Introduction

Drilling specialists often need a subsurface pore pressure guide to have safe and efficient drilling. Abnormal pressures (especially overpressure) are serious hazards that can be avoided by knowing the subsurface pore pressure before drilling (Zhang 2011). In addition to drilling hazards, understanding the distribution of overpressure, which origins from disequilibrium compaction of sediments, and determining a high-pressure zone are the important parameters in hydrocarbon exploration. Lots of research has been carried out on the mechanisms of fluid pressure in hydrocarbon reservoirs (Sun et al. 2019, 2021). Concerning fluid flow mechanisms inside shale reservoirs, Sun et al. (2021) investigated a molecular simulation of methane flow behavior through a realistic organic shale matrix under displacement pressure. Generally, the most widely used pore-pressure-estimation methods are based on two categories. These methods are based on empirical relationships derived from statistical data or based on laboratory measurements and rock physics models. However, most of these methods employ seismic data for this purpose. One of the earliest methods of pore pressure estimation is the outcome of studies performed by Pennebaker (1968). These studies showed the deviation of the pore pressure indicator from the normal compaction trend line. Effective stress methods: It works through Terzaghi’s principle, in which the difference between the total confining stress and the pore pressure controls the sediments' compaction. This method is made up of three steps. (1) Effective stress calculation from pore pressure indicators, (2) bulk density, calculation of the overburden stress. (3) The difference between overburden stress and effective stress gives pore fluid pressure (Das and Mukherjee 2020; Radwan and Sen 2021). Bowers (1995) calculated the effective stress using measured shale pore pressure and overburden pressure (Terzaghi and Peck 1948) and analyzed the corresponding acoustic velocities using the well data in the Gulf of Mexico. Bowers’s method is recognized as an effective stress-based approach in which the buried depth and acoustic velocity establish a relationship and are used to estimate the pressure. Consequently, the two mechanisms of loading and non-consolidation are considered. Concerning this relationship, the effective stress outside the velocity inversion zones is calculated from the intact curve. For the zones of velocity inversion case, the well logs should be utilized to determine the correct velocity inversion diagram. Holbrook et al. (2005) presented the porosity-dependent effective stress to predict pore pressures. To estimate the pore pressure using porosity data, Heppard et al. (1998) proposed an empirical relation for porosity similar to Etonʹs (1972) method. Zhang (2011) obtained an empirical relation to estimating the pore pressure by using porosity. Raiga-Clemenceau et al. (1988) presented a relation to estimating porosity via p-wave velocity (time-average). They replaced this relation with Zhang’s (2011) relation, and they could estimate the pore pressure using either time average or p-wave velocity. Wiley et al. (1956) estimated porosity through an experimentally optimized mean-time relationship. In which, the mean porosity relation of two derived equations is used to calculate the pore pressure and its slope. Unfortunately, pore pressure prediction in carbonate reservoirs using conventional methods is still a challenging task that results in high uncertainties, and so far there is no specific method generally accepted. The effects of chemical process and cementation post diagenesis on porosity are more important than the mechanical compaction in most carbonate rocks, so the conventional pore pressure prediction methods implicitly or explicitly using the normal compaction trend fails to give reliable results (Wang and Wang 2015). Banik et al. (2014) show that the pore pressure can be mainly a function of a single variable (acoustic impedance). They suggested the function σp = f (Ip) is continuous and since the deposition process in different basins or even different regions of a similar basin may be different, they suggest that the parameters defining the function or even the form of the function itself would possibly want to be adjusted in each different zone of interest.

In carbonate reservoirs in field applications, conventional pore pressure prediction methods such as Eaton and Bowers are still used which exposes the well drilling engineering to high risks. The Asmari reservoir is one of the famous Iranian carbonates reservoirs. The main purpose of the current study is to predicate the pore pressure of the Asmari Formation based on acoustic impedance using both seismic data and available well logs located in one of the Iranian oil fields. The workflow of the proposed method is schematically shown in Fig. 1. This figure provides a schematic overview of the prediction of pore pressure directly from the acoustic impedance.

Fig. 1
figure 1

The workflow illustrates the steps involved in the pore pressure processes algorithm

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Geological setting

We assess the performance of the pore pressure estimation via acoustic impedance for one of Iran’s oil fields. The study oil field is in the dip section of the north of Dezful embayment and about 60 km distance from south Ahwaz. This oil field is constrained to the Ahvaz oil field from northwest and west is close to Ab-Teymor oil field and from the northeast is a juxtaposition with the Shadegan oil field. This oil field was explored by drilling the MI-1 well number in 1963, and production from this field was started in 1974. The oil reservoir is under the saturated state, and the primary depth of oil–water contact has been estimated at 2272 m. This oil field comprises Asmari and Bangestan. The Asmari Formation lithology include sandstone, limestone, and dolomite which consists of a double porosity system consisting of fractures and matrices.

Based on the different formationsʹ surface maps, the Asmari Formation is an asymmetric anticline with a mild 0°–10° slope in the northwest–southeast direction. Depending on the Zagros regionʹs natural structure, the angle of inclination in the north wing is greater than about 5–6 degrees in the southern flank (Alavi 2004). Although part of the oil and gas are produced from porous layers, most of the production is related to fractures and open cracks. Due to the low slope of the anticlines in Khuzestan, the expansion of the areas with tensile fractures is low, and most of the oil production is from the matrix. This is also true of the Asmari Formation in the studied field. The lithology of the Asmari Formation includes sandstone, shale, and carbonate. Studies of this formation show that this formation has only limited joints and cracks and no fractures. Therefore, it can be concluded that the limited fractures in the carbonate layers do not affect the production of this reservoir (For more information see Alavi 2004).

Methodology

The pore pressure estimation method using acoustic impedance

As mentioned, the estimation of subsurface pore pressure as the pre-drilling analysis is an essential stage in hydrocarbon exploration. Nowadays, methods based on the seismic velocity and a density function are among the most widely used approaches in pore pressure estimation, in which overpressure zone estimation is of interest (Sayers et al. 2002). Of note pore and overpressure, estimation demands the availability of two parameters (velocity and density) in the desired range. Accurate density modeling is one of the most essential tasks in reservoir modeling. However, the accuracy of the density model is often challenged when seismic data alone are inverted for density. Similarly, geostatistical models also face challenges in terms of accuracy, as they frequently require adjustment of the model parameters for each reservoir rock type. The density and velocity are derived from the acoustic impedance of seismic data, in which its conversion to both density and velocity suffers from uncertainties resulting in low resolution estimates. This drawbacks of the conventional approach directly transmit uncertainty into the pressure prediction stage. Furthermore, acoustic impedance resulting from the seismic inversion is much higher with confidence than density (Chatterjee and Yalamanchili 2017).

Rasolofosaon and Tonellot (2011) investigate the impedance-based method for the quantitative evaluation of effective stress and detection of overpressures which shows that the pore pressure can be estimated as a continuous function of acoustic impedance. Consequently, simplifies the process of estimating pore pressure. This method is directly based on acoustic impedance, thereby eliminates the errors due to conversion step.

The Bowers method (Bowers 1995, 2002) showed how pressure could be derived from acoustic impedance as well as estimated density and velocity data based on the cognitive dependence of acoustic velocity on pressure.

$$ v = v_{0} + a\sigma_{e}^{b} $$
(1)
$$ \sigma_{e} = (\frac{{v - v_{0} }}{a})^{\frac{1}{b}} $$
(2)

where, σe denotes effective stress, and V0, a, and b are constant parameters.

Bowers (1995) proposed a certain amount of these parameters for his desired marine basin, while these parameters were determined through calibration with pore pressure data at wells in each region.

Multiplying both sides of Eq. 1 with density reads:

$$ Ip = Ip_{0} + c\sigma_{e}^{b} $$
(3)
$$ \sigma_{e} = (\frac{{Ip - Ip_{0} }}{c})^{\frac{1}{b}} $$
(4)

which leads to a new equation in terms of acoustic impedance. Banik et al. (2013) described Eqs. 3 and 4 as the modified Bowers’ method for effective stress. Likewise, the overburden pressure \({{\varvec{\sigma}}}_{{\varvec{O}}{\varvec{v}}}\), generally referred to as the density of sediments on the formation, and can be written as:

$$ \sigma_{Ov} = g\sum\limits_{j} {\rho_{j} \Delta h_{j} = } g\sum\limits_{j} {Ip_{j} \Delta t_{j} } $$
(5)

where, ρj and Ipj are impedance and density; g is the gravitational constant and \({\Delta {\varvec{t}}}_{{\varvec{j}}}\) and \({\Delta {\varvec{h}}}_{{\varvec{j}}}\) are the sample intervals in the time and depth domain, respectively. In Eq. 5, the summation performed over all samples collected from the considered formation.

Conventional pore pressure evaluation is primarily based on Terzaghi’s (and Biotʹs) effective stress principle which states that total vertical stress (σov) (or overburden stress) is equal to the sum of the effective vertical stress (σe) and the formation pore pressure (σp). The Biot’s effective stress coefficient (Biot and Willis 1957) alters the effective stress principle, in which effective stress principle eventually is given by:

$$ \sigma_{e} = \sigma_{Ov} - \alpha \sigma_{p} $$
(6)

where the Biot’s stress coefficient of (α) usually is to be 1.because, in sedimentary rocks, this coefficient is assumed to be the same. Banik et al. (2014) show that the pore pressure is mainly a function of a single variable (acoustic impedance). They suggested the function σp = f (Ip) is continuous and since the deposition process may be different in different basins or even different regions (of a similar basin). They suggest that the parameters defining the function or even the form of the function itself would possibly need to be adjusted for each different zone of interest. The function f can be linear or even nonlinear with higher-order polynomials of inverse acoustic impedance in a region with complicated sedimentation processes. In other words, the form of the function and its related parameters need to be adjusted in each region, which is similar to the effective Bower’s pressure ratio and velocity ratio, Eq. 1. The parameters V0, a, and b are varying by region and basin.

Banik et al. (2014) used the subsequent function successfully to convert the inverted acoustic impedance at once into pore-pressure volumes for the deepwater GOM Walker Ridge, Green Canyon, and Keathley Canyon areas:

$$ \sigma_{p} = A + B/I_{p} $$
(7)

where, A and B are transform parameters that need to be determined via the calibration of Eq. 7 in the well located in the same basin with a geologic environment similar to the target area. Banik et al. (2014) calculated effective stress, overburden pressure, and pore pressure directly from the acoustic impedance as well as velocity and density methods (Eaton 1972; Bower 1995) using the above equations for a deep well. They showed that both methods give practically similar results. The pore pressure estimation obtained from the usage of direct impedance can reduce errors due to the conversion of impedance to the density and velocity, and calculate the pore pressure only as a simple single-variable function. In what follows, we utilize the direct impedance method to predict pore pressure in a carbonate reservoir case study.

Available data of the oil filed study

We used 3D migrated seismic data related to the Khami group. The data used in this study are a part of the complete seismic data set of this region, shown in Fig. 2a. As mentioned above, this fieldʹs reservoir ranges from Asmari to the Pabdeh horizon. Seismic horizons can be selected using the information from the wells found in the wells. In Fig. 2b, two important seismic horizons, the Asmari and Pabdeh horizons are shown in the seismic cube.

Fig. 2
figure 2

The seismic cube used in this study. a An inline, a crossline, and a z-slice as an instance. b The Asmari (upper) and Pabdeh (lower)

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There are total 60 wells in this field. Due to the range of the reservoir and information related to acoustic, density logs, check shot data, and other information needed, 15 wells were used in this study, in which we used 10 wells due to the completeness of the required information for inversion. As mentioned in the previous sections, to estimate the pressure and calibrate the relationships between pressure and acoustic impedance in this area, the wellbore pressure data are required, RFT data. The remained well logs were used to calibrate the pressure estimation relationship.

Acoustic impedance inversion

One of the important steps in seismic data analysis is to establish the relationship between seismic data and the geological information of the studied area (Avseth et al. 2005). Correlation between well logs and seismic data results in identifying and making a correlation between the seismic horizon and the stratigraphy of the reservoir. One of the essential steps before seismological analysis is the depth conversion of the seismic data. The seismic data are recorded in the time domain, however, the well logs are in-depth domain. The common method of interpreting seismic data is to use a synthetic seismogram to match well logs with seismic data. In this study, we used the sonic and density logs of one of the vertical wellbores. Of note that before conducting any evaluation, corrections and depth evaluation of the synthetic seismogram are performed based on check shot and well logs (Rahimi and Riahi 2020). One of the usual methods is to compare seismic traces with the synthetic seismogram at the vertical well locations. In this study, we select the geological horizons (Asmari and Pabdeh Horizon) at the well and then connect the wells to each well by picking the horizon.

Acoustic impedance is the outcome of post-stack inversion. The post-stack seismic inversion technique is the most common approach where high-resolution models of the subsurface are generated, and the effect of the wavelet is eliminated (Chen and Sidney 1997).

There are many post-stack seismic inversion algorithms, in this present study, a Model-based inversion (MBI) algorithm is used. MBI is based on the convolution theory. A simple initial acoustic impedance model is convolved with the wavelet to obtain a synthetic trace compared with the actual seismic trace. (Mallick 1995). Then, this acoustic impedance model is changed iteratively until the resulting synthetic traces are in good agreement with the actual traces. Figure 3 shows the flowchart of the seismic inversion method used in this study.

Fig. 3
figure 3

Flowchart of post-stack seismic inversion used in this study

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In this study, the inversion is performed using Hampson Russell software. After estimating an optimal wavelet and building, the initial model inversion is performed by the model-based deterministic method. This method is performed with the Generalized Linear Inversion (GLI) algorithm. The model is modified through iterations until the calculated synthetic traces match with the real seismic traces. The algorithm also adds user-defined constraints such that the updated model lies in a lower–upper bound range. The constraints and user-defined parameters used in this method are reported in Table 1.

Table 1 Parameters and constraints used in our inversion modeling
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Figure 4 shows a schematic of the major steps performed in the Inversion process and inversion results for well No. 13 is shown in Fig. 5.

Fig. 4
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Schematic of the steps involved in the inversion process

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Fig. 5
figure 5

seismic inversion results in well No. 13. Left curve (red) is overlaid with log-calculated impedance (blue). A Zero-phase wavelet extracted from the seismic cube is shown in the next track. Using impedance from inversion result, reflectivity extracted then convolved with wavelet to generate synthetic trace (red). High correlation with real seismic trace can be interpreted as inversion accuracy. The correlation between the real seismic trace (black) and the synthetic trace (red) is 0.99 at this well location

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Pore pressure estimation

So far, a cube of acoustic impedance in the reservoir area has been calculated using the seismic inversion process. We continue our analysis to estimate the pore pressure using a direct relation from acoustic impedance, which was discussed in the previous section. To calculate the pore pressure in the study area, we need to find a relationship between calculated acoustic impedance and pore pressure. Here, the following function is used to convert impedance to pore pressure.

$$ \sigma_{p} = a + {\raise0.7ex\hbox{$b$} \!\mathord{\left/ {\vphantom {b {(1 + c \times Ip)}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${(1 + c \times Ip)}$}} $$
(8)

where, the coefficients a, b, and c are the constant coefficients that must be determined from the calibration of Eq. 8 using pressure data in available wells in this region; Ip and σp are inverted acoustic impedance and pore pressure, respectively.

Having the acoustic impedance determined, pore pressure estimation only requires constant coefficients. The constant coefficients of a, b, and c can be obtained by calibrating this relation at the locations of the wells in which the pressure and impedance values are known. It should be emphasized that to validate and determine the accuracy of the obtained relationship, the predicted pressure from this relation must be compared with the actual pressure. For this purpose, from 5 wells with RFT pressure data available, 3 wells were used to calibrate Eqs. 8, and 2 wells were used for validation and quality control of the method. Therefore, the three wellsʹ pressure and impedance are used to calibrate Eq. 8. The crossplot of RFT pressure data versus acoustic impedance for each sample of the three wells is shown in Fig. 6.

Fig. 6
figure 6

The crossplot of RFT versus acoustic impedance for three wells

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In this step, the MATLAB code and curve-fitting tool are used to calibrate Eq. 8 and determine the constant coefficients. For this purpose, by plotting the RFT via impedance, the desired relation based on relation 8 was obtained, as shown in Fig. 7a. After determining the desired equation, the best line that provides the best fit to the data is obtained based on the minimizing least squares error. Figure 8b shows the coefficients of the best-fitted line.

Fig. 7
figure 7

curve-fitting tool. a Insert the desired equation as an arbitrary equation to fit the curve onto the data. b The best fit line to the data is based on the equation shown above

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Fig. 8
figure 8

a The error bars of each sample data from the fitted line showing each dataʹs error. b Fixed coefficients a, b and c after calibration and curve-fitting on data

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Figure 8 shows the obtained errors for each sample (distance from the fitted line) and constant coefficients.

After determining the constant coefficients, a simple single-variable equation was obtained wherein the pore pressure is directly dependent only on the inverted acoustic impedance. So, by inserting the inverted impedance value in this calibrated equation can be achieved pore pressure at each point.

Figure 9 shows the acoustic impedance output from the inversion process.

Fig. 9
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Acoustic impedance cube

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The pressure cube is obtained from the acoustic impedance cube in the same interval, as shown in Fig. 10.

Fig. 10
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Pressure cube calculated from acoustic impedance with the calibrated parameters

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Fig. 11
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Comparison of estimated pore pressure values (blue line) and RFT values at a) well 8 and b well 55

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Quality control of impedance pressure relation

Like other modeling processes, quality control of the calculated pore pressure cube seems necessary to validate the results. To this end, wells No. 8 and No.55 that were unused in the procedure is used to assess the pore pressure accuracy.

Accordingly, the pore pressure values obtained using the calibrated pressure-impedance relation is compared with the actual pressure values measured at the wells. Figure 10 shows the results of this comparison which indicates the reliability of the obtained pore pressure model. In this figure, the red dots show the actual measured pressure value at the wells as a function of depth. The blue dots (and continues blue line) show the pore pressure values from the direct pressure-impedance relationship, respectively (Fig. 11).

Here, we performed an uncertainty analysis technique to further evaluate our results. The Standard Error (SE) can be used as a guide to interpreting the possible sampling error. It shows how close the estimate based on sample data might be to the value that would have been taken from the whole population. Confidence intervals use the standard error to derive a range in which expected the true value is likely to lie. A 95% confidence level is frequently used. If a cross plot of samples is drawn and the mean of each calculated, 95% of the means would be expected to fall within the range of two standard errors above and two below the mean of these means. Figure 12 shows the cross plot of acoustic impedance and pore pressure values with error bars and 95% confidence intervals.

Fig. 12
figure 12

The uncertainty analysis of predicted pore pressure using direct acoustic impedance. The X and Y axes are acoustic impedance and pressure values, respectively. Also, the solid and dashed lines are linear regression and 95% confidence intervals, respectively

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Discussion of the results of the estimated pore pressure

In the process of predicting a parameter, it may seem sufficient to test the results alone with the test data. Here, by excluding pressure data for wells No. 8 and No. 55 from all stages of the project and comparing the results obtained for these two wells with the true pore pressure values, the estimation was validated. Since the direct estimation of pressure on acoustic impedance is considered in this study, validation of inversion and impedance determination results is also important. In the section on the determination of acoustic impedance from inversion, a quantitative analysis of inversion results was performed. Here, too, it is appropriate to review the results qualitatively. The 2D cross section of inverted impedance is shown in Fig. 13. In terms of interpretability and qualitative results, it is expected that at the top and bottom of the reservoir (Asmari and Pabdeh horizon) there are two layers of high acoustic impedance, as shown in Fig. 13, with the highest acoustic impedance in this interval. There are two horizons, and within the reservoir interval, as expected, the acoustic impedance value is low, thus showing the interpretability of the acoustic impedance results and the inversion accuracy.

Fig. 13
figure 13

a Base map of the study area showing well locations. Note: the seismic crossline that was discussed is in thick blue (Crossline 403). b Inversion of acoustic impedance in cross-section 403

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Figure 14a shows the pore pressure corresponding to the desired cross section (Crossline 403). It is to be expected that pore pressure increase in the reservoir. In other words, it is expected that there exists an overpressure interval between two low-pressure confined layers as shown in Fig. 14. Figure 14b shows a Z slice in the reservoir interval. It can be seen that pressure decrease in the central part of this section, which is also consistent with the geological report of the field, where the ratio of hydrocarbon potential to total rock on both sides of the reservoir is better than the middle part of it. Another validation of the results is the pressure estimation.

Fig. 14
figure 14

a Pore pressure obtained from the acoustic impedance of the crossline 403. b Pore pressure obtained from acoustic impedance at Z slice in time scale

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In Fig. 15, a 3D image of the cube obtained for pore pressure is shown: Inline, Crossline, and Z Slice in time scale.

Fig. 15
figure 15

3D volume obtained for pore pressure in three directions inline, crossline, and Z slice in time scale

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Therefore, in terms of interpretability and qualifications, the results are accepted by qualitative and interpretative analysis. Figure 14 shows that a zone of overpressure between the two low-pressure layers of the reservoir. This result can be of great help in determining reservoir boundaries as well as in planning for drilling trajectory for new wells. As a complementary study, the results show a decrease in pressure in this sectionʹs central part relative to the surrounding area. This feature is consistent with the oil fieldʹs geological report, where the hydrocarbon potential of total rock on both sides of the reservoir is higher than the middle part of the reservoir. Besides, the pore pressure estimation results in this study can determine the weight of the drilling mud window required in the drilling design of new wells. Concerning the pore pressure cube obtained in this study, it can be concluded that less drilling mud reservoir is needed in the central part of the Z slice section. Other results that may be plausible in this study are observations of high-pressure regions where their pressure is close to the reservoir pressure, which may indicate reservoir leakage and fluid migration in these areas. The formation pressure assessment regulates the weight of the drilling mud and studies the reservoir, and fluid migration is required.

Concluding remarks

Pore pressure estimation is an important issue in improving oil field exploration and drilling projects. The commonly used pressure estimation methods are methods that calculate pore pressure from estimated velocity equations, such as Eaton’s method, which requires both the density and velocity information obtained by acoustic impedance inversion, and suffer from conversion errors. In this study, we directly estimated pore pressure based on an impedance-based method to directly predict pore pressure using the seismically derived acoustic impedance and prevent the additional errors and uncertainty due to converting acoustic impedance into density and velocity cubes. We assessed this method for one of Iran’s oil fields. We obtained an acoustic impedance cube from seismic data inversion, using a simple transformation function to convert the acoustic impedance to pore pressure, and calibrate this relationship with multi-well pressure data. The results are evaluated qualitatively and quantitatively. The measured pressure from RFT in the two wells No. 8 and No. 55, in which their data were not involved in any computation steps, has acceptable accuracy compared to our predicted pore pressure in these wells. In terms of interpretability and qualifications, results are also accepted by qualitative and interpretative analysis. The results show a zone of overpressure between the two low-pressure layers of the reservoir. This result can be of great help in determining reservoir boundaries as well as in planning for drilling trajectory for new wells. The results of this study reflect that acoustic impedance data through the integrated post-stack inversion scheme are needed for impedance-based pore-pressure prediction.