An intelligent heuristic-clustering algorithm to determine the most probable reservoir model from pressure–time series in underground reservoirs

Abstract

Precise characterization of underground reservoirs requires accurate calculations of the reservoir’s petrophysical data and accurate selection of the mathematical model governing the reservoir’s dynamic. In this study, we develop a novel heuristic-clustering algorithm, namely GA–DBSCAN–KMEANS, that can be applied over pressure transient data to assess the true reservoir model out of a pool of candidates. In this algorithm, each specific reservoir model is considered a subpopulation in the GA (genetic algorithm). Then, the simultaneous optimization of all the reservoir models is sought using the proposed hybrid algorithm. During the optimization process, the population size of different models will be either decreased, increased, or unchanged based on the average quality match obtained for each model. A combined DBSCAN (density-based spatial clustering of applications with noise)–KMEANS clustering scheme is used to increase the population size for the best reservoir model in each iteration of the GA. The accuracy of the proposed algorithm was verified using several synthetic data and a real field case obtained from the open literature. The tested data were collected from different types of reservoir models, including homogeneous reservoirs, matrix-fracture dual-porosity reservoirs, and fault-limited reservoirs. For uncertainty analyses and to test the performance of the algorithm under large numbers of initializations, Monte Carlo simulations were conducted. Results of the Monte Carlo simulations unveiled high values of P10, P50, and P90 for the probability of the true reservoir model and low values of these statistics for the false reservoir models. This shows that the outcome of the proposed algorithm is not affected by the initial randomization of the solution subspaces; hence, the developed algorithm is a reliable tool in determining the most probable reservoir model from transient well testing data.

Appendix A: Laplace solution of the reservoir models

In this section, we provide the mathematical formula for the pressure solution of the reservoir models used in this study. The solutions can be found in the Laplace domain in (Ershaghi et al. 1993), and we have used the numerical Stehfest Laplace inverse algorithm to generate the solutions in the time domain. For all the reservoir models, $\bar{P}_{\text{wD}}$ is defined as follows after considering skin and wellbore storage into the wellbore pressure model:

$$ \bar{P}_{\text{wD}} = \frac{{s\bar{P}_{\text{D}} + S}}{{s\left[ {1 + sC_{\text{D}} \left( {s\bar{P}_{\text{D}} + S} \right)} \right]}}, $$

(A-1)

In the above equation, s is the Laplace variable, S is the skin factor, and $C_{\text{D}}$ is the dimensionless wellbore storage coefficient. In addition, $\bar{P}_{\text{D}}$ is the pressure response without skin and wellbore storage and is different for each reservoir model. In the following, we have provided the different formulations used for $\bar{P}_{\text{D}}$ for each reservoir model. In all the following equations, $r_{\text{D}}$ is the dimensionless radius that we are investigating for the pressure. Since we are interested in wellbore pressure, $r_{\text{D}}$ will be equal to 1.0 in all the equations.

1.1 Model 1: Infinite acting homogeneous reservoir (Ershaghi et al. 1993)

$$ \bar{P}_{\text{D}} = \frac{{K_{0} \left( {r_{\text{D}} \sqrt s } \right)}}{{s\sqrt s K_{1} \left( {\sqrt s } \right)}}, $$

(A-2)

where $K_{0}$ is the modified Bessel function of the second kind with zero degree and $K_{1}$ is the modified Bessel function of the second kind with the first degree; s is the Laplace parameter.

1.2 Model 2: Infinite acting dual-porosity reservoir with PSS interporosity flow (Ershaghi et al. 1993)

$$ \bar{P}_{\text{D}} = \frac{{K_{0} \left( {r_{\text{D}} \sqrt {sf\left( s \right)} } \right)}}{{s\sqrt {sf\left( s \right)} K_{1} \sqrt {sf\left( s \right)} }}, $$

(A-3)

$$ f\left( s \right) = \frac{{\omega \left( {1 - \omega } \right)s + \lambda }}{{\left( {1 - \omega } \right)s + \lambda }} $$

where $K_{0}$ is the modified Bessel function of the second kind with zero degree and $K_{1}$ is the modified Bessel function of the second kind with the first degree; s is the Laplace parameter, and $r_{\text{D}}$ is the dimensionless radius. $\lambda$ and $\omega$ are interporosity flow term and matrix storativity ratio, respectively.

1.3 Model 3: Infinite acting homogeneous reservoir with linear fault (Ershaghi et al. 1993)

$$ \bar{P}_{\text{D}} = \frac{{K_{0} \left( {r_{\text{D}} \sqrt s } \right) + K_{0} \left( {d_{\text{D}} \sqrt s } \right)}}{{s\sqrt s K_{1} \left( {\sqrt s } \right)}}, $$

(A-4)

where $K_{0}$ is the modified Bessel function of the second kind with zero degree and $K_{1}$ is the modified Bessel function of the second kind with the first degree; s is the Laplace parameter; $d_{\text{D}}$ is the dimensionless distance from well to the fault; and $r_{\text{D}}$ is the dimensionless radius.

1.4 Model 4: Infinite acting dual-porosity reservoir (PSS interporosity flow) with linear fault (Ershaghi et al. 1993)

$$ \bar{P}_{\text{D}} = \frac{{K_{0} \left( {r_{\text{D}} \sqrt {sf\left( s \right)} } \right) + K_{0} \left( {d_{\text{D}} \sqrt {sf\left( s \right)} } \right) }}{{s\sqrt {sf\left( s \right)} K_{1} \sqrt {sf\left( s \right)} }}, $$

(A-5)

where $K_{0}$ is the modified Bessel function of the second kind with zero degree and $K_{1}$ is the modified Bessel function of the second kind with the first degree; s is the Laplace parameter; $d_{\text{D}}$ is the dimensionless distance from well to the fault; and $r_{\text{D}}$ is the dimensionless radius. The function $f\left( s \right)$ is the same as defined before for model 2.

1.5 Model 5: Infinite acting homogeneous reservoir with angular 45° fault (Ershaghi et al. 1993)

$$ \bar{P}_{\text{D}} = \frac{{K_{0} \left( {r_{\text{D}} \sqrt s } \right) + s\sqrt s K_{1} \left( {\sqrt s } \right) + \mathop \sum \nolimits_{i = 1}^{7} K_{0} \left( {d_{{{\text{D}}i}} \sqrt s } \right)}}{{s\left\{ {s\sqrt s K_{1} \left( {\sqrt s } \right) + C_{\text{D}} s\left[ {K_{0} \left( {r_{\text{D}} \sqrt s } \right) + S\sqrt s K_{1} \left( {\sqrt s } \right) + \mathop \sum \nolimits_{i = 1}^{7} K_{0} \left( {d_{{{\text{D}}i}} \sqrt s } \right) } \right]} \right\}}}, $$

(A-6)

where $K_{0}$ is the modified Bessel function of the second kind with zero degree and $K_{1}$ is the modified Bessel function of the second kind with the first degree; s is the Laplace parameter. $d_{{{\text{D}}i}}$ is the dimensionless distance defined to the ith imaginary well, and $r_{\text{D}}$ is the dimensionless radius. $C_{\text{D}}$ is the dimensionless wellbore storage coefficient.

1.6 Model 6: Infinite acting dual-porosity reservoir (PSS interporosity flow) with angular 45° fault (Ershaghi et al. 1993)

$$ \bar{P}_{\text{D}} = \frac{{K_{0} \left( {r_{\text{D}} X} \right) + sXK_{1} \left( X \right) + \mathop \sum \nolimits_{i = 1}^{7} K_{0} \left( {d_{{{\text{D}}i}} X} \right)}}{{s\left\{ {XK_{1} \left( X \right) + C_{\text{D}} s\left[ {K_{0} \left( {r_{\text{D}} X} \right) + SXK_{1} \left( X \right) + \mathop \sum \nolimits_{i = 1}^{7} K_{0} \left( {d_{{{\text{D}}i}} X} \right)} \right]} \right\}}}, $$

(A-7)

$$ X = \sqrt {sf\left( s \right)} $$

All the parameters are the same as model 5, except for $f\left( s \right)$ which is the same defined before for model 2.

1.7 Model 7: Finite acting homogeneous reservoir (no flow boundary) (Ershaghi et al. 1993)

$$ \bar{P}_{\text{D}} = \frac{{K_{1} \left( {R_{\text{eD}} \sqrt s } \right) I_{0} \left( {r_{\text{D}} \sqrt s } \right) + I_{1} \left( {R_{\text{eD}} \sqrt s } \right) K_{0} \left( {r_{\text{D}} \sqrt s } \right)}}{{s\sqrt s \left[ {I_{1} \left( {R_{\text{eD}} \sqrt s } \right) K_{1} \left( {\sqrt s } \right) - K_{1} \left( {R_{\text{eD}} \sqrt s } \right) I_{1} \left( {\sqrt s } \right)} \right]}}, $$

(A-8)

where $K_{0}$ is the modified Bessel function of the second kind with zero degree and $K_{1}$ is the modified Bessel function of the second kind with the first degree. Similarly, $I_{0}$ is the modified Bessel function of the first kind with zero degree and $I_{1}$ is the modified Bessel function of the second kind with first degree. Also, s is the Laplace parameter, $r_{\text{D}}$ is the dimensionless radius, and $R_{\text{eD}}$ is the dimensionless radius at the reservoir boundary.

1.8 Model 8: Finite acting dual-porosity reservoir with PSS interporosity flow (Ershaghi et al. 1993)

$$ \bar{P}_{\text{D}} = \frac{{K_{1} \left( {R_{\text{eD}} \sqrt {sf\left( s \right)} } \right) I_{0} \left( {r_{\text{D}} \sqrt {sf\left( s \right)} } \right) + I_{1} \left( {R_{\text{eD}} \sqrt {sf\left( s \right)} } \right) K_{0} \left( {r_{\text{D}} \sqrt {sf\left( s \right)} } \right)}}{{s\sqrt {sf\left( s \right)} \left[ {I_{1} \left( {R_{\text{eD}} \sqrt {sf\left( s \right)} } \right) K_{1} \left( {\sqrt {sf\left( s \right)} } \right) - K_{1} \left( {R_{\text{eD}} \sqrt {sf\left( s \right)} } \right) I_{1} \left( {\sqrt {sf\left( s \right)} } \right)} \right]}}, $$

(A-9)

All parameters are the same defined in model 7, except for the $f\left( s \right)$ function which is the same defined before for mode 2.