The workflow for pore pressure prediction based on prestack multiwave parameters (Fig. 1) is described below.
Step 1: Obtain n (n ≥ 3) limited-offset stacks of PP and PS (converted wave) after amplitude-preserving processing, and align them; subsequently, perform the prestack PP-PS simultaneous joint inversion to obtain the elastic parameter data of \(P_{{imp}}^{T}\) (P-wave impedance), \(S_{{imp}}^{T}\) (S-wave impedance), and \({\rho ^{\text{T}}}\)(density) in the time domain.
Step 2: Correct the velocity model based on the velocity spectrum according to the calibrated well-logging velocity model. Then, convert the elastic parameter data obtained above into the burial depth domain using the corrected velocity model with the ground surface as the datum. Therefore, \(P_{{imp}}^{D}\) (P-wave impedance), \(S_{{imp}}^{D}\) (S-wave impedance), and \({\rho ^{\text{D}}}\) (density) in the measured depth domain are obtained.
The time‒depth conversion presented above is different from that of conventional methods. Specifically, the datum of the conventional time‒depth conversion is the sea level, and thus, the obtained depth is the altitude. In contrast, the datum of the presented time‒depth conversion is the ground surface, and the depth here refers to the measured depth. Accordingly, the converted data are in the measured depth domain. This step lays the foundation for estimating the overburden pressure.
Step 3: Calculate the overburden pressure \({P_{ov}}\) using the obtained density data.
Specifically, integrate \({\rho ^{\text{D}}}\)in the depth domain along the burial depth to obtain the overburden pressure:
$${P_{ov}}=\int_{0}^{{{H_0}}} {{\rho ^{\text{D}}}\left( H \right)} gdh$$
1
where \({H_0}\) is the depth of the observation point from the ground surface; g is the gravitational acceleration; and is the burial depth.
Step 4: Calculate the effective stress of the rock framework \({P_e}\) according to the P- and S- wave impedance and density.
In the seismic formation pressure Model I (SFPM-I) proposed by Liu10, the effective stress of the rock framework is expressed as:
$${P_e}=\frac{{1{\text{+}}\sigma }}{{3{\text{-}}3\sigma }} \cdot \rho \cdot {V_p}^{2} \cdot {{\Delta H} \mathord{\left/ {\vphantom {{\Delta H} H}} \right. \kern-0pt} H}$$
2
where \(\sigma\) is Poisson's ratio; \({V_p}\)is the P-wave velocity; \(\rho\)is the density; and\({{\Delta H} \mathord{\left/ {\vphantom {{\Delta H} H}} \right. \kern-0pt} H}\)is the compaction deformation per unit thickness.
Liu defines \(\sigma\) in Eq. 2 as a constant (equal to 0.12), which makes Eq. 2 a univariate quadratic function of the P-wave velocity\({V_p}\). However, in this research \(\sigma\) is kept a variable and Eq. 2 is rewritten as:
$${P_e}={{\left( {P{{_{{imp}}^{{\text{D}}}}^2} - \frac{4}{3}S{{_{{imp}}^{{\text{D}}}}^2}} \right)} \mathord{\left/ {\vphantom {{\left( {P{{_{{imp}}^{{\text{D}}}}^2} - \frac{4}{3}S{{_{{imp}}^{{\text{D}}}}^2}} \right)} {{\rho ^{\text{D}}}}}} \right. \kern-0pt} {{\rho ^{\text{D}}}}} \cdot {{\Delta H} \mathord{\left/ {\vphantom {{\Delta H} H}} \right. \kern-0pt} H}$$
3
Then, the effective stress of the rock framework \({P_e}\) is calculated using Eq. 3.
Step 5: Calculate the pore pressure \({P_p}\) by subtracting the rock framework effective stress from the overburden pressure:
$${P_p}={P_{ov}} - {{\left( {P{{_{{imp}}^{{\text{D}}}}^2} - \frac{4}{3}S{{_{{imp}}^{{\text{D}}}}^2}} \right)} \mathord{\left/ {\vphantom {{\left( {P{{_{{imp}}^{{\text{D}}}}^2} - \frac{4}{3}S{{_{{imp}}^{{\text{D}}}}^2}} \right)} {{\rho ^{\text{D}}}}}} \right. \kern-0pt} {{\rho ^{\text{D}}}}} \cdot {{\Delta H} \mathord{\left/ {\vphantom {{\Delta H} H}} \right. \kern-0pt} H}$$
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where \({{\Delta H} \mathord{\left/ {\vphantom {{\Delta H} H}} \right. \kern-0pt} H}\) is a variable dependent on various factors, such as depth, lithology, and sedimentary burial history, which are, in most cases, difficult to calculate directly. The solution of this research is as follows. First, based on the measured pressure of multiple wells with available acoustic full-waveform well logging data, calculate the overburden pressure profiles of these wells; then, substitute the computed \({P_{ov}}\), and measured \({P_p}\), \(P_{{imp}}^{{\text{D}}}\), \(S_{{imp}}^{{\text{D}}}\), and \({\rho ^{\text{D}}}\) into Eq. 4 to determine \({{\Delta H} \mathord{\left/ {\vphantom {{\Delta H} H}} \right. \kern-0pt} H}\) at these wells; finally, perform the second-order polynomial fitting between \({{\Delta H} \mathord{\left/ {\vphantom {{\Delta H} H}} \right. \kern-0pt} H}\)and H via the least square method based on the measured formation pressure data and the calculated \({{\Delta H} \mathord{\left/ {\vphantom {{\Delta H} H}} \right. \kern-0pt} H}\) of the wells:
$${{\Delta H} \mathord{\left/ {\vphantom {{\Delta H} H}} \right. \kern-0pt} H}=a{H^2}+bH+c$$
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where a, b and c are the fitting coefficients.
Based on the built second-order polynomial relationship, the P- and S-wave impedances, and the density, the effective stress of the rock framework is calculated using Eq. 3.
The modified pore pressure presented in this research is:
$${P_p}={P_{ov}} - {{\left( {P{{_{{imp}}^{{\text{D}}}}^2} - \frac{4}{3}S{{_{{imp}}^{{\text{D}}}}^2}} \right)} \mathord{\left/ {\vphantom {{\left( {P{{_{{imp}}^{{\text{D}}}}^2} - \frac{4}{3}S{{_{{imp}}^{{\text{D}}}}^2}} \right)} {{\rho ^{\text{D}}}}}} \right. \kern-0pt} {{\rho ^{\text{D}}}}} \cdot \left( {a{H^2}+bH+c} \right)+D$$
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where D is an empirical constant specific to the area of interest.
The corresponding pore pressure coefficient formula k can be written as:
$$k={{\left( {{P_{ov}} - {{\left( {P{{_{{imp}}^{{\text{D}}}}^2} - \frac{4}{3}S{{_{{imp}}^{{\text{D}}}}^2}} \right)} \mathord{\left/ {\vphantom {{\left( {P{{_{{imp}}^{{\text{D}}}}^2} - \frac{4}{3}S{{_{{imp}}^{{\text{D}}}}^2}} \right)} {{\rho ^{\text{D}}}}}} \right. \kern-0pt} {{\rho ^{\text{D}}}}} \cdot \left( {a{H^2}+bH+c} \right)+D} \right)} \mathord{\left/ {\vphantom {{\left( {{P_{ov}} - {{\left( {P{{_{{imp}}^{{\text{D}}}}^2} - \frac{4}{3}S{{_{{imp}}^{{\text{D}}}}^2}} \right)} \mathord{\left/ {\vphantom {{\left( {P{{_{{imp}}^{{\text{D}}}}^2} - \frac{4}{3}S{{_{{imp}}^{{\text{D}}}}^2}} \right)} {{\rho ^{\text{D}}}}}} \right. \kern-0pt} {{\rho ^{\text{D}}}}} \cdot \left( {a{H^2}+bH+c} \right)+D} \right)} {{P_w}}}} \right. \kern-0pt} {{P_w}}}$$
7
where \({P_w}\) is the hydrostatic pressure.
Ultimately, the calculated pore pressure can be corrected according to the pressure measurements obtained from drilling, after which the pore pressure (coefficient) can be mapped.