Page 239 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Using Nanofluids to Control Fines Migration in Porous Systems 213

where L 1 and L 2 are the functions of derivatives of two variables; λ 1 and
λ 2 are the line combination coefficients. If the directed derivatives of C 1
and C 2 will be collinear, it is necessary that:

dt D λ 1 B 1 1 λ 2 B 2 λ 1 D 1 1 λ 2 D 2
5 5 5 σ (A.3)
dx D λ 1 A 1 1 λ 2 A 2 λ 1 C 1 1 λ 2 C 2
In matrix form, the system of Eq. (A.3) can be written as:

A 1 σ 2 B 1 A 2 σ 2 B 2 λ 1

5 0 (A.4)
C 1 σ 2 D 1 C 2 σ 2 D 2 λ 2

If there are nontrivial values of λ 1 and λ 2 , it becomes an eigenvalue
problem; the determinate of coefficient matrix should be zero. Therefore,
two characteristic directions can be obtained as:

ð A 1 D 2 2A 2 D 1 1B 1 C 2 2B 2 C 1 Þ6
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ð
ð
6
σ 5 ð A 1 D 2 2A 2 D 1 1B 1 C 2 2B 2 C 1 Þ 24 A 1 C 2 2A 2 C 1 Þ B 1 D 2 2B 2 D 1 Þ
2 A 1 C 2 2A 2 C 1 Þ
ð
(A.5a)
If the discriminant is a positive number,
2
ð A 1 D 2 2A 2 D 1 1B 1 C 2 2B 2 C 1 Þ 2 4 A 1 C 2 2 A 2 C 1 Þ B 1 D 2 2 B 2 D 1 Þ . 0
ð
ð
(A.5b)
This quadratic equation has two real roots and there are two families of
characteristics C 1 and C - presented in form of αðx D ; t D Þ 5 const and
βðx D ; t D Þ 5 const, which are called as characteristic parameters. When the dis-
criminant is zero or negative, the system is parabolic and elliptic, accordingly.
Since there are two characteristic directions for hyperbolic system of
equations, there will be two different family of characteristics generated
by two distinct values. The subscript 6 in Eq. (A.5) indicates the corre-
sponding terms to two roots of Eq. (A.4). The characteristic lines form a
curvilinear map that serves as the possible solution route; the unique solu-
tion for different sets of initial and boundary conditions can be obtained
along these characteristics lines.
Total derivatives of (C 1 , C 2 )in x D 2 ϕ domain will be only functions
of ζ along each characteristic, αðx D ; t D Þ and βðx D ; t D Þ,
@C 1 @ϕ
@C 1 @x D
C 1;ξ 5 1 5 C 1;x D D;ξ 1 C 1;t D D;ξ
x
t
@x D @ξ @t D @ξ
@C 2 @ϕ (A.6)
@C 2 @x D
C 2;ξ 5 1 5 C 1;x D D;ξ 1 C 2;t D D;ξ
t
x
@x D @ξ @t D @ξ