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THEORETI CAL RESERVOI R MODELSTHEORETI CAL RESERVOI R MODELS

T I M ET I M E AREA OFAREA OFI NTERESTI NTEREST

MODELSMODELS

EARLY TI MEEARLY TI ME

NEAR

WELLBORE

MI DDLE TI MEMI DDLE TI ME RESERVOI R

LATE TI MELATE TI MERESERVOI R

BOUNDARI ES

Wellbore storage and Skin Infinite conductivity vertical fracture

Finite conductivity vertical fracture Partial penetrating (limited entry) well Horizontal well

Homogeneous Double porosity

Double permeability Radial composite Linear composite

Infinite lateral extent

Single boundary Wedge (two intersecting boundaries) Channel (two parallel boundaries)

! Sealing! Constant pressure

! Sealing

! Constant pressure

! Sealing

! Constant pressure

! No boundary

Circular boundary

Composite rectangle

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Early Time Models

Area of InterestArea of Interest::NEAR W ELLBORENEAR W ELLBORE

(1) W ellbo re s to r age and Sk in W e ll bo re s to r age and Sk in

(2) I n f i n i t e co n d u ct i v i t y I n f i n i t e co n d u ct i v i t y ve r t i ca l f r act u r e ve r t i ca l f r act u r e

(3) Fin i t e conduc t i v i t y Fin i t e cond uc t i v i t y ve r t i ca l f r act u r e ve r t i ca l f r act u r e

(4) Par t i al pene t r a t i n g Par t i al pene t r a t i n g

( l im i t e d en t r y ) w el l ( l im i t e d en t r y ) w el l

(5) Hor i zon t a l w e l l Hor i zon t a l w e l l

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Assumptions

A well is generally characterized by a constant W.B.S. which governs theproduction due to wellbore fluid decompression/compression when the wellis opened or closed in.

Log - log response

Both the pressure and the derivative curves follow a straight line of unitslope (n = 1 ) until the pressure disturbance is in the wellbore (pure wellborestorage). Afterwards, the derivative passes through a hump until thewellbore effects become negligible.

Parameter: C, wellbore storage constant;

S, formation permeability damage (skin)

In case of multiphase flow at the wellbore it is possible to have a changingWBS option

The magnitude depends upon the type of completion (surface/downholeshut-in)

(1) W ellbo re s to r age and Sk in W e l l bo re s to r age and Sk in

Early Time Models

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log plog p'

log t

surface flowrate

sandface flowrate

drawdown

q

time

surface flowrate

sandface flowrate

build-up

q

time

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Assumptions

The well intercepts a single vertical fracture plane. The flowlines pattern is

orthogonal to the fracture and the transient pressure response defines al i near f l owl i near f l ow in the reservoir. The well is at the center of the fracture andthere are no p losses along the fracture length.

Log - log responseThe pressure and the derivative curves are parallel and they both follow astraight line with slope equal to n = 0 .5n = 0 .5 . The derivative pressure valuesare half of the pressure values.

Parameter: x f , fracture half length

Specialized plot

The linear flow has no particular shape on a semi-log plot. It is onlydetected on the specialized plot p - v s- ( t ) 0 .5

(2) I n f i n i t e conduc t i v i t y ve r t i ca l f r act u r e I n f i n i t e conduc t i v i t y ve r t i ca l f r act u r e

Early Time Models

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Xf

log t

n on o pp losses a long th e f rac tu r e leng t hlosses a long t he f r act u r e leng t h

log p

log p'

1/2

LinearL inear

f lowf low

Early Time Models

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(3) Fin i t e condu ct i v i t y ve r t i ca l f r act u r e Fin i t e condu ct i v i t y ve r t i ca l f r act u r e

Assumptions

The well intercepts a single vertical fracture plane. The flowlines pattern is

orthogonal to the fracture and along the fracture length. The transientpressure response defines b i l i near f l owb i l i near f l ow in the reservoir. The well is at

the center of the fracture and there are p losses along the fracturelength.

Log - log responseThe pressure and the derivative curves are parallel and they both follow astraight line with slope equal to n = 0 .2 5n = 0 .2 5 . Afterwards, the response startsto be linearlinear with slope n = 0 .5n = 0 .5 . Bilinear flow is a very early time featureand it is often masked by WBS effects.

Parameter: x f , fracture half length ;

x f x w , fracture conductivity

Specialized plot

The linear flow has no particular shape on a semi-log plot. It is only

detected on the specialized plot p - v s- ( t ) 0 .25

Early Time Models

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Xf

pp losses a long t he f r actu re leng t hlosses a long t he f r actu re leng t h

log p

log p'

log t

Bi l inearf low

Linearf low

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Assumptions

The well produces from a perforated interval smaller than the totalproducing interval. This produces spherical or hemispherical flowdepending on the position of the opened interval with respect to the upperand lower boundaries.

Log - log response

At very early times a first rad ia l f l owrad ia l f l ow , relative to the perforated interval,may establish. This is often masked by WBS effects. Then spher i ca l f l owspher i ca l f l owdevelops and, correspondingly, the derivative curve exhibits a n = - 0 .5slope. Eventually, later on, the r ad ia l f l owrad ia l f l ow in the full formation isachieved.

Parameter: k z/ k r , vertical to radial permeability ratio;

S, permeability damage (skin) relative to the perforatedinterval

Specialized plotThe spherical flow has no particular shape on a semi-log plot. It is only

detected on the specialized plot p - v s- ( t ) 0 .5

(4) Par t i al pene t r a t i ng w e ll Pa r t i al pene t r a t i ng w e ll

Early Time Models

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log p

log p'

log t

-1/2

Spher ica lf l ow

Early Time Models

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Impact of anisotropy on spherical flow

log plog p'

log t

Early Time Models

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(5) Hor i zon t a l w e l l Hor i zon t a l w e ll

Assumptions

The well is strictly horizontal and the vertical or slanted section is not

perforated. There is no flow parallel to the horizontal well. Both the top andthe bottom of the formation are sealing.

Log - log response

At first r ad ia l f l owrad ia l f l ow may establish in a plane orthogonal to the horizontal

well with an anisotropic permeability k = ( k zk r ) 0 .5 . When the top/bottomboundaries are reached, l i nea r f l owl inea r f l ow with a n = 0 . 5n = 0 . 5 slope is achieved.Later on, ho r i zon t a l r ad ia l f l owho r i zon t a l r ad ia l f l ow develops in the formation.

Parameter: k z/ k r , vertical to radial permeability ratio;

L, producing horizontal well length;

S, formation permeability damage (skin);

formation k r h

Specialized plotThe radial flow regimes can be analyzed on a semi-log plot. The linear flow

regime is only detected on the specialized plot p - v s - ( t ) 0 .5 .

Early Time Models

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RADI AL FLOW( H or i zon ta l l i ne)

log p

log p'

log t

EARLY RADI AL FLOW

( H o r i zon ta l l i ne )

LI NEAR FLOW

( 1 / 2 sl op e)

Early Time Models

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Area of InterestArea of Interest::

RESERVOI RRESERVOI R

(1) Homogeneous Homogeneous

(2) Doub le po ros it y Doub le po ros it y

(3) Doub le pe rm eab i l i t y Doub le pe rm eab i l i t y

(4) Rad ia l com pos i te Rad ia l com pos i te

(5) Linear com posi te Linear com posi te

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(1) Homogeneous Homogeneous

Assumptions

The reservoir is homogeneous, isotropic and has constant thickness.

Log - log response

At early times the pressure response is under the influence of WBS effects(n=1). When infinite acting radial flowinfinite acting radial flow ( I .A.R.F) is established in the

formation, the pressure derivative stabilizes and follows a horizontal line.

Parameters: formation k h ;

S, formation permeability damage (skin)

Specialized plot

On a semi-log plot (Horner plot) the points corresponding to the horizontaltrend of the derivative follow a straight line of slope m .

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log t

Horner time

Pre

ssure

I .A.R.F.

log p

log p'

I .A.R.F.

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(2) Doub le po ros it y Doub le po ros it y

Assumptions

Two distinct porous media are interacting in the reservoir: the m a t r i x

b locks , with high storativity and low permeability and the f i ssu ressys tem, with low storativity and high permeability.

Main points:

The fissures system is assumed to be uniformlydistributed throughout the reservoir

The matrix is not producing directly into the wellbore, but

only into the fissures

Only the fissure system provides the total mobility, butthe matrix blocks supply most of the storage capacity.

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Parameters definition

Total porosityTotal porosity, t : t = f+ m (0.01

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contrast between the parameters of the matrix and fissures (, k)

communication degree between matrix and fissures (interface skin)

A doub le po r osi t y r espon se depends upon :

Two types of flow regimes from matrix to fissures are considered:

a) Rest r i c ted f l ow cond i t i ons (pseudo steady state regime: Sk i n > 0 )The matrix response is slower

b) Un r e st r i ct ed f l o w co n d i t i o n s (transient regime: Sk i n = 0 )The matrix response is faster

Middle Time Models

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Doub le poros it y : r est r i ct ed f l ow cond i t i ons ( S> 0 )Doub le po rosi t y : r est r i ct ed f low cond i t i ons ( S> 0 )

In this model, also called pseudo steady interporosity flow, it isassumed that the fissures are partially plugged and that the flow from

the matrix is restricted by a skin damage at the surface of the blocks.

Log - Log response

Three different regimes can be observed during welltest:

1 ) At early times only the fissures flow into the well. The contribution of thematrix is negligible. This corresponds to the hom ogeneous behav io r o ft he f i ssu r e sys t em .

2 ) At intermediate times the matrix starts to produce into the fissures untilthe pressure tends to stabilize. This corresponds to a t r a n si t i o n f l o wr e g i m e .

3 ) Later, the matrix pressure equalizes the pressure of the surroundingfissures. This corresponds to the h o m o g en e o u s b e h av i o r o f t h e t o t a lsy s t e m ( m a t r i x a n d f i ssu r e s) .

Middle Time Models

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FISSURES

FEEDING MATRIX

P

ressure

H o r n er t i m e

log p

log p'

log t

Middle Time Models

ddl d l

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Wellbore storage effect on fissure flow identification

log plog p'

log t

Middle Time Models

Middl Ti M d l

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Doub le po rosi t y : un rest r i ct ed f low cond i t i ons ( S= 0 )Doub le po rosi t y : un rest r i ct ed f low cond i t i ons ( S= 0 )

In this model, also called transient interporosity flow, it is assumed that

there is no skin damage at the surface of the matrix blocks. The matrixreacts immediately to any change in pressure in the fissure system andthe first fissure homogeneous regime is often not seen.

Log - Log response

Only two different regimes can be observed during the welltest:

1 ) At early times, both the matrix and the fissure are producing, butpressure change is faster in the fissures than in the matrix.This

corresponds to a t r a n si t i on f l ow r e g im e.

2 ) Later, the matrix pressure equalizes the pressure of the surroundingfissures.

This corresponds to the h o m o g en e o u s b eh a v i o r o f t h e t o t a l sy s t e m( m a t r i x a n d f i ssu r e s ) .

Middle Time Models

Middl Ti M d l

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slabs

log p

log p'

log t

( k h ) 2 = 1 / 2 ( k h ) 1 ( k h ) 1

Middle Time Models

Middl Ti M d l

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(3) Doub le pe rm eab i l i t y Doub le pe rm eab i l i t y

Assumptions

Stratified reservoirs, where layers with different characteristics can be

identified and grouped as two distinct porous media, are interacting withtheir own permeability and porosity. The double - permeability behavior isobserved when cross f low establishes in the reservoir between the twoporous media (main layers).

Main points :

In each homogeneous layer the flow is radial.

In multilayer reservoirs the h i g h k layers are grouped by

convention into Layer 1 while Layer 2 describes thelo w k or tighter zones.

The two layers can produce either simultaneously orseparately into the well.

Crossflow always goes from the lower K layer to the higher Klayer.

Middle Time Models

Middle Time Models

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TotalTotalKh : (kh) tot = (kh)1 + (kh)2

Mobility rat ioMobility rat io : defines the contribution of the high K layer to the total Kh= (kh)1/ [(kh)1 + (kh)2 ]if ==== 1there is doub le

Storativity ratioStorativity ratio,: defines the contribution of the high K layer to thetotal storativity = [hCt]1/ [(hCt)1+ (hCt)2]

I nterlayer crossflowI nterlayer crossflow,, : defines the effect of vertical crossflow between layers = Arw

2/[(kh)1 + (kh)2]

if ====0 there is no cross f low

where A defines the vertical resistance to flow and isfunction of the vertical permeability, k z betweenlayers.

Middle Time Models

Middle Time Models

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Doub le pe rm eab i l i t y w i t h i n t e r l aye r Doub le pe rm eab i l i t y w i t h i n t e r l aye r c ross f lowc ross f low

Anywhere in the reservoir, the interlayer crossfIow is proportional to the

pressure difference between the two layers.

Log - Log response

Three different regimes can be observed during the welltest:

1 ) At early times, the layers are producing independently and the behaviorcorresponds to two layers w i t h o u t cr o s sf l o w .

2 ) At intermediate times, when the fluid flow between the layers isactivated, the pressure response follows a t r a n si t i on f l ow r e g im e.

3 ) Later, the pressure equalizes in the two layers. This corresponds to thehom ogeneous behav io r o f t he to t a l sys tem .

Middle Time Models

Middle Time Models

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LAYER 1

LAYER 2

(kh)1

(kh)2

(kh)1> (kh)

2

log p

log p'

log t

No crossflow if = 0

Middle Time Models

Middle Time Models

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(4) Rad ia l com pos i te ( Rad ia l com pos i te ( ln tln t/ 2 )/ 2 )

Assumptions

The well is at the center of a circular homogeneous zone of radius r i (i nne r

reg ion ), communicating with an infinite homogeneous reservoir (o u t e rreg ion ). The inner and the outer zones have different reservoir and/or fluid

properties. There is no pressure loss at the radial interface ri.

This R.C. model is characterized by a change in mobility and storativity inthe radial direction.


" Mobility Ratio, M : M = (kh/)1/(kh/)2

" Storativity Ratio, D : D = (hCt )1/(hCt )2

Log - log response

The two reservoir regions are seen in sequence:

1) The pressure behavior describes the homogeneous regime in the

inner region (kh/)12) After a transition, a second homogeneous regime is achieved in theouter region (kh/)2

Middle Time Models

Middle Time Models

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log plog p'

log t

(kh)1

Middle Time Models

Middle Time Models

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(5) Linear com posi te ( no Linear com posi te ( no ln tln t/ 2 )/ 2 )

Assumptions

The well is in a homogeneous infinite reservoir, but in one direction there isa change in reservoir and/or fluid properties. There is no pressure loss atthe linear interface L1This L.C. model is characterized by a change in mobility and storativity inthe linear direction.

Parameters definition# Mobility Ratio, M : M = (kh/)1/(kh/)2

# Storativity Ratio, D : D = (hCt )1/(hCt)2

Log - log response

The two reservoir regions are seen in sequence:

1) The pressure behavior describes the homogeneous regime in the inner

region (Kh/)1

2) After a transition, a second homogeneous regime is achieved in theouter region. The average mobility of the two zones is defined as:

[(kh/)1+(kh/)2]/2

Middle Time Models

Middle Time Models

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-1

(kh)1

log p

log p'

log t

Middle Time Models

Late Time Models

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Area of InterestArea of Interest::

RESERVOI RRESERVOI RBOUNDARI ESBOUNDARI ES

(1) I n f i n i t e la t er a l ex t en t (1) I n f i n i t e la t er a l ex t en t I n f i n i t e la t er a l ex t en t

Late Time Models

(2) Sing le bou nd ary (2) Sing le bou nd ary Sing le bou nd ary

(3) W e d g e

( i n t e r s ect i ng b oun da r i es)

(3) W e d g e W e d g e

( i n t e r sect i ng bou nda r i es)( i n t e r sect i ng bou nda r i es)

(4) Channel

( pa ra l l el boun dar i es)

(4) Channel Channel

( pa ra l l el boun dar i es)( pa ra l l el boun dar i es)

(5) Ci rcu la r bou nd ary (5) Ci rcu la r bou nd ary Ci rcu la r bou nd ary

(6) Com pos i te rect ang le (6) Com pos i te rect ang le Com pos i te rect ang le

Late Time Models

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Assumptions

One linear fault, located at some distance from the producing well, limits the

reservoir extension in one direction (sea l ing ), or provides a pressuresupport in one direction (w a t e r d r i v e co n s t a n t p r e ssu r e).

Parameters

Boundary distance from the well, d

Log - log response

Before the boundary is reached the reservoir response shows infinitehomogeneous behavior (I.A.R.F.). Two possible cases may exist:

1 ) seal i ng fau l t : after the boundary is felt the reservoir behavior isequivalent to an infinite system with a permeability half of the initialresponse permeability. On the Horner plot, the presence of a sealingboundary is shown by the doubled straight line slope : m 2 = 2 m 1

2 ) co n st a n t p r e ssu r e : the water drive support produces a constant wellpressure response. After the first radial flow regime, the derivative dropswith slope n = - 1 .

Sing le bou nd ary Sing le bou nd ary Sing le bou nd ary

Late Time Models

Late Time Models

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Pressure

Horner time

m 2 = 2 m 1

m 2 = 0 m 1

log p

log p'

log t

(kh)1

(kh)2= 1/2 (kh)

1

-1

Late Time Models

Late Time Models

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Assumptions

Two intersecting boundaries, sealing or constant pressure, located at somedistance from the producing well, limit the reservoir extension in two

directions. The intersection angle is always less then 180. The well isin any position between the two barriers.

Parameters: Distances from well to boundaries, d 1 and d 2

Intersection angle: = 2 [ [ [ [m 1/ m 2 ]]]]

Log - log response

Before the boundaries are reached the reservoir response shows the

first infinite homogeneous behavior (I.A.R.F.) with a permeability ofk 1.The radial flow duration is a function of the location of the well betweenthe two boundaries. Two cases may exist:

1 ) t w o sea l in g f au l t s: when both the boundaries are reached, thereservoir behavior is equivalent to an infinite system with a

permeability: k2 = (/2)

k12 ) co n st a n t p r e ssu r e : If one (or both) of the boundaries is water drive,

the pressure stabilizes and the derivative drops.

I n t e r sect i ng bounda r i es ( W edge)I n t ersect i ng bounda r i es ( W edge)I n t e rsect i ng bounda r i es ( W edge)

Late Time Models

Late Time Models

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WELL CENTERED

WELL OFF-CENTERED2

12

1

d 1

d 2

(kh)1

(kh)2=1/2(kh)1

(kh)3=/2(kh)1

log p

log p'

log t

Late Time Models

Late Time Models

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Assumptions

Two parallel boundaries, sealing or constant pressure, located at some

distance from the producing well, limit the reservoir extension in twoopposite directions. In the other directions the reservoir is of infiniteextent. The well is in any position between the two boundaries.

Parameters: Boundary distances from the well, d 1 and d 2

Log - log response

Before the boundaries are reached the reservoir response shows infinitehomogeneous behavior (I.A.R.F.). The radial flow duration is a

function of the location of the well in the channel. Two possible casesmay exist:

1 ) t w o sea l in g f au l t : when the boundaries are reached, a linear flow

regime ( n = 0 .5 ) establishes. The linear flow is detected on the

specialized plot p - v s- (t ) 0 .52 ) co n st a n t p r e ssu r e : If one (or both) of the boundaries is water

drive, the pressure stabilizes and the derivative drops.

Para l le l bou nd ar ies ( Chan ne l )Para l le l bou nd ar ies ( Chan ne l )Para l le l bou nd ar ies ( Chan ne l )

Late Time Models

Late Time Models

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1 2WELL CENTERED W ELL OFF- CENTERED

1

2

log p

log p'

log t

(kh)1

(kh)2 = 1/2 (kh)1

1/2

Late Time Models

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Closed r eserv o i r ( com pos i te boun dar ies)Closed r eserv o i r ( com pos i te boun dar ies)

Assumptions

The closed system behavior is characteristic of bounded reservoirs. Only

the rectangular reservoir shape is here considered and each side can beeither a sealing barrier, a constant pressure boundary or at infinity (i.e.:no boundary).

Parameters:Boundaries distances from the well

d 1 , d 2 , d 3 , d 4

Log - log response

Before the boundaries are reached the reservoir response first shows theinfinite homogeneous behavior (I.A.R.F.). The radial flow duration is afunction of the location of the well inside the rectangular area. Dependingupon the type of the existing barriers, boundaries can be:

1 ) se al in g f a u l t s : The effect of each sealing fault is seen according to

its distance from the well. If all the sealing boundaries are reached, aclosed system is then defined and pseudo steady-state conditions apply(i.e.: the flowing pressure is linearly proportional to time ).

Late Time Models

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A closed system is characterized by a loss of pressure (dep le t i on) in thereservoir, expressed as :

p = p i - p av g

The pressure behavior of closed systems is totally different duringdrawdown and build-up periods:

d r aw d ow n der iv at i v e : when all the sealing boundaries are reachedboth the pressure and the derivative curvefollow a unit slope (n = 1 ) straight line.

On the specialized Cartesian plot p - v s - t i m e ,

the flowing pressure is a linear function oftime.

bu i l d -up de r i va t i ve : when all the sealing boundaries are reached

the reservoir pressure tends to stabilize at theaverage reservoir pressure p av g and, as aconsequence, the derivative curve drops.

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r e

log p

log p'

log t

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2 ) Cons tan t p ressu r e : If any of the sides acts as a constant pressureboundary, due to water drive support, the log-log pressure curve tends to stabilize and thederivative drops.

Only the sealing faults closer to the well may befelt but, when the effect of pressure supportstart to act, any other sealing boundary ismasked.

Because no depletion is present in this case, thepressure derivative trend is the same for boththe build-up and drawdown periods.

Late Time Models

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log plog p'

log t

1

2

3

4

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1

2

3

4

log plog p'

CONSTANT PRESSURE

SEALI NG

log t

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ENI_WTA_1