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THEORETICAL RESERVOIR MODELS THEORETICAL RESERVOIR MODELS
TIMETIME AREA OFAREA OFINTERESTINTEREST
MODELSMODELS
EARLY TIMEEARLY TIME NEARWELLBORE
MIDDLE TIMEMIDDLE TIME RESERVOIR
LATE TIMELATE TIMERESERVOIR
BOUNDARIES
• 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
Early Time Models
Area of InterestArea of Interest::
NEAR WELLBORENEAR WELLBORE
(1) Wellbore storage and SkinWellbore storage and Skin
(2) Infinite conductivity Infinite conductivity vertical fracture vertical fracture
(3) Finite conductivity Finite conductivity vertical fracture vertical fracture
(4) Partial penetratingPartial penetrating
(limited entry) well (limited entry) well
(5) Horizontal wellHorizontal well
AssumptionsA well is generally characterized by a constant W.B.S. which governs the production due to wellbore fluid decompression/compression when the well is opened or closed in.
Log - log responseBoth the pressure and the derivative curves follow a straight line of unit slope (n=1) until the pressure disturbance is in the wellbore (pure wellborestorage). Afterwards, the derivative passes through a hump until the wellbore 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 changing WBS option• The magnitude depends upon the type of completion (surface/downhole shut-in)
(1) Wellbore storage and SkinWellbore storage and Skin
Early Time Models
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆t
surface flowrate
sandface flowrate
drawdown
q
time
surface flowrate
sandface flowrate
build-up
q
time
Early Time Models
AssumptionsThe well intercepts a single vertical fracture plane. The flowlines pattern is orthogonal to the fracture and the transient pressure response defines a linear flowlinear flow in the reservoir. The well is at the center of the fracture and there are no ∆p losses along the fracture length.
Log - log responseThe pressure and the derivative curves are parallel and they both follow a straight line with slope equal to n = 0.5n = 0.5. The derivative pressure values are half of the pressure values.
Parameter: xf, fracture half length
Specialized plot The linear flow has no particular shape on a semi-log plot. It is only detected on the specialized plot ∆∆∆∆p - vs- (∆∆∆∆t) 0.5
(2) Infinite conductivity vertical fracture Infinite conductivity vertical fracture
Early Time Models
Xf
log ∆∆∆∆t
nono ∆∆∆∆∆∆∆∆pp losses along the fracture lengthlosses along the fracture length
log ∆∆∆∆p
log ∆∆∆∆p'
1/2
Linear Linear flowflow
Early Time Models
(3) Finite conductivity vertical fracture Finite conductivity vertical fracture
AssumptionsThe well intercepts a single vertical fracture plane. The flowlines pattern is orthogonal to the fracture and along the fracture length. The transient pressure response defines bilinear flowbilinear flow in the reservoir. The well is at the center of the fracture and there are ∆p losses along the fracture length.
Log - log responseThe pressure and the derivative curves are parallel and they both follow a straight line with slope equal to n = 0.25n = 0.25. Afterwards, the response starts to be linearlinear with slope n = 0.5n = 0.5. Bilinear flow is a very early time feature and it is often masked by WBS effects.
Parameter: xf, fracture half length ;xfx 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 - vs- (∆∆∆∆t) 0.25
Early Time Models
Xf
∆∆∆∆∆∆∆∆pp losses along the fracture lengthlosses along the fracture length
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
Bilinear flow
Linear flow
Early Time Models
AssumptionsThe well produces from a perforated interval smaller than the total producing interval. This produces spherical or hemispherical flow depending on the position of the opened interval with respect to the upper and lower boundaries.
Log - log responseAt very early times a first radial flowradial flow, relative to the perforated interval, may establish. This is often masked by WBS effects. Then spherical flowspherical flowdevelops and, correspondingly, the derivative curve exhibits a n =- 0.5slope. Eventually, later on, the radial flowradial flow in the full formation is achieved.
Parameter: kz/kr, vertical to radial permeability ratio;S, permeability damage (skin) relative to the perforated interval
Specialized plot The spherical flow has no particular shape on a semi-log plot. It is only detected on the specialized plot ∆∆∆∆p - vs- (∆∆∆∆t) 0.5
(4) Partial penetrating well Partial penetrating well
Early Time Models
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
-1/2
Spherical flow
Early Time Models
Impact of anisotropy on spherical flow
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
Early Time Models
(5) Horizontal wellHorizontal well
AssumptionsThe 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 and the bottom of the formation are sealing.
Log - log responseAt first radial flowradial flow may establish in a plane orthogonal to the horizontal well with an anisotropic permeability k = (kzkr)0.5. When the top/bottom boundaries are reached, linear flowlinear flow with a n = 0.5n = 0.5 slope is achieved. Later on, horizontal radial flowhorizontal radial flow develops in the formation.
Parameter: kz/kr, vertical to radial permeability ratio;L, producing horizontal well length;S, formation permeability damage (skin); formation kr h
Specialized plot The radial flow regimes can be analyzed on a semi-log plot. The linear flow regime is only detected on the specialized plot ∆∆∆∆p - vs - (∆∆∆∆t) 0.5.
Early Time Models
RADIAL FLOW(Horizontal line)
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
EARLY RADIAL FLOW(Horizontal line)
LINEAR FLOW(1/2 slope)
Early Time Models
Area of InterestArea of Interest::
RESERVOIRRESERVOIR
(1) HomogeneousHomogeneous
(2) Double porosity Double porosity
(3) Double permeability Double permeability
(4) Radial composite Radial composite
(5) Linear compositeLinear composite
Middle Time Models
(1) HomogeneousHomogeneous
AssumptionsThe reservoir is homogeneous, isotropic and has constant thickness.
Log - log responseAt 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 kh; S, formation permeability damage (skin)
Specialized plot On a semi-log plot (Horner plot) the points corresponding to the horizontal trend of the derivative follow a straight line of slope m.
Middle Time Models
log ∆∆∆∆t
Horner time
Pres
sure
I.A.R.F.
log ∆∆∆∆p
log ∆∆∆∆p'
I.A.R.F.
Middle Time Models
(2) Double porosity Double porosity
AssumptionsTwo distinct porous media are interacting in the reservoir: the “matrix blocks“ , with high storativity and low permeability and the “fissures system“, with low storativity and high permeability.
Main points:
The “fissures system“ is assumed to be uniformly distributed 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, but the matrix blocks supply most of the storage capacity.
Middle Time Models
Parameters definition
� Total porosityTotal porosity, φφφφt : φt = φf + φm (0.01 < φf < 1%)
� TotalTotal kh : kh = (kh)f (fissure system only)
� Storativity ratioStorativity ratio, ωωωω: defines the contribution of the fissure system to the
total system ω = [φVCt]f / [(φVCt)f + (φVCt)m ]
(0.001< ω <0.1)
� Interporosity flowInterporosity flow, λλλλ: defines the ability of the matrix to flow into the fissuresλ = α rw
2 (Km / Kf ) ( 10-4 < λ < 10-9 )
where αααα is related to the geometry of the fissure network
Middle Time Models
contrast between the parameters of the matrix and fissures (φ, k)
communication degree between matrix and fissures (interface skin)
A double porosity response depends upon:
Two types of flow regimes from matrix to fissures are considered:
a) Restricted flow conditions (pseudo steady state regime: Skin > 0) The matrix response is slower
b) Unrestricted flow conditions (transient regime: Skin = 0)The matrix response is faster
Middle Time Models
Double porosity: restricted flow conditions (S>0) Double porosity: restricted flow conditions (S>0)
In this model, also called “pseudo steady interporosity flow“, it is assumed 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 responseThree different regimes can be observed during welltest:
1) At early times only the fissures flow into the well. The contribution of the matrix is negligible. This corresponds to the homogeneous behavior of the fissure system.
2) At intermediate times the matrix starts to produce into the fissures until the pressure tends to stabilize. This corresponds to a transition flow regime.
3) Later, the matrix pressure equalizes the pressure of the surrounding fissures. This corresponds to the homogeneous behavior of the total system (matrix and fissures).
Middle Time Models
FISSU
RES
FEEDING MATRIX
Pre
ssu
re
Horner time
ωωωω
λλλλ
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
Middle Time Models
Wellbore storage effect on fissure flow identification
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
Middle Time Models
Double porosity: unrestricted flow conditions (S=0) Double porosity: unrestricted flow conditions (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 matrix reacts immediately to any change in pressure in the fissure system and the first fissure homogeneous regime is often not seen.
Log - Log responseOnly two different regimes can be observed during the welltest:
1) At early times, both the matrix and the fissure are producing, but pressure change is faster in the fissures than in the matrix.This corresponds to a transition flow regime.
2) Later, the matrix pressure equalizes the pressure of the surrounding fissures.This corresponds to the homogeneous behavior of the total system (matrix and fissures).
Middle Time Models
slabs
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
(kh)2 = 1/2 (kh)1 (kh)1
Middle Time Models
(3) Double permeability Double permeability
AssumptionsStratified reservoirs, where layers with different characteristics can be identified and grouped as two distinct porous media, are interacting with their own permeability and porosity. The double - permeability behavior is observed when crossflow establishes in the reservoir between the two porous media (main layers).
Main points :
In each homogeneous layer the flow is radial.
In multilayer reservoirs the high k layers are grouped by convention into “Layer 1“ while “Layer 2“ describes the low k or tighter zones.
The two layers can produce either simultaneously or separately into the well.
Crossflow always goes from the lower K layer to the higher K layer.
Middle Time Models
Parameters definition
� TotalTotal Kh : (kh) tot = (kh)1 + (kh)2
� Mobility ratioMobility ratio κκκκ : defines the contribution of the high K layer to the total Kh
κ = (kh)1 / [(kh)1 + (kh)2 ]if κ κ κ κ = = = = 1 there is double φφφφ
� Storativity ratioStorativity ratio, ϖϖϖϖ: defines the contribution of the high K layer to the
total storativity ω = [φhCt]1 / [(φhCt)1+ (φhCt)2]
� Interlayer crossflowInterlayer crossflow,, λ λ λ λ : defines the effect of vertical crossflow between layers
λ = A rw2/[(kh)1 + (kh)2]
if λλλλ ====0 there is no crossflow
where A defines the vertical resistance to flow and is function of the vertical permeability, kz between layers.
Middle Time Models
Double permeability with interlayer Double permeability with interlayer crossflow crossflow
Anywhere in the reservoir, the interlayer crossfIow is proportional to the pressure difference between the two layers.
Log - Log responseThree different regimes can be observed during the welltest:
1) At early times, the layers are producing independently and the behavior corresponds to two layers without crossflow.
2) At intermediate times, when the fluid flow between the layers is activated, the pressure response follows a transition flow regime.
3) Later, the pressure equalizes in the two layers. This corresponds to the homogeneous behavior of the total system.
Middle Time Models
LAYER 1
LAYER 2
(kh)1
(kh)2
(kh)1 > (kh)2
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
No crossflow if λ λ λ λ = 0
Middle Time Models
(4) Radial composite (Radial composite (lntlnt/2) /2)
AssumptionsThe well is at the center of a circular homogeneous zone of radius ri (inner region), communicating with an infinite homogeneous reservoir (outer region). 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 in the radial direction.
Parameters definition
" Mobility Ratio, M : M = (kh/µ)1/(kh/µ)2
" Storativity Ratio, D : D = (φhCt )1/(φhCt )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/µ)12) After a transition, a second homogeneous regime is achieved in the
outer region (kh/µ)2
Middle Time Models
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
(kh)1
Middle Time Models
(5) Linear composite (no Linear composite (no lntlnt/2)/2)
AssumptionsThe well is in a homogeneous infinite reservoir, but in one direction there is a change in reservoir and/or fluid properties. There is no pressure loss at the linear interface L1
This L.C. model is characterized by a change in mobility and storativity in the linear direction.
Parameters definition# Mobility Ratio, M : M = (kh/µ)1/(kh/µ)2
# Storativity Ratio, D : D = (φhCt )1/(φhCt)2
Log - log responseThe two reservoir regions are seen in sequence:
1) The pressure behavior describes the homogeneous regime in the inner region (Kh/µ)1
2) After a transition, a second homogeneous regime is achieved in the outer region. The average mobility of the two zones is defined as: [(kh/µ)1+(kh/µ)2]/2
Middle Time Models
-1
(kh)1
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
Middle Time Models
Area of InterestArea of Interest::
RESERVOIR RESERVOIR BOUNDARIESBOUNDARIES
(1) Infinite lateral extent(1) Infinite lateral extentInfinite lateral extent
Late Time Models
(2) Single boundary(2) Single boundarySingle boundary
(3) Wedge(intersecting boundaries)
(3) WedgeWedge(intersecting boundaries)(intersecting boundaries)
(4) Channel (parallel boundaries)
(4) Channel Channel (parallel boundaries)(parallel boundaries)
(5) Circular boundary(5) Circular boundaryCircular boundary
(6) Composite rectangle(6) Composite rectangleComposite rectangle
AssumptionsOne linear fault, located at some distance from the producing well, limits the reservoir extension in one direction (sealing), or provides a pressure support in one direction (water drive constant pressure).
ParametersBoundary distance from the well, d
Log - log responseBefore the boundary is reached the reservoir response shows infinite homogeneous behavior (I.A.R.F.). Two possible cases may exist:1) sealing fault : after the boundary is felt the reservoir behavior is equivalent to an infinite system with a permeability half of the initial response permeability. On the Horner plot, the presence of a sealing boundary is shown by the doubled straight line slope : m2 = 2m1
2) constant pressure : the water drive support produces a constant well pressure response. After the first radial flow regime, the derivative drops with slope n = -1.
Single boundarySingle boundarySingle boundary
Late Time Models
Pres
sure
Horner time
m2 = 2 m1
m2 = 0 m1
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
(kh)1
(kh)2 = 1/2 (kh)1
-1
Late Time Models
AssumptionsTwo intersecting boundaries, sealing or constant pressure, located at some
distance from the producing well, limit the reservoir extension in two directions. The intersection angle θθθθ is always less then 180°. The well is in any position between the two barriers.
Parameters: Distances from well to boundaries, d1 and d2
Intersection angle: θ θ θ θ = 2π [π [π [π [m1/ m2]]]]
Log - log responseBefore the boundaries are reached the reservoir response shows the first infinite homogeneous behavior (I.A.R.F.) with a permeability of k1. The radial flow duration is a function of the location of the well between the two boundaries. Two cases may exist:
1) two sealing faults: when both the boundaries are reached, the reservoir behavior is equivalent to an infinite system with a permeability: k2 = (θ /2π) k1
2) constant pressure: If one (or both) of the boundaries is water drive, the pressure stabilizes and the derivative drops.
Intersecting boundaries (Wedge) Intersecting boundaries (Wedge) Intersecting boundaries (Wedge)
Late Time Models
WELL CENTERED
WELL OFF-CENTERED2
12
1θθθθ
d1
d2
(kh)1
(kh)2=1/2(kh)1
(kh)3=θ/2π(kh)1
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
Late Time Models
AssumptionsTwo parallel boundaries, sealing or constant pressure, located at some
distance from the producing well, limit the reservoir extension in two opposite directions. In the other directions the reservoir is of infinite extent. The well is in any position between the two boundaries.
Parameters: Boundary distances from the well, d1 and d2
Log - log responseBefore the boundaries are reached the reservoir response shows infinite
homogeneous behavior (I.A.R.F.). The radial flow duration is a function of the location of the well in the channel. Two possible cases may exist:
1) two sealing fault: when the boundaries are reached, a linear flow regime (n = 0.5) establishes. The linear flow is detected on the specialized plot ∆∆∆∆p - vs- (∆∆∆∆t) 0.5
2) constant pressure: If one (or both) of the boundaries is water drive, the pressure stabilizes and the derivative drops.
Parallel boundaries (Channel)Parallel boundaries (Channel)Parallel boundaries (Channel)
Late Time Models
1 2WELL CENTERED WELL OFF-CENTERED
1
2
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
(kh)1
(kh)2 = 1/2 (kh)1
1/2
Late Time Models
Closed reservoir (composite boundaries) Closed reservoir (composite boundaries)
AssumptionsThe closed system behavior is characteristic of bounded reservoirs. Only the rectangular reservoir shape is here considered and each side can be either a sealing barrier, a constant pressure boundary or at infinity (i.e.: no boundary).
Parameters : Boundaries distances from the well d1, d2, d3, d4
Log - log responseBefore the boundaries are reached the reservoir response first shows the infinite homogeneous behavior (I.A.R.F.). The radial flow duration is a function of the location of the well inside the rectangular area. Depending upon the type of the existing barriers, boundaries can be:
1) sealing faults : The effect of each sealing fault is seen according to its distance from the well. If all the sealing boundaries are reached, a closed system is then defined and pseudo steady-state conditions apply(i.e.: the flowing pressure is linearly proportional to time ).
Late Time Models
A closed system is characterized by a loss of pressure (depletion) in the reservoir, expressed as :
∆∆∆∆p = pi - pavg
The pressure behavior of closed systems is totally different during drawdown and build-up periods:
drawdown derivative : when all the sealing boundaries are reached both the pressure and the derivative curve follow a unit slope (n = 1) straight line. On the specialized Cartesian plot p-vs-time,the flowing pressure is a linear function of time.
build-up derivative : when all the sealing boundaries are reached the reservoir pressure tends to stabilize at the average reservoir pressure pavg and, as a consequence, the derivative curve drops.
Late Time Models
re
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
Late Time Models
2) Constant pressure : If any of the sides acts as a constant pressure boundary, due to water drive support, the log-log pressure curve tends to stabilize and the derivative drops.
Only the sealing faults closer to the well may be felt but, when the effect of pressure support start to act, any other sealing boundary is masked.
Because no depletion is present in this case, the pressure derivative trend is the same for both the build-up and drawdown periods.
Late Time Models
log ∆∆∆∆p
log ∆∆∆∆p'
log ∆∆∆∆t
1
2
3
4
Late Time Models
1
2
3
4
log ∆∆∆∆p
log ∆∆∆∆p'
CONSTANT PRESSURE
SEALING
log ∆∆∆∆t
Late Time Models
