Dake L.P. Fundamentals of reservoir engineering

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DARCY'S LAW AND APPLICATIONS 111

and substituting this in equ. (4.6) gives the real gas potential as

RT Zdp

Φ= +

(4.19)

But, since

RT Z dp

ddpgdzgdz

Φ= + = +

(4.20)

then the gradient of the gas potential in the flow direction is simply

d1dpdz

dl dl dl

(4.21)

and Darcy's equation for linear flow is again

kd kdp dz

dl dl dl

µµ

æö

=− =− +

ç÷

èø

(4.12)

The above merely illustrates that real gas flow can be described using precisely the

same form of equations as for an incompressible liquid.

4.6 DATUM PRESSURES

An alternative way of expressing the potential of any fluid is

Φ = p +

where

is the psi-potential and has the units-potential per unit volume. Using this

function, Darcy's law becomes

kA d kA d

dl dl

ρψ

µµ

=− =−

(4.22)

The

potential is also frequently referred to as the "datum pressure", since the

function represents the pressure at any point in the reservoir referred to the datum

plane, as illustrated in fig. 4.3.

DARCY'S LAW AND APPLICATIONS 112

+ z

arbitrary datum plane (z=z)

= p

g (z

− z

)

= p

g (z

− z

)

, z

)

, z

)

Fig. 4.3 Referring reservoir pressures to a datum level in the reservoir, as datum

pressures (absolute units)

Suppose pressures are measured in two wells, A and B, in a reservoir in which an

arbitrary datum plane has been selected at z = z

. If the pressures are measured with

respect to a datum pressure of zero, then as shown in fig. 4.3, the calculated values of

and

are simply the observed pressures in the wells referred to the datum plane,

i.e.

= (absolute pressure)

+ (gravity head)

In a practical sense it is very useful to refer, pressures measured in wells to a datum

level and even to map the distribution of datum pressures throughout the reservoir. In

this way the potential distribution and hence direction, of fluid movement in the

reservoir can be seen at a glance since the datum pressure distribution is equivalent to

the potential distribution.

4.7 RADIAL STEADY STATE FLOW; WELL STIMULATION

The mathematical description of the radial flow of fluids simulates flow from a reservoir,

or part of a reservoir, into the wellbore.

For the radial geometry shown in fig. 4.4, flow will be described under what is called

the steady state condition. This implies that, for a well producing at a constant rate q;

dp/dt = 0, at all points within the radial cell. Thus the outer boundary pressure p

and

the entire pressure profile remain constant with time. This condition may appear

somewhat artificial but is realistic in the case of a pressure maintenance scheme, such

as water injection, in which one of the aims is to keep the pressure constant. In such a

case, the oil withdrawn from the radial cell is replaced by fluids crossing the outer

boundary at r = r

DARCY'S LAW AND APPLICATIONS 113

pressure

q = cons

p = constant

Fig. 4.4 The radial flow of oil into a well under steady state flow conditions

In addition, for simplicity, the reservoir will be assumed to be completely homogeneous

in all reservoir parameters and the well perforated across the entire formation

thickness.

Under these circumstances, Darcy's law for the radial flow of single phase oil can be

expressed as

kA dp

(4.23)

Since the flow rate is constant, it is the same across any radial area, A = 2πrh, situated

at distance r from the centre of the system. Therefore, equ. (4.23) can be expressed as

2rkhdp

and separating the variables and integrating

wf w

qdr

2kh r

òò

where p

is the conventional symbol for the bottom hole flowing pressure. The

integration results in

pp ln

2kh r

−=

(4.24)

which shows that the pressure increases logarithmically with respect to the radius, as

shown in fig. 4.4, the pressure drop being consequently much more severe close to the

well than towards the outer boundary. In particular, when r = r

then

ewf

pp ln

2kh r

−=

(4.25)

When a well is being drilled it is always necessary to have a positive pressure

differential acting from the wellbore into the formation to prevent inflow of the reservoir

fluids. Because of this, some of the drilling mud will flow into the formation and the

DARCY'S LAW AND APPLICATIONS 114

particles suspended in the mud can partially plug the pore spaces, reducing the

permeability, and creating a damaged zone in the vicinity of the wellbore.

The situation is shown in fig. 4.5, in which r

represents the radius of this zone. If

pressure

skin

∆

Fig. 4.5 Radial pressure profile for a damaged well

the well were undamaged, the pressure profile for r < r

would be as shown by the

dashed line, whereas due to the reduction in permeability in the damaged zone,

equ. (4.25) implies that the pressure drop will be larger than normal, or that p

will be

reduced. This additional pressure drop close to the well has been defined by van

Everdingen

skin

2kh

∆= (4.26)

in which the ∆p

skin

is attributed to a skin of reduced permeability around the well and S

is the mechanical skin factor, which is just a dimensionless number. This definition can

be included in equ. (4.25) to give the total steady state inflow equation as

ewf

pp ln S

2kh r

æö

−= +

ç÷

èø

(4.27)

in which it can be seen that if S is positive then p

the pressure drawdown,

contains the additional pressure drop due to the perturbing effect of the skin.

Since equ. (4.27) is frequently employed by production engineers, it is useful to

express it in field units rather than the Darcy units in which it was derived. The reader

should check that this will give

ewf

qB r

p p 141.2 In S

kh r

æö

−= +

ç÷

èø

(4.28)

in which the geometrical factor 2π has been absorbed in the constant. This equation is

frequently expressed as

DARCY'S LAW AND APPLICATIONS 115

ewf

qoilrate(stb/d)

p p pressure drawdown (psi)

7.08 10 kh

Bln S

−

æö

ç÷

èø

(4.29)

where the PI, or Productivity Index of a well, expressed in stb/d/psi, is a direct measure

of the well performance.

One of the aims of production engineering is to make the PI of each well as large as is

practically possible, consistent with the economics of doing so. This is termed well

stimulation. The ways in which a well can be stimulated can be deduced by considering

how to vary the individual parameters in equ. (4.29) so as to increase the PI. The

various methods are

summarised below.

a) Removal of Skin (S)

Before making any capital expenditure to remove a positive mechanical skin, it is first

necessary to check that the formation has in fact been damaged during drilling. This

can best be done by performing a pressure buildup test, which is normally carried out

as routine, immediately after completing the well. The manner in which S can be

calculated in the analysis of such a test is detailed in Chapter 7. sec. 7.

If it is determined that S is positive, the formation damage can be reduced by acid

treatment. The type of acid used depends on the nature of the reservoir rock and the

type of plugging materials which must be removed. If the formation is limestone,

treatment with hydrochloric acid will invariably remove the skin because of the solubility

of the rock itself. In sandstone reservoirs, in which the rock matrix is not soluble,

special, so-called, mud acids are used. As a result of a successful acid job, the skin

factor can be reduced to zero or may even become negative.

b) Increasing the effective permeability (k)

As noted al ready, due to the logarithmic increase of pressure with radius, the main

part of the pressure drawdown occurs close to the well. Therefore, if the effective

permeability in this region of high drawdown can be increased, the productivity can be

considerably enhanced. This can be achieved by hydraulic fracturing, in which high

fluid pressures maintained in the wellbore will induce vertical fractures in the formation.

Once the fractures have been initiated, they can be propagated deep into the formation

by increasing the wellbore pressure and injecting a suitable fracturing fluid, carrying

granular propping agents. In carbonate reservoirs the same effect can be achieved by

fracture-acidising.

c) Viscosity reduction (µ)

If the oil viscosity is very high, the flow rate in the reservoir will be correspondingly low,

and the time scale attached to the recovery will be greatly extended. The viscosity can

be siginificantly reduced by raising the temperature of the oil, a typical viscosity-

temperature relation being shown in fig. 4.6(a). The thermal stimulation process

DARCY'S LAW AND APPLICATIONS 116

applied to effect this viscosity reduction is steam soaking. Steam is injected into the

reservoir and, in the simple model shown in fig. 4.6(b), extends to a radius r

, the

magnitude of which is primarily a function of the amount of steam injected, usually

several thousand tons, over a period of several days. During injection heat is lost in the

wellbore and to the cap and base rock, but since steam is used, these losses are

reflected as a reduction in latent heat and therefore take place without a significant

change of temperature.

150

hot zone

(b)

(a)

150 T (°F)

400

2 cp

(cp)

Fig. 4.6 (a) Typical oil and water viscosities as functions of temperature, and

(b) pressure profile within the drainage radius of a steam soaked well

Following injection, the well is opened on production and the cold oil crossing into the

heated annular region has its viscosity greatly reduced and consequently the PI is

increased. A typical steam soak production rate, in comparison to the unstimulated

rate, is shown in fig. 4.7. There is an initial surge in production followed by a steady

decline as the temperature in the hot zone is reduced, due to the continual loss of heat

to the cap and base rock, as a function of time, and the removal of heat with the

produced fluids. When the production rate declines towards the unstimulated rate, the

cycle is repeated.

steam soak

production

unstimulated production

time (yrs)

oil

rate

(stb/d)

123

100

Fig. 4.7 Oil production rate as a function of time during a multi-cycle steam soak

DARCY'S LAW AND APPLICATIONS 117

The flow equations during the initial and later part of a steam soak cycle will be

described in Chapter 9, sec. 6 and Chapter 6, sec. 4, respectively, since they serve as

an interesting example of the flexibility of radial flow equations.

d) Reduction of the oil formation volume factor (B

)

As already described in Chapter 2, sec. 4; B

(rb/stb) can be minimised by choosing

the correct surface separator, or combination of separators.

e) Reduction in the ratio r

/ r

Since r

/ r

appears as a logarithmic term, it has little influence on the PI and alteration

of the ratio by, for instance, underreaming the wellbore to increase r

, is seldom

considered as a means of well stimulation.

f) Increasing the well penetration (h)

It was assumed in deriving equ. (4.29) that the well was completed across the total

formation thickness and thus the flow was entirely radial. If the well is not fully

penetrating, there is a distortion of the radial flow pattern close to the well giving rise to

an additional pressure drawdown. This is generally accounted for by using the full

formation thickness in equ. (4.29) and including the effect of partial penetration as an

additional skin factor. The method of calculating this additional skin is described in

Chapter 7, sec. 9. Increasing the well penetration, if possible, will obviously increase

the Pl but in many cases wells are deliberately completed over a restricted part of the

reservoir to avoid excessive gas or water production from individual sands, or to

prevent coning.

The methods for stimulating the production of a well, described in this section, do not

necessarily increase the ultimate oil recovery from the reservoir, but rather, reduce the

time in which the recovery is obtained. As such, they are generally regarded as

acceleration projects which speed up the production, thus having a favourable effect on

the discounted cash flow.

There are exceptions. For instance, if a well has stopped producing, then any

stimulation which results in oil production can be regarded as increasing the recovery.

These methods, however, should be distinguished from the enhanced recovery

techniques, described in sec. 4.9, in which the reservoir is energised to increase the

recovery. In stimulation there is frequently no net energy increase in the reservoir. In

steam soaking, for instance, heat energy is supplied to the reservoir and is

subsequently lost during the production cycle; as opposed to continuous steam drive,

in which the aim is to keep the steam in the reservoir thus increasing the total energy of

the system.

4.8 TWO-PHASE FLOW: EFFECTIVE AND RELATIVE PERMEABILITIES

In describing Darcy's law, it has so far been assumed that the permeability is a rock

property which is a constant, irrespective of the nature of the fluid flowing through the

pores. This is correct (with the noted exception of gas flow either at low pressures or

very high rates) provided that the rock is completely saturated with the fluid in question,

DARCY'S LAW AND APPLICATIONS 118

and results from defining k in equ. (4.8) as the permeability, rather than the K in

equ. (4.3), the latter having a dependence on the fluid properties. The permeability so

defined is termed the absolute permeability.

If there are two fluids, such as oil and water, flowing simultaneously through a porous

medium, then each fluid has its own, so-called, effective permeability. These

permeabilities are dependent on the saturations of each fluid and the sum of the

effective permeabilities is always less than the absolute permeability. The saturation

dependence of the effective permeabilities of oil and water is illustrated in fig. 4.8(a). It

is conventional to plot both permeabilities as functions of the water saturation alone

since the oil saturation is related to the former by the simple relationship S

= 1−S

Considering the effective permeability curve for water, two points on this curve are

known. When S

= S

, the connate or irreducible water saturation, the water will not

flow and k

= 0. Also, when S

= 1 the rock is entirely saturated with water and k

= k,

the absolute permeability. Similarly for the oil, when S

= 0 (S

= 1) then k

= k and,

when the oil saturation decreases to S

, the residual saturation, there will be no oil flow

and k

= 0. In between these limiting values, for both curves, the effective permeability

functions assume the typical shapes shown in fig. 4.8(a). The main influence on the

shapes of the curves appears to be the wettability, that is, which fluid preferentially

adheres to the rock surface

. Although it is difficult to quantify this influence, the

permeability curves can be measured in laboratory experiments for the wettability

conditions prevailing in the reservoir

absolute

permeability

1-S

k′

1-S

Fig. 4.8 (a) Effective and (b) corresponding relative permeabilities, as functions of the

water saturation. The curves are appropriate for the description of the

simultaneous flow of oil and water through a porous medium

The effective permeability plots can be normalised by dividing the scales by the value

of the absolute permeability k to produce the relative permeabilities

ro w rw w

k(S )

k(S)

k(S) andk(S)

(4.30)

DARCY'S LAW AND APPLICATIONS 119

The plots of k

and k

, corresponding to the effective permeability plots of fig. 4.8(a),

are drawn in fig. 4.8(b). Both sets of curves have precisely the same shape, the only

difference being that the relative permeability scales have the range zero to unity.

Relative permeabilities are used as a mathematical convenience since in a great many

displacement calculations the ratio of effective permeabilities appears in the equations,

which can be simplified as the ratio of

ow ro w row

ww rw w rww

k (S ) k k (S ) k (S )

k(S) k k (S) k (S)

In figs. 4.8(a) and (b) the parts of the curves for water saturations below S

= S

and

above S

= 1 - S

are drawn as dashed lines because, although these sections of the

plots can be determined in laboratory experiments, they will never be encountered in

fluid displacement in the reservoir, since the practical range of water saturations is

≤ S

≤ 1—S

The maximum relative permeabilities to oil and water that can naturally occur during

displacement are called the end-point relative permeabilities and defined as

(fig. 4.8(b)),

ro ro w wc

kk(atSS)

′

and

rw rw w or

kk(atS1S)

′

==− (4.31)

Sometimes the effective permeability curves are normalised in a different manner than

described above, by dividing the scales of fig. 4.8(a) by the value of k

= S

) =

k G

′

, the maximum effective permeability to oil. The resulting curves are shown in

fig. 4.9.

1 S

Fig. 4.9 Alternative manner of normalising the effective permeabilities to give relative

permeability curves

DARCY'S LAW AND APPLICATIONS 120

In this case, the normalised relative permeability curves are defined as

ow ww

ro w rw w

ow wc ow wc

k(S) k(S)

K(S) andK (S)

k(S S ) k(S S )

(4.32)

To describe the simultaneous flow of oil and water in the reservoir, applying Darcy's

law, the absolute permeability k, implicitly used in the earlier sections of this chapter,

must be replaced by the effective permeabilities k

) and k

) respectively. Using

the alternative methods of normalising the effective permeability curves, the required

permeabilities can be expressed as either

) = kk

) or k

) = k

= S

) K

) (4.33)

and

) = kk

) or k

) = k

= S

) K

)

both interpretations, naturally, giving the same values of the effective permeabilities. As

already mentioned, in many equations describing the displacement of one immiscible

fluid by another it is the ratio of effective permeabilities which is required, and from equ.

(4.33) this can be expressed as

ro ro

rw rw

(4.34)

To complicate matters further, in the literature, it is not normal to distinguish between

the two ways of presenting relative permeability curves by assigning one of them a

capital letter; both interpretations are denoted by the symbol k

. In this text, the relative

permeabilities used will be those obtained by normalising the effective permeability

curves with the absolute permeability (fig. 4.8(b)).

Relative permeabilities are measured in the laboratory by studying the displacement of

oil by water (or gas) in very thin core plugs, in which it is safe to assume that the fluid

saturations are uniformly distributed with respect to thickness. Therefore, these

laboratory-measured, or rock-relative permeability relationships, can only be used

directly to describe flow in a reservoir in which the saturations are also uniformly

distributed with respect to thickness. In the majority of practical cases, however, there

is a non-uniform water saturation distribution in the vertical direction which is governed

by capillary and gravity forces and, therefore, there must also be a relative permeability

distribution with respect to thickness. Because of this, the rock-relative permeabilities

can seldom be used directly in field displacement calculations.

Practically the whole of Chapter 10 is devoted to describing methods of generating

averaged (or pseudo) relative permeabilities, as functions of the thickness averaged

water saturation. These are used to describe the displacement of oil by water in a more

realistic fashion, taking account of the manner in which the fluid saturations are

distributed, with respect to thickness, as they simultaneously move through the

reservoir.