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Darcy's law for multiphase flow

Morris Muskat et al.[1][2] developed the governing equations for multiphase flow (one vector equation for each fluid phase) in porous media as a generalisation of Darcy's equation (or Darcy's law) for water flow in porous media. The porous media are usually sedimentary rocks such as clastic rocks (mostly sandstone) or carbonate rocks.

Nomenclature
Symbol Description SI Units
subscript: phase a, vector component
vector component of directional contact surface between two grid cells m2
directional contact surface between two (usually neighboring) grid cells m2
unit vector along 3rd axis (z is a reminder here: 3 is z-direction) 1
acceleration of gravity m/s2
acceleration of gravity with direction m/s2
absolute permeability as a 3x3 tensor m2
relative permeability of phase a= w, o, g fraction
directional relative permeability (i.e. 3x3 tensor) fraction
pressure Pa
volumetric flux (Darcy velocity) through grid cell contact surface m/s
volumetric flow rate through grid cell contact surface m3/s
pore (fluid) flow velocity m/s
Darcy (fluid) velocity along axis m/s
Darcy (fluid) velocity m/s
gradient operator m−1
dynamic viscosity Pa s
mass density kg/m3
 where a = w, o, g

The present fluid phases are water, oil and gas, and they are represented by the subscript a = w,o,g respectively. The gravitational acceleration with direction is represented as or or . Notice that in petroleum engineering the spatial co-ordinate system is right-hand-oriented with z-axis pointing downward. The physical property that links the flow equations of the three fluid phases, is relative permeability of each fluid phase and pressure. This property of the fluid-rock system (i.e. water-oil-gas-rock system) is mainly a function of the fluid saturations, and it is linked to capillary pressure and the flowing process, implying that it is subject to hysteresis effect.

In 1940 M.C. Leverett[3] pointed out that in order to include capillary pressure effects in the flow equation, the pressure must be phase dependent. The flow equation then becomes

 where a = w, o, g

Leverett also pointed out that the capillary pressure shows significant hysteresis effects. This means that the capillary pressure for a drainage process is different from the capillary pressure of an imbibition process with the same fluid phases. Hysteresis does not change the shape of the governing flow equation, but it increases (usually doubles) the number of constitutive equations for properties involved in the hysteresis.

During 1951-1970 commercial computers entered the scene of scientific and engineering calculations and model simulations. Computer simulation of the dynamic behaviour of oil reservoirs soon became a target for the petroleum industry, but the computing power was very weak at that time.

With weak computing power, the reservoir models were correspondingly coarse, but upscaling of the static parameters were fairly simple and partly compensated for the coarseness. The question of upscaling relative permeability curves from the rock curves derived at core plug scale (which is often denoted the micro scale) to the coarse grids cells of the reservoir models (which is often called the macro scale) is much more difficult, and it became an important research field that is still ongoing. But the progress in upscaling was slow, and it was not until 1990-2000 that directional dependency of relative permeability and need for tensor representation was clearly demonstrated,[4][5] even though at least one capable method[6] was developed already in 1975. One such upscaling case is a slanted reservoir where the water (and gas) will segregate vertically relative to the oil in addition to the horizontal motion. The vertical size of a grid cell is also usually much smaller than the horizontal size of a grid cell, creating small and large flux areas respectively. All this requires different relative permeability curves for the x and z directions. Geological heterogeneities in the reservoirs like laminas or crossbedded permeability structures in the rock, also cause directional relative permeabilities. This tells us that relative permeability should, in the most general case, be represented by a tensor. The flow equations then become

 where a = w, o, g

The above-mentioned case reflected downdip water injection (or updip gas injection) or production by pressure depletion. If you inject water updip (or gas downdip) for a period of time, it will give rise to different relative permeability curves in the x+ and x- directions. This is not a hysteresis process in the traditional sense, and it cannot be represented by a traditional tensor. It can be represented by an IF-statement in the software code, and it occurs in some commercial reservoir simulators. The process (or rather sequence of processes) may be due to a backup plan for field recovery, or the injected fluid may flow to another reservoir rock formation due to an unexpected open part of a fault or a non-sealing cement behind casing of the injection well. The option for relative permeability is seldom used, and we just note that it does not change (the analytical shape of) the governing equation, but increases (usually doubles) the number of constitutive equations for the properties involved.

The above equation is a vector form of the most general equation for fluid flow in porous media, and it gives the reader a good overview of the terms and quantities involved. Before you go ahead and transform the differential equation into difference equations, to be used by the computers, you must write the flow equation in component form. The flow equation in component form (using summation convention) is

 where a = w, o, g  where = 1,2,3

The Darcy velocity is not the velocity of a fluid particle, but the volumetric flux (frequently represented by the symbol ) of the fluid stream. The fluid velocity in the pores (or short but inaccurately called pore velocity) is related to Darcy velocity by the relation

 where a = w, o, g

The volumetric flux is an intensive quantity, so it is not good at describing how much fluid is coming per time. The preferred variable to understand this is the extensive quantity called volumetric flow rate which tells us how much fluid is coming out of (or going into) a given area per time, and it is related to Darcy velocity by the relation

 where a = w, o, g

We notice that the volumetric flow rate is a scalar quantity and that the direction is taken care of by the normal vector of the surface (area) and the volumetric flux (Darcy velocity).

In a reservoir model the geometric volume is divided into grid cells, and the area of interest now is the intersectional area between two adjoining cells. If these are true neighboring cells, the area is the common side surface, and if a fault is dividing the two cells, the intersection area is usually less than the full side surface of both adjoining cells. A version of the multiphase flow equation, before it is discretized and used in reservoir simulators, is thus

 where a = w, o, g

In expanded (component) form it becomes

 where a = w, o, g

The (initial) hydrostatic pressure at a depth (or level) z above (or below) a reference depth z0 is calculated by

 where a = w, o, g

When calculations of hydrostatic pressure are executed, one normally does not apply a phase subscript, but switch formula / quantity according to what phase is observed at the actual depth, but we have included the phase subscript here for clarity and consistency. However, when calculations of hydrostatic pressure are executed one may use an acceleration of gravity that varies with depth in order to increase accuracy. If such high accuracy is not needed, the acceleration of gravity is kept constant, and the calculated pressure is called overburden pressure. Such high accuracy is not needed in reservoir simulations so acceleration of gravity is treated as a constant in this discussion. The initial pressure in the reservoir model is calculated using the formula for (initial) overburden pressure which is

 where a = w, o, g

In order to simplify the terms within the parenthesis of the flow equation, we can introduce a flow potential called the -potential, pronounced psi-potential, which is defined by

 where a = w, o, g

It consists of two terms which are absolute pressure and gravity head. To save computing time the integral can be calculated initially and stored as a table to be used in the computationally cheaper table-lookup. Introduction of the -potential implies that

 where a = w, o, g

The psi-potential is also frequently called the "datum pressure", since the function represents the pressure at any point in the reservoir after being transferred to the datum plane / depth z0. In practical engineering work it is very useful to refer pressures measured in wells to a datum level or to map the distribution of datum pressures throughout the reservoir. In this way the direction of fluid movement in the reservoir can be seen at a glance since the datum pressure distribution is equivalent to the potential distribution. Two simple examples will clarify this. A reservoir may consists of several flow units that are separated by tight shale layers. Fluid from one reservoir or flow unit can enter a fault at one depth and exit the fault in another reservoir or flow unit at another depth. Likewise can fluid enter a production well in one flow unit and exit the production well in another flow unit or reservoir.

The multiphase flow equation for porous media now becomes

 where a = w, o, g

This multiphase flow equation has traditionally been the starting point for the software programmer when he/she starts transforming the equation from differential equation to difference equation in order to write a program code for a reservoir simulator to be used in the petroleum industry. The unknown dependent variables have traditionally been oil pressure (for oil fields) and volumetric quantities for the fluids involved, but one may rewrite the total set of model equations to be solved for oil pressure and mass or mole quantities for the fluid components involved.[7]

The above equations are written in SI units and we are assuming that all material properties are also defined within the SI units. A result of this is that above versions of the equations do not need any unit conversion constants. The petroleum industry applies a variety of units, of which a least two have some prevalence. If you want to apply units other than SI units, you must establish correct unit conversion constants for the multiphase flow equations.

Conversion of units Edit

The above equations are written in SI units (short SI) suppressing that the unit D (darcy) for the absolute permeability is defined in non-SI units. That is why there are no unit-related constants. The petroleum industry doesn't use the SI units. Instead, they use a special version of SI units that we will call Applied SI units, or they use another set of units called Field units which has its origin from US and UK. Temperature is not included in the equations, so we can use the factor-label method (also called unit-factor method) which says that if we have a variable/parameter with unit H, we multiply this variable/parameter by a conversion constant C and then the variable gets the unit G that we want. This means that we apply the transformation H*C = G, and the non-SI effect of the definition of permeability is included in the conversion factor C for permeability. The transformation H*C = G apply for every spatial dimension so we concentrate on the main terms, neglecting the signs, and then complete the parenthesis with the gravity term. Before we start the conversion, we notice that both the original (single phase) the flow equation of Darcy and the generalized (or extended) multiphase flow equations of Muskat et al. are using reservoir velocity (volume flux), volume rate and densities. The units of these quantities are given a prefix r (or R) in order to distinguish them from their counterparts at standard surface conditions which gets a prefix s (or S). This is especially important when we convert the equations to Field units. The reason for going into details in the seemingly simple topic of unit conversion, is that many people make mistakes when doing unit conversions.

Now we are ready to start the conversion work. First, we take the flux version of the equation and rewrite it as

   where a = w, o, g

We want to place the composite conversion factor together with the permeability parameter. Here we note that our equation is written in SI units, and that the group of variables/parameters (hereafter called parameters for short) on the right-hand side constitute a dimensionless group. Now we convert each parameter and collect these conversions into a single conversion constant. Now we note that our list with conversion constants (the C's) goes from applied unit to SI units, and this very common for such conversion lists. We therefore assume that our parameters are entered in applied units and convert them (back) to SI units.

   where a = w, o, g

Notice that we have removed relative permeability which is a dimensionless parameter. This composite conversion factor is called Darcy's constant for the flux formulated equation, and it is

 

Since our parameter group is dimensionless in base SI units, we don't need to include the SI units in the units for our composite conversion factor as you can see in the second table. Next, we take the rate version of the equation and rewrite it as

   where a = w, o, g

Now we convert each parameter and collect these conversions into a single conversion constant.

   where a = w, o, g

Notice that we have removed relative permeability which is a dimensionless parameter. This composite conversion factor is called Darcy's constant for the flux formulated equation, and it is

 

The pressure gradient and the gravity term are identical for the flux and the rate equations, and will, therefore, be discussed only once. The task here is to have a gravity term that is consistent with the applied units ("H-units") for the pressure gradient. We must, therefore, place our conversion factor together with the gravity parameters. We write "the parenthesis" in SI units as

   where a = w, o, g

and rewrite it as

   where a = w, o, g

Now we convert each parameter and collect these conversions into a single conversion constant. First, we note that our equation is written in SI units, and that the group of parameters on the right-hand side constitute a dimensionless group. We, therefore, assume that our parameters are entered in applied units and convert them (back) to SI units.

   where a = w, o, g

This gives the composite conversion factor for the consistency-conversion as

 

Since our parameter group is dimensionless in SI units, we don't need to include the SI units in the units for our composite conversion factor as you can see in the second table.

This is it for the analytical equations, but when the programmer transform the flow equation into a finite difference equation and further into a numerical algorithm, they are eager to minimize the number of computational operations. Here is an example with two constants that can be reduced to one by the fusion

 

Using industry units, the flux version of the flow equation in vector form becomes

   where a = w, o, g

and in component form it becomes

   where a = w, o, g  where   = 1,2,3

Using industry units, the rate version of the flow equation in vector form becomes

   where a = w, o, g

and in component form it becomes

   where a = w, o, g

Conversion of units is a fairly rare activity, even for technical professionals, but that is also the reason why people forget how to do it correctly.

See also Edit

References Edit

  1. ^ Muskat M. and Meres M.W. 1936. The Flow of Heterogeneous Fluids Through Porous Media. Paper published in J. Appl. Phys. 1936, 7, pp 346–363. https://dx.doi.org/10.1063/1.1745403
  2. ^ Muskat M. and Wyckoff R.D. and Botset H.G. and Meres M.W. 1937. Flow of Gas-liquid Mixtures through Sands. Published in Transactions of the AIME 1937, 123, pp 69–96. SPE document ID is SPE-937069-G. https://dx.doi.org/10.2118/937069-G
  3. ^ Leverett M.C. 1941. Capillary behaviour in porous solids. Paper presented at the Tulsa meeting of AIME October 1940. Published in Transactions of the AIME 1941, 142, pp 159–172. SPE document ID is SPE-941152-G. https://dx.doi.org/10.2118/941152-G
  4. ^ Pickup G.E and Sorbie K.S. 1996. The Scaleup of Two-Phase Flow in Porous Media Using Phase Permeability Tensors. Paper SPE-28586 first presented at the SPE Annual Technical Conference & Exhibition held in New Orleans, US, 25–28 September 1994. Paper SPE-28586-PA published in SPEJ December 1996. https://dx.doi.org/10.2118/28586-PA
  5. ^ Kumar A.T.A. and Jerauld G.R. 1996. Impacts of Scaleup on Fluid Flow from Plug to Grid Block Scale in Reservoir Rock. Paper SPE/DOE 35452 presented at the 1996 SPE/DOE 10th Symposium on Improved oil Recovery held in Tulsa, Oklahoma, USA, 21–24 April 1996. Paper SPE-35452-MS published by SPE 1996. https://dx.doi.org/10.2118/35452-MS
  6. ^ Kyte J.R. and Berry D.W. 1975. New Pseudo Functions to Control Numerical Dispersion. Paper SPE 5105 first presented at SPE-AIME 49th Annual Fall Meeting held in Houston October 6–9, 1974. Paper SPE-5105-PA published by SPEJ August 1975 pp 269-275. https://dx.doi.org/10.2118/5105-PA
  7. ^ Young L.C. and Stephenson R.E. 1983. A Generalized Compositional Approach for Reservoir Simulation. Paper SPE 10516 first presented at SPE Reservoir Symposium held in New Orleans, January 31 - February 3, 1982. Paper SPE-10516-PA first published by SPEJ October 1983, vol 23, no 05, pp 727-742; Transaction AIME 275; now by ResearchGate; https://www.researchgate.net/publication/244956766 and by https://dx.doi.org/10.2118/10516-PA

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This article may be too technical for most readers to understand Please help improve it to make it understandable to non experts without removing the technical details January 2017 Learn how and when to remove this template message Morris Muskat et al 1 2 developed the governing equations for multiphase flow one vector equation for each fluid phase in porous media as a generalisation of Darcy s equation or Darcy s law for water flow in porous media The porous media are usually sedimentary rocks such as clastic rocks mostly sandstone or carbonate rocks Nomenclature Symbol Description SI Unitssubscript phase a vector component s displaystyle sigma A s displaystyle A sigma vector component s displaystyle sigma of directional contact surface between two grid cells m2A displaystyle mathbf A directional contact surface between two usually neighboring grid cells m2e z 3 displaystyle mathbf e z 3 unit vector along 3rd axis z is a reminder here 3 is z direction 1g displaystyle g acceleration of gravity m s2g displaystyle mathbf g acceleration of gravity with direction m s2K displaystyle mathbf K absolute permeability as a 3x3 tensor m2K r a displaystyle K ra relative permeability of phase a w o g fractionK r a displaystyle mathbf K ra directional relative permeability i e 3x3 tensor fractionP a displaystyle P a pressure Paq a displaystyle mathbf q a volumetric flux Darcy velocity through grid cell contact surface m sQ a displaystyle Q a volumetric flow rate through grid cell contact surface m3 sv a displaystyle mathbf v a pore fluid flow velocity m su a s displaystyle u a sigma Darcy fluid velocity along axis s displaystyle sigma m su a displaystyle mathbf u a Darcy fluid velocity m s displaystyle nabla gradient operator m 1m a displaystyle mu a dynamic viscosity Pa displaystyle cdot sr a displaystyle rho a mass density kg m3u a m a 1 K r a K P r a g displaystyle mathbf u a mu a 1 K ra mathbf K cdot left nabla P rho a mathbf g right where a w o gThe present fluid phases are water oil and gas and they are represented by the subscript a w o g respectively The gravitational acceleration with direction is represented as g displaystyle mathbf g or g z displaystyle g nabla z or g e z 3 displaystyle g mathbf e z 3 Notice that in petroleum engineering the spatial co ordinate system is right hand oriented with z axis pointing downward The physical property that links the flow equations of the three fluid phases is relative permeability of each fluid phase and pressure This property of the fluid rock system i e water oil gas rock system is mainly a function of the fluid saturations and it is linked to capillary pressure and the flowing process implying that it is subject to hysteresis effect In 1940 M C Leverett 3 pointed out that in order to include capillary pressure effects in the flow equation the pressure must be phase dependent The flow equation then becomes u a m a 1 K r a K P a r a g displaystyle mathbf u a mu a 1 K ra mathbf K cdot left nabla P a rho a mathbf g right where a w o gLeverett also pointed out that the capillary pressure shows significant hysteresis effects This means that the capillary pressure for a drainage process is different from the capillary pressure of an imbibition process with the same fluid phases Hysteresis does not change the shape of the governing flow equation but it increases usually doubles the number of constitutive equations for properties involved in the hysteresis During 1951 1970 commercial computers entered the scene of scientific and engineering calculations and model simulations Computer simulation of the dynamic behaviour of oil reservoirs soon became a target for the petroleum industry but the computing power was very weak at that time With weak computing power the reservoir models were correspondingly coarse but upscaling of the static parameters were fairly simple and partly compensated for the coarseness The question of upscaling relative permeability curves from the rock curves derived at core plug scale which is often denoted the micro scale to the coarse grids cells of the reservoir models which is often called the macro scale is much more difficult and it became an important research field that is still ongoing But the progress in upscaling was slow and it was not until 1990 2000 that directional dependency of relative permeability and need for tensor representation was clearly demonstrated 4 5 even though at least one capable method 6 was developed already in 1975 One such upscaling case is a slanted reservoir where the water and gas will segregate vertically relative to the oil in addition to the horizontal motion The vertical size of a grid cell is also usually much smaller than the horizontal size of a grid cell creating small and large flux areas respectively All this requires different relative permeability curves for the x and z directions Geological heterogeneities in the reservoirs like laminas or crossbedded permeability structures in the rock also cause directional relative permeabilities This tells us that relative permeability should in the most general case be represented by a tensor The flow equations then become u a m a 1 K r a K P a r a g displaystyle mathbf u a mu a 1 mathbf K ra cdot mathbf K cdot left nabla P a rho a mathbf g right where a w o gThe above mentioned case reflected downdip water injection or updip gas injection or production by pressure depletion If you inject water updip or gas downdip for a period of time it will give rise to different relative permeability curves in the x and x directions This is not a hysteresis process in the traditional sense and it cannot be represented by a traditional tensor It can be represented by an IF statement in the software code and it occurs in some commercial reservoir simulators The process or rather sequence of processes may be due to a backup plan for field recovery or the injected fluid may flow to another reservoir rock formation due to an unexpected open part of a fault or a non sealing cement behind casing of the injection well The option for relative permeability is seldom used and we just note that it does not change the analytical shape of the governing equation but increases usually doubles the number of constitutive equations for the properties involved The above equation is a vector form of the most general equation for fluid flow in porous media and it gives the reader a good overview of the terms and quantities involved Before you go ahead and transform the differential equation into difference equations to be used by the computers you must write the flow equation in component form The flow equation in component form using summation convention is u a s m a 1 K r a s b K b g g P a r a g e z g displaystyle u a sigma mu a 1 K ra sigma beta K beta gamma left nabla gamma P a rho a g e z gamma right where a w o g where s displaystyle sigma 1 2 3The Darcy velocity u a displaystyle mathbf u a is not the velocity of a fluid particle but the volumetric flux frequently represented by the symbol q a displaystyle mathbf q a of the fluid stream The fluid velocity in the pores v a displaystyle mathbf v a or short but inaccurately called pore velocity is related to Darcy velocity by the relation v a ϕ 1 q a ϕ 1 u a displaystyle mathbf v a phi 1 mathbf q a phi 1 mathbf u a where a w o gThe volumetric flux is an intensive quantity so it is not good at describing how much fluid is coming per time The preferred variable to understand this is the extensive quantity called volumetric flow rate which tells us how much fluid is coming out of or going into a given area per time and it is related to Darcy velocity by the relation Q a A u a displaystyle Q a mathbf A cdot mathbf u a where a w o gWe notice that the volumetric flow rate Q a displaystyle Q a is a scalar quantity and that the direction is taken care of by the normal vector of the surface area and the volumetric flux Darcy velocity In a reservoir model the geometric volume is divided into grid cells and the area of interest now is the intersectional area between two adjoining cells If these are true neighboring cells the area is the common side surface and if a fault is dividing the two cells the intersection area is usually less than the full side surface of both adjoining cells A version of the multiphase flow equation before it is discretized and used in reservoir simulators is thus Q a m a 1 A K r a K P a r a g displaystyle Q a mu a 1 mathbf A cdot mathbf K ra cdot mathbf K cdot left nabla P a rho a mathbf g right where a w o gIn expanded component form it becomes Q a m a 1 A s K r a s b K b g g P a r a g e z g displaystyle Q a mu a 1 A sigma K ra sigma beta K beta gamma left nabla gamma P a rho a g e z gamma right where a w o gThe initial hydrostatic pressure at a depth or level z above or below a reference depth z0 is calculated by P a P a 0 z 0 z r a z g z d z displaystyle P a P a0 int limits z 0 z rho a left z right g left z right dz where a w o gWhen calculations of hydrostatic pressure are executed one normally does not apply a phase subscript but switch formula quantity according to what phase is observed at the actual depth but we have included the phase subscript here for clarity and consistency However when calculations of hydrostatic pressure are executed one may use an acceleration of gravity that varies with depth in order to increase accuracy If such high accuracy is not needed the acceleration of gravity is kept constant and the calculated pressure is called overburden pressure Such high accuracy is not needed in reservoir simulations so acceleration of gravity is treated as a constant in this discussion The initial pressure in the reservoir model is calculated using the formula for initial overburden pressure which is P a P a 0 g z 0 z r a z d z displaystyle P a P a0 g int limits z 0 z rho a left z right dz where a w o gIn order to simplify the terms within the parenthesis of the flow equation we can introduce a flow potential called the ps displaystyle psi potential pronounced psi potential which is defined by ps a P a g z 0 z r a z d z displaystyle psi a P a g int limits z 0 z rho a left z right dz where a w o gIt consists of two terms which are absolute pressure and gravity head To save computing time the integral can be calculated initially and stored as a table to be used in the computationally cheaper table lookup Introduction of the ps displaystyle psi potential implies that ps a P a r a g z displaystyle nabla psi a nabla P a rho a g nabla z where a w o gThe psi potential is also frequently called the datum pressure since the function represents the pressure at any point in the reservoir after being transferred to the datum plane depth z0 In practical engineering work it is very useful to refer pressures measured in wells to a datum level or to map the distribution of datum pressures throughout the reservoir In this way the direction of fluid movement in the reservoir can be seen at a glance since the datum pressure distribution is equivalent to the potential distribution Two simple examples will clarify this A reservoir may consists of several flow units that are separated by tight shale layers Fluid from one reservoir or flow unit can enter a fault at one depth and exit the fault in another reservoir or flow unit at another depth Likewise can fluid enter a production well in one flow unit and exit the production well in another flow unit or reservoir The multiphase flow equation for porous media now becomes Q a m a 1 A K r a K ps a displaystyle Q a mu a 1 mathbf A cdot mathbf K ra cdot mathbf K cdot nabla psi a where a w o gThis multiphase flow equation has traditionally been the starting point for the software programmer when he she starts transforming the equation from differential equation to difference equation in order to write a program code for a reservoir simulator to be used in the petroleum industry The unknown dependent variables have traditionally been oil pressure for oil fields and volumetric quantities for the fluids involved but one may rewrite the total set of model equations to be solved for oil pressure and mass or mole quantities for the fluid components involved 7 The above equations are written in SI units and we are assuming that all material properties are also defined within the SI units A result of this is that above versions of the equations do not need any unit conversion constants The petroleum industry applies a variety of units of which a least two have some prevalence If you want to apply units other than SI units you must establish correct unit conversion constants for the multiphase flow equations Conversion of units EditThe above equations are written in SI units short SI suppressing that the unit D darcy for the absolute permeability is defined in non SI units That is why there are no unit related constants The petroleum industry doesn t use the SI units Instead they use a special version of SI units that we will call Applied SI units or they use another set of units called Field units which has its origin from US and UK Temperature is not included in the equations so we can use the factor label method also called unit factor method which says that if we have a variable parameter with unit H we multiply this variable parameter by a conversion constant C and then the variable gets the unit G that we want This means that we apply the transformation H C G and the non SI effect of the definition of permeability is included in the conversion factor C for permeability The transformation H C G apply for every spatial dimension so we concentrate on the main terms neglecting the signs and then complete the parenthesis with the gravity term Before we start the conversion we notice that both the original single phase the flow equation of Darcy and the generalized or extended multiphase flow equations of Muskat et al are using reservoir velocity volume flux volume rate and densities The units of these quantities are given a prefix r or R in order to distinguish them from their counterparts at standard surface conditions which gets a prefix s or S This is especially important when we convert the equations to Field units The reason for going into details in the seemingly simple topic of unit conversion is that many people make mistakes when doing unit conversions Now we are ready to start the conversion work First we take the flux version of the equation and rewrite it as 1 K g P a m a u a displaystyle 1 frac K nabla gamma P a mu a u a nbsp where a w o gWe want to place the composite conversion factor together with the permeability parameter Here we note that our equation is written in SI units and that the group of variables parameters hereafter called parameters for short on the right hand side constitute a dimensionless group Now we convert each parameter and collect these conversions into a single conversion constant Now we note that our list with conversion constants the C s goes from applied unit to SI units and this very common for such conversion lists We therefore assume that our parameters are entered in applied units and convert them back to SI units C k K C g C p P a C m m a C u u a C f K g P a m a u a K g P a m a u a S I 1 displaystyle frac C k KC nabla nabla gamma C p P a C mu mu a C u u a C f frac K nabla gamma P a mu a u a left frac K nabla gamma P a mu a u a right SI 1 nbsp where a w o gNotice that we have removed relative permeability which is a dimensionless parameter This composite conversion factor is called Darcy s constant for the flux formulated equation and it is C f C k C C p C m C u C k C C m C u C p displaystyle C f frac C k C nabla C p C mu C u frac C k C nabla C mu C u C p nbsp Since our parameter group is dimensionless in base SI units we don t need to include the SI units in the units for our composite conversion factor as you can see in the second table Next we take the rate version of the equation and rewrite it as 1 A K g P a m a Q a displaystyle 1 frac AK nabla gamma P a mu a Q a nbsp where a w o gNow we convert each parameter and collect these conversions into a single conversion constant C A A C k K C g C p P a C m m a C Q Q a C r A K g P a m a Q a A K g P a m a Q a S I 1 displaystyle frac C A AC k KC nabla nabla gamma C p P a C mu mu a C Q Q a C r frac AK nabla gamma P a mu a Q a left frac AK nabla gamma P a mu a Q a right SI 1 nbsp where a w o gNotice that we have removed relative permeability which is a dimensionless parameter This composite conversion factor is called Darcy s constant for the flux formulated equation and it is C r C A C k C C p C m C Q C k C C A C m C Q C p displaystyle C r frac C A C k C nabla C p C mu C Q frac C k C nabla C A C mu C Q C p nbsp The pressure gradient and the gravity term are identical for the flux and the rate equations and will therefore be discussed only once The task here is to have a gravity term that is consistent with the applied units H units for the pressure gradient We must therefore place our conversion factor together with the gravity parameters We write the parenthesis in SI units as g P a r a g displaystyle nabla gamma P a rho a g nbsp where a w o gand rewrite it as 1 r a g g P a displaystyle 1 frac rho a g nabla gamma P a nbsp where a w o gNow we convert each parameter and collect these conversions into a single conversion constant First we note that our equation is written in SI units and that the group of parameters on the right hand side constitute a dimensionless group We therefore assume that our parameters are entered in applied units and convert them back to SI units C r r a C g g C g C p P a C p d g r a g g P a r a g g P a S I 1 displaystyle frac C rho rho a C g g C nabla nabla gamma C p P a C pdg frac rho a g nabla gamma P a left frac rho a g nabla gamma P a right SI 1 nbsp where a w o gThis gives the composite conversion factor for the consistency conversion as C p d g C r C g C C p displaystyle C pdg frac C rho C g C nabla C p nbsp Since our parameter group is dimensionless in SI units we don t need to include the SI units in the units for our composite conversion factor as you can see in the second table This is it for the analytical equations but when the programmer transform the flow equation into a finite difference equation and further into a numerical algorithm they are eager to minimize the number of computational operations Here is an example with two constants that can be reduced to one by the fusion C c g C p d g g displaystyle C cg C pdg cdot g nbsp Using industry units the flux version of the flow equation in vector form becomes u a C f m a 1 K r a K P a C p d g r a g displaystyle mathbf u a C f mu a 1 mathbf K ra mathbf K cdot left nabla P a C pdg rho a mathbf g right nbsp where a w o gand in component form it becomes u a s C f m a 1 K r a s b K b g g P a C p d g r a g e z g displaystyle u a sigma C f mu a 1 K ra sigma beta K beta gamma left nabla gamma P a C pdg rho a g e z gamma right nbsp where a w o g where s displaystyle sigma nbsp 1 2 3Using industry units the rate version of the flow equation in vector form becomes Q a C r m a 1 A K r a K P a C p d g r a g displaystyle Q a C r mu a 1 mathbf A cdot mathbf K ra cdot mathbf K cdot left nabla P a C pdg rho a mathbf g right nbsp where a w o gand in component form it becomes Q a C r m a 1 A s K r a s b K b g g P a C p d g r a g e z g displaystyle Q a C r mu a 1 A sigma K ra sigma beta K beta gamma left nabla gamma P a C pdg rho a g e z gamma right nbsp where a w o gDarcy petroleum basic conversions Symbol dimensions Description Applied SI units conversion SI units Field units conversion SI unitssubscript phase a Industry factor C Academia Industry factor C Academiasubscript vector component s displaystyle sigma nbsp g displaystyle gamma nbsp you have multiply by to get you have multiply by to getm a displaystyle m a nbsp M mass kg 1 kg lb 4 535924E 01 kgt displaystyle t nbsp T time d 8 640000E 04 s d 8 640000E 04 sx y z or x s displaystyle x sigma nbsp L position also X Y Z or X s displaystyle X sigma nbsp etc m 1 m ft 3 048000E 01 mL or D displaystyle Delta nbsp x D displaystyle Delta nbsp y D displaystyle Delta nbsp z L length also D displaystyle Delta nbsp X D displaystyle Delta nbsp Y D displaystyle Delta nbsp Z or D x s displaystyle Delta x sigma nbsp or D X s displaystyle Delta X sigma nbsp etc m 1 m ft 3 048000E 01 mV r e s displaystyle V res nbsp L3 reservoir volume rm 1 rm rb 1 589873E 01 rmv a s displaystyle v a sigma nbsp L T pore fluid flow velocity at reservoir conditions rm d 1 157407E 05 rm s rft d 3 527778E 06 rm sq a s displaystyle q a sigma nbsp L T volumetric flux Darcy velocity at reservoir conditions rm d 1 157407E 05 rm s rft d 3 527778E 06 rm su a s displaystyle u a sigma nbsp L T Darcy fluid velocity at reservoir conditions rm d 1 157407E 05 rm s rft d 3 527778E 06 rm sQ o displaystyle Q o nbsp L3 T surface volumetric flow rate of oil at standard conditions sm3 d 1 157407E 05 sm3 s stb d 1 840131E 06 sm3 sQ g displaystyle Q g nbsp L3 T surface volumetric flow rate of gas at standard conditions sm3 d 1 157407E 05 sm3 s Msft3 d 3 277413E 04 sm3 sQ a displaystyle Q a nbsp L3 T reservoir volumetric flow rate through contact surface rm3 d 1 157407E 05 rm3 s rb d 1 840131E 06 rm3 sm a displaystyle mu a nbsp M LT dynamic viscosity cP 1 000000E 03 Pa displaystyle cdot nbsp s cP 1 000000E 03 Pa displaystyle cdot nbsp sA s displaystyle A sigma nbsp L2 contact surface area between two grid cells or into borehole m2 1 m2 ft2 9 290304E 02 m2K r a displaystyle K ra nbsp 1 relative permeability fraction 1 fraction fraction 1 fractionK g s displaystyle K gamma sigma nbsp L2 absolute permeability mD 9 869233E 16 m2 mD 9 869233E 16 m2 s displaystyle nabla sigma nbsp 1 L gradient operator for s displaystyle sigma nbsp axis m 1 1 m 1 ft 1 3 280840E 00 m 1P a displaystyle P a nbsp ML T2 L2 pressure bar 1 000000E 05 Pa psi 6 894757E 03 Par a displaystyle rho a nbsp M L3 mass density kg m3 1 kg m3 lb rb 1 601846E 01 M m3d u a s d t displaystyle du a sigma dt nbsp L T2 acceleration m s2 1 m s2 ft s2 3 048000E 01 m s2Darcy petroleum conversion constants Symbol dimensions Description Value using Applied SI units Value using Field units Value using SI unitsC f displaystyle C f nbsp 1 Darcy constant for volumetric flux Darcy velocity u 8 527017E 03 c P r m d b a r m D m 1 displaystyle frac cP rm d bar mD m 1 nbsp 6 328287E 03 c P r f t d p s i m D f t 1 displaystyle frac cP rft d psi mD ft 1 nbsp 9 869233E 09 P a s r m s P a m 2 m 1 displaystyle frac Pa s rm s Pa m 2 m 1 nbsp C r displaystyle C r nbsp 1 Darcy constant for volumetric rate Darcy rate Q 8 527017E 03 c P r m 3 d b a r m D m displaystyle frac cP rm 3 d bar mD m nbsp 1 127116E 03 c P r b d p s i m D f t displaystyle frac cP rb d psi mD ft nbsp 9 869233E 09 P a s r m 3 s P a m 2 m displaystyle frac Pa s rm 3 s Pa m 2 m nbsp C p d g displaystyle C pdg nbsp 1 Conversion constant for gravity term 1 000000E 05 b a r m k g m 3 m s 2 displaystyle frac bar m kg m 3 m s 2 nbsp 2 158399E 04 p s i f t l b f t 3 f t s 2 displaystyle frac psi ft lb ft 3 ft s 2 nbsp 1 P a m k g m 3 m s 2 displaystyle frac Pa m kg m 3 m s 2 nbsp g displaystyle g nbsp L T2 Standard acceleration of gravity 9 806650E 00 m s2 3 217405E 01 ft s2 9 806650E 00 m s2C c g displaystyle C cg nbsp L T2 fused conversion and acceleration of gravity constants 9 806650E 05 b a r m k g m 3 displaystyle frac bar m kg m 3 nbsp 6 944445E 03 p s i f t l b f t 3 displaystyle frac psi ft lb ft 3 nbsp 9 806650E 00 P a m k g m 3 displaystyle frac Pa m kg m 3 nbsp Conversion of units is a fairly rare activity even for technical professionals but that is also the reason why people forget how to do it correctly See also EditPetroleum reservoir Reservoir engineering Reservoir modeling Reservoir simulation Black oil equations Conversion of units includes tables of conversion factors Dimensionless numbers in fluid mechanics Fermi problem used to teach dimensional analysis Rayleigh s method of dimensional analysis Similitude model an application of dimensional analysis System of measurement Units of measurement List of dimensionless quantities Orders of magnitude numbers Dimensional analysis Normalization statistics and standardized moment the analogous concepts in statistics Buckingham p theoremReferences Edit Muskat M and Meres M W 1936 The Flow of Heterogeneous Fluids Through Porous Media Paper published in J Appl Phys 1936 7 pp 346 363 https dx doi org 10 1063 1 1745403 Muskat M and Wyckoff R D and Botset H G and Meres M W 1937 Flow of Gas liquid Mixtures through Sands Published in Transactions of the AIME 1937 123 pp 69 96 SPE document ID is SPE 937069 G https dx doi org 10 2118 937069 G Leverett M C 1941 Capillary behaviour in porous solids Paper presented at the Tulsa meeting of AIME October 1940 Published in Transactions of the AIME 1941 142 pp 159 172 SPE document ID is SPE 941152 G https dx doi org 10 2118 941152 G Pickup G E and Sorbie K S 1996 The Scaleup of Two Phase Flow in Porous Media Using Phase Permeability Tensors Paper SPE 28586 first presented at the SPE Annual Technical Conference amp Exhibition held in New Orleans US 25 28 September 1994 Paper SPE 28586 PA published in SPEJ December 1996 https dx doi org 10 2118 28586 PA Kumar A T A and Jerauld G R 1996 Impacts of Scaleup on Fluid Flow from Plug to Grid Block Scale in Reservoir Rock Paper SPE DOE 35452 presented at the 1996 SPE DOE 10th Symposium on Improved oil Recovery held in Tulsa Oklahoma USA 21 24 April 1996 Paper SPE 35452 MS published by SPE 1996 https dx doi org 10 2118 35452 MS Kyte J R and Berry D W 1975 New Pseudo Functions to Control Numerical Dispersion Paper SPE 5105 first presented at SPE AIME 49th Annual Fall Meeting held in Houston October 6 9 1974 Paper SPE 5105 PA published by SPEJ August 1975 pp 269 275 https dx doi org 10 2118 5105 PA Young L C and Stephenson R E 1983 A Generalized Compositional Approach for Reservoir Simulation Paper SPE 10516 first presented at SPE Reservoir Symposium held in New Orleans January 31 February 3 1982 Paper SPE 10516 PA first published by SPEJ October 1983 vol 23 no 05 pp 727 742 Transaction AIME 275 now by ResearchGate https www researchgate net publication 244956766 and by https dx doi org 10 2118 10516 PA Retrieved from https en wikipedia org w index php title Darcy 27s law for multiphase flow amp oldid 1016063422, wikipedia, wiki, book, books, library,

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