Darcy’s Experiment:

: apparent flow velocity, cm/s

: flow rate, cm3/s

: cross sectional area, cm2

: system length, cm

: differential pressure in L-direction, atm

: permeability, Darcy

: fluid viscosity at system T, cp

Darcy’s Law:

Reservoir rock is typically composed of consolidated sand grains (sandstone) or

carbonate rock with fissures. The network of pores defines a complicated system of

varying pore shape, diameter and orientation. Useful predictions about the flow within

the reservoir can be obtained by modelling the network as an array of cylindrical pores of

fixed diameter.

In the simplest situation we consider steady, one-dimensional flow within an

undersaturated oil reservoir. The following assumptions can be made regarding the flow:

1) Since the oil is undersaturated, a gas phase is not present. For this simplified

analysis, it is assumed that no water is present. Therefore, the flow has a single-

phase.

2) The undersaturated oil is assumed to behave as an incompressible fluid.

3) Flow occurs only in the radial direction and is therefore one-dimensional. In

addition, the characteristics of reservoir rock are assumed constant for all

orientations.

4) The flow is assumed to be at steady state.

5) The pore size is small such that the Reynolds Number is small and the flow is

laminar. Conversely, the pore size is large compared to the mean free path of a

molecule such that the no-slip assumption applies.

6) The fluid exhibits Newtonian behavior.

Consider the case of a one-dimensional (z-wise) flow through a single pore:

We can write the differential mechanical energy balance for this flow:

The wall shear stress for laminar flow with a Newtonian fluid is:

Combining the equations:

Darcy’s Law is typically expressed as a partial differential equation:

One dimensional in “Z” direction

The superficial velocity, Us , is defined as the average fluid velocity over a unit cross-

sectional area perpendicular to the coordinate axis of interest:

Us is not the physical velocity of the fluid.

It is only used to represent the velocity form of the volumetric flow rate.

A relationship between the interstitial and superficial velocities can be obtained using the

porosity:

Finally, we can combine the Mechanical Energy Balance and Darcy’s Law equations to

obtain an equation for permeability:

Volumetric Flow equation for the Simplified System:

Volumetric flow towards the well will be considered positive by convention. Consider

the following simplified well:

For this analysis, we assume that the variation in flow in the vertical and azimuthal

direction is negligible compared to the radial direction:

Darcy’s law for a horizontal, one-dimensional flow in radial coordinates can be written

as:

Consider the area defined by radius ‘r’:

Based on our definition for positive flow:

Combining with Darcy’s Law we obtain:

This equation can be integrated to give the volumetric flow equation:

For pressure at any radius r based on rw (the well radius):

As discussed earlier, the typical oil field units are:

qo = STB/day

k = md {millidarcies}

re, rw, h = ft

Pe, Pwf = psi

μ = cP {centipoise}

Using these units we obtain:

And in terms of the pressure equation:

Example calculation:

Determine the drainage radius (re) and the radius at which ½ of the pressure drop in the

reservoir occurs for the following data:

Pe = 3000 psi Pwf = 600 psi h = 10 ‘

k = 100 md = 1 cP rw = 4”

qo = 1860 STB/day Bo = 1.2 RB/STB

re = 680 ft

r½P = 15 ft

Observations: The radius where half of the pressure drop occurs (here 15 ft) is a

benchmark used in industry. For the example considered, the remaining half of the

pressure drop occurs over a radial distance of 665 ft. This emphasizes the importance of

ensuring no damage occurs to the formation region closest to the well. It should be noted

that the entire flow from the reservoir is concentrated in the small area associated with

the wellbore region, ultimately increasing the magnitude of the pressure loss (where ql is

constant but us varies with cross-sectional area).

Reservoir Pressure Curve for Example

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

radius (ft)

Res

ervo

ir P

ress

ure

(p

si)

re

Pwf

Pe

rw

Reservoirs with Non-Homogeneous Permeability:

Vertically Non-Homogeneous Permeability:

Radially Non-Homogeneous Permeability:

Permeability variation in the near wellbore region:

The flow rate from reservoir is very sensitive to variation of permeabilities in the near

wellbore region.

Three approaches to address this point:

1) Skin effect

2) Effective wellbore radius

3) Hawkin’s formula

local permeability < natural reservoir permeability Û damaged formation

local permeability > natural reservoir permeability Û stimulated formation

Skin Effect:

Since the pressure drop within the well is a logarithmic function of radial distance from

the well, the reservoir region closest to the well is a strong determinant of the magnitude

of the overall well pressure drop. We can define an ideal pressure curve which is

associated with a uniform permeability throughout the reservoir. Deviations from the

ideal radial pressure profile in the near well-bore region can be incorporated into the flow

equation using an additive ‘pressure drop’ term with a parameter known as the skin

effect, S.

This term can be combined with the ideal pressure drop term to give the following

equations:

At the drainage radius: (the limiting radius for the region of constant pressure boundary)

In oilfield units:

The skin effect pressure drop can have either a positive or negative value:

Positive S and P s : This is an undesirable effect since it reduces the flow from the well.

There are several causes that result in a positive skin effect value:

partial well completion (perforation height less than reservoir height)

inadequate number of perforations

generation of a gas fraction during flow through the well (causing a localised

restriction of the flow)

damage to the natural reservoir rock in the near well-bore region

Negative S and Ps : Negative values are desirable and can be obtained through:

matrix stimulation in the near well-bore region resulting in higher permeability than

in the natural formation

induced fracturing in the reservoir with the effect of reducing the resistance to the

flow within the reservoir.

Typical Values: S = 5 (highly damaged well)

S = -5 (highly stimulated well)

Effective Wellbore Radius:

Previously, we defined the concept of treating the pressure change associated with the

skin effect as an additive pressure drop term giving the following equation for overall

reservoir pressure drop:

Consider the group:

Then

And: r’w = effective wellbore radius

In effect, we model the well such that the skin effect pressure loss is incorporated into the

overall reservoir pressure drop equation by modifying the wellbore radius. It is important

to remember that this is a model which provides the correct overall pressure drop. The

effective well diameter does not represent the actual diameter in the well. Based on the

typical conditions we obtain the following effective wellbore diameters:

Damaged well r’w <rw

Stimulated well r’w >rw

Hawkin’s Formula:

Hawkins (1956) developed a formula which incorporated parameters more relevant to the

effected near well-bore region: rs represents the radius of the region, and ks represents the

average permeability of the effected region.

Consider the following system with the associated pressure profile:

We define two pressure drop values relative to the limiting skin pressure Ps:

Ideal: based on the permeability of the natural reservoir rock, k

Real: based on actual permeability in the near well-bore region, ks:

Subtracting the real pressure loss from the ideal we obtain the skin effect pressure drop:

Substituting with our previous equation for the skin effect pressure drop:

This approach attempts to model the relative effects of permeability impairment and the

penetration of the damage.

Productivity Index (J):

A key concern of the production engineer is to maximize the production rates (profit)

through the use of the minimum pressure drop (cost). This concept is captured in the

parameter called productivity index.

overall pressure drop = drawdown = driving force = Pe – Pwf = {psi}

production rate = volumetric flow at surface = qo {STB/day}

productivity index:

For a fixed drainage pressure Pe:

Plot pwf versus qo, draw a straight line, the slope is (-1/J) and intercept is pe

For fixed fluid and reservoir characteristics, the index line is followed and the production

rate can be increased by reducing Pwf, the pressure at the wellbore. This can be

accomplished by incurring additional costs. For example, mechanical lift (pumping)

could be introduced.

For a specific reservoir, J can be increased by:

- reducing the viscosity (ex. miscible or thermal recovery methods) or

- reducing s (ex. stimulation or fracturing)