@article{072ab1b554394435b0fb7923221c1ac1,

title = "New solution to the pressure transient equation in a two-layer reservoir with crossflow",

abstract = "This paper presents a new mathematical model and an analytical solution to the pressure transient equation of a uniform-flux which fully penetrates the vertical well in a two-layer petroleum reservoir with crossflow. This new model and solution provide an accurate and fast tool to (1) evaluate a vertical well performance in a two-layer reservoir; (2) estimate the effects of formation properties on pressure behavior at the locations both far away from and around the well. The solution is obtained from the model by using Laplace transform, double Fourier transform, and Green's functions method. When all the corresponding parameters are identical in each layer, the proposed solution for a two-layer reservoir is naturally reduced to the solution for a single layer reservoir. Based on the analytical solution, the pressure of the vertical well, which produces with a constant rate in the two-layer reservoir, can be examined in detail. More importantly, a dimensionless crossflow coefficient is proposed to describe the crossflow in a two-layer reservoir quantitatively. It is verified that the crossflow coefficient is not constant, but a function of the time and distance away from the wellbore. In the numerical experiments, the pressure response is observed in three stages: (1) the first one is similar to the commingled system because the crossflow is very weak at the beginning of the production; (2) the second one is the transition stage when the crossflow becomes stronger with the production; (3) the third stage is close to the performance of the homogeneous system when the crossflow coefficient approaches zero.",

keywords = "Crossflow, Pressure behavior, Two-layer reservoir",

author = "Jing Lu and Botao Zhou and Rahman, {Md Motiur} and Xiaoming He",

note = "Funding Information: This paper was presented at the SPE Reservoir Characterization and Simulation Conference and Exhibition held in Abu Dhabi, UAE, 14–16 September 2015 (SPE175630). The authors would like to thank Society of Petroleum Engineers and the senior management of Khalifa University of Science and Technology at Abu Dhabi, UAE for their continuous support and for permission to publish this work. Appendix The Laplace transform is a widely used integral transform, but an analytical inversion of a Laplace domain solution is usually difficult to obtain; thus, a numerical inversion method must be used. There are several numerical algorithms in literature that can be used to perform the Laplace inversion. The frequently used inversion algorithm is Zakian{\textquoteright}s method [ 7–9 ], which is fast and easy to implement. Zakian{\textquoteright}s method approximates the time domain function using the following series of weighted evaluations of domain function: (A.1) f t = 2 t ∑ j = 1 n Re K j F α j t where F(s) is Laplace transform of f(t), and s = α j ∕ t in Eq. (A.1) , Re is the real part operator for a complex number. The constants K j and α j for n = 5 are given in Table A.1 . In this paper, we use Eq. (A.1) and the constants K i and α i given in Table A.1 to perform the Laplace inversion. Funding Information: This paper was presented at the SPE Reservoir Characterization and Simulation Conference and Exhibition held in Abu Dhabi, UAE, 14?16 September 2015 (SPE175630). The authors would like to thank Society of Petroleum Engineers and the senior management of Khalifa University of Science and Technology at Abu Dhabi, UAE for their continuous support and for permission to publish this work. Publisher Copyright: {\textcopyright} 2018 Elsevier B.V.",

year = "2019",

month = dec,

day = "15",

doi = "10.1016/j.cam.2018.05.065",

language = "British English",

volume = "362",

pages = "680--693",

journal = "Journal of Computational and Applied Mathematics",

issn = "0377-0427",

publisher = "Elsevier",

}