Academia Journal of Scientific Research 5(2): 034-038, February 2017 DOI: 10.15413/ajsr.2016.0152 ISSN 2315-7712 ©2017 Academia Publishing

Research Paper

Numerical Investigation of Reservoir Problems

Accepted 17th October, 2016 ABSTRACT In numerical simulation of reservoir problems, the correctness of simulation is of great importance. Thus, validation with experimental and/or analytical results must be done. The aim of this paper is to present a mathematical framework and its corresponding coupled with finite element discretization for fully saturated porous media. The model is developed as a coupled displacement-pressure formulation in which the porous medium is composed of a soil skeleton and water in the pores. The considered problem in this paper is the reservoir filled with oil subjected to uniform pressure at the top. The consequent mathematical model involves equations of mass and momentum balance for the whole system. Interesting outcomes are achieved from the numerical simulation and results thoroughly discussed. Key words: Numerical simulation, multiphase modelling.

INTRODUCTION In numerical simulation of geotechnical problems, it is important to consider both the mass transport and flux equations. This part of geomechanics considers the transport problems in aquifers, displacements due to soil consolidation and design of safe containers etc. Simulation of such problems using the finite elements includes the full interaction of the pore pressure with the soil skeleton. It is to be stated that these models based on fully coupled formulation require the simultaneous solution of fluid flow equation and equilibrium equations in terms of displacements. Thus, the degrees of freedom per node correspond to both displacement and pore pressures.

The soil medium has various components such that it can be also called a multiphase body. It follows that observed at a macroscopic scale soil media can be considered as mixtures (Alcoverro, 2003). A detailed description of the porous media is not the scope of this paper although, consolidation problem was taken into consideration. The consolidation problem of a soil layer is known since the work of Terzaghi (1943) and Biot (1955). In classical fully saturated consolidation, the boundary conditions are drained during the entire calculation while the flow of water out of the region of interest is considered. In the work of Zienkiewicz and Shiomi (1984), it is shown that any constitutive relations of the soil

skeleton can be defined incrementally since it is assumed that changes in the pore pressure do not cause any strain change in the porous solid material. In the literature different versions of effective stress principles are proposed (de Boer and Ehlers, 1990). Therefore, the effective stress principle in saturated soils is considered following the work of Verruijt (2010).

In the present work, the analysis of water flow through saturated soil, coupled with the mechanical behavior of the soil skeleton is considered. A mathematical framework assuming porous medium in which the voids of the medium are filled with water is considered and the solution of the partial differential equation system is solved using the finite element method. The numerical model involves both momentum and mass balance equations. GOVERNING EQUATIONS In saturated porous media, the governing equations considering the dynamic problem are the mass balance and the momentum conservation equations. Following the work of Oettl et al. (2004), the momentum balance and are mass conservation equations of fluid saturated medium

Kemal Edip, Mihail Garevski, Vlatko Sheshov, Julijana Bojadjieva and Toni Kitanovski Institute of Earthquake Engineering and Engineering Seismology,Skopje, Macedonia. *Corresponding author. E-mail: kemal@pluto.iziis.ukim.edu.mk

Academia Journal of Scientific Research; Edip et al. 035

Table 1: Parameters for the oil reservoir.

Young’s modulus 1.44 × 104 MPa

Poisson’s ratio 0.2

Biot’s coefficient 0.79

Kw 1.23 × 104 Mpa

Solid density 2000 kg/m3

Oil density 940 kg/m3

Porosity 0.2

Permeability 2 x 10-13 m2

Kinematic viscosity 1.2 x 10-4 m2/s

Figure 1: Model of an oil reservoir.

combined yielding the following two equations:

10

s f

n np

K K

u v

(1)

u σ b (2)

In Equation 1, the term ů stands for the solid velocity, Ks and Kf denote the bulk modulus of solid and fluid phases. In Equation 2, the term ü is the acceleration of the solid phase, b is the body force and σ is the Cauchy stress tensor. The term ρ

denotes the density of the medium and can be written as:

1 s fn n

(3)

In Equation 3, the terms ρw and ρs denote the densities of water and solid phases. The effective stress tensor is given as:

' p σ σ I (4)

The relative fluid velocity v is governed by the Darcy’s law and can be written as:

f f

f

pg

k

v b u

(5)

Where k is the permeability tensor, g is the gravity andp

is the gradient of pressure.

In order to derive the finite element matrices, the weighted residual method is applied to equation 5. The displacement vector u of the solid skeleton and the pore pressure p are chosen as the basic variables of the problem. The finite element discretization in space is presented by the following system of equations:

0 0

0

sw u

T

sw ww ww w

K C fu u

C P H fp p

(6)

In Equation 6, the matrix Csw stands for coupling between solid and water phases. The matrix P stands for compressibility and matrix H for permeability. More detailed explanation for the derivation of the constituent matrices can be found in the work of Edip (2013).

Simulation of the reservoir

In this particular example, a filled oil reservoir is simulated. In numerical simulation, the fluid phase is considered to be oil with the parameters (Table 1). Oil reservoirs are composed of porous media which include both solid phase and pores filled with oil. The pore pressure and strain in the solid have interaction with each other. Any change of the pore pressure is affected by the strain of solid and vice versa. Figure 1 shows the porous sample.

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Ver

tica

l d

efo

rma

tio

n (

m)

Figure 2. Comparison of numerical vertical displacement with experimental values.

The dimensions are 2 m × 3 m. A uniform load is applied

at the top of the reservoir. The applied uniform load is of 4000 kPa. The horizontal motions of the sides are restricted. The vertical displacement at the bottom is considered to be zero. Under the undrained condition, fluid is not allowed to flow through the boundaries. The parameters of this sample are given in Table 1.

Numerical results of the vertical displacement and pore pressure are shown in figures along with the analytical solutions as given in the work of Zheng et al. (2003). The vertical displacement and pore pressure is distributed along the element. After the initial undrained loading, the top of the sample is opened such that the pore pressure at the top becomes zero so that the oil can flow through the upper boundary, while the load remains constant at 4000 kPa. The pressure and displacement change with time is simulated using the three phase model. Figure 2 shows the results and comparisons plotted. The red dots present the

numerical results while the solid lines present the experiment performed at the MIT laboratories as illustrated in the work of Zheng et al. (2003). As it is seen from Figure 2, as time goes to infinity the state of the sample settles down.

Figure 3 shows the pore pressure comparison performed. It is to be stated that the relation between the pore pressure and displacement in the analyzed case is time dependent and the dependence of time is properly solved considering the Darcy’s flow (Edip, 2013).

From Figure 3, it is seen that the pore pressure is decreased as time passes. This is a logical consequence since there is expected to be a flow of liquid state through the drained boundary conditions which is at the top of the model.

The comparison of the numerical and experimental results show good correspondence with each other. This is due to the fact that in numerical modelling the interaction

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Wa

ter

pre

ssu

re (

kp

a)

Figure 3. Comparison of pore pressure with experimental values.

between displacement and pore pressure is simultanously considered. Namely, the change in the pore pressure has effects on the displacement of the solid phase. CONCLUSION In this paper, a numerical model concerning the porous medium is presented by considering the interaction between the solid displacement and pore pressure in porous medium. The model is applied to a plane strain consolidation of a saturated reservoir subjected to uniform loading at the top of the soil layer. The investigation in this particular paper showed that the simulation of reservoir is correctly performed. As can be seen from the results the coupling of pores pressure with displacement leads to similar results obtained from experiments. It might be conluded that this model is a good base for further development of the coupled approach for unsatured modelling of soil media.

REFERENCES Alcoverro J (2003) The effective stress principle. Mathematical and Computer Modelling. 37(5–6): p. 457-467. Biot MA (1955). Theory of elasticity and consolidation for porous

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effective stresses. acta mechanica. 83(1-2): p. 77-92. Edip K (2013). Development of three phase model with finite and infinite

elements for dynamic analysis of soil media, Ss. Cyril and Methodius: Institute of Earthquake Engineering and Engineering Seismology.

Edip K, Garevski M, Sesov V. Numerical simulation of wave propagation in soil media.

Oettl G, Stark R, Hofstetter G (2004) Numerical simulation of geotechnical problems based on a multi-phase finite element approach. Comput. Geotech. 31(8): p. 643-664.

Terzaghi K (1943). Theoretical soil mechanics. Verruijt A (2010). Theory of Consolidation, in An Introduction to Soil

Dynamics, Springer Netherlands. p. 65-90. Zheng YB, Robert B, Daniel R (2003). Reservoir Simulation with the Finite

Element Method Using Biot Poroelastic Approach, in Earth Resources Laboratory Industry Consortia Annual Report;2003-112003: Massachusetts Institute of Technology. Earth Resources Laboratory.

Zienkiewicz OC, Shiomi T (1984). Dynamic behaviour of saturated porous

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media; The generalized Biot formulation and its numerical solution. Int. J.

numer. Anal. methods in geomech. 8(1): p. 71-96.

Cite this article as: Edip K, Garevski M, Sheshov V, Bojadjieva J, Kitanovski T (2017). Numerical Investigation of Reservoir Problems. Acad. J. Sci. Res. 5(2): 034-038. Submit your manuscript at: http://www.academiapublishing.org/journals/ajsr