Aplikasi Kisi Gas Automata Untuk Memperkirakan Efektifitas Porositas dan Model Penghalang Permeabilitas Pada Segitiga Dengan Variasi Ketinggian Lattice Gas Automata Applications to Estimate Effective Porosity and Permeability Barrier Model of the Triangle with a Height Variation

Penelitian ini bertujuan untuk menghitung porositas efektif (фeff) dan permeabilitas (k) menggunakan model segitiga dengan variasi tinggi yaitu 3, 4, 5, 6 dan 7 cm. Perhitungan porositas dan permeabilitas yang efektif dilakukan dengan menggunakan model Lattice Gas Automata (LGA), yang diimplementasikan dengan bahasa pemrograman Delphi 7.0. Untuk model segitiga penghalang dengan tinggi 3, 4, 5, 6 dan 7 cm, nilai porositas efektif dan permeabilitas, masing-masing: фeff (T1) = 0,1690, k (T1) = 0 , 001339 pixel2; фeff (T2) = 0,1841, k (T2) = 0,001904 pixel2; фeff (T3) = 0,1885, k (T3) = 0,001904 pixel2; фeff (T4) = 0,1938, k (T4) = 0001925 pixel2; dan фeff (T5) = 0,2053, k (T5) = 0,002400 pixel2. Dari hasil simulasi, diperoleh tinggi segitiga akan berpengaruh signifikan terhadap nilai porositas efektif dan permeabilitas. Pada segitiga lebih tinggi, menyebabkan tabrakan model aliran fluida LGA mengalami lebih banyak hambatan untuk penghalang, sehingga porositas efektif dan permeabilitas menurun. Sebaliknya, jika segitiga lebih rendah, menyebabkan tabrakan model aliran fluida LGA mengalami lebih sedikit hambatan untuk penghalang, sehingga porositas efektif dan permeabilitas meningkat.


Introduction
There are two common ways the study of fluid. The first is to take a macroscopic viewpoint that describes the fluid as a continuum. The second uses microscopic point of view that illustrates the interaction between the particles in a fluid. Fluid has a characteristic length scale. On a macroscopic scale, the characteristics associated with a channel width or diameter of obstacles or it can also measure the vortex. At the micro scale, these characteristics are determined by particle displacement distance before the collision, or the mean free path. On a micro scale, the mean free path for the liquid fluid is much smaller than gas (Bimo. BB, 2009). With the *corresponding Author: halau_uddin@yahoo.com http://www.jurnal.unsyiah.ac.id/JAcPS advancement in the field of numerical computation, simulation of fluid flow have been done, although they found some difficulty in the number of lattice, and numerical stability (Koponen, 1998). Some relationships between reservoir parameters have been able to be explained by numerical modeling, one of which is a method of Lattice Gas Automata (LGA).
LGA is a variation of cellular automata system, the lattice as a medium. In this research, physical modeling done for validation mechanisms of fluid flow in porous medium. The purpose of this research is to get a comprehensive understanding of the mechanisms of fluid flow in porous medium with the actually. Permeability is a reservoir rock properties to be able to pass the liquid through the pore are interconnected, without destroying the particle forming or the rock frame work Henry Darcy has introduced a simple equation to calculate the velocity laminar flow of a viscous fluid in a porous medium written as: dx dp η k q  (1) Where q is flow rate per cross-sectional area is expressed in centimeters per second, k is hydraulic permeability,  is viscosity of the fluid, and dx dp is gradient of pressure. Equation (1) in the discipline geohydrology can be modified to: Where k f is coefficient of seepage, h is head height difference of head, and l is length of the medium. Thus it is clear that the permeability is k expressed in Darcy. Definitions API for 1 Darcy is a porous medium has a permeability of 1 Darcy, if the liquid-phase the viscosity 1 centipoise flowing at a speed of 1 cm/sec through the cross-sectional area of 1 cm 2 at a hydraulic gradient of the atmosphere (76.0 mm Hg) per centimeter and if the liquid is entirely filling the medium.
From the above definition does not explain the relationship between permeability and porosity. Actually there is no relationship between permeability and porosity. Always porous permeable rock, but on the contrary, the porous rock is not necessarily permeable. This is because rock has a higher porosity of the pore is not necessarily related to each other. On the contrary it can be seen that the porosity is independent of particle size and permeability is a direct function of the grain size (Koesoemadinata, R.P, 1980).

The equation microdynamic and macrodynamic LGA
Rules and medium collisions in the LGA is a triangular lattice. The following is a mathematical formulation to describe fluid flow from the cellular system LGA method: Particles moving in a hexagonal lattice as Boolean variable n i (x,t), the value will be equal to 1 if there is a particle and 0 if no particle moving from position x to position x+c i . Operator delta (Δ) is the collision operator that describes changes in the value n i (x,t). These collisions can be valuable operator 0, 1 or -1.
If there is no change in the number of particles i as a result of collision events, where the number of particles before and after the collision is equal to the value of ∆ I = 1. The particles move from position x to position x + c i , particles moving at a speed unit in the direction: where i are 1, 2, 3, …, 6. The particles collisions in the medium and the gas grid must meet the law of conservation of mass, it's condition: and meet the law of conservation of momentum using the relationship (6), the equation micro-dynamic for all directions i, conservation of mass becomes: Meanwhile, to get the equations of conservation of momentum is obtained by multiplying the equation (3) with c i , Equation (7) and (8) described the evolution of mass and momentum in Boolean terrain and can be considered as a mass and momentum balance equation of the lattice gas systems.
LGA macro-dynamic equation obtained by looking at the case in Figure 1 which illustrates the evolution that occurs in the system. In the picture depicted a region A, of the lattice surrounded by the line S. Equation (8) can be written as: The left side of the equation is identical to equation finite difference and right sections are discrete statement of integral surface. By stating Σ< n i > as the average number of particles for all components i in one group and assumed <n i >(x,t) changes slowly over space and time. Microscopic conservation of mass can be stated to be: so that the equation (10) can be written in the form: where α component of velocity c i otherwise c iα . The above description also applies to the momentum flux, so the law of conservation of momentum equation can be written as: The above equation can be written in a simpler form to becomes: To declare the equation (12) and (13), defined variable density physics: and density of momentum: with substituting equation (12) and (13) to (11) obtained the continuity equation: by defining the momentum flux tensor for the LGA as: then the macroscopic momentum equation becomes: In the case of a low speed, tensor Π αβ can be expanded becomes (Frisch U, dkk., 1986): where   is the elasticity tensor. Finally, from the equation (16) can be approached equation that approximates the shape of real, namely: Equation (19) is similar to the Euler equations for the case of compressed streams. While A and B are two free elastic modules that can be obtained from the population of an average <n i > ( Frisch U, et al., 1986).

Equation permeability on the porosity of the rock cracks
An understanding of the pattern of fluid flow in the cracks is very important to do. In exploration and exploitation, whether it's oil or to search for ground water, information about the pattern of cracks can give us a fluid movement. Thus it is possible to predict the position of the fluid is located. (Halauddin, 2003). Simple cracks pattern shown in Figure 1.
Permeability is calculated by equation (22), (Koponen, et al, 1996): Where eff  is effective porosity medium, c is coefficient Kozeny and S is specific surface area. S is calculated by an equation (23), (Dullien, 1992) ) With R 0 is the hydraulic radius.

Research Methodology
There are five pieces cracks with barrier model of the triangle to be simulated based on the high varying. These cracks are make based on the high varying are 3, 4, 5, 6 and 7 cm, but it,s of the triangle is fixed that is 4 cm. Profile of five such cracks are as shown in Figure 2.

Results and Discussion
The simulation was performed with a program language Borland Delphi 7.0., with a timestep constant at 1000 time-step. Files saved with the extention (*.md3). There are several parameters directly known only after running the data, are: 1. In Notepad notes form the magnitude of total porosity and effective porosity and permeability values. 2. In the form of bmp.image, obtained a graph of porosity, a graph of permeability versus timestep, as well as illustrations of fluid flow *corresponding Author: halau_uddin@yahoo.com http://www.jurnal.unsyiah.ac.id/JAcPS through cracks samples for each variation of the angle at the time before and after running data. While the permeability results LGA models shown detail in Table 1.
Calculating of porosity and permeability for cracks model with high barrier of the triangle for 3 cm, time-step of duration are 1000 shown in Fig. 3.   In Figure 6, showed magnitude of effective porosity value and permeability for 4 cm with high barrier of the triangle, with of value 0,1841 and 0,001664 pixel 2 , respectively.  In Figure 8, showed magnitude of effective porosity value and permeability for 5 cm with high barrier of the triangle, with of value 0,1885 and 0,001904 pixel 2 , respectively.      Lattice Gas Automata (LGA) model can calculating total porosity value, effective porosity and permeability versus time-step using the barrier model of the triangle with a high varying. Total porosity value, effective porosity and permeability influenced by high barrier of the triangle. If barrier of the triangle high, influenced as linier towards to rise total porosity value, effective porosity and permeability. In case to caused that current of simulation fluid at running data just direction as horizontal causing the collision fluid with model as barrier of triangle according to high barrier of the triangle model.