ABSTRACT
The total fluid mass in a porous volume can be computed using its density distribution, as it was done in section 2.1.3; in any differential porous volume dV:
fluid mass in dV: dMf = ρf dV = ϕ ρf dV ⇒ Mf = ∫ VB
ϕ ρf (r, t)dV
r = (x, y, z) is the position vector of a fluid particle in V = ϕVB; t ≥ 0. (4.1)
This equation presupposes that the pore fluid density depends on both space and time. If the mass of the fluid is conserved, it must remain constant under motion. Thus, the advective or total derivative (eq. 2.4) of the previous integral (4.1) is zero (eq. 2.6a):
DMf Dt
= D Dt
ϕ ρf (r, t)dV = ∫ VB
( ∂(ϕ ρf )
∂t + ∇ · (ϕ ρf vf )
) dV = 0 (4.2)
This equation is the integral form of the principle of conservation of fluid mass in a porous rock, valid for any part of the rock continuum. Vector vf = (vx, vy, vz) is the field velocity of the fluid particles. Figure 4.1 shows a schematic representation of a differential porous rock volume with bulk modulus VB and pore volume V. The volume VB can be any arbitrary portion of the porous rock. For example let VB be a differential volume (REV) VB ≈ dV , consequently the continuous integrand of equation (4.2) must be identical to zero:
∂(ϕ ρf )
∂t + ∇ · (ϕ ρf vf ) = D(ϕ ρf )Dt + (ϕ ρf ) ∇ · vf = 0 (4.3)
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Formula (4.3) is also known as the continuity equation for the fluid, which expresses the same principle of conservation of mass in the form of a partial differential equation. Under appropriate conditions, this law can be related directly to the volumetric deformation of the fluid (see section 2.1.3 for further details):
D(ϕ ρf )
Dt = −(ϕ ρf ) ∇ · vf ⇒ ∇ · vf = −1
ϕ ρf
D(ϕ ρf )
Dt ; but Vf ρf = Mf
⇒ −Vf ϕMf
D
Dt
( ϕMf Vf
) = −Vf DDt (Vf )
D Vf Dt
) = 1
( DVf Dt
)
Therefore:
with: εf = VfVf = ∇ · uf , vf = ∂uf
∂t ⇒ ∇ · vf = 1Vf
( DVf Dt
) = D εf
Dt (4.4)
Hence, the divergence of the fluid velocity is equal to the rate of change of the fluid volume Vf . The variable εf is defined as the volumetric deformation (dilatation or compaction) of the fluid phase. If the fluid and the porosity do not change with time:
∂(ϕ ρf )
∂t = 0 ⇒ ∇ · (ϕ ρf vf ) = 0 (4.5)
If the fluid is incompressible and the porosity is constant:
ϕ ρf ∇ · (vf ) = 0 ⇒ ∇ · vf = 0 (4.6)
All previous equations are different forms of the same principle of conservation of fluid mass. The last formula (4.6) is the simplest form of the continuity equation. If there is extraction or injection (qf ) of fluid mass in one or more parts of the reservoir:
( ∂(ϕ ρf )
∂t + ∇ · (ϕ ρf vf )
) dV =
qf (r, t)dV
⇔ ∂(ϕ ρf ) ∂t
+ ∇ · (ϕ ρf vf ) = qf [
kg
m3s
] (4.7)
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f f the injection of fluid into the reservoir (source: qf > 0) at the specific locations where the wells are drilled.
