flujo de fluidos en medios porosos 2013

FLUJO DE FLUIDOS EN MEDIOS POROSOS

Ley de Darcy 1856

“La velocidad del flujo de un líquido através de un medio poroso, debido ala diferencia de presión, esproporcional al gradiente de presiónen la dirección del flujo”.

Q representa la tasa o volumen de flujo hacia abajo, a través de un cilindro conarena empacada, que tiene una sección transversal A y una longitud L, h1 y h2representan la altura sobre un punto de referencia sobre un manómetro,colocado a la entrada y salida respectivamente, y representa la columnahidráulica en el punto 1 y 2. K es una constante de proporcionalidad y seencontró, que era una característica del empaque de la arena.

Forma generalizada de la ley de Darcy(*) Sistema horizontal lineal

Clasificación de los sistemas de flujo en yacimientoSe clasifican de acuerdo:

1. Tipos de fluidos en el reservorio

2. Regimen de flujo

3. Geometría del reservorio

4. Número de fases fluyendo en el reservorio

5. Angulo de inclinación

Tipos de fluidos en el reservorioThe isothermal compressibility coefficient is essentially thecontrolling factor in identifying the type of the reservoir fluid.In general, reservoir fluids are classified into three groups:

• Incompressible fluids• Slightly compressible fluids• Compressible fluids

Tipos de fluidos en el reservorio

Incompressible FluidsAn incompressible fluid is defined as the fluid whose volume (ordensity) does not change with pressure.Incompressible fluids do not exist; this behavior, may be assumedin some cases to simplify the derivation and the final form ofmany flow equations.Slightly Compressible FluidsThese “slightly” compressible fluids exhibit small changes involume, or density, with changes in pressure.It should be pointed out that crude oil and water systems fit intothis category.Compressible FluidsThese are fluids that experience large changes in volume as afunction of pressure. All gases are considered compressible fluids.

Régimen de flujoThere are basically three types of flow regimes that must berecognized in order to describe the fluid flow behavior andreservoir pressure distribution as a function of time.

•Steady-state flow

•Unsteady-state flow

•Pseudosteady-stateflow

Régimen de flujoSteady-State FlowThe flow regime is identified as a steady-state flow if the pressureat every location in the reservoir remains constant, i.e., does notchange with time.

Mathematically, this condition is expressed as the rate of changeof pressure p with respect to time t at any location i is zero.

In reservoirs, the steady-state flow condition can only occur whenthe reservoir is completely recharged and supported by strongaquifer or pressure maintenance operations.

Régimen de flujoUnsteady-State Flow - transient flowIs defined as the fluid flowing condition at which the rate of change ofpressure with respect to time at any position in the reservoir is not zeroor constant.

This definition suggests that the pressure derivative with respect to timeis essentially a function of both position i and time t.

Pseudosteady-State-Flow;semisteady-state-flow;quasisteady-state-flowWhen the pressure at different locations in the reservoir is declininglinearly as a function of time, i.e., at a constant declining rate, the flowingcondition is characterized as the pseudosteady-state flow.

Mathematically, this definition states that the rate of change of pressurewith respect to time at every position is constant.

Geometría del reservorio

Most reservoirs have irregular boundaries and a rigorousmathematical description of geometry is often possible only withthe use of numerical Simulators. The flow geometry may berepresented by one of the following flow geometries:

• Radial flow• Linear flow• Spherical and hemispherical flow

Geometría del reservorioRadial FlowIn the absence of severe reservoir heterogeneities, flow into oraway from a wellbore will follow radial flow lines from a substantialdistance from the wellbore. Because fluids move toward the wellfrom all directions and coverage at the wellbore, the term radialflow is given to characterize the flow of fluid into the wellbore.

Geometría del reservorioLinear FlowOccurs when flow paths are parallel and the fluid flows in asingle direction. In addition, the cross sectional area to flowmust be constant.A common application of linear flow equations is the fluidflow into vertical hydraulic fractures.

Geometría del reservorioSpherical and Hemispherical FlowDepending upon the type of wellbore completionconfiguration, it is possible to have a spherical orhemispherical flow near the wellbore. A well with a limitedperforated interval could result in spherical flow in the vicinityof the perforations. A well that only partially penetrates thepay zone, could result in hemispherical flow.

Geometría del reservorio

Número de fases fluyendo en el reservorio

There are generally three cases of flowing systems:

• Single-phase flow (oil, water, or gas)

• Two-phase flow (oil-water, oil-gas, or gas-water)

• Three-phase flow (oil, water, and gas)

Angulo de inclinación

FLUJO CONTINUO

The applications of the steady-state flow to describe theflow behavior of several types of fluid in different reservoirgeometries are:

• Linear flow of incompressible fluids• Linear flow of slightly compressible fluids• Linear flow of compressible fluids• Radial flow of incompressible fluids• Radial flow of slightly compressible fluids• Radial flow of compressible fluids• Multiphase flow

FLUJO CONTINUO

Linear Flow of Incompressible FluidsIt is assumed the flow occurs through a constant cross-sectional area A, where both ends are entirely open to flow. Itis also assumed that no flow crosses the sides, top, or bottom.

q = flow rate, bbl/dayk = absolute permeability, mdp = pressure, psiaμ = viscosity, cpL = distance, ftA = cross-sectional area, ft2

FLUJO CONTINUO

Linear Flow of Slightly Compressible Fluids

qref = flow rate at a reference pressure, bbl/dayp1 = upstream pressure, psip2 = downstream pressure, psik = permeability, mdμ = viscosity, cpc = average liquid compressibility, psi−1

FLUJO CONTINUOLinear Flow of Compressible Fluids (Gases)For a viscous gas flow in a homogeneous-linear system, the real-gas equation-of-state can be applied to calculate the number ofgas moles n at pressure p, temperature T, and volume V.

It is essential to notice that gas properties z and μg are a very strong functionof pressure. The gas properties must be evaluated at the average pressure.

Qsc = gas flow rate at sc, scf/dayk = permeability, mdT = temperature, °Rμg = gas viscosity, cpA = cross-sectional area, ft2L = total length of the linear system, ft

FLUJO CONTINUO

Radial Flow of Incompressible FluidsIn a radial flow system, all fluids move toward the producingwell from all directions. Before flow can take place, however,a pressure differential must exist. Thus, if a well is to produceoil, which implies a flow of fluids through the formation tothe wellbore, the pressure in the formation at the wellboremust be less than the pressure in the formation at somedistance from the well.

The pressure in the formation at the wellbore of a producingwell is know as the bottom-hole flowing pressure (flowingBHP, pwf).

FLUJO CONTINUORadial Flow of Incompressible Fluids

Qo = oil, flow rate, STB/daype = external pressure, psipwf = bottom-hole flowing pressure, psik = permeability, mdμo = oil viscosity, cpBo = oil formation volume factor, bbl/STBh = thickness, ftre = external or drainage radius, ftrw = wellbore radius, ft

FLUJO CONTINUO

Radial Flow of Compressible Gases (Solución aproximada)

Qg = gas flow rate, Mscf/daype = external pressure, psipwf = bottom-hole flowing pressure, psik = permeability, mdh = thickness, ftre = drainage radius, ftrw = wellbore radius, ftμg = gas viscosity, cpT = temperature, °R

FLUJO CONTINUO

Radial Flow of Compressible Gases (Solución exacta)

m(pe) = real gas potential as evaluated from 0 to pe, psi2/cpm(p)= real gas potential as evaluated from 0 to Pwf, psi2/cpk = permeability, mdh = thickness, ftre = drainage radius, ftrw = wellbore radius, ftQg = gas flow rate, Mscf/day

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Conceptos de PermeabilidadRepresenta la facilidad con que los fluidos se desplazan a través delmedio poroso.

No existe relación de proporcionalidad entre ф y K.

La K se mide en Darcys. En la industria se emplea el milidarcy,equivalente a 0,001 darcy. Las rocas pueden tener K que van desde0,5 hasta 3.400 md.

Las normas API para determinar (K) de las rocas definen k como “elrégimen de flujo en ml por seg de un fluido de 1 cp de viscosidadque pase a través de una sección de 1 cm2 de roca, bajo ungradiente de presión de una atm (760 mm Hg.) por cm2, y encondiciones de flujo viscoso”.

Conceptos de permeabilidad

Permeabilidad absoluta (k). Es aquella que se mide cuando unfluido satura 100 % el espacio poroso. Normalmente, el fluidode prueba es aire o agua.

Permeabilidad efectiva (ke = kr*k). Es la medida de la k a unfluido que se encuentra en presencia de otro u otros fluidosque saturan el medio poroso. La Ke es función de la saturaciónde fluidos, siempre las Ke son menores que la K absoluta.

Permeabilidad relativa (Kr=ke/k). Es la relación existenteentre la Ke y K. Esta medida es muy importante en ingenieríade yacimientos, ya que da una medida de la forma como unfluido se desplaza en el medio poroso. La sumatoria de laspermeabilidades relativas es menor de 1.0.

Permeabilidad relativaA la Sor o a la Swc se tiene que ke ≈ kabs. Si un 2-3 % de faseno-mojante se introduce, esta se mete a los poros grandes yobstaculiza el flujo de la. Si los poros fueran iguales, no habríaobstáculos.

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Combinación de permeabilidadesFlujo lineal capas en serie

Combinación de permeabilidadesFlujo lineal capas en paralelo

Combinación de permeabilidadesFlujo radial capas en serie

Combinación de permeabilidadesFlujo radial capas en paralelo

EFECTO KLINKENBERG

EFECTO DE LA COMPRESIBILIDAD

DAÑO - SKIN

GRADIENTES

flujo de fluidos en medios porosos 2013

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