The first important experiments of fluid flow through porous media, were reported by Dupuit in 1854, using water-filters. His results showed that the pressure drop across the filter is propor-tional to the water filtration velocity. In 1856 Henry Darcy proved that flow of water through sand filters, obeys the following relationship:
q = K · Ah
∆l, (5.1)
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where h is a difference in manometer levels, i.e. hydrostatic height difference, A is cross-sectional area of the filter,
∆l is thickness of the filter in the flow direction and K is a proportionality coefficient.
In Darcy’s experimental results, as in Eq. (5.1), viscosityµ, was not included because only water filters were investigated and hence, the effects of fluid density and viscosity had no real experimental significance.
Experiments repeated after Darcy, have proved that if the manometric level, h, is kept constant, the same flow rate (or flow velocity) is measured, irrespective of the orientation of the sand filter (see Fig. 5.1).
Datum plane
θ= 90o 0 <o θ< 90o θ= 90o θ
I II III
Figure 5.1: Orientation of the sand filter with respect to the direction of gravitation.
The pressure difference across the sand filter in Fig. 5.1, for the 3 cases are given,
I : ∆pI=ρg(h −∆l), II : ∆pII=ρg(h −∆l · sinθ), III : ∆pIII=ρgh,
where∆l is the thickness or length of the sand filter in the flow direction.
Since the water velocity is proportional to the manometric level (observation made by Darcy), the flow velocity is proportional to,
v∝(∆p +ρg∆z),
where∆z is the elevation in the gravitational field. (∆z accounts for the inclined flow direction relative to horizontal flow.)
If the sand filter is made longer, a reduced flow velocity is expected and similarly if the water is replaced by a fluid of higher viscosity, a reduced flow velocity is expected.
v∝ 1 µ
∆p +ρg∆z
∆l .
The proportionality, above, can be replaced by equality, by introducing a proportionality coefficient k,
v = k µ
∆p +ρg∆z
∆l , (5.2)
where k is the permeability.
The pressure at any point along the flow path is related to a reference height or datum plane z0, where ∆z = z − z0 and e.g. z0= 0 at a level where the reference pressure is 1 atm. A pressure difference∆(p +ρgz) = (p +ρgz)2− (p +ρgz)1will create a fluid flow between the two points, unless the pressure p is equal to the static pressure −ρgh. In these cases no flow is expected and static equilibrium is established, as observed in any reservoir where the fluid pressure increases with depth.
Fluid flow in a porous rock is therefore given by the pressure potential difference∆(p + ρgz), i.e. the sum of pressure difference and elevation in the gravity field. In a historical context, the pressure potential has been associated with the energy potential (energy pr. mass) and the following definition has been used,
Φdef= p ρ+ gz.
Substituting the pressure potential difference∆Φin Eq. (5.2), one can rewrite the equality equation based on Darcy’s deduction,
q = Ak µρ∆Φ
∆l
where k is the permeability of the porous medium (filter, core sample/plug, etc.),µ is the viscosity of the fluid and l is the length of the porous medium in the direction of flow andΦis the pressure potential. The flow rate q = dV /dt, is volume pr. time.
The Darcy’s law in differential form is, q = Ak
For linear and horizontal flow (parallel to the x-axis) of incompressible fluid, the elevation is constant, i.e. dz/dx = 0, and Dracy’s law is written,
q = −Ak µ
d p
dx, (5.4)
where the minus sign "-", in front of the pressure gradient term, compensates for a negative pressure gradient in the direction of flow (since fluids move from high to low potential). Ve-locity and flow rate are pr. definition positive parameters (see the example below).
At this point it is important to notice that the permeability, k, is introduced in Eqs. (5.4) and (5.3), as a proportionality constant and not as a physical parameter. The permeability does pr. definition, not carry any characteristic information about the porous medium. When
permeability is related to the transport capability of the porous medium, as often is the case in practical situations, the fact that this information about the porous medium is missing in Eq. (5.4), is often overlooked. The proportionality constant k, called permeability, describes not only the porous medium transport capability, as such, but represents all information about the porous medium etc., which is otherwise not described by any of the other parameters in Eq. (5.4).
Example: Linear horizontal core flow
The minus sign "-" in the horizontal flow equation Eq. 5.4 is justified by consider-ing linear core flow.
Let’s assume a constant liquid flow rate q, through a core sample, as shown in Fig. 5.2. The pressures p1, p2and the positions x1, x2are labelled according to
Figure 5.2: Horizontal flow in a core sample.
Assuming a homogeneous porous medium and integration from position 1 to 2, the pressure term is written as follows,
d p
where p1> p2 in positive flow direction. Since x2 obviously is larger than x1, the value of d p/dx is pr. definition negative, i.e. the minus sign "-" is needed to balance the equation.
.
The fluid velocity related to the cross-section area A is called the superficial (i.e. filtration) or bulk velocity, and the linear flow velocity is written,
u = q A= −k
µ d p
dx. (5.5)
The real velocity of fluid flow in the pores is called the interstitial (true) velocity, vporeand is necessarily higher than the bulk velocity, since the flow cross-section area is, on average,φ times smaller than the bulk cross-section A. The directions of pore flow are inclined relative to the general flow direction and a characteristic inclination angleα is assigned to describe this effect. This effect will increase the pore velocity even more, as illustrated in Fig. 5.3. If, in addition, the porous medium contains a residual saturation of a non-flowing phase, e.g. a connate water saturation Swc, the pore flow velocity is affected through the reduction of the
flow cross-section area. The sum of these effects will cause the pore flow velocity to become considerably higher than the bulk velocity,
vpore= q
Figure 5.3: Pore flow velocity in a porous medium.
Experimental tests from different porous rocks have shown that an average inclination an-gle,α' 36oand that this angle may vary between 12oto 45o. If a typical porosity of 25% and a connate water saturation of 10% are assumed, then the pore velocity will be about 7 times higher than the bulk velocity.
Example: Linear inclined core flow
When the direction of flow is inclined, with an angleθ to the horizontal flow direction, the gravitational force has to be considered, since the fluids are moving up or down in the gravitational field.
In order to keep a constant flow rate q, through a porous medium of length∆l, a pressure difference∆p is applied. See Fig. 5.4.
l
Figure 5.4: Core flow at a dip angleθto the horizontal axis
Flow at an angle to the horizontal direction is described by Eq. (5.3), where the minus sign is describing linear flow,
q = −Ak µ
d(p +ρgz) dl .
z is the elevation in the gravitational field and from Fig.5.4 it’s evident that z = l sinθ, where l is the direction of flow. The flow equation becomes,
q = −Ak µ
d p dl − Ak
µρg sinθ. Integration from position 1 to 2, gives
q + Ak
µρg sinθ
∆l = Ak µ∆p.
The pressure difference is given,
∆p = µ∆l
Ak q +ρg∆l sinθ, where horizontal linear flow is∆pθ=0= q(µ∆l)/(Ak).
In order to maintain a constant flow rate through the core sample, the pressure difference needs to be adjusted relative to the inclination angle (dip angle). In a up-dip situation, as in Fig. 5.4, the pressure difference has to be larger relative to the horizontal case, since the fluid is pushed upwards in the gravitational fields, i.e.,
0 ≤θ< 90o ⇒ ∆p ≥∆p0
−90o<θ< 0 ⇒ ∆p <∆p0
.