# LAWS OF FILTRATION

**Lecture Outline**

- 3.1. Velocity of filtration in a porous medium
- 3.2. Darcy’s linear law of filtration
- 3.3. Law of filtration in a fractured medium

## VELOCITY OF FILTRATION IN A POROUS MEDIUM

When studying filtration, it is convenient to put apart the sizes and the form of pores and to admit that the fluid goes like a continuous medium filling all the volume of porous medium, including the space occupied with a skeleton of rock.

Let’s assume that the volumetric flow rate Q of a fluid goes through surface F of porous medium:

*Q=*P _{p},* (3.1)

where w is the real average velocity of a liquid; *F* is the area of pores.

The area of pores *F _{p}* is connected to a full area of cross-sectional surface

*F*through clearance (see 2.2). Assuming equality of clearance and porosity we obtain:

*mF.*

The value:

*и* = iv *m* (3.2)

is called filtration velocity and this velocity defines a square-average flow of a fluid. Since *m <* 1, then the velocity of a filtration always less than real average velocity.

The physical meaning of introducing filtration velocity lies in the fact that when doing this some fictitious flow is introduced. Its flow rate through any cross-section is equal to the real flow rate, the fields of pressure of fictitious flow and of real flow are the same, and hydraulic resistance of a fictitious flow is equal to the hydraulic resistance of a real one.

## Darc/s linear law of filtration

In 1856 the French engineer Darcy has discovered the main law of a filtration establishing linear relation between loss of pressure *- H _{2}* and the volumetric flow rate

*Q*of liquid flow in a tube with cross-sectional area F filled with porous medium (figure 3.1).

The head for an incompressible liquid H is determined by: where *z* is the height of position; *ply* is the piezometric height; *у* is the specific weight; *и* is the velocity of liquid motion.

As filtration velocity is usually small, the head *H* is presented in the form:

Я = г + -.

*Y*

Darcy’s law is as follows

_{Q=c}EiHL_{Ft}_{ (}3._{3)}

jL/

where c is the factor of proportionality (factor of a filtration), its dimension *is LIT.*

*Figure 3.1. The schematic of an inclined formation*

Darcy’s law shows that a linear relation exists between the head loss and the flow rate of a liquid. We shall write down it in the differential form, taking into account relation (3.4):

u=Q/F,

(3-4)

*dH u=c--.*

*d s*

In the vector form it is as follows:

*U = -c gradH,* (3.5)

where s is the distance along an axis of a curvilinear tube of a flow.

The factor of filtration c is a joint characteristic of medium and liquid. To take into account the properties of various liquids and the properties of rock we will write Darcy’s law in the following form:

*kv д H u = —---,*

or

*г] d s*

*к d p u =---,*

(3.6)

*T] d s*

where *h* is the coefficient of dynamic viscosity; *к* is the permeability of porous medium.

In SI [?]=m^{2}. In the mixed system when [/?]=kgF/cm^{2}, [^] =0,01g/cm*s= =lcP, [s] = 1cm, [u] = cm / s, к is measured in Darcy, i.e. [k] = D. And for c we obtain:

_{С}Л

Permeability of sandy reservoirs usually is in the limits *к* = 100 1000 mD, and for clays the permeability is considerably smaller.

Permeability is defined by geometrical structure of the porous medium,

i.e. the sizes and the form of particles and system of their packing. A lot of attempts were done to establish the dependence of permeability on these characteristics, relying on Poiseuille’s law for laminar flow in pipes and on Stokes’ law for flow around particles for various schematics of porous medium. As real rocks do not comply with the frameworks of these geometrical models, the theoretical evaluations of permeability are unreliable. That is why the permeability is usually determined experimentally.

## Law of filtration in a fractured medium

In fractured formations the velocity of filtration is linked with average velocity through fracturing:

*u=m _{f}w.*

Average velocity is expressed through gradient of pressure under Boussinesq formula. The flow through fractures is interpreted as flow between two flat parallel plates:

* ^{S}f dp *w = —-—

12?7 *dl*

Using previously obtained relations, we shall obtain a linear law of filtration in fractured media:

ot _{f}Ff dp

(3-7)

*и - ———*

12?7 *dl*

By analogy with Darcy’s law, the permeability of fractured mediums is equal to:

*a _{f}F_{f}d *

^{f}12

(3.8)

If a fractured-porous medium is deformable then its permeability changes with the change of pressure:

* ^{k}f=k_{f}l-p* (p

_{0}-p)J , (3.9)

where *P** is a parameter that depends on elastic properties of a fractured medium and geometry of fractures (see formula (2.4)).

In fractured rocks, linear law can be violated at large velocities of filtration because of considerable influence of inertial forces. Critical values of Reynolds’ number depend on fracture roughness: for smooth fractures *Re _{cr}* is 500, and for the rough ones the value of this number is up to 0,4. For rough fractures the expression for Reynolds’ number is:

Re =---

v *m*

*tn*

**Lecture conclusion**

The lecture describes the most important characteristics of filtration i.e. the law of filtration. Law of filtration gives the link between the velocity of filtration and distribution of pressure in a formation. Various types of filtration laws and limits of their applicability are given.

**Test questions**

- 1. How filtration flow is determined?
- 2. Describe physical sense of filtration flow.
- 3. Give formulation of Darcy’s law.
- 4. Give formulation of Darcy’s law in a differential form.
- 5. What is gradient of pressure?
- 6. Describe filtration law in a fractured medium.