Diffusitivity eqn

Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow

The diffusivity equation for fluid flow in terms of dimensionless variable is

(1)

The initial and outer boundary conditions are

(2)

(3)

D

D

D

D

D2D

D2

rP

rP

r1

rP

∂∂

=∂∂

×+∂

∂

( ) 00,rP DD =( ){ } 0t,rP DDD

rlim

D

=∞→

while the inner boundary condition is

(4)

and

(5)

Eq. (4) states that the dimensionless wellbore unloading rate plus the dimensionless sand face flow rate must equal unity, the surface flow rate.

The wellbore unloading or storage constant, , is that defined by van Everdingen and Hurst. That is,

1)r∂P∂

(-dt

dPC 1r

D

D

D

wDD D

==

1rD

DDwD D

)]r∂P∂

(S-P[P ==

DC

2w

D hcr2C

Cπφ

= (6) , , C represents the volume of wellbore fluid unloading or storage, cc/atm. Storage may be by virtue of either compressibility or changing liquid level.

Eq. (5) introduces a steady state skin effect and, thus, a pressure drop at the sand face which is proportional to the sand face flow rate. Note from eq. (4) that

(7)

where q is the constant surface flow rate, and qsf is the sand face flow rate. Finally, the dimensionless flowing pressure, PwD, is the same as used previously by

D

wDD1r

D

Dsf

dtdP

C-1)r∂P∂

(-q

qD

== =

)t(p DD

μπ

=q

)p-p(kh2P wfi

wD 2wt

D rckt

tφμ

=

vn Everdingen and Ramey. Thus, PwD represents the pressure within the wellbore, while PD represents pressures on the formation side of the skin effect.

Solution of the Equation

Solution of eqs. (1) through (5) follows readily using the Laplace transformation. The transform of the dimensionless flowing pressure may be written as

(8)

[ ])}]p(KpS)p(K{pC)p(Kp[p

)p(KpS)p(KPL

10D1

10wD ++

+=

where K0 and K1 are modified Bessel functions of the second kind of zero and unit orders

Another related solution has appeared in the petroleum literature. By using the superposition principle and representing the well as a continuous line source, the Laplace transform of the dimensionless well pressure may be shown to be:

(9)

[ ]}]S)p(K{pC1[p

S)p(KPL

0D

0wD ++

+=

Eq. (8) can be written as:

(10)

A semi log plot of the function versus

is shown in fig.1 for various values of the skin S.

It can be observed that the various curves merge into a single straight line as decreases for a given S value.

In practice, however, if one consider the dimensionless pressure (PwD) range associated with each skin value, one

]

)p(Kp)p(K

S

1pC[p

1]P[L

1

0D

wD

++

=

)p(Kp)p(K

S1

0+S2-pe

S2-pe

finds that the function plotted in fig. 1 is always in the linear portion of the curve at that skin. As a result, the function can always be approximated by the

semi-log straight line, i.e. by

Eq. (10) thus becomes:

(11)

)p(Kp)p(K

S1

0+

S2-pe2

lnγ

]

pe2

ln

1pC[p

1≈]P[L

S2-

D

wD

γ

+

By changing the variable p into z:

z = p CD

(12)

The dimensionless pressure only depends upon tD/CD and e2S. This justifies a posteriori the choice of these dimensionless variables by Gringarten, et al. to construct the type curves of Fig. 2

]

eC/z2

ln

1zC[z

1≈)]z(P[L

S2D

D

wD

γ

+

Fig. 2 Wellbore storage and skin type curves in a homogeneous reservoir (Gringarten et al; 1979)

Definition

Suppose f (t) be a function of t defined for t > 0. Then the

integral , if it exists, is a function of s, say, F(s). The

function F(s) is called the Laplace transform of f (t) and is denoted by L {f (t)}. Thus

Laplace transforms of derivatives

If, L {f (t)} = F (s), then L {f' (t)} = s F(s) – f (0), provided

∫∞

0

st dt)t(fe

{ } ∫∞

0

st- dt)t(fe)s(F)t(fL ==

0)t(feLt st-

∞→t=

Then the general formula for nth derivative is written as

L {fn (t)} = sn F(s) - sn-1f (0) – sn-2 f (0) - …..- f n-1(0).

Laplace transform of integrals

If, L {f (t)} = F (s), then This is used to calculate the integral of the any function.

Inverse Laplace transforms

If L {f (t)} = F(s), then f (t) is called the inverse Laplace transform of F(s) and is denoted by f (t) = L-1 {F (s)}.

).s(Fs1

du)u(f∫t

0

=

Stehfest Algorithm

It has been pointed out that in the search for analytical solution to the diffusivity equation a second order linear partial differential equation the Laplace transform f (s) of the solution f (t) has to be inverted. For situations where this inversion is either very difficulty or impossible Stehfest has presented an algorithm by values of t, i.e. given the solution in Laplace space f (s) the real time solution f (t) at a given value of t = T can be evaluated. The Stehfest algorithm is as

( ) ∑N

1iia i

T2ln

fVT

2lnTf

==

Where, fa (T) = numerical approximation to the function value at T i.e. f (T)

=Laplace transform of f (t) Vi = a set of predetermined coefficients which

depend on N (an even integer).The weighting factors, Vi, are determined from the formula:

It was shown by the Stehfest that the optimum value of N depends on the Number of digits carried in the computation. For example for N = 12 the values of Vi are given by

)s(f

∑]

2N

,imin[

21i

k

2/Ni

2N

i

)!i-k2()!k-i()!1-k(!k)!k-2N

(

)!k2(k)1-(V

+=

=

I ViI Vi

1 -0.01666666666666 7 -3891705.533308

2 16.01666666666 8 7053286.333279

3 -1247.000000002 9 -8005336.499933

4 27554.33333318 10 5552830.499949

5 -263280.8333323 11 -2155507.19998

6 1324138.699994 12 359251.1999968

The computed inverse transforms are accurate to at least four decimal places in most cases.

When the Stehfest algorithm is employed to evaluate the real time function PD (tD) from its Laplace transform PD (s), it is computed at specific values of tD sufficient to cover the range required.

The PD function can then be plotted vs. tD on Cartesian, semi-log or log-log scales and its characteristics investigated.

If required it can be represented by some convenient numerical approximation, e.g. polynomial, cubic spline, rational function or interpolated between the tabulated values.

Pressure Transient data Analysis of Two Porosity System

The development of the double porosity model has, in fact, followed the same pattern as that of the homogeneous model, for which the “basic” model was first introduced (by Theis in the 1930’s), then outer boundary conditions, and finally, inner boundary conditions when early time analysis was developed in late 1960’s.

The concept of a double porosity medium as representative of a fissured reservoir was first introduced by Barenblatt, et al. Their model assumed the existence of two porous regions of distinctly different porosities and permeabilities within the formation.

One region (the fissures) had a high conductivity and carried the reservoir fluid to the well, whereas the other region (the blocks) had a low conductivity and fed liquid only to the fissures, acting as a uniformly distributed source.

As a consequence, each point in space was associated with two pressures, namely:

(1) the average liquid pressure, pf, in the fissures in the vicinity of the point and

(2) the average liquid pressure, pm, in the porous blocks in the vicinity of that some point. Another assumption was that the flow of fluid from blocks to fissures was occurring under pseudo-steady state conditions:

Partial Differential Equations

The basic partial differential equations for fluid flow in a two-porosity system were presented by Warren and Root in 1963. The model was extended by Mavor and Cinco_Ley to include wellbore storage and skin effect. The fundamental partial differential equations are

(1)

(2)

D

Df

D

DM

D

Df

D2D

Df2

t∂P∂

t∂P∂

)-1(r∂P∂

r1

r∂P∂

ω+ω=×+

( )λ=ω DMDfD

DM P-Pt∂

P∂)-1(

( )fmm p-pkqμα

=

where ω is the dimensionless fracture storage parameter

(3)

and λ is the dimensionless matrix fracture permeability ratio:

(4)

α is the inter-porosity flow shape factor in ft-2, and and tD are defined in the nomenclature.

mf

f

)Vc()Vc()Vc(φ+φ

φ=ω

2w

f

m rkk

α=λ

DP

A complete mathematical definition requires additional equations which represent the appropriate initial and boundary conditions. For a system at constant pressure, the initial condition is;

(5)

The inner boundary condition in the case of constant producing pressure is:

(6)

where S is the skin factor. The two outer boundary condition are considered: an infinitely large reservoir and closed outer boundary. For an infinitely large reservoir the

0)0,r(P DDf =

1)r∂P∂

(S-P 1rD

DfDf D

==

condition is:

(7)

For closed outer boundary, the condition is:

(8)

The dimensionless flow rate into the wellbore is given by:

(9)

and qD is defined in the nomenclature.

0)t,r(P DDDf∞→r

limD

=

0r∂P∂

eDD rrD

Df ==

1rD

DfDD D

)r∂P∂

(-)t(q ==

The cumulative production is related to the flow rate by:

(10)

The above equations define completely the statement of the problem. Compared to a homogeneous system, the parameters ω and λ define the behavior of a two-porosity system. ω is the ratio of the storage capacity of the fracture to the total capacity of the medium.

The parameter λ reflects the intensity of the fluid transfer between the matrix and the fractures, and is representative of the geometry of the system as included in the shape factor, α.

DD

D qdt

)Q(d=

Method of Solution

A common method for solving eqs. (1) and (2) under the conditions given by eqs. (5)-(9) is to use the Laplace transformation

The equations are transformed into a system of ordinary differential equations which can be solved analytically. The resulting solution in the transformed space is a function of the Laplace variable, s, and the space variable rD.

To obtain the solution in real time and space, the inverse Laplace transform is used. This can be done several ways.

The simplest method is to look for the function and the inverse in well known tables for Laplace transforms.

In the present work, the inverse was found by using an algorithm for approximate numerical inversion of the Laplace space solution. This algorithm was presented by Stehfest in 1970 and has been used with success by many authors.

(11)

where

[ ])}])p(pf(K)p(pfS))p(pf(K{pC))p(pf(K)p(fp[p

])p(pf[K)p(pfS])p(pf[KPL

10D1

10wD ++

+=

λ+ωλ+ωω

=p)-1(p)-1(

)p(f

Eq. (11) can be approximated in all practical cases by:

(12)

In eq. (12), on the other hand, the Laplace variable z is defined with respect to the ratio of the dimensionless time to the dimensionless wellbore storage constant. This ratio is independent of storativity and is simply written as:

Warren and Root’s f function is now given by:

]}e)C/()z(zf

2{lnz[z

1)z(P

1-S2

mfD

Df

+γ+

=

Ctkh

000295.0Ct

D

D Δ×

μ×=

mfD

mfD

)C(z)-1()C(z)-1(

)z(f+

+

λ+ωλ+ωω

= (13)

At early times (z→∞) the f function is equal to ω and eq. (12) becomes

At late times (z→0), the f function is equal to unity and eq. (12) reduces to:

]}e)C/(z

2{lnz[z

1)z(P

1-S2

mfD

Df

+ωγ+

=

]}e)C/(z

2{lnz[z

1)z(P

1-S2

mfD

Df

+γ+

=

At intermediate times, the f function reduces to λ(CD)f+m and eq. (12) becomes:

]}e

2{lnz[z

1)z(P

1-S2-

Df

λγ+

=

Our investigation of this problem indicates that the pressure response of the double porosity reservoir with transient block to fissure flow is indeed very different from that with pseudo-steady state flow.

It is possible to show that the pressure solution for transient flow from blocks to fissures is given as

for horizontal slab blocks, and

for spherical blocks.

λωωλ

+ω=p)-1(3

tanhp3

)-1()p(f

]1-p)-1(15

cothp)-1(15

[p5

1)p(f

λω

λωλ

+ω=

At intermediate times, during the transition period, f(p) is no longer constant, but becomes equivalent to:

for slab, and for spheres

For the transient inter-porosity flow model, with slab shaped matrix and with fracture skin:

where

p3λ

p53λ

])atanh(aS1

)atanh(a[

p3)p(f

F+λ

+ω=

λω

=p)-1(3

a

For the transient inter-porosity flow model with spherically shaped matrix with fracture skin:

where

]}1-)bcoth(b{S1

1-)bcoth(b[

p5)p(f

F+λ

+ω=

λω

=p)-1(15

b

Wellbore storage and skin

Decline Curve Analysis Using Type Curves for Dual Porosity System

Transient Rate Solutions

Two types of two porosity systems are considered: the unbounded reservoir and the closed, bounded reservoir. The solutions for the unbounded system have appeared elsewhere in the literature. The solutions for the bounded, closed system are given below.

Infinite Outer Boundary

(1))})p(pf(K)p(pfS))p(pf(K{p

))p(pf(K)p(pf)p(q

10

1D +

=

When the well has zero skin above equation reduces to

(2)

This equation is for a finite radius constant pressure source in an infinite system.

Three flow period are now considered: early time, intermediate time, and late time. At early time, t→0, p→∞, f→ω, and eq. (2) inverts to:

(3)

))p(pf(pK))p(pf(K)p(pf

)p(q0

1D =

2/1-DD ]

t[q

ωππ

=

At intermediate time, λ, controls the flow, f→λ/p, and eq. (2) inverts to:

(4)

At late time, t→∞, p→0, f→1 and eq. (2) inverts to:

(5)

When the well has a +ve skin, at early time eq. (1) inverts to: (6)

The intermediate time solution with +ve skin is:

)(K)(K

q0

1D λ

λλ=

]809.0t)[ln2/1(1

qD

D +=

S1

qD =

])(KS)(K[)(K

q10

1D λλ+λ

λλ=

(7)

The late time solution with +ve skin is;

(8)S]809.0t)[ln2/1(1

qD

D ++=

Closed Outer Boundary

The solution for the dimensionless flow rate, qD, in Laplace space is given by;

)]}r)p(pf(I))p(pf(K-))p(pf(I)r)p(pf(K[p)p(pf-))p(pf(K)r)p(pf(I))p(pf(I)r)p(pf(K{p)])p(pf(I)r)p(pf(K-))p(pf(K)r)p(pf(I[)p(pf

qeD111eD10eD10eD1

1eD11eD1D +

=