Introduction

Predicting pore pressures allows Exploration and Production projects to plan and drill oil and gas wells successfully if the optimal structure of the well is designed, saving costs and time and avoiding drilling hazards. During the drilling operation, the upper formations keep the pore pressure, and below the maximum pressure, the formation withstands without fracturing (Chen 2019), providing information to model hydrocarbon migration in project evaluation (Satti et al. 2015). An accurate prediction guarantees reliable information (Wang et al. 2023). It is a must to know how different overpressure mechanisms affect rock properties and determine what causes overpressure (Luo et al. 2019; Lin et al. 2018). In complex geological conditions, the cause of overpressure is hard to identify (Carcione and Helle 2002), which decreases the prediction reliability.

Bowers (2002), Eaton (1975), Hassan and Bay (2019), and Sayers (2006) employed methods supported by the relation between compressive wave velocity and adequate stress estimate pore pressure in hydrocarbon clastic reservoirs by using loggings acquired while Drilling (LWD) or wireline logging (WL). However, this approach only applies to shale rather than calcareous formations because carbonates do not compact uniformly with depth as shale does. Weakley (1990) used sonic velocity trends to overcome this situation to estimate pore pressures in carbonate if formations comprise carbonate, sand, and shale. The points where the sonic trends change identify shaly intervals and lithology tops, and the lateral shifting of the sonic trends at each lithology change provides a continuous interval velocity profile. The technique obtained outstanding results when applied to 20 wells with pore pressures ranging from 10 pounds/gallon to 18 pounds/gallon (ppg) and up to 7010 m in depth. In another attempt, Azadpour and Manaman (2015) applied a modified Eaton’s method to determine sonic velocity trends for each lithology and Weakley’s technique to create a pore pressure profile, using LWD data of a carbonate oil reservoir to estimate pore pressure successfully. In a similar approach, Rashidi et al. (2015) varied the compaction exponent of Eaton’s method to match the variation in the sonic velocity trends to obtain compaction exponents of 3.0 for depths up to 2600 m in shale formations and 0.7 for depths higher than 2600 m in carbonate formations. The results of previous investigations confirm the reliability of Weakley’s method. Despite their promising results, they argue for carrying out additional research to modify qualitative or quantitative methods. Pore pressure prediction methods still need to be improved in carbonate reservoirs; therefore, an enormous recent study was conducted without getting a solution due to the complexity of carbonates (Sharifi et al. 2023).

On the one hand, relations between rock physics and seismic velocity analysis allow estimating pore pressure from seismic interval velocity as interval velocity, and on the other one, seismic data inversion of pre- and post-stack data provides results confirmed by petrophysical analysis of detrital formations (Carcione and Helle 2002; Sayers et al. 2006). Seismic inversion is required to transform high-frequency well log data to low-frequency data, replacing a rapidly varying, fine-layered model with an equivalent and more homogeneous model, which Liner (2014) estimated from high-frequency logs using the Backus (1962) averaging based on Hooke’s law for Vertical Transverse Isotropy media (Li et al. 2020). Nowak and Heppard (2010) observed that the acoustic impedance of pressured shale increases with burial depth as shale compacts, relating impedance to pore pressure, so they inverted seismic data for shale impedance and turned it directly to pore pressure. However, all pore pressure calculations depended on the predictability of shale and clay rocks, not on sandstone, siltstone, marl, limestone, and other rock types. Wagner et al. (2013) applied post-stack inversions that supplied separated velocity and density profiles that confirmed wells of detrital formations geology. Banik et al. (2013) predicted pore pressure before drilling directly from the acoustic impedance inverted from pre-stack data instead of from inverted wave velocity and density profiles, avoiding adding error and uncertainty in the prediction. The estimated pore pressure is like the one obtained using velocity and density but was done more efficiently. The comparison of results with well logs suggests that pore pressure is a continuous function of acoustic impedance. They consider clastic formations but not calcareous rocks. Dewantari et al. (2018) obtained acoustic impedance and P-wave velocity from a post-stack seismic inversion and estimated pore pressure in depth, for which they determined Normal Compaction Trend (NCT) in each formation, including a carbonate one using Eaton’s method and well logs to calibrate the NCTs. However, post-stack seismic inversion assumes that the stacked section equals a zero offset, which is generally false, providing an unreliable output.

Noah et al. (2019) estimated interval seismic velocities in depth from stacking velocities by the DIX formula (5) and density in depth by Gardner’s formula, using them to generate the overburden pressure and NCT velocity, and with Eaton’s equation, predict the pore pressure before drilling, which were like those estimated using well logs. The geology in the area did not include calcareous formations. Different to the earlier, Bahmaei and Hosseini (2020) modified sonic logs using the velocity of the checkshot to remove density from the inverted acoustic impedance to convert the velocity to a pore pressure using Bowers’ ratio. The results of the pore pressure model were validated using pore pressure data obtained by the Modular Formation Dynamic (MDT) testing technique. As known, pore pressure prediction by well logging provides insight only in the vicinity of the wellbore. Mahmood et al. (2021) and Abbey et al. (2021) quantified the uncertainty in pore pressure prediction from 3D seismic data by calibrating it with well logs using Eaton’s methods for pore pressure prediction using seismic velocities and well logs. The results suggest that the uncertainty in pore pressure prediction is higher where the dominant lithology is shale and does not include carbonates. Overall, the uncertainty in pore pressure prediction from seismic data varies between 10 and 30% in the studied area, even though this research used density and velocity extracted from inverted acoustic impedance. Pore pressure prediction methods face severe challenges in carbonate reservoirs, so recent research is still being conducted. Although many of them have not gotten promising results yet due to the complexity of carbonates (Sharifi et al. 2023), notwithstanding the most significant effort to predict carbonate pore pressure concentrates on research using well log (Guo et al. 2023; Makarian et al. 2023; Rai et al. 2022), including new approaches based on Artificial Intelligence techniques (Delavar and Ramezanzadeh 2023). The above references highlight the importance of continuing research on predicting pore pressure, especially in calcareous formations containing 50% of the world’s oil reserves.

Despite all cited results, pore pressure prediction methods still need to improve in carbonate reservoirs, so enormous recent research was conducted without getting a beyond-doubt solution due to the complexity of carbonates (Sharifi et al. 2023). This research estimated the pore pressure of a calcareous formation using the acoustic impedance produced by the pre-stack inversion of a seismic line from the Middle Magdalena Valley Basin, which includes overpressured calcareous formations. The pore pressure distribution is reliable, matches overpressure in calcareous rocks near the well, and estimates anomalous pressure in zones distant from the well. The method requires facies identification and, through simplified steps, calculates the overload stress and the normal compaction trend velocity. The method uses a pore pressure equation dependable on acoustic impedance derived from the Bowers' method, whose basin-dependent parameters are calibrated with the Weakley's pressure profile without needing the density extracted from the acoustic impedance. On the other hand, the calibration of the well logs with the normal compaction trend velocities estimated by the Eaton and Bowers' methods provided the parameters A and B, while the Weakley's method provided the U. The comparison of the pore pressures estimated in calcareous rocks by the previous methods indicated that the Weakley's method was the best reliable. On the other hand, the overpressures predicted by this method near the well in the calcareous formations are very close to those observed during drilling, contrary to pressures estimated by the Eaton and Bowers' methods.

Geological framework

The study area is in the Middle Magdalena Valley Basin of Colombia, MMVB (Fig. 1a). The basin corresponds to a depression-intermountain-oriented SW–NE, limited by complex system faults (Roncancio and Martinez 2011) between the Central Cordillera and the Oriental Cordillera, and the Magdalena River north. The fault system comprises thrust systems to the eastern and western influenced by pre-Cretaceous igneous–metamorphic basement affecting sedimentary sequence. Within the basin, more exactly in the center of it, some seismic surveys are being carried out using a seismic line along the well (Fig. 1b); the seismic section and well interpretation include the reflection surfaces and some lithology markers (tops): Esmeraldas, Umir Discordance, and La Luna–Galembo Formations (Fig. 1c). The first formation corresponds to sediments deposited in continental to marginal environments during intracratonic rifts along the Triassic and Jurassic periods. The second one was settled on fluvial and coastal environments during the extensional phase, associated with a retro-arc rift, and the third one, the Cretaceous–Paleocene, was deposited under marine conditions during tectonic thermal subsidence (Gaona-Narvaez et al. 2013). According to the petroleum system classification, the source rock is especially La Luna Formation, consisting of the Salada, Pujamana, and Galembo units (Roncancio and Martinez 2011). The lower part of the Umir formation comprises shaly layers with carbonates and micas units, while the upper part forms shale with fine-grained sandstone to siltstone intercalations. The Esmeralda Formation presents micaceous fine-grained sandstones and siltstones interspersed with layers of black shale mottled and layers of carbon at the top. The upper Paleogene–Neogene sequence comprises The Mugrosa, Colorado, and Royal Formations, with the Mugrosa Formation characterized by shales with intercalations of fine-grained sandstones. At the same time, the Colorado Formation comprises sandstone toward the base, claystone, siltstone, and shale toward the middle and top. On top of the sequence, the Real Group mainly includes conglomerate, sandstone, and claystone conglomerates.

Fig. 1
figure 1

a Geographic location of the Middle Magdalena Valley Basin (Colombia); b 2D stacked section; and c stratigraphic column of the Middle Magdalena Valley Basin of 2D seismic lines location in the De Mares area

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Pore pressure prediction theory

Shale index \(({I}_{{\text{sh}}})\) based on gamma ray log:

$$(I_{\text{sh}}) = \left({\text{GR}}_{\log} - {\text{GR}}_{\min} \right)/\left({\text{GR}}_{\max} - {\text{GR}}_{\min} \right),$$
(1)

where \({{\text{GR}}}_{{\text{min}}}\) corresponds to the minimum gamma ray, \({({\text{GR}}}_{{\text{max}}})\) corresponds to the maximum gamma ray, and \({({\text{GR}}}_{{\text{log}})}\) is the gamma ray reading of formation. Sediment compaction diminishes the shale porosity under the overburden stress \(({\sigma }_{v})\), and according to Eaton (1975), the normal compressional trend velocity in shale \(({V}_{{\text{NCT}}})\) at depth z is:

$${V}_{{\text{NCT}}}\left(z\right)= {V}_{0}+{A\sigma }_{n}^{B}\left(z\right),$$
(2)

where \({V}_{0}\) is the near-surface sonic velocity, and \({\sigma }_{n}\) is the difference between \({\sigma }_{v}\) and the normal pore pressure \({p}_{{\text{p}}}\) (hydrostatic pressure). A and B are basin-dependent parameters estimated when the \({V}_{{\text{NCT}}}\) matches the sonic velocity of shale \(V\), and in an overpressurized pore area, \(V\) separates from \({V}_{{\text{NCT}}}\). According to Bowers (1995) \({\sigma }_{{\text{e}}}/{ \sigma }_{n}={\left(V/{V}_{{\text{NCT}}}\right)}^{X}\) with the lithology-dependent exponent \(X=3\), the pore pressure becomes:

$${p}_{{\text{p}}}\left(z\right)={ \sigma }_{v} \left(z\right)-\left[{ \sigma }_{v} \left(z\right)-{ p}_{n}\left(z\right)\right]{\left[V\left(z\right)/{V}_{{\text{NCT}}}\left(z\right)\right]}^{3},$$
(3)

where \({\sigma }_{{\text{e}}}\) is the vertical effective stress.

In turn, the Bowers (1995) estimates the curve associated to undercompaction:

$$V\left(z\right)={V}_{ml}+A{\sigma }_{{\text{e}}}^{B},$$
(4)

where \({V}_{{\text{ml}}}\) is the mudline velocity (1520 m/s or 5000 ft/s), and A and B are the basin-dependent parameters calibrated with logs. From Eq. (4), with \({{ \sigma }_{{\text{e}}}=\sigma }_{{\text{max}}}\):

$${\sigma }_{{\text{max}}}={([{ V}_{{\text{max}}}-{V}_{{\text{ml}}}]/A)}^{1/B},$$
(5)

where \({V}_{{\text{max}}}\) is the maximum \(V\) before overpressure deflection. Next, Bowers estimated the discharge curve associated with fluids expansion:

$${V}_{d}\left(z\right)={V}_{{\text{ml}}}+A{\left({{ \sigma }_{{\text{max}}}\left[{ \sigma }_{e}\left(z\right)/{ \sigma }_{{\text{max}}}\right]}^\frac{1}{U}\right)}^{B},$$
(6)

where \(U\) measures sediment plasticity and ranges from 3 to 8 in clastic sediments, and when \(U\) = 1, Eq. (6) becomes Eq. (5). Finally, the pore pressure becomes:

$${p}_{{\text{p}}}={ \sigma }_{v}-{([{V}_{d}(z)-{V}_{{\text{ml}}}]/A)}^{1/B},$$
(7)

with \({V}_{d}\left(z\right)\) estimated with Eq. (6). Despite the above, Eaton and Bowers’ methods are unreliable because carbonate rocks do not compact uniformly with depth like shale. Weakley (1990) applied a different sonic velocity trend approach to face the carbonate rock challenge. He formerly determined lithology tops using the gamma ray log and drew sonic velocity trends in each lithological section. To circumvent the \({V}_{{\text{NCT}}}\) discontinuities where lithology changes, he constructed a continuous \({V}_{{\text{NCT}}}\) by laterally displacing the overlying \({V}_{{\text{NCT}}}\) section to match its bottom with the top of the underlying \({V}_{{\text{NCT}}}\) section. Normal velocity trend lines were then drawn through the normally pressed and compacted sections. Each \({V}_{{\text{NCT}}}\) is associated with a pore pressure estimated by integrating the bulk density and calculation of the overburden gradient. This information is used to solve Eq. (8) and create an overlay for the area.

$$X={\text{log}}\left[\left({G}_{0}-{G}_{{\text{p}}}\right)/\left({G}_{0}-{G}_{{\text{n}}}\right)\right]/{\text{log}}({\Delta t}_{{\text{n}}}/{\Delta t}_{0}),$$
(8)

where \({G}_{0}\) is the overburden gradient, \({G}_{{\text{p}}}\) is the pore pressure gradient, \({G}_{{\text{n}}}\) is the normal pore pressure gradient, \({\Delta t}_{0}\) is the log sonic traveltime, and \({\Delta t}_{{\text{n}}}\) is the normal sonic traveltime. Equation (8) calculates X, which corresponds to U in Eq. (6), although it usually equals 3. In another approach, Banik et al. (2013) multiplied Bowers’ Eq. (4) by the formation density to express it in terms of the acoustic impedance \({I}_{p}\) and the acoustic impedance of the mud column \({I}_{p{\text{o}}}:\)

$${I}_{p}\left(z\right)={I}_{p{\text{o}}}+C{\sigma }_{e}^{B}.$$
(9)

The algebraic manipulation of Eq. (9) leads to the pore pressure:

$$p_{{\text{p}}} \left( z \right) = Ao + Bo / \left( {1 + CoI_{p} \left( z \right)} \right),$$
(10)

Ao, Bo, and Co are basin-dependent parameters, and in sedimentary rocks, Bo usually equals one and is lower than one when porosity is low (Tang et al. 2015).

Methodology

The methodology is summarized in Fig. 2 estimating the porosity of the pores along the well with the methods of Eaton, Bowers, and Weakley, obtaining the acoustic impedance distribution in the subsurface using pre-stack seismic inversion, and estimating the pore pressure with the equation Banik pore pressure, previously calibrated with the Weakley's pressure profile. To do this, we used density, gamma ray, and sonic logs from a 4910-well located in the Middle Magdalena Valley Basin, which drilled Neogene to Cretaceous rocks, along with seismic logs from a 2D line acquired in the vicinity of the well. The logs must be compatible with the seismic data, so they were transformed from the Kilo-Hertz band to the seismic bandwidth using the Backus method (Backus 1962). Using Eaton’s approach, we calculated Ish, Pp, and Pn with Eq. (2) and VNCT with Eq. (3). Those zones with Ish greater than 0.8 were then considered shale zones; the estimated Sv up to 4900 m depth was 17,000 psi (20 ppg) with a gradient of 0.465 psi/feet, and VNCT was calculated along the interval 460–976 m, where pressure behaved normally during the drilling. The calibration of the VNCT curve along the 460–976 m interval estimated A and B parameters for the Middle Magdalena Valley Basin. Then, we calculated the pore pressure with the Bowers' method, using the unloading curves, which calibrated the Bowers' constants for the Middle Magdalena Valley Basin. Then, we calculated Ish, Sv, Pn, and the curve with Eq. (4), Smax with Eq. (5), the unloading curve with Eq. (6), and the Pp with Eq. (7). Next, we applied the Weakley's method, which is based on Eaton’s concepts of VNCT with X set to 3, although it differs in unconventional reservoirs. We identified the shale units using gamma ray (GR) and marked the unit tops on sonic traveltime (DT), density (ρ), and gamma ray (GR) profiles. We determined the normal compaction trend of the shale units in each formation and drew velocity trend lines in the sonic record from the top of each shale unit to the top of the next shale unit, respecting the sonic velocity of each shale interval; next, we obtained a relatively continuous interval velocity profile, by shifting the trend line where the lithology changed, joining the last value of interval velocity in the previous lithological section with the first value in the next. The result obtained was a relative interval sonic traveltime DT, and with Weakley’s Eq. (8), we calculated X and, subsequently, the pore pressure with Eq. (7). In the next step, we obtained Ip through the pre-stack seismic inversion of the 2D seismic line and calculated Se using Eq. (9) with the seismic impedances of the trace closest to the well. Next, we calculated Sv by multiplying GDT by the summation of Ip along this seismic trace, with g and DT equal to 9.8 m/s and 2 ms, respectively. \({A}_{o}, {B}_{o}\) and \({C}_{o}\) parameters of Eq. (10) were determined from calibration with pore pressure estimated by the Weakley's method in the well. Finally, we used these parameters with Eq. (10) to calculate the pore pressure in the subsurface.

Fig. 2
figure 2

Flowchart of the methodology for determining an estimated pressure porosity from impedance

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Results and discussion

The well log set comprises the sonic velocity (Vp), density (ρ), gamma ray (GR), and spontaneous potential (SP) logs, used to calculate the porosity ϕ, pore pressure (Pp), vertical stress (Sv), and Index shale (Ish) logs (Fig. 3). The Ish greater than 0.8 identifies shale intervals in Colorado (456–975 m), Esmeraldas (2286–2560 m), and Mugrosa (975–2286 m) Formations, the Umir Unconformity (2560–2383 m), in the Simití (3383–4588 m), and the limestone of La Luna Formation (2834–3383 m). On the other hand, Ish below 0.8 corresponds to sandy units with intercalations of siltstones and clays. Unlike the formations above, the Rosablanca, the La Luna–Galembo Member, and the La Luna–Salada Member contain calcareous units.

Fig. 3
figure 3

Well logs uploaded by Hampson and Russell program, from left to right: depth (m) and Formation tops, Track 1. P-wave velocity (red) and its smoothed log using Backus (blue). Track 2. Density (green) and Backus-smoothed density (yellow). Track 3. Gamma ray (green) and smoothed with Backus (light green). Track 4. Porosity (orange). Track 5. S-wave velocity (pink). Track 7. Pore pressure (blue) and vertical stress (red). Track 8. Shale volume

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The behavior of the Vp and Pp curves indicates possible overpressure zones in the Esmeraldas Formation (2300 m), Umir Unconformity (2600 m), Pujamaná shaly unit of the La Luna Formation (3400 m), and in the Simití Formation (3800 m). The Smax was approximately 17,000 psi (20 ppg) at 4900 m with an Sv gradient between 1 and 1.06 psi/ft. The first, second, and third tracks of Fig. 3 show the original Vp, ρ, and GR records sampled at 0.3 m and superimposed on them, with a smoother appearance, and those transformed to the seismic bandwidth.

On the other hand, Fig. 4a–c shows the sonic velocity records (blue dots) with spatial sampling of 10, 20, and 0.3 m with the respective VNCT curves (green line), where the coincidence of the three VNCT curves indicates that the smoothing technique does not affect its estimation. The VNCT curve calibrated in the 457–975 m normally pressured interval, with A = 14.7 and B = 0.72 for the MMVB, shows sonic velocity deviation from VNCT after 2900 m depth, indicating overpressure of a possible compaction event (2286 m) in the basal part of the Esmeraldas Formation. This deviation increases at the top of the Umir Unconformity (2560 m), suggesting more than one overpressure mechanism in the Cretaceous units of the basin due to low compaction, fluid expansion, and tectonism.

Fig. 4
figure 4

a Compaction trend smoothed with Backus and λ = 10 m; b compaction trend smoothed with Backus and λ = 20 m; c compaction trend calculated with sonic logs. In all three curves, the separation of the well data takes place in the overpressure zones

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Figure 5a, b contains Eaton’s Pp sampled every 20 m and 10 m, and Fig. 5c contains the Bowers’ Pp sampled every 0.3 m. Figure 5 shows a possible compaction event (2286 m) in the basal part of the Esmeraldas Formation. The Pp shows gas events (orange triangle) in the Umir Unconformity (2560 m), with a lower mud weight (blue line), and two gas influx events (2404 and 2482 m) in the Esmeralda Formation, detected during borehole conditioning (orange triangle). Pp's in the Real Group and the upper part of the Colorado Formation, higher than the weight of the mud, are associated with the low consolidation of shallow formations. In contrast, the middle and basal parts of the Colorado Formation show normal pressure. The Mugrosa Formation is an intercalation of sand and sandstone with few clay intervals, where an inflow event (1368 m) required raising the mud weight to 17 ppg to avoid gas bubbles reaching the surface. The overpressure events (green dots) in the calcareous La Luna Formation do not coincide with the estimated Eaton's Pp, and in the Rosablanca and La Luna Formations, where dolomites and limestone predominate, the scarce shale intervals did not allow calculating Eaton's Pp. Bowers’ Pp is similar to Eaton’s due to its applicability in rocks with low permeability and high porosity, except in the Colorado Formation, where Bowers’ Pp is lower due to low compaction. Bowers’ Pp’s do not coincide with overpressure events identified during drilling in the Rosablanca, La Luna–Galembo Member, and La Luna–Salada Member Formations containing calcareous deposits.

Fig. 5
figure 5

Pore pressure calculated by Eaton’s method using a well log data; b sonic-smoothed and λ = 20 m; and c sonic-smoothed and λ = 10 m. The three correlated pore pressure curves identify overpressure events. The calibration constants for the MMV, A = 14.7 and B = 0.72

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The Bowers' constants calibrated for the MMVB were A = 0.2, B = 0.85, and U = 0.65. Figure 5a shows the points where abrupt changes in sonic DTn occur. Figure 6b shows the recalibrated continuous sonic trend line constructed according to the Weakley's method (red line) and the sonic normal compaction trend estimated with the Eaton's method. At 3397 m depth, Pp, Sv, and Pn of 14.1, 20, and 9388 ppg and traveltimes DT and DTn through the La Luna–Pujamaná Formation of 320 us/m and 155 us/m allow estimating an X equal to 0.718. At the top of the La Luna Formation–Galembo Member values of Pp, Sv, and Pn of 16.4, 20.4, and 8.9 ppg, with Δto and Δtn of 100.64 and 66.81 us/ft, provide an estimated X of 2.57. Table 1 contains points A, B, C, D, E, F, G, and H and the traveltimes DTn across each unit with pore overpressure estimated with the Weakley's method fluctuating between 13 and 19 ppg. Figure 6c shows Pp’s estimated with the Weakley's method and those obtained by the Eaton's method. However, the Weakley's Pp matches the mud weight curve except in the middle zone of the La Luna–Pujamaná, the La Luna–Salada, Simití Formations, and the upper part of the Tablazo Formation, where it is less than the weight of mud. In contrast, as Fig. 5c shows, Eaton’s Pp does not match the mud weight.

Fig. 6
figure 6

a Sonic log of shale intervals (blue) and NCT curve by formation (red); b NCT profile (green) and continuous superposition of NCT curves of shale intervals (red); and c pore pressure calculated with the exponent X estimated by Weakley’s correlation (yellow), pore pressure calculated with the Eaton's method (red), and mud weight (blue)

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Table 1 Pore pressure at 2850 m in the La Luna–Galembo Formation by Weakley's approach
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Figure 7a shows the acoustic impedance and density logs (red line) obtained from the seismic inversion and the Backus-smoothed logs (blue line), and the high correlation between them indicates the inversion reliability. On the other hand, yellow arrows indicate the reflectors in the synthetic seismogram and the CDP record. The reflectors at 1080, 1470, 1700, 1960, and 2026 ms correspond to the tops of the Esmeraldas Formation, the Umir Unconformity, and the Galembo, Pujamaná, and Salada Members of the La Luna Formation. The variation of the amplitude of the reflectors with the offset coincides in both seismograms.

Fig. 7
figure 7

a Seismic inversion results (red), initial inversion model (Black), and well logs smoothed with Backus (blue); b there is a good match between reflectors of seismic (black) and synthetic data (red)

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Figure 8 shows the Ip and Ish profiles tying the subsurface Ip distribution. In the Esmeraldas Formation at 1360 ms, the color varies from red to orange (9000 < Ip < 7000 (m/s) (g/cc)); in the Umir Discordance, the colors range from red to yellow (8000 < Ip < 6000 (m/s) (g/c)). Similar behaviors are evident in the Simiti and La Luna–Pujamaná Formations. Meanwhile, the Esmeraldas Formation (1500 ms) shows strong impedance contrasts associated with clay formations. In the Galembo Member–La Luna Formation, from 1900 ms, and in the La Luna–Salada at 2150 ms, calcareous rocks predominate, with greater hardness and lower porosity, so the impedances tend to become higher.

Fig. 8
figure 8

Trace of acoustic impedance resulting from the seismic inversion. This impedance decreases in shale rocks, possibly overpressured due to increased porosity and reduced density. Evidence of overpressure is visible in the basal part of the Esmeraldas Formation, the Umir Discordance, and the La Luna Formation’s shale intervals

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Figure 9 shows in Track 1 curves of mud weight (blue line) and Sv (yellow line) and curve Pp calculated with the seismic impedance (black line), in Track 2 the smoothed Backus impedance (red), and in Track 3 with the Backus-smoothed NCT. The Ip curve (purple line) separation from the NCT curve takes place at 1240 ms in the Esmeralda Formation and 1470 ms in the Umir unconformity, reinforcing the idea of predicting pore overpressure from seismic impedances. Figure 10 shows a high correlation between the pore pressure calculated from the sonic registration and pore pressure calculated from acoustic impedances, and the colors coincide in the overpressured areas. Figure 10 indicates that the seismic interpretation, stratigraphic and structural knowledge, and the Banik constants of the basin, calculated by the Weakley's method, allow the prediction of pore pressures.

Fig. 9
figure 9

The pore pressure curve from acoustic impedances (black) and mud weight (blue) shows similar behavior (Track 1). Acoustic impedance resulting from seismic inversion (Track 2). Acoustic impedance in shale intervals (purple) and normal compaction trend curve TD (red). The regions where both curves separate are possible overpressure zones (Track 3). Shale volume profile, where values greater than 0.8 indicate shale intervals (Track 4)

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Fig. 10
figure 10

Pore pressure distribution estimated from inverted acoustic impedances of the 2D pre-stack section. The overpressure zones are at the top of the Umir Discordance (1570–1750 ms), where pressure increases from 14.5 to 15.7 pp, and the other, toward the top of the La Luna–Galembo Formation (1750 ms), with pore pressure rising from 15.7 to 16.5 ppg. The last zones are in the limestone intervals of the Pujamaná and the Salada Members of the La Luna Formation (blue), with pore pressure between 15.8 and 16.6 ppg

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Conclusions

The impedance provided by pre-stack inversion is used to estimate the subsurface pore pressure distribution in calcareous formations, determine the appropriate mud weight, and optimize the drilling design. This research applied the Eaton, Bowers, and Weakley methods to well logs from the La Luna carbonate Formation of the Colombian MMV, establishing Eaton’s constant and Weakley’s exponent for this formation and demonstrating a more reliable prediction by the Weakley's method. The research presents a method that simplifies the calculation of pore pressure using acoustic impedance in an equation derived from Bowers and calibrated in Weakley’s pressure profile. The overpressures estimated by the method near the wellbore are very close to those observed during drilling, in contrast to those estimated by the Eaton and Bowers' methods, indicating the most reliable performance. Application of the method to a seismic line provided a reliable subsurface pore pressure distribution that matched the overpressures observed in the well and the predicted overpressures in distant zones.