pge 361 lecture 8 rock permeability average [compatibility mode]

Permeability Averages

Average absolute permeabilityg p y

Weighted-average permeabilityH i bilit Harmonic-average permeability

Geometric-average permeability

Average permeability correlations

Timur (1968) The Morris-Biggs Equation (1967)


Average Absolute Permeabilityg y It is rare to encounter a homogeneous reservoir in actual practice.

the reservoir contains distinct layers, blocks, or concentric rings of y , , g

varying permeabilities.

Also, because smaller-scale heterogeneities always exist, core Also, because smaller scale heterogeneities always exist, core

permeabilities must be averaged to represent the flow characteristics

of the entire reservoir or individual reservoir layers (units). y ( )

The proper way of averaging the permeability data depends on how

permeabilities were distributed as the rock was deposited. p p


Weighted-average permeability: P1 P2flow

Weighted-average permeability:

This averaging method is used to g gdetermine the average permeability of layered-parallel beds with different permeabilities.

Consider the case where the flow system is comprised of three parallel layers (with equal width) that are separated from one another by thin impermeable barriers, i.e., no cross flow,

h i th Fias shown in the Figure.



The flow from each layer can be calculated by applying Darcys equation in a linear form to give:Darcys equation in a linear form to give:

Layer 1Pwhk The total flow rate from the entire system

LPwhkQ 111 =

Layer 2Pwhk

Q tavg=

The total flow rate from the entire system

LPwhkQ 222 =

Layer 2L

QT =where, L

Pwhk

Layer 3QT = total flow ratekavg = average permeability for the entire modelw = width of the formation

LPwhkQ 333 = P = P1 - P2hT = total thickness



The total flow rate QT is equal to the sum of the flow rates through each layer=Q1+Q2+Q3rates through each layer Q1+Q2+Q3

Combining the above expressions gives:

PwhkPwhkPwhkPwhkL

PwhkL

PwhkLPwhk

LPwhk

Q 332211TavgT ++==

Or: 332211Tavg hkhkhkhk ++=n==++= n

n

jjjhk

1332211avg h

hkhkhkk=

n

jjh

1

Tg h



A similar layered system with variable layers width Assuming no

P1 flow P2

variable layers width. Assuming no cross-flow between the layers, the average permeability can be g p yapproximated in a manner similar to the previous derivation to give:

== == n

n

jjj

n

n

jjjj Akwhk

11avgk

==

n

jj

n

jjj Awh

11

avg

L

where, Aj = cross-sectional area of layer jwj = width of layer j


Harmonic-Average Permeability

Permeability variations can occur laterally in a reservoir as well as inthe vicinity of a well bore.the vicinity of a well bore.

Consider this figure which shows an illustration of fluid flow through a series combination of beds with different permeabilities.

For a steady-state flow the flow For a steady-state flow, the flow rate is constant and the total pressure drop P is equal to the sum of the

d h b dpressure drops across each bed, or:

Q

PPTT = = PP11 + + PP22 + + PP33 k1 k2 k3



Substituting for the pressure drop by applying Darcys equation gives:equation, gives:

LQLQLQLQP T 321 ++==n

AKAKAKAKP

avgT

321

++==

=== n 1j

jT

avg L

L

LLLLk

=

++ n1j j

j

3

3

2

2

1

1g

kL

kL

kL

kL

where, Lj = length of each bedkj = absolute permeability of each bed



For radial systems:

)/l (= n weavg )/rln(r)/rln(rk

=

1j j

1jj

k)/rln(r

=1j jThis relationship can be used as a basis for estimating a number of useful quantities inestimating a number of useful quantities in production work (For example, theeffects of mud invasion, acidizing, or well shooting can be estimatedshooting can be estimatedfrom it).


Geometric-Average Permeability

Warren and Price (1961) illustrated experimentally that the most probable behavior of a heterogeneous formation most probable behavior of a heterogeneous formation approaches that of a uniform system having a permeability that is equal to the geometric average. The geometric

i d fi d th ti ll b th f ll i average is defined mathematically by the following relationship:

n ))l (k(h

=

=

n

j

1jjj

avg

h

))ln(k(hexpk If the thicknesses (hj) of all core samples are the same:

=1j jh1 Examplen

n321avg )...k..........kk(kk = p

Rock PermeabilityAbsolute Permeability Correlations:T l d i i l h d h TiTwo commonly used empirical methods are the Timur equation and the Morris-Biggs equation.

Timur (1968):Timur (1968) d th f ll i i f ti ti thTimur (1968) proposed the following expression for estimating the permeability from connate water saturation and porosity:

2

4.48.58102k = 2wcS

8.58102k

Rock Permeability The Morris-Biggs Equation:M i d Bi (1967) t d th f ll i tMorris and Biggs (1967) presented the following two expressions for estimating the permeability if oil and gas reservoirs: 23 reservoirs:For an oil reservoir: 23

S62.5k

=

wcS For a gas reservoir:

23 where, k = absolute permeability, Darcy = porosity fraction3

S2.5k

=

= porosity, fractionSwc = connate-water saturation, fraction

wcS Example

Conversion Factorsffor

Oilfield UnitsOilfield Units

Darcys Law - Darcy Unitsy y Linear (1-D) flow of an incompressible fluid

( )pAkq = where,

q cm3/s

( )pL

q = k darcies A cm2

p atm cp L cm

The Darcy a derived unit of permeability, defined to make this equation coherent (in Darcy units)

Darcys Law - Oilfield Unitsy

Linear (1-D) flow of an incompressible fluid( ) p

( )pAkCq = where,

q bbl/D

( )pL

q = q bbl/D k millidarcies A ft2

p psia p psia cp L ft

The approach demonstrated will be to convert each term back to Darcy units, restoring the coherent equation, then collecting th i f t t bt i th ilfi ld it t t Cthe conversion factors to obtain the oilfield unit constant, C

Darcys Law - Oilfield Unitsy

( )p[psia]L[ft][cp]

]A[ftk[md]0.001127q[bbl/D]2

=

The unit of the constant is defined from the above equation

[ ][ p]

][psia][md][ft[ft][cp][bbl/D]0.0011271C 2=

We were able to cancel leaving the units of C as shown above because,

[cm][cp]/s][cm3 ][atm][cm

[cm][cp]/s][cm11[d] 2

REFERENCES:REFERENCES:Ahmed, Tarek : Reservoir Engineering Handbook-Ch. 4: Fundamental of rock properties, Second Edition, Gulf Professional Publishing, 2001.

pge 361 lecture 8 rock permeability average [compatibility mode]

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