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The flow of two or more immiscible fluids in porous media is ubiquitous particularly in oil industry. This includes secondary and tertiary oil recovery, CO2 sequestration, etc. Accurate predictions of the development of these processes are important in estimating the benefits, e.g., in the form of increased oil extraction, when using certain technology. However, this accurate prediction depends to a large extent on two things; the first is related to our ability to correctly characterize the reservoir with all its complexities and the second depends on our ability to develop robust techniques that solve the governing equations efficiently and accurately. In this work, we introduce a new robust and efficient numerical technique to solving the governing conservation laws which govern the movement of two immiscible fluids in the subsurface. This work will be applied to the problem of CO2 sequestration in deep saline aquifer; however, it can also be extended to incorporate more cases. The traditional solution algorithms to this problem are based on discretizing the governing laws on a generic cell and then proceed to the other cells within loops. Therefore, it is expected that, calling and iterating these loops several times can take significant amount of CPU time. Furthermore, if this process is done using programming languages which require repeated interpretation each time a loop is called like Matlab, Python or the like, extremely longer time is expected particularly for larger systems. In this new algorithm, the solution is done for all the nodes at once and not within loops. The solution methodology involves manipulating all the variables as column vectors. Then using shifting matrices, these vectors are sifted in such a way that subtracting relevant vectors produces the corresponding difference algorithm. It has been found that this technique significantly reduces the amount of CPU times compared with traditional technique implemented within the framework of Matlab.

SPE Disciplines:

El-amin, Mohamed Fathy (KAUST) | Negara, Ardiansyah Kusuma (King Abdullah University of Science and Technology) | Salama, Amgad (King Abdullah U of Science & Tech) | Sun, Shuyu

The problem of coupled structural deformation with two-phase flow in porous media is solved numerically using cellcentered finite difference (CCFD) method. In order to solve the system of governed partial differential equations, the implicit pressure explicit saturation (IMPES) scheme that governs flow equations is combined with the the implicit displacement scheme. The combined scheme may be called IMplicit Pressure-Displacement Explicit Saturation (IMPDES). The pressure distribution for each cell along the entire domain is given by the implicit difference equation. Also, the deformation equations are discretized implicitly. Using the obtained pressure, velocity is evaluated explicitly, while, using the upwind scheme, the saturation is obtained explicitly. Moreover, the stability analysis of the present scheme has been introduced and the stability condition is determined.

**Introduction **Coupling of solid deformation with fluid flow in porous media is one of the most challinging numerical issues in reservoir engineering applications such as CO2 sequestration and enhanced oil recovery. Several schemes; such as, fully implicit, IMplicit-EXplicit (IMEX), operator splitting, and IMplicit Pressure Explicit Saturation (IMPES); are widely used to solve the time-dependent partial differential equations that govern fluid flow in porous media. However, only few works address the issues related to mathematical and numerical modeling of the coupled flow-deformation problem. Furthermore, IMPES scheme is widely used for simulating the problem of two-phase flow in porous media. In spite of the fact that fully implicit method [1-5] is computationally expensive, it is unconditionally stable. The IMEX method [6-8] is more stable compares to the fully implicit method because it considers the linear terms implicitly and solves the other terms explicitly. The IMEX scheme is used to solve the ordinary differential equations resulting from the spatial discretization of the time-dependent partial differential equations. The operator splitting method [9-11] is used to simplify the original problem into a simpler form by the time-lag dimension. Furthermore, the iterative operator splitting [11, 12] has been considered as a part of the step of the iteration in the fully implicit method. The sequential method [13], on the other hand, is a modified version of the

IMPES method since the saturation is also evaluated implicitly. Comparing to the fully implicit method, the computational cost of the sequential method is less; therefore it is suitable for the large size models where stability becomes an important consideration.

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