[Reservoir System] Water Influx From Aquifer
CONTENTS OF PRESENTATION
Introduction to Water Water Influx Classification Classifi cation of Water Influx
Water Influx Models
1
Introduction To Water Influx
• Nearly all hydrocarbon reservoirs are surrounded by water-bearing rocks called aquifers. • Aquifers larger than reservoirs (oil or gas), is small in size the effect is negligible. • Aquifer supports the pressure of reservoir due to water influx from the aquifer. • Water influx occurs as the reservoir pressure decline during production. • Pressure support depends on : size of aquifer, shape of aquifer, and the permeability of the aquifer. 2
CONTENTS OF PRESENTATION
Introduction to Water Water Influx Classification Classifi cation of Water
Degree of Pressure Maintenance
Influx
Water Influx Models
Outer Boundary Condition Flow Regimes Flow Geometries 3
Degree of Pressure Maintenance
• Active Water Drive Rate of Water Influx = reservoir total production rate
• Partial Water drive • Limited Water Drive
4
Outer Boundary Condition • Infinite System The effect of the pressure changes at the aquifer can never be felt at the outer boundary. This boundary is for all intents and purposes at a constant pressure equal to initial reservoir pressure. • Finite System The aquifer outer limit is affected by the influx into the oil zone and that the pressure at this outer limit changes with time 5
FLOW REGIMES
• Steady State • Semisteady/Pseudosteady State • Unsteady State
6
FLOW GEOMETRIES
• Edge Water Drive • Bottom Water Drive • Linear Water Drive
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CONTENTS OF PRESENTATION
Introduction to Water
Pot Aquifer Model
Influx Classification of Water
Material Balance Model
Influx
Steady State Model Water Influx Models
Unsteady State Model Pseudosteady State Model 8
Pot Aquifer Model • Simplest model that based on basic definition of compressibility. • A drop in the reservoir pressure, due to the production of fluids, causes the aquifer water to expand and flow into reservoir. • The compressibility is defined mathematically as :
∆ ∆ • Applying to the aquifer gives : ( )
9
Pot Aquifer Model (Cont’d)
Radial Aquifer Geometries 10
Pot Aquifer Model (cont’d) • Assuming aquifer shape is radial, then :
ℎϕ 5.615
• The reservoir is not circular in nature, so need modification that is to include the fractional enroachment angle f, and give: •
( ) Where : 360 360
11
MATERIAL BALANCE MODEL • Assumption : a. Aquifer respons instantaneous to pressure changing at reservoir b. Time independent
Using Havlena Odeh method to simplify the calculation Procedures : a. Determine the reservoir condition b. Using Least Square Method (Regression) c. Determine the We based on the graph 12
STEADY STATE MODEL
Schiltuis’s Model
Steady State Model Hurst’s Modified Model
13
Schiltuis’s Model • Rate of Water Influx is proportional to Pressure drop dWe dt
( pi p)
• Pi is assumed constant dWe dt
k ' ( pi p)
*k’ is a water influx constant t
We k ' ( pi
p )dt
0
14
Schiltuis’s Model (cont’d) Calculate K’ from Darcy and Superposition Darcy : dWe 2 kh( pe po ) q
If
k '
dt
360 Bw ln( ) r o r e
0.00708kh STB / day / psi r e Bw ln( ) 360 r o
In this case, it’s similar to productivity index definition to describe well performance : t
We k ' ( pe 0
t
po )dt k ' pt , STB t 0
15
Hurst’s Modified Model • In Schiltuis’s Model, the problem is that the as the water drained from aquifer, the aquifer drainage radius ra will increase as the time increases. • Hurst (1943) proposed that “apparent” aquifer radius ra would increase with time. • Therefore, the dimensionless radius may be replaced with a time dependent function, as :
• 0. 0 0708 ℎ ( ) ln()
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Hurst’s Modified Model (cont’d) • Simplified form:
( ) ln()
In terms of We
− ()
∆ or ()
• a and C are two unknowns contant, must determined from reservoir-aquifer pressure and water influx historical data. To determine is based on simplified form as a linear relationship.
−
ln() or − ln ln 17
Hurst’s Modified Model (cont’d) Ln (t)
(Pi – P)/ew
Graphical determination of C and a 18
Unsteady State Model
Van EverdingenHurst’s Model
Edge Water Drive Bottom Water Drive
Carter-Tracy’s Model
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Van Everdingen-Hurst Model • Edge Water Drive Van Everdingen and Hurst assumed that the aquifer is characterized by : Uniform thickness Constant permeability Uniform porosity Constant rock compressibility Constant water compressibility • • • • •
Using the dimensionless diffusivity equation for radial system to determine water influx
2 pD 1 pD pD 2 t D r D r D r D 20
Van Everdingen-hurst Model (cont’d) Constant Terminal Pressure Condition Initial conditions : p = pi for values of radius r Outer boundary conditions : For an infinite aquifer p = pi at r = ∞
r a
Water influx
r R Reservoir
Aquifer
For a bounded aquifer
=0 at r = r a
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Van Everdingen-hurst Model (cont’d) the dimensionless diffusivity equation 2 p D 1 pD pD 2 t D r D r D r D tD
0.0002637 k t
r D
C t r R 2
Solution :
qD (t D )
pD
r R
pi p pi p wf
q 2 k h p
tD
dt t qD ( t D ) dt D q dt dt D 2 k h p 0 0
r
W eD ( t D )
ct r R 2
k
We
2 k h p 22
Van Everdingen-hurst Model (cont’d) The Water influx is then given by : We BpWeD
with
r e
r R
Water influx
Reservoir
B= 1.119 Ф c t r e 2 h f Aquifer
B WeD We
Δp
= water influx constant, bbl/psi = dimensionless water influx = cumulative water influx, bbl = pressure drop at the boundary, psi
θ 360
23
Van Everdingen-Hurst’s Model (Cont’d) Solution WeD in tabulated and graphical forms
Van Everdingen-hurst Model (cont’d) • Since pressure drop are assumed to occur at different times • To determine total water influx is using Principle of superposition
Δp1 = p0 - 0.5 (p0 + p1) = 0.5 (p0 – p1) Δp2 = 0.5 (p0 + p1) - 0.5 (p1 + p2) = 0.5 (p0 – p2) Δp3 = 0.5 (p1 + p2) - 0.5 ( p2 + p3) = 0.5 (p1 – p3)
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Van Everdingen-hurst Model (cont’d)
Δ Δ
Δ
Δ
n1
B p j Q( t Dn t Dj ) j 0
We
A
n1
tD
B Δp j Q A(t n t j ) j 0
0.006327 k
c t r R2
A
kt
c t r R2
A t
0.0002637 k
c t r R2 25
Van Everdingen-hurst Model (cont’d) • Bottom Water Drive Van Everdingen-Hurst is not adequate to describe the vertical water encroachment in bottom-water-drive system Coats(1962) modified the diffusivity equation to account for the vertical flow by including an additional term in the equation
2p 1 p 2p Ct p F 2 2 r r k r z 0.0002637 k t Fk =
WOC Oil
kv kh
Oil Water
K v = vertical permeability Bottom water drive K h = horizontal permeability Fk = ratio of vertical to horizontal permeability
Water
radial flow
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Van Everdingen-hurst Model (cont’d) Allard
and Chen (1988) introduced a newly dimensionless variable ZD (dimensionless vertical distance)
ℎ 1/2
Where, h = aquifer thickness, ft
Allard
and chen tabulated the values of WeD as a function of rD, tD and ZD
27
Van Everdingen-Hurst’s Model (Cont’d) The Water influx is then given by : We BpWeD
with B= 1.119 Ф c t r e 2 h f
Solution WeD in tabulated forms
Carter-Tracy’s Model • To reduce the complexity of water influx calculations, Carter-Tracy (1960) proposed a calculation technique that does not require superposition and allows direct calculation of water influx. • Assumptions : Constant terminal rate case Finite and infinite aquifer Radial flow
29
Carter-Tracy’s Model (cont’d)
rate
q1
q2 q3
q0 t0
t1
t2
t3
time
Constant Rate Graph for Carter-Tracy 30
Carter-Tracy’s Model (cont’d) For constant rate • Dimenesionless water influx , then
. or t . ϕ 0.00633 ∗ If ., then ∗
• For describing constant rate graph, so :
∗ ∗ +∗ 31
Carter-Tracy’s Model (cont’d) • To simplified :
−
∗ + Or
= −
−
=
=
∗ + ∗ + −
∗ + =
32
Carter-Tracy’s Model (cont’d) • Assumed that the characteristics of the pressureinflux response in the first part is unknown. • The value of accumulatife water influx from i to j have to be calculate. • If i = j -1
∗ − − −
Written in the form of integral convolution
− ∆(λ)′ −
λ
λ is the dummy variable of integration
33
Carter-Tracy’s Model (cont’d) • Combine the two previous equations, using Laplace Transform, Carter-Tracy got :
− −
∆ − ′ − ′
B = the Van Everdingen-Hurst water influx constant Δ Pn = Pi – Pn
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Carter-Tracy’s Model (cont’d) • Determine the PD For infinite acting aquifer, Edwardson et al
. 370.529 137.582 5.69549 328.834265.488 . ′ Where . 716.441 46.7.984 270.038 71.0098 . . 1296.86 1204.73 618.618 538.072 142.41 . 35
Carter-Tracy’s Model (cont’d) Approximation
for tD >100 :
0.5 0.80907 ′ 1 2
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PSEUDOSTEADY STATE MODEL Fetkovich’s Model • Fetkovich (1971) develop a method of describing the approximate water influx behaviour of a finite aquifer for radial and linear geometries. • This method does not require the use of superposition. • Based on the premise that the productivity index concept will adequately describe water influx from a finite aquifer into a hydrocarbon reservoir. • This method neglects the effect of any transient priod 37
Fetkocivh’s Model (cont’d) • Finite aquifer but big enough (r e > 3xrR ) r e
Water influx
r R Reservoir
Aquifer
Finite aquifer but big enaough re > 3 x rR 38
Fetkocivh’s Model (cont’d) • Two simple equation: Inflow equation
and Material balanace based on compressibility The max possible water influx if P = 0 : Combining equation 1 1
• •
a
39
Fetkocivh’s Model (cont’d) • Differentiating the previous equation respect to time, and the result :
• Subtituting the above equation to inflow equation
40
Fetkocivh’s Model (cont’d)
at ∆ C is evaluated in intial condition as − Subtituting inflow equation − 1 −
• t=0
•
We =
0;
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Fetkocivh’s Model (cont’d) • General equation of Fetkovich for the nth time period:
Where :
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