A stochastic well-test analysis on transient pressure data using iterative ensemble Kalman filter

Abstract

Accurate estimation of the reservoir parameters is crucial to predict the future reservoir behavior. Well testing is a dynamic method used to estimate the petro-physical reservoir parameters through imposing a rate disturbance at the wellhead and recording the pressure data in the wellbore. However, an accurate estimation of the reservoir parameters from well-test data is vulnerable to the noise at the recorded data, the non-uniqueness of the obtained match, and the accuracy of the optimization algorithm. Different stochastic optimization methods have been applied to this address problem in the literature. In this study, we apply the recently developed iterative ensemble Kalman filter in the context of well-test analysis to infer reservoir parameters from the noisy recorded data. Since the introduction of the ensemble Kalman filter (EnKF) by Evensen in 1994 as a novel method for data assimilation, it has had enormous impact in many application domains because of its robustness and ease of implementation, and numerical evidence of its accuracy. While the objective of the standard EnKF approaches is to approximate the statistical properties of geological parameters conditioned to observation, via an ensemble, the objective of the iterative ensemble Kalman methods is to approximate the solution of inverse problems using a deterministic derivative-free iterative scheme. We conducted three case studies of the application of the iterative ensemble Kalman methods for a well-test analysis of a homogenous reservoir model, a dual-porosity heterogeneous system, and a faulted discontinuous reservoir. We demonstrated that the convergence occurs very rapidly almost at the first iterations contrary to the well-known particle swarm optimization algorithm. The maximum relative error for the simulated cases is below 15%, which belongs to the skin factor. Low relative error, narrowed uncertainty range over time, and excellent graphical match obtained between the simulated derivative data and the generated curve by using the iterative EnKF verify the robustness of the developed algorithm even in dealing with complex heterogeneous models.

Appendix: Well pressure behavior at the Laplace domain

Solution for the well pressure at the Laplace medium is provided at this section for different reservoir systems studied at this paper.

1.1 Infinite acting homogenous reservoir [69]

$$\bar{p}_{\text{wD}} \left( s \right) = \frac{{k_{0} \left( {\sqrt s } \right) + S\sqrt s k_{1} \left( {\sqrt s } \right)}}{{s\left\{ {\sqrt s k_{1} \left( {\sqrt s } \right) + s{\text{C}_{D}}\left[ {k_{0} \left( {\sqrt s } \right) + S\sqrt s k_{1} \left( {\sqrt s } \right)} \right]} \right\} }},$$

(15)

where S and C_D are, respectively, skin factor and the dimensionless wellbore storage coefficient; s is the Laplace parameter; k₀ and k₁ are, respectively, modified Bessel functions of second type with orders zero and one.

1.2 Infinite acting dual-porosity reservoir with PSS interporosity flow [70, 71]

$$\bar{p}_{\text{wD}} \left( s \right)\,=\,\frac{{k_{0} \left( {\sqrt {sf\left( s \right)} } \right) + S\sqrt {sf\left( s \right)} k_{1} \left( {\sqrt {sf\left( s \right)} } \right)}}{{s\left\{ {\sqrt {sf\left( s \right)} k_{1} \left( {\sqrt {sf\left( s \right)} } \right) + sC_{\text{D}} [k_{0} \left( {\sqrt {sf\left( s \right)} } \right) + S\sqrt {sf\left( s \right)} k_{1} \left( {\sqrt {sf\left( s \right)} } \right)]} \right\}}},$$

(16)

where f(s) is defined by the following equation:

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

(17)

ω is the fracture storativity ratio and λ stands for the interporosity flow coefficient representing how strong is the communication between the matrix and the fracture system. Other parameters are the same as for Eq. 15.

1.3 Infinite acting homogenous reservoir with a linear fault [72]

$$\bar{p}_{\text{wD}} \left( s \right) = \frac{{k_{0} \left( {\sqrt s } \right) + S\sqrt s k_{1} \left( {\sqrt s } \right) + k_{0} \left( {2d_{\text{D}} \sqrt s } \right)}}{{s\left\{ {\sqrt s k_{1} \left( {\sqrt {s} } \right) + sC_{\text{D}} \left[ {k_{0} \left( {\sqrt s } \right) + S\sqrt s k_{1} \left( {\sqrt s } \right)} \right]} \right\}}},$$

(18)

where d_D is the fault dimensionless distance defined by $d_{\text{D}} = \frac{{L_{f} }}{{r_{w} }}$.