A New Analytical Solution of the Triple-Porosity Model for History Matching and Performance Forecasting in Unconventional Oil Reservoirs

SPE Journal ◽  
2018 ◽  
Vol 23 (06) ◽  
pp. 2060-2079 ◽  
Author(s):  
Kaixuan Qiu ◽  
Heng Li
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
History Matching ◽  
Oil Reservoirs ◽  
Production Data ◽  
Numerical Inversion ◽  
Unconventional Oil ◽  
The Laplace Transform ◽  
Porosity Model

Summary Production data from hydraulically fractured horizontal wells in ultralow-permeability reservoirs have been observed to exhibit a half-slope straight line on a log-log plot of rate vs. time. Using these observations, many analytical models of linear flow have been developed to analyze the well performance in these reservoirs. However, it is relatively complicated to solve these models with the Laplace transform and numerical inversion. In this paper, a new analytical solution of a triple-porosity model is derived for unconventional oil reservoirs. The partial-differential equations (PDEs) are transformed into ordinary-differential equations (ODEs) by integration, instead of the Laplace transform. A rate-vs.-time solution in real-time space can be obtained, bypassing the numerical inversion for the Laplace transform. Our model is validated by comparison with both the numerical-model and Laplace-transform solutions. The physical meanings of the model parameters are analyzed through sensitivity analysis. The new model is then applied to field-production data for history matching and forecasting. Besides matching the production data, the pore volume (PV) of the matrix and fracture media can be estimated, which helps quantitative analysis of unconventional reservoirs.

History Matching and Rate Forecasting in Unconventional Oil Reservoirs With an Approximate Analytical Solution to the Double-Porosity Model

2015 ◽  
Vol 19 (01) ◽  
pp. 070-082 ◽  
Author(s):  
B. A. Ogunyomi ◽  
T. W. Patzek ◽  
L. W. Lake ◽  
C. S. Kabir
Keyword(s):  
Analytical Solution ◽  
History Matching ◽  
Horizontal Wells ◽  
Double Porosity ◽  
Unconventional Reservoirs ◽  
Production Data ◽  
Time Space ◽  
Double Porosity Model ◽  
Fractured Horizontal Wells ◽  
Porosity Model

Summary Production data from most fractured horizontal wells in gas and liquid-rich unconventional reservoirs plot as straight lines with a one-half slope on a log-log plot of rate vs. time. This production signature (half-slope) is identical to that expected from a 1D linear flow from reservoir matrix to the fracture face, when production occurs at constant bottomhole pressure. In addition, microseismic data obtained around these fractured wells suggest that an area of enhanced permeability is developed around the horizontal well, and outside this region is an undisturbed part of the reservoir with low permeability. On the basis of these observations, geoscientists have, in general, adopted the conceptual double-porosity model in modeling production from fractured horizontal wells in unconventional reservoirs. The analytical solution to this mathematical model exists in Laplace space, but it cannot be inverted back to real-time space without use of a numerical inversion algorithm. We present a new approximate analytical solution to the double-porosity model in real-time space and its use in modeling and forecasting production from unconventional oil reservoirs. The first step in developing the approximate solution was to convert the systems of partial-differential equations (PDEs) for the double-porosity model into a system of ordinary-differential equations (ODEs). After which, we developed a function that gives the relationship between the average pressures in the high- and the low-permeability regions. With this relationship, the system of ODEs was solved and used to obtain a rate/time function that one can use to predict oil production from unconventional reservoirs. The approximate solution was validated with numerical reservoir simulation. We then performed a sensitivity analysis on the model parameters to understand how the model behaves. After the model was validated and tested, we applied it to field-production data by partially history matching and forecasting the expected ultimate recovery (EUR). The rate/time function fits production data and also yields realistic estimates of ultimate oil recovery. We also investigated the existence of any correlation between the model-derived parameters and available reservoir and well-completion parameters.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

scholarly journals Multi-term fractional integro-differential equations in power growth function spaces

2021 ◽  
Vol 24 (3) ◽  
pp. 739-754
Author(s):  
Vu Kim Tuan ◽  
Dinh Thanh Duc ◽  
Tran Dinh Phung
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Growth Function ◽  
Function Spaces ◽  
Power Growth ◽  
The Laplace Transform

Abstract In this paper we characterize the Laplace transform of functions with power growth square averages and study several multi-term Caputo and Riemann-Liouville fractional integro-differential equations in this space of functions.

Analytical solutions of citrate–phosphate coupled model of rice (Oryza sativa L.) roots

2020 ◽  
Vol 13 (07) ◽  
pp. 2050061
Author(s):  
Huiping Zhang ◽  
Shuyue Wang ◽  
Zhonghui Ou
Keyword(s):  
Oryza Sativa ◽  
Analytical Solution ◽  
Laplace Transform ◽  
Oryza Sativa L ◽  
Coupled Model ◽  
Inverse Laplace Transform ◽  
Citrate Solution ◽  
Separation Variable ◽  
The Laplace Transform ◽  
Citrate Phosphate

The citrate secreted by the rice (Oryza sativa L.) roots will promote the absorption of phosphate, and this process is described by the Kirk model. In our work, the Kirk model is divided into citrate sub-model and phosphate sub-model. In the citrate sub-model, we obtain the analytical solution of citrate with the Laplace transform, inverse Laplace transform and convolution theorem. The citrate solution is substituted into the phosphate sub-model, and the analytical solution of phosphate is obtained by the separation variable method. The existence of the solutions can be proved by the comparison test, the Weierstrass M-test and the Abel discriminating method.

scholarly journals Numerical inversion of the Laplace transform

1999 ◽  
Vol 110 (23) ◽  
pp. 11176-11186 ◽  
Author(s):  
Bruno Hüpper ◽  
Eli Pollak
Keyword(s):  
Laplace Transform ◽  
Numerical Inversion ◽  
The Laplace Transform
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