C. Dong et al. / Natural Science 2 (2010) 1030-1034

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ble; 4)The fluid flow obeys Darcy’s Law; 5)The flow is

under an isothermal condition. On the basis of these as-

sumptions, and utilizing the continuity equation and flow

equation, the pressure equation of streamline model can

be established as follows:

twv 0cow

0PDqP

(1)

where λo, λw are, respectively, mobility of oil and water,

μm2/Pa·s; λt is total mobility, μm2/Pa·s; γo, γw are, re-

spectively, gravities of oil and water, dimensionless; qvo,

qvw are volume flow rate of producer and injector, m3/s;

qv is the total flow rate of injection or production wells,

m3/s; D is the depth from the reference level, its direc-

tion is the same to gravity acceleration, m; pcow is the

capillary force between oil phase and water phase, Pa.

Eq.1 can be solved by finite difference method. And

the pressure distribution, i.e. pressure field, can be de-

rived.

According to the pressure field, streamline tracing is

implemented by using Pollock method which is defined

with the following assumptions: the velocity field in a

grid cell as a linear interpolation, and the velocity in

each direction is only a function of the coordinate in that

direction.

In a 2D grid system, with the pressure at the face, the

potential at the face of the grid cell can be calculated.

Then the velocity in each direction is derived, the TOF

can be calculated also. The face at which the streamline

exits is the one corresponding to the minimum positive

exit time. The exit point will be the entry point of the

adjacent grid cell, repeating this calculation process until

the result converges to the grid cell which a production

well is in. Connecting the exit points in chronological,

an integrated streamline can be derived.

In a 3D system, the exit points can be obtained by

utilizing Pollock method to calculate the values in the

z-direction. By connecting these points, production and

injection wells, the streamline can be derived.

The advantages of this method: being analytical; and

it constructs a velocity field that satisfies the flux con-

servation condition.

2.2. Saturation Field

After the pressure filed is solved and the streamlines are

traced by utilizing finite difference method, the next step

is to solve the saturation along the streamlines by intro-

ducing the time-of-flight concept. The time-of-flight is

defined as the time required for a particle to travel a dis-

tance along the streamline,

0

1

x

t

d

u

(2)

In streamline tracing, the time for streamlines across

each grid cell, Δte,i , can be calculated using Pollock

method. The time of flight can be expressed as follow:

,

1

nblocks

ei

i

t

(3)

where nblocks designates the number of grid cells when

a particle travel a distance as s; Δte,i is the time for a par-

ticle to travel through the i’th cell, s.

A. Distribution of volumetric flux of streamline.

Tracing streamline from the cell of injection well to the

cell of production well. For a grid cell with source or

sink, as the streamline is not linear in segments, the

streamline generalizes from the face of the grid cell in-

stead of generalizing from the centre of the grid cell.

For simplicity, distribution of flux is based on chang-

ing the number of streamlines with different flow rate of

injection well, and the volumetric flux along each

streamline is constant. The higher injection flow rate, the

more streamlines. Similarly, much more streamlines can

be traced in a high flow rate region.

In grid cell (i,j,k) with a source, the volumetric flux at

the interface (i ± 1/2,j,k) of grid cell (i,j,k) and grid cell (i

± 1,j,k) is Qi ± 1/2,j,k. The volumetric flux along each

streamline is qsli ± 1/2,j,k at the interface(i ± 1/2,j,k). The

number of streamlines generalized from this face is ni ±

1/2,j,k can be expressed as follow:

12, ,12, ,12, ,ijk ijkslijk

nQq

(4)

where Qi ± 1/2,j,k is given as:

12, ,12, ,, ,1, ,i jkijkijkijk

QTXPP

(5)

where TXi ± 1/2,j,k represents the transmissibility in the

x-direction of oil phase and water phase between the

grid cell (i ± 1,j,k) and the grid cell (i,j,k); Pi,j,k is the

pressure in the grid cell (i,j,k).

Similarly, the number of streamlines on other faces

can be obtained.

B. Establishing the saturation equation in streamline

model. Considering capillary effect, the fluid flow equa-

tion can be expressed as follow:

itw 0cow

vPDP

(6)

Combining with saturation equation and mass con-

servation equation, the saturation equation of streamline

model for oil-water two phases can be derived:

w

t

wwo

ww cowvw

τt

S

S

DPq

(7)