Reservoir characterisation using hybrid optimisation of genetic algorithm and pattern search to estimate porosity and impedance volume from post-stack seismic data: A case study

Abstract

In the current study, a seismic inversion based on a hybrid optimisation of genetic algorithm (GA) and pattern search (PS) is carried out. The GA is an approach to global optimisation technique that always converges to the global optimum solution but takes much time to converge. On the other hand, the PS is a local optimisation technique and can converge at local or global optimum solution depending on the starting model. If these two techniques are used together (here termed hybrid optimisation), they can enhance one's benefit and reduce the drawbacks of others. The present study developed a methodology to combine GA and PS in a single flowchart and utilise seismic reflection data exclusively to predict porosity and impedance volume in inter-well regions. The algorithms are initially tested on synthetically created data based on the wedge model, the coal coking model, and the 1D convolution model. The performance of the algorithm is remarkably acceptable, according to the error analysis and statistical analysis between the inverted and the anticipated results. After that, the field post-stack seismic data from the Blackfoot field, Canada, is transformed into impedance and porosity using a developed hybrid optimisation technique. The inverted/predicted sections show very high-resolution subsurface information with impedance varying from 6000 to 14000 m/s×g/cc and porosity varying from 5 to 40% in the region. The error decreases from 1.0 to 0.5 for impedance inversion, whereas it varies from 1.4 to 0.5 for porosity inversion within 3000 iterations, which cannot be achieved by a single optimisation technique. The findings also demonstrated a sand channel (reservoir) anomaly with low impedance (6000–9000 m/s×g/cc) and high porosity (12–20%) in between 1040 and 1060 ms time intervals. This study provides evidence that subsurface parameters like acoustic impedance or porosity may be promptly and affordably determined using seismic inversion based on hybrid optimisation. The developed methodology is very helpful in finding subsurface parameters in a limited time and cost, which cannot be achieved only by global or local optimisation.

Appendices

Appendix A

1.1 A.1 Mathematical formulation for impedance prediction

The acoustic impedance can be estimated from the hybrid optimisation as follows:

• 1. Select desired seismic and well-log data as input.

• 2. Convert depth to time so that the well-log (depth domain) and seismic data (time domain) can be used together.

• 3. Select a desired genetic operator and implement it to get the initial population that contains the acoustic impedance vector.

• 4. Calculate reflectivity from the impedance using equation (1).

• 5. Calculate synthetic trace using the following formula.

  $$Synthetic \; trace \; \left(t\right)= w \times r = \int^{\infty}_{0}r\left(n\right)w\left(n-t\right)dn,$$

  (A.1)

where $t$ is time and $n$ is the data index number. When using sampled data, the linear discrete convolution's integral transforms into a sum and can be written as follows.

$$Synthetic \; trace \; \left(t\right)= w \times r = \sum_{n=0}^{n=N}r\left(n\right)w\left(n-t\right ).$$

(A.2)

• 6. Calculate the RMS error between synthetic data and the input seismic data using equation (2).

• 7. Modify the initial population to reduce RMS error as much as possible within a limited time interval by repeating steps 3–6.

• 8. The output of this step will be acoustic impedance (say ${AI}_{0}$) that satisfies equation (2).

• 9. Use a pattern search algorithm by choosing the initial model $({AI}_{0}$) as input.

• 10. Construct a pattern vector and create the mesh.

• 11. Calculate the RMS error using equation (2).

• 12. Identify the best solution based on the RMS error and modify the new solution based on the best solution.

• 13. Check whether the new solution is better than the existing best solution. Based on these, expand or contract mesh.

• 14. Repeat steps 9–12 to terminate the program and get the desired acoustic impedance.

The above RMS error is just one of many ways to estimate error, and whether it represents the ‘actual’ error depends on the specific problem and data. Here, the RMS error calculates the square root of the mean of the squared differences between predicted values and actual values. It quantifies how far, on average, the predictions are from the true values. In this optimisation and model-fitting procedures, RMS error serves as the objective function to minimise. In such cases, minimising the RMS error can lead to model parameters that produce predictions that are closest, on average, to the observed values. That is RMS error is chosen here which is very close to the actual error.

Appendix B

2.1 B.1 Mathematical formulation for porosity prediction

Porosity, a measurement of the amount of pore space within a rock, is frequently used as a sign of a reservoir's capacity to hold fluids. In general, a rock's impedance reduces as its porosity and fluid saturation rise. Although there are instances where more porosity can lead to higher impedance, the relationship between porosity and impedance is not always clear-cut (Feng et al. 2020).

Nevertheless, to calculate the porosity for post-stack data that is based on the link between AI and porosity, the information on acoustic impedance based on well logs can be used. In this case, the goal is to provide a method for utilising AI reflectivity to forecast porosity for a related sand channel. To compute porosity from density logs, a well-log analysis technique called density porosity is commonly used. Given below is the relationship for calculating bulk density from the density log (Serra 1983).

$$\rho =\phi \times{\rho }_{f}+\left(1-\phi \right){\rho }_{ma},$$

(B.1)

where $\phi, \, { \rho }_{f}\,{ {\text\,{and}}\, \rho }_{ma}$ indicate porosity fluid and matrix density, respectively. To calculate porosity from density logs, we select 1.1 g/cc for fluid density and 2.65 g/cc for matrix density. Further, a velocity (v) formula for porous rock has been proposed by Wyllie et al. (1956) as follows:

$$\frac{1}{v}=\frac{\phi }{{v}_{f}}+\frac{1-\phi }{{v}_{ma}},$$

(B.2)

where ${v}_{f}$ and ${v}_{ma}$ stand for fluid and matrix velocity, respectively. Further, Rasmussen and Maver (1996) provided an AI of the porous rock as follows:

$${\text{log}}\left(AI\right)={\text{log}}\left({\rho }_{ma} \times {v}_{f}\right)-{\text{log}}\left(\frac{\phi }{1-\phi }\right) .$$

(B.3)

Equation (B.3) can be approximated by equations (B.1 and B.2) since the matrix density and velocity are significantly greater than the fluid density and velocity. Further, a model connecting AI and porosity is provided by Rasmussen and Maver (1996) and is shown below:

$${\text{log}}\left(AI\right)={\text{log}}\left({AI}_{0}\right)+q\, {\text{log}}\left(\frac{\phi }{1-\phi }\right) .$$

(B.4)

This is a straight-line equation with slope $q$ and intercept ${AI}_{0}$. According to observations, the value of the log(${AI}_{0}$) in equation (B.4) has very little impact relative to $q \, {\text{log}}\,\left(\frac{\phi }{1-\phi }\right)$, therefore we can disregard it (Yilmaz 2001; Kearey et al. 2002; Chatterjee et al. 2016). The relationship between acoustic impedance and reflection coefficient, also known as impedance reflectivity, can be expressed mathematically using the following equation (Rasmussen and Maver 1996).

$${r}_{AI}=\frac{1}{2}\left[{\text{log}}({AI}_{i+1})-{\text{log}}\left({AI}_{i}\right)\right].$$

(B.5)

The formula below can also be used to define porosity reflectivity (Rasmussen and Maver 1996).

$${r}_{\phi }=\frac{1}{2}\left[\left({\text{log}}\left(\frac{{\phi }_{i+1}}{1-{\phi }_{i+1}}\right)\right)-{\text{log}}\left(\frac{{\phi }_{i}}{1-{\phi }_{i}}\right)\right].$$

(B.6)

The porosity reflectivity and the acoustic reflectivity can be connected (while ignoring smaller parameters) using equations (B.3–B.6) (Rasmussen and Maver 1996; Das and Chatterjee 2016; Shankar et al. 2021). The relationship between porosity reflectivity and AI reflectivity can be described as follows:

$${r}_{AI}=q{r}_{\phi }.$$

(B.7)

To calculate the slope, q, or the correlation coefficient between acoustic impedance (AI) and porosity reflectivity, a linear regression analysis can be performed. Once the linear regression line is fitted, the slope represents the correlation factor or the coefficient of the AI variable. It indicates the rate of change or the strength of the relationship between AI and porosity reflectivity.

By the analogy of equation (B.7), we can understand the porosity wavelet can be defined as the multiplication of the AI wavelet and the correlation coefficient $q$. As a result, the correlation coefficient will be multiplied by the AI wavelet (calculated from density and velocity logs) to produce the porosity wavelet. The seismic response or signature connected to changes in porosity in the subsurface is represented by the porosity wavelet. It is usually obtained from well-log data or empirical observations, and it is specifically intended to capture the properties of porosity reflectivity.