location should be honoured by generating such shale layer at neighbouring data location. Variogram-based simulation algorithms always exactly honour hard data values via the use of kriging, which is an exact interpolation method (Lima 2005). Soft data which refers to data sets obtained from indirect measurements, e.g. seismic sections, represents the “filtered” view of the subsurface heterogeneity (Arpat 2005). Since the filtered view is not the exact physical response, it is usually approximated by a mathematical model, which often times is referred to as “forward model”. Soft data sets are usually integrated using variogram-based methods (Deutsch and Journel 1998).

### Estimation algorithm

All regression algorithms, including kriging, full cokriging, kriging with an external drift model or collocated cokriging with a Markov-type cross-covariance model, are low-pass filters which tend to yield an over-smoothed image of the actual spatial variability of the primary data \(z_{1} .\) Such characteristics are suited for static volumetric calculations such as mapping of reservoir top, but may under-represent extreme values (conditional bias) when applied to dynamic modelling, e.g. spatial distribution of permeability modelling (Onyekwelu 2013). It may also remove certain patterns of spatial connectivity arising from barriers and flow paths for such primary data as permeability. To overcome these shortfalls, stochastic simulation (i.e. full-pass) algorithms that reproduce full spectrum (i.e. covariance) of spatial variability can be applied. The sequential Gaussian simulation with collocated cokriging algorithm provides a more reliable approach since it involves sequential steps in drawing alternatives, equiprobables and joint realizations of the random variable component from random function model. The various realizations represent possible images of the spatial distribution of regionalized variable over the domain. Each realization represents the properties that have been employed in developing the model. Details of this method can be found in Cekirge et al. (1981), Xu et al. (1992), Deutsch (1992).

## Location and geology

^{2}in area of land extending towards the Atlantic Ocean (Kulke 1995). Wu and Bally (2000) classified the NDB as a classical shale tectonic province due to the presence of over-pressured shales and shale diapiric structures associated with the area. The overall sediment volume in the Niger Delta is ~ 500,000 km

^{3}(Hospers 1965) with ~ 10 km sedimentary thickness around the depocenters (Kaplan et al. 1994).

The NDB formed along a failed arm of the triple junction system (aulacogen) was initially developed in the late Jurassic following the breakup of the Gondwana into the African and South American plates (Burke 1972; Whiteman 1982). The southwestern coast of Nigeria and Cameroon which harbours two of the arms formed the West African passive continental margin, while the third failed arm developed to form the Benue Trough (Lehner and De Ruiter 1977). During Cretaceous to Tertiary time, synrift sediments were accumulated within the basin with the Albian age sediments being the oldest dated sediments. Several episodes of transgression and regression led to the deposition of marine and marginal marine sediments and carbonates (Doust and Omatsola 1990). During the Early Santonian—Late Cretaceous, the occurrence of basin inversion marked the end of the synrift phase. Renewed subsidence resulting from the separation of the continents paved the way for the sea to transgress into the Benue Trough. Progradation of the Niger Delta clastic wedge continued during the Mid-Cretaceous into the depocenter located on top of the deformed continental margin at the spot where the triple junction was situated. During the Late Cretaceous, sediment progradation was interrupted by episodes of transgression (Whiteman 1982).

## Materials and methods

*X*

_{obs}is observed values and

*X*

_{model}is modelled values at time/place

*i*.

## Results and discussion

Summary of reservoir tops and bases and thicknesses at each well

Well number | Reservoir | |||
---|---|---|---|---|

K200 (ft) | K300 (ft) | |||

Top | Base | Top | Base | |

Well_001 | 8370 | 8450 | 8470 | 8670 |

Well_002 | 8230 | 8380 | 8450 | 8590 |

Well_003 | 8190 | 8300 | 8400 | 8660 |

Well_004 | 8200 | 8300 | 8430 | 8620 |

Well_005 | 8190 | 8360 | 8430 | 8620 |

Well_006 | 8530 | 8710 | 8800 | 8900 |

Well_007 | 8240 | 8310 | 8380 | 8480 |

Well_008 | 8270 | 8310 | 8430 | 8520 |

Well_009 | 8050 | 8120 | 8190 | 8320 |

Well_010 | 8420 | 8460 | 8560 | 8690 |

Maximum | 8530 | 8710 | 8800 | 8900 |

Minimum | 8050 | 8120 | 8190 | 8320 |

Ranges of some petrophysical properties

Reservoir | Petrophysical property ranges | |||
---|---|---|---|---|

Total porosity | Permeability (mD) | Shale volume | Water saturation | |

K200 | 0.25–0.35 | ≥ 100.0 | 015–0.25 | 0.27–0.29 |

K300 | 0.15–0.28 | ≥ 100.0 | 0.10–0.20 | 0.28–0.30 |

## Conclusion

## Notes

### Acknowledgements

We acknowledge Shell Petroleum Development Company (SPDC) Nigeria Limited for providing the data used for this study. The authors are also grateful to the Director, Directorate of Petroleum Resources for the strong recommendation to SPDC Nigeria limited authorizing the release of their data for this work. We also acknowledge with thanks the University of Calabar, Calabar, for providing the tools and environment for research work.

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