Introduction

The simulation of a reservoir and prediction of its performance requires an accurate description of the reservoir is necessary. The characteristics parameters like porosity and permeability are essential for determining the hydraulic flow unit (HFU) and the prediction of the reservoir performance. The concept of HFU has been developed to estimate permeability in the cored zone and then use their result in the un-cored zone. Depending upon the nature and availability of data, various methods are utilized to estimate HFU. Some of these are gamma-ray, flow zone indicator, cumulative flow and storage capacity curve, stratigraphic modified Lorenz plot, etc. (Zargari et al. 2013). Carbonate reservoirs are as crucial as sandstone since they contribute 60% of the world’s hydrocarbon containment. Characterization of carbonates is a challenging task owing to their composition. Calcite is composed of calcium carbonate, and it is chemically reactive and more prone to diagenesis than silica. Carbonates in their early stage of deposition exhibit high porosity and permeability. Dissolution and re-precipitation are often seen in carbonates with time. Due to this reason, carbonates mostly exhibit secondary porosity and permeability (Abdulelah et al. 2018). Diagenesis caused heterogeneities in carbonate reservoirs, which makes it tough to characterize carbonate reservoirs (Bagci and Akbas 2007). Thus, making it challenging to comprehend the storage and flow capacity. This heterogeneity resulted from depositional setting and post-depositional diagenesis. Apart from causing the problem in characterization, it also causes variation in pore size, grain size, and cementation, thereby damaging the direct relationship between porosity and permeability. Carbonate reservoirs exhibit a high level of heterogeneity at every scale, from core to field. The formation’s discretization is essential in terms of similar flow properties to manage heterogeneity in reservoir modeling. Recently, the interest in extending the rock-typing approaches is increasing, intending to identify the potential layers in complex lithology like carbonates. The hydraulic flow unit is also described as a reservoir section where the factors controlling the fluid flow are different from the rest of the reservoir. The factors that influence fluid flow are pore-throat geometrical attributes like mineralogy and texture (Amaefule et al. 1993). Various combinations of these factors lead to distinct rock flow units having similar fluid transport properties. Therefore, an HFU is included in several rock facies, depending on their depositional environment and mineralogical content. The hydraulic flow unit is an interval with specific specified properties, which has noticeable development in the reservoir. This property is considered a practical tool for grouping reservoir rocks and characterizing heterogeneity using porosity and permeability relationships (Masoud et al. 2020). Many attempts were made to describe carbonates using both geological and petrophysical properties in the recent past. The most appropriate and widely used method is the hydraulic flow unit (HFU). The classification of carbonate reservoirs into various HFUs helps in delineating distinct reservoir zones where its unique petrophysical parameter characterizes each flow zone. These flow zones help for a better understanding of the heterogeneity of the carbonate reservoir. HFU focuses on clustering similar fluid pathways in the reservoir where the lithofacies’ concept works on the distribution of various lithologies (Ebanks 1987). HFU is generally determined using rock quality index (RQI), flow zone indicator (FZI), and the void ratio (Abed 2014a; Bize-Forest et al. 2014). Flow zone indicator is dependent on the geological framework, characteristic of rock type, and pore-throat geometry of the rock. FZI is determined by reservoir quality index (RQI) and normalized porosity (φz) (Fujii and Fujimoto 1996). One of the parameters for estimating the thickness of the pay zone is permeability. In wells, where core data is unavailable, well logs are used to estimate rock mass permeability (Abed 2014b). The hydraulic flow unit methodology is a modification of the Kozeny-Carmen equation and the concept of mean hydraulic radius (Bize-Forest et al. 2014). In this method, it was considered that the reservoir rock to be made of a stack of capillary tubes. Poiseuille’s and Darcy’s Laws provide a relationship between porosity and permeability (Fahad and Holditch 2000).

The literature survey’s summary indicated significant importance and influence of characteristic parameters for the prediction of hydraulic flow in the carbonate reservoir. Therefore, in this investigation, the porosity, permeability, and grain density of the carbonate reservoir in Oilmax Mumbai, India, were determined. Based on the porosity and permeability values, the normalized porosity or pore volume-to-grain volume ratio infraction (φz), hydraulic flow unit (HFU), flow zone indicator (FZI), and reservoir quality index (RQI) were calculated to predict the hydraulic flow and feasibility of oil production in the study area.

Methodology

A total of thirty-two core samples data was collected from well A at the burial depth varied from 1618.86 to 1634.14 m from the Oilmax, Mumbai, India (Tables 2 and 3). The data in Tables 2 and 3 represents the average value of the characteristic parameters. The collected data were processed to determine the flow zone indicator (FZI), reservoir quality index (RQI), normalized porosity (φz), and hydraulic flow unit (HFU). This paper focuses on determining the above parameters and understanding the flow zone in the carbonate reservoir based on the data collected from the Oilmax, Mumbai India; therefore, the testing images are not provided.

The flow characteristics of the reservoir were determined by the flow zone indicator (FZI). This is one of the critical parameters to establish the relationship between the reservoir’s petrophysical properties at well log level. The flow zone in the reservoir represented using the term FZI. The mathematical representation of FZI (Onuh et al. 2017) is:

$$ \mathrm{FZI}=\frac{RQI}{\varnothing z} $$
(1)

where FZI is the flow zone indicator, RQI is the reservoir quality index, and φz is the normalized porosity

The reservoir quality index (RQI) is an important parameter for quantifying the reservoir’s flow zone. The RQI (Awad and Saleh 2020) is generally determined with the help of FZI. The mathematical representation of FZI is:

$$ \mathrm{RQI}=0.031\ \sqrt{\left(\frac{K}{\upvarphi \mathrm{e}\ }\right)} $$
(2)

where RQI is the reservoir quality index, K is the permeability (mD), and φe is the effective porosity (%)

Normalized porosity (φz) was calculated as (Enaworu et al. 2016):

$$ \varnothing z=\kern0.5em \frac{\varnothing e}{1-\varnothing e} $$
(3)

The NORMSINV function (Z value) calculates the inverse of the standard normal cumulative distribution function for a supplied probability value. The syntax of the function is NORMSINV (probability). The supplied probability is the probability value (between 0 and 1), at which the inverse of the standard normal cumulative distribution function is calculated (ExcelFunctions.net 2008-2021). The normal distribution is a continuous probability function that is given by the formula (ExcelFunctions.net 2008-2021):

$$ f\left(x,\mu, \sigma \right)=\frac{1}{\sqrt{2\pi}\sigma }{e}^{-\frac{{\left(x-\mu \right)}^2}{2{\sigma}^2}} $$
(4)

where μ is the mean of the distribution, σ2 is the variance, and x is the independent variable for which the function is evaluated. The cumulative normal distribution function is given by the integral, from -∞ to x, of the normal probability density function.

The data calculated based on the NORMSINV (probability) is shown in Table 1.

Table 1 The data calculated based on the NORMSINV (probability) function
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Lithology of the study area

The study area comes under the Mumbai offshore basin, India. The basin is located on the western continental shelf of India (Goswami et al. 2007). The lithology is principally of dark gray shales with subordinate sands. Carbonate fancies (Devgarh Formation) along the southern edge of Mumbai high are similar to foraminiferal-algal banks. The Mumbai platform has shale interbeds in limestone (Goswami et al. 2007). The limestone of the study area was micrite, biomicrite, and chalky type (Gupta et al. 1981). The thickness of limestone varies from shelf to deeper regimes. Shelf margined has thinner carbonates but is deposited at significant depth. Clastic sedimentation with carbonate bands is found in Diu Formation. The fancies within the Daman Formation are of significance as they host a handsome volume of hydrocarbon. In the early Miocene, thick carbonate sedimentation happened in Bombay high and its western part. The sediments are of fine grain in nature and clayey carbonate in composition (Goswami et al. 2007). The location map is shown in Fig. 1 (Kumar 2009).

Fig. 1
figure 1

Location map showing study area (Kumar 2009)

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Results and discussions

Reservoir characterization was carried out using core testing data obtained from a well in Oilmax, Mumbai, India. The core testing data is represented in Table 2. The porosity and permeability were determined using the porosimeter and permeameter at the laboratory scale. The data was collected from Oilmax, Mumbai, for the research purpose. The porosity and permeability varied from 0.3 to 20.5% and 0.002 to 1.484 mD at the burial depth from 1618.86 to 1634.14 m. The grain density ranged from 2.61 to 2.7 g/cc for the sample obtained from 1618.86- to 1634.14-m depth. The abrupt permeability reported in various samples (e.g., samples 2, 4, 6) is due to the possible occurrence of fracture, cemented matrix, and abundance of clay. The porosity range in samples indicates the reservoir’s porous nature, while the permeability was found significantly less. Few of the samples showed permeability less than 0.001 mD. The lesser values of permeability indicate a poor flow zone in the carbonate reservoir.

Table 2 Characterization results of carbonate reservoir in Oilmax, Mumbai
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The correlation between porosity and permeability was established, as shown in Fig. 2. The outlier was removed to develop the correlation. The prediction of permeability based on the porosity data was obtained using the trend line shown in Fig. 2. The scattering of data was observed in the correlation between the porosity and permeability, indicating porosity’s insufficiency alone to predict the variation in permeability. The trend line equation showed the poor linear and power correlation with R2 of 0.44. The poor correlation is attributed to the insufficiency of the porosity alone to predict permeability and the requirement of some other input parameters for permeability prediction.

Fig. 2
figure 2

Relation of permeability with porosity (limestone)

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In order to quantify the flow zone, further studies were carried out, and normalized porosity (φz), reservoir quality index (RQI), flow zone indicator (FZI), and hydraulic flow unit (HFU) were calculated based on the values given in Table 2. A similar technique was used elsewhere (El Sayed et al. 2021; El Sayed et al. 2020).

Hydraulic flow unit (HFU) method

The general method of identifying hydraulic flow units (HFUs) is based on the flow zone indicator (FZI), which is calculated based on the values of φz and RQI. The increase in the product of φz and RQI results in the quality of the reservoir. The φz and RQI were calculated as per Eqs. (2) and (3). The results of the computed values are represented in Table 3. The FZI was calculated as per Eq. (1) and represented in Table 4. The optimum number of HFUs was obtained by the correlation between φz and RQI (Fig. 3). The similar FZI values lie on the straight line of the log-log-scale plot of RQI and φz. The other FZI values scattered from the straight line. This indicated the similar pore throat as well as flow zones in the samples. Based on Fig. 2, all the data fall in seven rock types and, hence, confirmed the seven HFUs. The FZI varied from 0.071971 to 6.024804 at a depth of 1618.86 to 1634.14 m for HFU1 to HFU7, respectively. Each of the HFUs exhibited uniform rock characteristics with a uniform fluid flow zone. The HFUs also exhibited the porous nature of the carbonate rock. The HFUs with higher FZI values showed an improved quality of fluid flow zone in the rock’s porous structure.

Table 3 Calculated data of normalized porosity (φz) and reservoir quality index (RQI)
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Table 4 FZI Values calculated for the probability plot method
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Fig 3
figure 3

Log-log graph between φz and RQI

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The probability plot is easy to predict and understand than that of the histogram; therefore, the probability plot between the FZI and Z values was developed, as shown in Fig. 4. NORMSINV function was used to calculate the probability of inverse normal cumulative distribution (Z value), which has a mean and standard deviation. The distribution of data forming a distinct straight line indicated each of the HFUs. Hence, it is possible to obtain the separate HFUs for each set of the data. The distortion of the straight line to some degree was observed due to the superposition effect. In this plot, the regression coefficient for all HFUs varied from 0.65 to 0.97, indicating a poor to approximately good flow zone in the reservoir. The HFU4, HFU5, and HFU7 showed more or less similar flow zone as the slopes are almost equal. The probability plot confirmed the highly porous nature of the reservoir with a good flow zone in most samples.

Fig. 4
figure 4

Log graph between FZI and cumulative probability function (Z value)

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The correlation between the porosity and permeability classified by FZI for different HFUs is represented in Fig. 5. A classified and satisfactory relationship with R2 varied from 0.93 to 1 was observed between the porosity and permeability correlations obtained by the FZI curve for each HFUs. A non-linear and predictable relationship was obtained in Fig. 5. The non-linearity in the correlation between porosity and permeability is due to the reservoir’s highly porous nature. This is also important to know that a high porous reservoir is may not necessarily be highly permeable. The correlation coefficient was used to predict the significant deviation in the characteristic properties in the reservoir. Figure 4 showed that HFU 3 with an R2 value of 0.93 indicated more deviation than the other. The overall observation indicated the highly porous nature of the carbonate reservoir in the study area. The HFU1 showed poor flow compared to the other HFUs, while HFU2 showed good flow because of the highest R2 value, i.e., 1. Other HFUs like HFU3 HFU7 indicated near to good flow zone in the carbonate reservoir of the study area.

Fig. 5
figure 5

Semi-log graph between porosity and permeability

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Conclusions

The following conclusions were drawn for the carbonate reservoir at Oilmax, Mumbai, India, based on the data collected and the statistical analysis.

  • The variation in porosity and permeability was observed from 0.3 to 20.5% and 0.002 to 1.484 mD at the burial depth from 1618.86 to 1634.14 m, respectively.

  • The grain density varied from 2.61 to 2.7 g/cc for the sample obtained from 1618.86- to 1634.14-m depth.

  • The porosity range varied from 0.3 to 20.5% indicates the reservoir’s porous nature, while the permeability was found significantly less. Few of the samples showed permeability less than 0.001 mD. The lesser values of permeability indicate a poor flow zone in the carbonate reservoir.

  • The poor correlation between porosity and permeability showed the insufficiency of the porosity alone to predict permeability and the requirement of some other input parameters for permeability prediction.

  • The φz and RQI varied from 0.070663812 to 0.257861635 and 0.029803 to 0.254999 at a depth of 1618.86 to 1634.14 m. The values were used for the prediction of the FZI using a log-log plot.

  • Each of the HFUs exhibited uniform rock characteristics with a uniform fluid flow zone. The HFUs also revealed the porous nature of the carbonate rock. The HFUs with higher FZI values showed an improved quality of fluid flow zone in the rock’s porous structure.

  • The HFU4, HFU5, and HFU7 indicated more or less similar flow zone as the slopes were almost equal. The probability plot confirmed the highly porous nature of the reservoir with a good flow zone.

  • The HFU1 showed poor flow compared to the other HFUs, while HFU2 showed good flow because of the highest R2 value, i.e., 1.