Trazadores
Segunda presentacin terica:
Transporte de un trazador pasivo (conservativo)
2014
El proceso de transporte motorizante del movimiento de un trazador es la CONVECCION.El transporte convectivo del trazador ocurre simplemente porque el trazador se encuentra presente en una
corriente fluida (por ej. agua de inyeccin). Simplemente el trazador se deja llevar por la corriente (como una
hoja que cae desde un rbol, sobre el ro).
Transporte de un trazador pasivo (conservativo)
Se denomina densidad de flujo convectivo (Jc) al producto:
vCJc Sus unidades son:
segm
gr
seg
m
m
gr
**
23
Sin embargo ocurren fenmenos a nivel poro que determinan que no todas las molculas de trazador se muevan de forma uniforme (con la misma velocidad).
Si focalizamos nuestra atencin en los conductos (poros + gargantas porales) encontraremos que las velocidades son mayores en las zonas centrales que cerca de las paredes. Asumiendo un modelo de tubo,
tendremos esquemticamente el siguiente mecanismo:
Transporte de un trazador pasivo (conservativo)
Adems del proceso anterior (que adelanta molculas respecto a otras), encontraremos que los tubos se encuentran interconectados entre si, constituyendo redes tortuosas. Como consecuencia, molculas que ingresan juntas al medio poroso (por ej. un testigo de corona) podrn bifurcarse al presentarse caminos alternativos.
Transporte de un trazador pasivo (conservativo)
tubo
En consecuencia, si inyectamos un batch de trazador en la entrada del testigo (de concentracin Co),
observaremos su avance por la conveccin, pero este avance no ser uniforme sino que habr molculas
adelantadas respecto al promedio y otras atrasadas. Este fenmeno se conoce como DISPERSION
HIDRODINAMICA y depende del grado de heterogeneidad que presenta la roca. Usualmente los parmetros
asociados al ancho de la distribucin de concentracin del trazador se utilizan como medida de la intensidad del fenmeno de dispersin hidrodinmica.
Transporte de un trazador pasivo (conservativo)
2
Parmetro indicativo
del grado de dispersin:
v.t
Numerosos estudios de la dispersin
hidrodinmica de trazadores en rocas petroleras
avalan considerar a este fenmeno como una
especie reforzada de difusin molecular. Es decir. la dispersin del trazador estara
gobernada por la ley de Fick de la difusin
molecular.
La nube de trazador (inicialmente rectangular o pulso cuadrado) avanzar por conveccin de manera que su centro de masa se habr desplazado una distancia:
Transporte de un trazador pasivo (conservativo)
2
tvx .
v.t
Simultneamente, desde el centro de masa, las molculas
experimentarn un proceso del tipo difusivo, por el cual
algunas de ellas harn saltos difusivos hacia atrs y otras
hacia delante, en acuerdo con la ley de Fick, esto es:
tDL.2
El otrora coeficiente de difusin molecular es ahora
el coeficiente de dispersin hidrodinmica DL.
Complementariamente, siguiendo las indicaciones
de la ley de Fick, el flujo dispersivo ser:
x
CDJ Ld
segm
gr
m
m
gr
seg
m
**
2
32
Nuevamente las unidades resultan:
Transporte de un trazador pasivo (conservativo)
Considerando el ingreso y salida de los flujos
convectivos y dispersivos en un elemento de volumen:
Ax
CD
x
L
A
x
CD
xx
L
xAV .
AvCx
AvCxx
producirn una variacin temporal de la
masa de trazador en tal elemento, dada por:
t
CxA
..
x
Considerando un elemento muy delgado y un
intervalo de tiempo pequeo, de manera que
los cocientes de incrementos finitos se
puedan convertir en derivadas, se obtiene la
ecuacin diferencial denominada CDE:
2
2
i
Lx
CD
x
Cv
t
C
Resultados experimentales de dispersin sobre testigos
vvd
DL
8.1
Zona de inters petrolero
Transporte de un trazador pasivo (conservativo)
En la zona de inters:
Se denomina dispersividad y es una escala de longitud para la cual se alcanza la homogeneidad
Acerca de la homogeneidad..
Supongamos que deseamos calcular la densidad de un
material compuesto de esferas de hierro (densidad 7.6
g/cm3) y de plstico (densidad 0.7 g/cm3).
Si tomamos una muestra pequea (dimetro d1) podra
ocurrir que no incluya ninguna esfera de hierro o que
incluya muchas en relacin a las de plstico. Ello depender de donde se site el tomamuestras.
Si aumento el tamao de las muestras, lograr valores
mas uniformes de densidad, independientemente de
donde ubique el tomamuestras
hierro plstico
d1
d2
Claramente la densidad mas representativa del material es 3.2 g/cm3.
Pero evidentemente est involucrada una cuestin de escala
d1 d2
En el caso de la dispersividad, el carcter dispersivo de la roca
tambin depende del tamao de muestra analizada.
Si estamos inyectando trazadores en testigos (bereas o roca real del
reservorio) encontraremos una escala de algunos milmetros (la
dispersividad) por sobre la cual muestras diferentes del mismo
testigo, mostrarn el mismo carcter dispersivo. Es decir, el ancho de
la nube de trazador crecer de acuerdo a:
con
siendo la escala medida.
tDL.
vDL
Dispersividades dependientes de la escala del ensayo
22
.ix
Cv
x
Cv
t
C
Incluyendo la ley empirica para DL resultar la CDE:
O, en terminos de la velocidad de Darcy:
2
2
.i
DDx
Cv
x
Cv
t
C
La resolucin de esta ecuacin difererencial permitir obtener las concentraciones
del trazador a lo largo del medio y su evolucion en el tiempo, esto es C(x,t)
Por supuesto antes deber especificarse como se ha efectuado la inyeccion del
trazador (lugar de inoculacin, forma, duracin, etc).
Si inyectamos una solucin trazadora de concentracion
Co, en x=0 en forma continua a lo largo del tiempo (para
t>0), en un testigo de corona por el cual est fluyendo un
caudal de agua limpia (iw), resultarn perfiles de
concentracion de trazador dados por la funcin:
2
2
.i
DDx
Cv
x
Cv
t
C
Testigo de corona
C
+x-x x=0
tD
tvxerfcCoC
L
tx4
5.01),(
Trazado durante el barrido de un testigo de corona
C(x,t)
Co
-x x=0 +x
t=t1
Un frente de solucin trazadora avanzar hacia la derecha
C(x,t)
Co
t
Si se toman muestras en la salida del testigo de corona y
se registran los valores de concentracion medidos en
funcion del tiempo, obtendremos:
t1
Cad(x,t)
1
VPI
Es usual expresar el registro anterior graficarnodo la relacion C/Co, en funcion del volumen
poral inyectado (referido al volumen poral del testigo)
1
Este experimento y esta forma de representacin son usuales para evaluar por ejemplo la retencin de polmeros en el diseo de tratamientos EOR.
-Co
Una inyeccion de trazador tipo pulso cuadrado se obtiene inyectando la solucion trazadora de
concentracion Co durante un tiempo acotado. Es decir, se inyecta agua limpia, luego solucion
trazadora y finalmente agua limpia otra vez. La C(x,t) para una inyeccion de trazador tipo pulso
cuadrado puede obtenerse de manera muy simple, restando dos soluciones del tipo:
tD
tvxerfcCoC
L
tx4
5.01),(
pero desfasadas en el tiempo!!
Interpretation of Interwell Tracers Tests
Simulation
Interwell passive tracer tests are a powerful tool for the evaluation of the
secondary recovery projects in oilfield reservoirs.
In these projects, water injected in injection wells pushes the oil to the producing wells in which
it is afterwards extracted.
Heterogeneity, anisotropy, unfavorable mobility ratios, fractures and faults conspire to reach
acceptable swept efficiencies.
Channeling of water between injecting and producing wells is the most common problem.
The interwell tracer tests permit us to detect these problems and then, as a consequence,
they facilitate the corrective decisions.
Also they permit us to determine some reservoir parameters, principally volumes, thickness
and permeabilities of the watered layers.
Obviously, for reaching the last objective it is necessary to assume a reservoir model.
Introduction
Introduction
In relation to the model choice we must consider that:
1. We need to model a reservoir, which is practically unknown.
2. We have only approximate values of thickness and permeabilities as references.
3. We don't know about heterogeneity (shape and extension).
4. Heterogeneity and unfavorable mobility relations cause unstable interfaces
between water and oil.
5. The dynamic conditions of the water injection can complicate the situation
(i.e., unbalanced flow rates and pressures)
For these reasons, we believe that it doesn't have sense to define a detailed
model when the level of uncertainty is so high.
So, a simple model contemplating the main features of the reservoir is appropriate.
Also, a simple model facilitates the fitting of parameters by optimization
Transport of a passive tracerSimplifying assumptions
1. Layered reservoir neglecting gravitational forces.
2. One phase is flowing (water)
3. Stationary pressures and velocities.
4. Darcy law is valid.
5. Superposition principle is valid.
02
2
2
2
Y
p
X
p
Transport of a passive tracer
Pressure and velocity fields
For an homogeneous and isotropic layer,
the pressure field satisfies:
However, a great percentage of the oil reservoirs of Argentina are not isotropic.
Many of they were dune fields in the past.
Dune fields show
typical corded structures,
perpendicular to the wind
dominant direction.
So the permeability must be
bigger in the direction of the
cords.
Several tracer tests performed
in aeolian oilfields of Argentina
reveal this aspect.
Transport of a passive tracer
The reservoirs
We wanted to consider the above factors of reservoir anisotropy.
So, we supposed two perpendicular directions, one for the Kmax
and one for the Kmin, for which, the Laplace equation becomes:
02
2
2
2
P
MIN
P
MAXy
pK
x
pK
NP
j
wj
j
MIN
MAXx
C
xxq
K
K
hv
12
1
NP
j
wj
j
MIN
MAXy
C
yyq
K
K
hv
12
1
We obtained the velocity components:
CosyySenoxxK
KSenoyyCosxxC wjwj
MIN
MAXwjwj
2
With,
Transport of a passive tracer
Pressure and velocity fields
NLCisenryy
rxx
iwIwIi
iwIwIi...,,2,1
cos
t
yvand
t
xv yx
Transport of a passive tracer
Streamlines computation
Being the velocities,
NLCii
2
We can consider the coordinates of the starting points:
NLCiyyxxr wPiwPiwP ...,,2,1,)()(22
Later, we solve the Eqs.1 satisfying the condition:
at the end of each streamline, in each producing well.
(1)
(2)
(3)
Transport of a passive tracer
Streamlines computation
Balanced Regular Five Spot
Isotropic case
Transport of a passive tracer
Streamlines computation
Balanced Regular Five Spot
Non Isotropic case
3MIN
MAX
K
K
o69
40.0
120
200
0 10 20 30 40 500
10
20
30
40
50
y (
m)
x (m)
40.0
120
200
0 10 20 30 40 500
10
20
30
40
50
y (
m)
x (m)
Transport of a passive tracer
Flux fronts
Balanced Regular Five Spot
Isotropic case
Fronts for 40, 120, 200 days.
Transport of a passive tracer
Mass transport
We consider that the convection dispersion equation (CDE) is valid on eachstreamtube. So, considering an square pulse injection, with:
2
2
2
2
1
1
0 22
1
22
1,
ssErfc
ssErfc
C
tsC
C(0,t) = C0, para 0 t tC(0,t) = 0, para t > tC(s,0) = 0, para s > 0
s
L dsv
svs0
2
22 12
We obtain the solution:
In which the hydrodynamic dispersion and the divergent convergent character of each streamtube may be considered making:
(4)
(5)
Transport of a passive tracer
Step and square pulse responses
r
i
d
Factors influence
on tracer records
The pattern: balanced five spot
We take as example a balanced five spot pattern, in which the
wells are distributed in an uniform way with identical flow rates.
Also, no faults and no anisotropy are presents.
P-1
I-1
Layer thickness influence
Well P-1
We can see how the layer thickness
controls the breakthrough, the pike
position and the broadness of the
tracer records.
Here:
qp = qi = 100 m3/d
m20
h = 1.0 m
h = 0.5 m
20.0
00.1wS
Dispersivity influence
Well P-1
Dispersivity controls the breakthrough,
the broadness and slightly the pike
position of the tracer records.
Here:
qp = qi = 100 m3/d
m5
h = 1.0 m
m10
m20
20.0
00.1wS
Anisotropy influence
Here, the pattern is a balanced five spot again,
in which the wells are distributed in an uniform
Way, with identical flow rates.
But now, the water contribution from injectors
along the 45 diagonal, it is more important
that those along the other diagonal (135).
o45
0.2min
max K
K
P-1
I-1
h = 1.0 m
m20
o45
0.2min
max K
K
0.1min
max K
K
Anisotropy influence
Well P-1
The anisotropy works in opposite way
that the dispersivity.
While increments of dispersivity anticipate
the breakthrough and pike position, giving
greater broadness, the anisotropy increments
anticipate the breakthrough and
pike position, but reducing the broadness.
Faults influence
The fault works reinforcing the water assistance in the producers
located in the same block of the injector, specially in those along the fault (i.e.: P-1).
P-1
I-1
Sealing fault
Faults influence
Well P-1
Without fault
With fault
The above fault works in similar way
that the anisotropy (on the well P-1).
The fault existence anticipate the breakthrough
and pike position, but reducing the broadness.
Fault + anisotropy: joint influence
Fault: yes
Anisotropy: no
Fault: yes
Anisotropy: yes
Fault: no
Anisotropy: no
The joint action of anisotropy and fault
(an often combination in the oilfields)
can to anticipate strongly the breakthrough and
pike position of the records.
Outside injectors influence
P-1
I-1
Occasionally, the lack of tracer (or the scarce production of tracer)
in a producer may be consequence of the outside injectors action.
For example, here we have considered that the water flow rates
of the injectors I-2, I-3 and I-4 are 200 m3/d, while the other
injector water flow rates are only 100 m3/d.
Outside injectors influence
Well P-1
Balanced case
unbalanced case
Validation of PORO outputs:
Interpretation of tracer records
from a laboratory scale experiment
performed in the CEA (Grenoble, Fr)
0.95
1.60
0.475
0.475
0.60
The experimental arrange
Overall dimensions
Water saturated sand layer
Overall dimensions:
Length: 1.60 m
Width: 0.95 m
Height: 0.80 m
Injector well in the centre
Four producers
0.27
0.35
0.320.32
Well
Well Well
Well Well 1
Well 2Well 3
Well 4
The experimental arrange
Tracer, well coordinates and flow rates
Layer thickness 0.80 m
Porosity 35%
Water saturation 100.00%
3 tracer experiments - Tracer: Rhodamine WT
Exp 1
Flowrate @ Injection well 107 mL/min
Flowrate @ each production well 26.8 mL/min
Injected mass 0.041 g
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
0.00
0.05
0.10
0.15
0.20
0.25
0.30
recdia
recdia
recdia
recdia
day after injection
Well 1
Well 2
Well 3
Well 4
Experimental tracer daily fractional recovery (TDFR)
injtracer
producerwatersampletracerrectracer
injtracer m
qC
t
m
mTDFR
.1
Cumulative recoveries
(at day 6.44):
Well 1: 0.17733
Well 2: 0.20178
Well 3: 0.2249
Well 4: 0.20657
Mesh: 0.81058
Simulation
In a first step, we obtained the simulated tracer records for the following conditions:
1. Water saturation ( Sw = 1)
2. Total layer thickness watered (h = 0.8 m )
3. Reported porosity (0.35)
4. Nominal water flow rates (injector: 107 mL/min and producers: 26.8 mL/min)
5. 81.06 % tracer recovery
6. No faults
7. No anisotropy
8. Potential flow conditions.
9. Dispersivity (will be estimated)
Tracer daily fractional recovery (TDFR)
Well 1 (experiment 1)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Days after injection
measured
simulated
35.0
005.0
mWe observe a little advancement
of the simulated record in relation
to the experimental one.
The same behavior was observed
in the other wells.
Tracer daily fractional recovery (TDFR)
Well 1 (experiment 1)
0.0 0.5 1.0 1.5 2.0 2.5
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Days after injection
measured
simulated
38.0
006.0
m The goodness of the fitting becameacceptable if we increase slightly
the dispersivity and porosity.
We consider that it may be possible
certain flexibility in the porosity
more that in the thickness
Tracer daily fractional recovery (TDFR)
Well 2 (experiment 1)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00
0.05
0.10
0.15
0.20
0.25
measured
simulated
day after injection
38.0
006.0
m
Tracer daily fractional recovery (TDFR)
Well 3 (experiment 1)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
measured
simulated
day after injection
38.0
006.0
m
Tracer daily fractional recovery (TDFR)
Well 4 (experiment 1)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
measured
simulated
day after injection
38.0
006.0
m
Comments
The PORO simulator have some problems for fitting:
1. The late breakthroughs of the tracer.
2. The long tails of the experimental records.
It may be consequence of the strongly diffusive character of the experimental tracer flux.
So, molecular diffusion may transport tracer between streamlines, retarding it in the faster
zones and also in the slower zones.
In a real oilfield tracer test we expect convective dominant flows.
Nevertheless, the PORO performance is acceptable in this lab scale experiment.
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