Trazadores 2

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).


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:


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.


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.


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:


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:


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


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.


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,


Pressure and velocity fields

NLCisenryy

rxx

iwIwIi

iwIwIi...,,2,1

cos

t

yvand

t

xv yx


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)



Balanced Regular Five Spot

Isotropic case




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)


Flux fronts


Isotropic case

Fronts for 40, 120, 200 days.


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)


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.



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



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



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



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.

Documents

Trazadores 2