- Email: [email protected]
Porosity and Permeability
4 Porosity and Permeability Porosity and permeability are measures of the storage capacity and flow capacity, respectively, in a porous medium. Many fundamental concepts are needed to understand storage and flow capacity. They include bulk volume, pore volume, net volume, porosity, porosity compressibility, transmissibility, and permeability. Averaging techniques for estimating a scalar value of permeability are presented in this chapter, but permeability is in general a tensor and can be approximated as a scalar in many cases. The tensor concept of directional permeability is discussed.
4.1
Bulk Volume and Net Volume
The bulk volume of a cell that is shaped like a parallelepiped, shown in Figure 4.1, is given by the triple scalar product ! ! ! VB ¼ U ðV W Þ !
!
ð4:1:1Þ
!
The vectors fU , V , W g are aligned along the axes of the parallelepiped and have magnitudes that are equal to the lengths of the sides of the parallelepiped. In Cartesian coordinates, we have the result VB ¼ Lx i^ Ly j^ Lz k^ ¼ Lx Ly Lz
ð4:1:2Þ
in terms of the lengths Lx , Ly , Lz of the sides of the Cartesian cell. The vectors ^ j, ^ k^ are unit vectors in Cartesian coordinates. i, Bulk volume VB of a Cartesian cell is the product of cross-sectional area A times gross thickness H: VB ¼ AH
ð4:1:3Þ
The area of a cell is the product of the x-direction cell length Dx and the y-direction cell length Dy; thus A ¼ Dx Dy. Gross thickness H is the cell length in the z-direction. Bulk volume is a measure of the gross volume in the system. It includes both rock volume and pore volume. To determine the volume of the system that is commercially significant, the gross volume must be adjusted by introducing the concept
Integrated Reservoir Asset Management. DOI: 10.1016/B978-0-12-382088-4.00004-9 Copyright # 2010 Elsevier Inc. All rights reserved.
50
Integrated Reservoir Asset Management
U V
W
Figure 4.1 Volume of a parallelepiped.
of net thickness. Net thickness h is the thickness of the commercially significant formation. The net-to-gross ratio Z is the ratio of net thickness h to gross thickness H: Z ¼ h=H, 0 Z 1
ð4:1:4Þ
Net thickness is always less than or equal to gross thickness.
4.2
Porosity and Grain Volume
Porosity is the fraction of a porous medium that is void space. The bulk volume VB of a porous medium is the sum of pore volume VP and grain volume VG; thus VB ¼ VP þ V G
ð4:2:1Þ
Porosity is the ratio of pore volume to bulk volume: f ¼ VP =VB
ð4:2:2Þ
Dividing Eq. (4.2.1) by VB and using the definition of porosity expresses the grain volume in terms of porosity as VG ¼1f VB
4.3
ð4:2:3Þ
Effective Pore Volume
Most porous media contain a fraction of pores that are not in communication with the flow path. This pore volume is ineffective. Effective pore volume ðVP Þeff is the interconnected pore volume that communicates with a well. Effective porosity is defined as the ratio of effective pore volume to bulk volume; thus feff
ðVP Þeff VB
ð4:3:1Þ
Porosity and Permeability
51
Unless stated otherwise, any further discussion using porosity will assume that the porosity of interest is effective porosity, and the subscript “eff ” is not written.
4.4
Porosity Compressibility
Porosity compressibility is a measure of the change in porosity f as a function of fluid pressure P. It is defined as cf ¼
1 df f dP
ð4:4:1Þ
If f0 is porosity at pressure P0 and f is porosity at pressure P, the integral of Eq. (4.4.1) yields the relationship 2
3
ðP
6 7 f ¼ f0 exp4 cf dP5
ð4:4:2Þ
P0
If porosity compressibility is constant with respect to pressure, the integral in Eq. (4.4.2) can be evaluated and gives f ¼ f0 exp cf DP
ð4:4:3Þ
where DP ¼ P – P0. The first-order approximation to Eq. (4.4.3) is f f0 1 þ cf DP ¼ f0 1 þ cf ðP P0 Þ
ð4:4:4Þ
Equation (4.4.4) is used in many reservoir flow simulators to calculate the change in porosity with respect to changes in fluid pressure. A first-order approximation is reasonable in many cases because reservoir rock compressibility is typically on the order of cf ¼ 3 106 =psia and a typical pressure change is on the order of P P0 1000 psia, so the bracketed term in Eq. (4.4.4) is on the order of 1.003. There are reservoirs where the approximation is not as valid because the product of rock compressibility and pressure change can be an order of magnitude larger.
4.5
Darcy’s Law and Permeability
The basic equation that describes fluid flow in porous media is Darcy’s law. Darcy found that flow rate was proportional to pressure gradient. Darcy’s equation for calculating volumetric flow rate q for linear, horizontal, single-phase flow is q ¼ 0:001127
KA DP m Dx
ð4:5:1Þ
52
Integrated Reservoir Asset Management
The units of the physical variables determine the value of the constant (0.001127) in Eq. (4.5.1). The constant 0.001127 corresponds to variables expressed in the following oil field units: q K A P m Dx
¼ volumetric flow rate (bbl/day) ¼ permeability (md) ¼ cross-sectional area (ft2) ¼ pressure (psi) ¼ fluid viscosity (cp) ¼ length (ft)
Figure 4.2 illustrates the terms in Darcy’s law for a cylindrical core of rock. The movement of a single-phase fluid through a porous medium depends on crosssectional area A that is normal to the direction of fluid flow, pressure difference DP across the length Dx of the flow path, and viscosity m of the flowing fluid. The minus sign indicates that the direction of fluid flow is opposite to the direction of increasing pressure: the fluid flows from high pressure to low pressure in a horizontal (gravity-free) system. The proportionality constant in Eq. (4.5.1) is permeability. If we rearrange Eq. (4.5.1) and perform a dimensional analysis, we see that permeability has dimensions of area (L2), where L is a unit of length: K¼
rate viscosity length ¼ area pressure
L3 time
force time L L2
¼ L2 force L2 L2
ð4:5:2Þ
The areal unit (L2) is physically related to the cross-sectional area of pore throats in rock. A pore throat is the opening that connects two pores. The size of a pore throat depends on grain size and distribution. For a given grain distribution, the cross-sectional area of a pore throat will increase as grain size increases. Relatively large pore throats imply relatively large values of L2 and correspond to relatively large values of permeability. Permeability typically ranges from 1 md (1.0 1015 m2) to 1 Darcy (1,000 md or 1.0 1012 m2) for commercially successful oil and gas fields. Permeability can be much less than 1 md in unconventional reservoirs such as tight gas and shale gas reservoirs. Advances in well stimulation technology and increases in oil and gas prices have improved the economics of low-permeability reservoirs. Area Flow q
Core
Length Δx
Figure 4.2 Darcy’s law.
Porosity and Permeability
53
Darcy’s law shows that flow rate and pressure difference are linearly related. The pressure gradient from the point of fluid injection to the point of fluid withdrawal is found by rearranging Eq. (4.5.1):
DP q m ¼ Dx 0:001127A K
4.5.1
ð4:5:3Þ
Superficial Velocity and Interstitial Velocity
Superficial velocity is the volumetric flow rate q in Darcy’s law divided by the cross-sectional area A⊥ normal to flow (Bear, 1972; Lake, 1989); thus u ¼ q=A⊥ in appropriate units. The interstitial, or “front,” velocity v of the fluid through the porous rock is the actual velocity of a fluid element as the fluid moves through the tortuous pore space. Interstitial velocity v is the superficial velocity u divided by porosity f, or v¼
u q ¼ f fA⊥
ð4:5:4Þ
Interstitial velocity is larger than superficial velocity since porosity is a fraction between 0 and 1.
4.5.2
The Validity of Darcy’s Law
Darcy’s law is valid when fluid flow is laminar. Laminar fluid flow represents one type of flow regime. Three types of flow regimes may be defined: laminar flow regime with low flow rate; inertial flow regime with moderate rate; and turbulent flow regime with high flow rate. Flow regimes are classified in terms of the dimensionless Reynolds number (Fancher and Lewis, 1933). The Reynolds number is the ratio of inertial (fluid momentum) forces to viscous forces. It has the form N Re ¼ 1488
rvD dg m
ð4:5:5Þ
where r vD dg m
¼ ¼ ¼ ¼
fluid density (lbm/ft3) superficial velocity (ft/sec) average grain diameter (ft) absolute viscosity (cp)
The flow regime is determined by calculating the Reynolds number. A Reynolds number that is low corresponds to laminar flow, and a high Reynolds number corresponds to turbulent flow. Govier (1978) presented the classification in Table 4.1 that expresses flow regime in terms of the Reynolds number.
54
Integrated Reservoir Asset Management
Table 4.1 Classification of Flow Regimes Flow Regime
Description
Laminar Inertial Turbulent
Low flow rates (NRe < 1) Moderate flow rates (1 < NRe < 600) High flow rates (NRe > 600)
The linear relationship between pressure gradient and rate in Darcy’s law is valid for many flow systems, but not all. Fluid flow in a porous medium can have a nonlinear effect that is represented by the Forchheimer equation (Govier, 1978). Forchheimer observed that turbulent flow in high-flow-rate gas wells has the quadratic dependence
2 DP q m q ¼ þ br Dx 0:001127A K A
ð4:5:6Þ
for fluid with density r and turbulence factor b. A minus sign and conversion unit is inserted in the first-order rate term on the right-hand side of Eq. (4.5.6) to be consistent with the rate convention used in Eq. (4.5.1). Equation (4.5.6) is called the Forchheimer equation. The nonlinear effect becomes more important in high-flowrate environments such as some gas wells and in hydraulic fracturing (Barree and Conway, 2005). Darcy’s law correctly describes laminar flow and may be used as an approximation of turbulent flow. Permeability calculated from Darcy’s law is less than true rock permeability at turbulent flow rates.
4.5.3
Radial Flow of Liquids
Darcy’s law for steady-state, radial, horizontal, single-phase liquid flow in a porous medium is Q¼
0:00708KhðPw Pe Þ mBlnðre =rw Þ
where Q ¼ liquid flow rate (STB/D) rw ¼ wellbore or inner radius (ft) re ¼ outer radius (ft) K ¼ permeability (md) h ¼ formation thickness (ft) Pw ¼ pressure at inner radius (psi) Pe ¼ pressure at outer radius (psi) m ¼ viscosity (cp) B ¼ formation volume factor (RB/STB)
ð4:5:7Þ
Porosity and Permeability
55
The formation volume factor that is in the denominator on the right-hand side of Eq. (4.5.7) converts volumetric flow rate from reservoir to surface conditions. The rate Q is positive for a production well {Pw < Pe} and negative for an injection well {Pw > Pe}. Different procedures may be used to estimate the outer radius re. The outer radius re is equated to the drainage radius of the well when analyzing the pressure at a well. Alternatively, the value of re in a reservoir flow model depends on the size of the grid block containing the flow rate term (Peaceman, 1978; Fanchi, 2006a). The flow rate is less sensitive to an error in the estimate of re than a similar error in a parameter like permeability because the radial flow calculation depends on the logarithm of re. It is therefore possible to tolerate larger errors in re than other flow parameters and still obtain a reasonable value for radial flow rate.
4.5.4
Radial Flow of Gases
Consider Darcy’s law in radial coordinates for a single phase: qr ¼ 0:006328
2prhK dPr m dr
ð4:5:8Þ
where the radial distance r increases as we move away from the well, and qr r h m K Pr
¼ ¼ ¼ ¼ ¼ ¼
gas rate (rcf/d) radial distance (ft) zone thickness (ft) gas viscosity (cp) permeability (md) reservoir pressure (psia)
The cross-sectional area in Eq. (4.5.8) is the cross-sectional area 2prh of a cylinder enclosing the wellbore. Let subscripts r and s denote reservoir and surface conditions, respectively. To convert from reservoir to surface conditions, we divide the gas rate at reservoir conditions by the gas formation volume factor; thus qs ¼
qr Bg
ð4:5:9Þ
where qs ¼ gas rate (scf/d) Bg ¼ gas formation volume factor (rcf/scf)
Gas formation volume factor Bg is a function of pressure P, temperature T, and gas compressibility factor Z from the real gas equation of state: Bb ¼
Ps Tr Zr Pr Ts Zs
ð4:5:10Þ
56
Integrated Reservoir Asset Management
The rate at surface conditions is found by substituting Eqs. (4.5.8) and (4.5.10) into (4.5.9) to get qs ¼ 0:03976
rhK Pr Ts Zs dPr m Ps Tr Zr dr
ð4:5:11Þ
and qs has units of scf/d. If we assume a constant rate, we can rearrange Eq. (4.5.11) and integrate from the inner radius to the outer radius to get ðre qs rw
dr re KhTs Zs ¼ qs ln ¼ 0:03976 rw Ps Tr r
P ðe
Pr dPr mZr
ð4:5:12Þ
Pw
Subscripts w and e denote values at the wellbore radius and external radius, respectively. Equation (4.5.12) can be written in a simpler form by introducing the real gas pseudopressure m(P): ðP mðPÞ ¼ 2 Pref
0
P 0 dP mg Z
ð4:5:13Þ
where Pref ¼ a reference pressure P0 ¼ a dummy variable of integration
The integrand of Eq. (4.5.13) has a nonlinear dependence on pressure. It is often necessary to solve the integral numerically because of the nonlinear dependence of gas viscosity and gas compressibility on pressure. Given m(P), the radial form of Darcy’s law becomes qs ¼ 0:01988
KhTs Zs
½mðPe Þ mðPw Þ re Ps Tr ln rw
ð4:5:14Þ
Specifying the standard conditions Zs ¼ 1 Ps ¼ 14.7 psia Ts ¼ 60 F ¼ 520 R
in Eq. (4.5.14) gives Darcy’s law for the radial flow of gas: qs ¼ 0:703
Kh
½mðPe Þ mðPw Þ re Tr ln rw
ð4:5:15Þ
Porosity and Permeability
57
Solving Eq. (4.5.15) for real gas pseudopressure at the external radius gives 1:422Tr mðPe Þ ¼ mðPw Þ
re ln rw
Kh
qs
ð4:5:16Þ
Equation (4.5.16) shows that m(Pe) is proportional to qs and inversely proportional to permeability.
4.5.5
Klinkenberg’s Effect
Klinkenberg found that the permeability for gas flow in a porous medium depends on pressure according to the relationship
b kg ¼ kabs 1 þ P
ð4:5:17Þ
where kg kabs P b
¼ ¼ ¼ ¼
apparent permeability calculated from gas flow tests true absolute permeability of rock mean flowing pressure of gas in the flow system Klinkenberg’s effect
The factor b is a positive constant for gas in a specific porous medium. Equation 1, we have the inequality kg kabs. (4.5.17) shows that when the factor ð1 þ b=PÞ approaches 1 and the apparent permeAs pressure increases, the factor ð1 þ b=PÞ ability to gas kg approaches the true absolute permeability of the rock kabs. The dependence of kg on pressure is called Klinkenberg’s effect and is attributed to the “slippage” of gas molecules along pore walls. The interaction between gas molecules and pore walls is greater at low pressures than at high pressures. Conversely, slippage along pore walls is greater at high pressures than at low pressures. At low pressures, the calculated permeability for gas flow kg may be greater than true rock permeability. Measurements of kg are often conducted with air and are not corrected for Klinkenberg’s effect. This should be considered when comparing kg with permeability estimates from other sources such as well tests.
4.5.6
Properties of Permeability
A micro scale measurement of grain size distribution shows that different grain sizes and shapes affect permeability. Permeability may be viewed as a mathematical convenience for describing the statistical behavior of a given flow experiment. In this context, transient testing gives the best measure of permeability over a large volume. Despite their importance to the calculation of flow, permeability and its distribution will not be known accurately. Seismic data can help define the distribution of
58
Integrated Reservoir Asset Management
permeability between wells if a good correlation exists between seismic amplitude and a rock quality measurement that includes permeability. Permeability depends on rock type. The two most common reservoir rock types are clastic reservoirs and carbonate reservoirs. The permeability in a clastic reservoir depends on pore size, which is seldom controlled by secondary solution vugs. Compacted and cemented sandstone rocks tend to have lower permeabilities than clean, unconsolidated sands. Productive sandstone reservoirs usually have permeabilities in the range of 10 to 1,000 md. The permeability in tight gas and coalbed methane reservoirs is less than 1 md. Carbonate reservoirs are generally less homogeneous than clastic reservoirs and have a wider range of grain size distribution. The typical matrix permeability in a carbonate reservoir tends to be relatively low. Significant permeability in a carbonate reservoir may be associated with secondary porosity features such as vugs and oolites. The presence of clay can adversely affect permeability. Clay material may swell on contact with fresh water, and the resulting swelling can reduce rock permeability by several orders of magnitude. Natural or manmade fractures can significantly increase flow capacity in both carbonate and clastic reservoirs. An extensive natural fracture system can provide high-flow-capacity conduits for channeling flow from the reservoir matrix to a wellbore. Naturally fractured reservoirs are usually characterized by relatively high-permeability, low-porosity fractures, and by a relatively low-permeability, high-porosity matrix. Most of the fluid is stored in the matrix, while flow from the reservoir to the wellbore is controlled by the permeability in the fracture system.
4.6
Permeability Averaging
Permeability averaging poses a problem in the estimation of a representative average permeability for use in Darcy’s equation. Permeability can be obtained from core plugs and well tests. Core plug permeability and well test permeability are measurements of permeability at different scales. Several techniques exist for estimating an average value of permeability. Some practical averaging techniques are presented below.
4.6.1
Parallel Beds—Linear Flow
Linear flow through parallel beds of differing permeability is shown in Figure 4.3. Pressure is constant at each end of the flow system, and total flow rate is the sum of the rates qi in each layer i: q¼
X i
qi
ð4:6:1Þ
Porosity and Permeability
59
q1 q2
~
~
qn
Figure 4.3 Beds in parallel.
Suppose layer i has length L, width w, net thickness hi, and permeability ki. Applying Darcy’s law for linear flow of a fluid with viscosity m gives kave ht wðp1 p2 Þ X ki hi wðp1 p2 Þ ¼ mL mL i
ð4:6:2Þ
where the sum is over all layers. After canceling common terms, we obtain the expression kave ht ¼
X
k i hi
ð4:6:3Þ
i
where ht ¼ total thickness kave ¼ an average permeability
Solving for kave gives X k i hi i X kave ¼ hi
ð4:6:4Þ
i
The average permeability for parallel flow through beds of differing permeabilities equals the thickness weighted average permeability. If each bed has the same thickness, kave is the arithmetic average.
4.6.2
Parallel Beds—Radial Flow
The average permeability for radial flow in parallel beds is the same relationship as linear flow—namely, the thickness weighted average
kave
X k i hi i ¼ X hi i
ð4:6:5Þ
60
Integrated Reservoir Asset Management
Q
Bed 1
Bed 2
Bed 3
Porous Medium
Figure 4.4 Flow in beds in series.
4.6.3
Beds in Series—Linear Flow
Figure 4.4 illustrates beds in series. The average permeability for beds in series is the harmonic average: X
kave
Li i X ¼ Li =ki
ð4:6:6Þ
i
where bed i has length Li and permeability ki.
4.6.4
Beds in Series—Radial Flow
Radial flow in beds in series treats beds as concentric rings around the wellbore. For a system with three beds, the average permeability for radial flow in beds in series is the harmonic average: kave ¼
lnðre =r w Þ lnðr e =r 2 Þ lnðr 2 =r 1 Þ lnðr 1 =r w Þ þ þ k3 k2 k1
ð4:6:7Þ
where rw ¼ the radius of the well re ¼ the radius to the outer ring
The radius re corresponds to the drainage radius of the system.
4.6.5
Random Flow
For permeability values distributed randomly, the average permeability is a geometric average: kave ¼ k1h1 k2h2 k3h3 . . . knhn
n P
1
hi i¼1
ð4:6:8Þ
Porosity and Permeability
61
where hi ¼ the thickness of interval i with permeability ki n ¼ the number of intervals
4.6.6
Permeability Averaging in a Layered Reservoir
The average permeability for a layered reservoir can be estimated using the following procedure: l
l
Determine the geometric average for each layer. Determine the arithmetic average of the geometric averages, weighted by the thickness of each layer.
Several other procedures exist for determining the average permeability of a layered reservoir. One method that can be applied with relative ease is to perform a flow model study using two models. One model is a cross-section model with all of the geologic layers treated as model layers. The other model is a single-layer model with all the geologic layers combined into a single layer. Flow performance from the cross-section model is compared with the flow performance of the single-layer model. The permeability in the single-layer model is adjusted until the performance of the single-layer model is approximately equal to the performance of the crosssection model. The resulting permeability is an “upscaled,” or average, permeability for the cross-section model.
4.7
Transmissibility
Flow between neighboring blocks is treated as a series application of Darcy’s law. We are concerned with the movement of fluids between two blocks such as those shown in Figure 4.5. If we assume that the conditions needed for Darcy flow are satisfied and simplify the problem by considering single-phase flow for a phase with formation volume factor B‘ , we have 1 Pi1 Pi Q ‘ ¼ KA c m‘ B‘ Dxi1 þ Dxi 2
ð4:7:1Þ
where Q ‘ ¼ the average volumetric flow rate of phase ‘ K ¼ the absolute permeability associated with a pressure drop from xi1 to xi Ac ¼ the cross-sectional area between xi1 and xi
To use Eq. (4.7.1), it is necessary to express the product KA c in terms of known variables—namely, xj, Kj, and Acj, where subscript j refers to blocks i1 and i.
62
Integrated Reservoir Asset Management
Pi -1 Pf Pi Ql x-axis Δxi−1
Δxi
Figure 4.5 Flow between blocks.
We begin by using Darcy’s law to write the flow rates through each shaded volume element. The flow rate of phase ‘ is 1 Pi1 Pf Q‘ ¼ Ki1 Ac, i1 m‘ B‘ Dxi1 2
ð4:7:2Þ
for block i1 and 1 P f Pi Q‘ ¼ Ki Ac, i m‘ B‘ Dxi 2
ð4:7:3Þ
for block i. The pressure Pfrefers to the pressure at the interface between block i1 and block i. Solving for Pi1 Pf and Pf Pi in Eqs. (4.7.2) and (4.7.3), respectively, and then adding the results yields 2
3 Dxi1 1 Dxi 1 6 2 m‘ B‘ 2 m‘ B‘ 7 7 Pi1 Pi ¼ Q‘ 6 4Ki1 Ac, i1 þ Ki Ac, i 5
ð4:7:4Þ
Inserting this expression into Eq. (4.7.1) and solving for KA c gives Dxi1 þ Dxi
Dxi1 Dxi þ Ki1 Ac, i1 Ki Ac, i
KA c ¼
ð4:7:5Þ
Substituting this expression back into Eq. (4.7.1) gives, after some simplification, the flow rate equation
1 2ðKAc Þi1 ðKAc Þi ðPi1 Pi Þ Q‘ ¼ m‘ B‘ Dxi1 ðKAc Þi þ Dxi ðKAc Þi1
ð4:7:6Þ
Porosity and Permeability
63
Equation (4.7.6) has the form 0 Q ‘ ¼ A‘, i1=2 ðPi1 Pi Þ
ð4:7:7Þ
0
where A‘, i1=2 is the Darcy phase transmissibility between blocks i1 and i. Notice that transmissibility depends on properties in both blocks i1 and i. A similar procedure is used to obtain transmissibility values for the y and z directions. Transmissibilities may be used to define sealing or partially sealing faults and to define high-permeability channels.
4.8
Measures of Permeability Heterogeneity
It is often useful to represent permeability heterogeneity with a single number. This number is referred to here as a measure of permeability heterogeneity. Several such measures exist (Lake and Jensen, 1989). The Dykstra-Parsons coefficient and the Lorenz coefficient are described in this section as illustrations. The procedure outlined in the following makes some simplifying assumptions that are not too restrictive in practice but make it possible to calculate permeability heterogeneity measures with relative ease. These measures may be used to verify that the permeability distribution used in a model has comparable heterogeneity to the permeability distribution observed in an analysis of field data.
4.8.1
The Dykstra-Parsons Coefficient
The Dykstra-Parsons coefficient can be estimated for a log normal permeability distribution as h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii VDP ¼ 1 exp lnðkA =kH Þ
ð4:8:1Þ
where kA is the arithmetic average kA ¼
n 1X ki n i¼1
ð4:8:2Þ
and kH is the harmonic average n 1 1X 1 ¼ kH n i¼1 ki
ð4:8:3Þ
The Dykstra-Parsons coefficient should be in the range 0 VDP 1. For a homogeneous reservoir, VDP ¼ 0 because kA ¼ kH . An increase in heterogeneity increases VDP. Typical values of the Dykstra-Parsons coefficient are in the range 0.4 VDP 0.9.
64
Integrated Reservoir Asset Management
As an example of a Dykstra-Parsons coefficient calculation, suppose we have a three-layer case with the following permeabilities: Layer 1: k ¼ 35 md Layer 2: k ¼ 48 md Layer 3: k ¼ 126 md
The arithmetic average is kA ¼
n 1X 1 ki ¼ ð126 þ 48 þ 35Þ ¼ 69:7 n i¼1 3
and the harmonic average is
n 1 1X 1 1 1 1 1 þ þ ¼ ¼ kH n i¼1 ki 3 126 48 35 or kH ¼ 52.3 md. Using these average values, we estimate the Dykstra-Parsons coefficient for a log normal permeability distribution to be h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii VDP ¼ 1 exp lnð69:7=52:3Þ ¼ 0:415
4.8.2
The Lorenz Coefficient
The Lorenz coefficient requires a bit more work than the Dykstra-Parsons coefficient. We begin by calculating the following quantities: X Xm n k h k h ; m ¼ 1, . . . , n ð4:8:4Þ Fm ¼ cum flow capacity ¼ i¼1 i i i¼1 i i and Hm ¼ cum thickness ¼
Xm
h i¼1 i
X
n
h; i¼1 i
m ¼ 1, . . . , n
ð4:8:5Þ
for n ¼ number of reservoir layers. Layers should be arranged in order of decreasing permeability; thus i ¼ 1 has thickness h1 and the largest permeability k1, while i ¼ n has thickness hn and the smallest permeability kn. By definition, cumulative flow capacity should be in the range 0 Fm 1, and cumulative thickness should be in the range 0 Hm 1 for 0 < m < n. The Lorenz coefficient is defined in terms of a plot of Fm versus Hm, shown in Figure 4.6. The Lorenz coefficient Lc is two times the area enclosed between the Lorenz curve ABC in the figure and the diagonal AC. The range of the Lorenz coefficient is 0 Lc 1. For a homogeneous reservoir, the Lorenz coefficient satisfies the equality Lc ¼ 0. An increase in heterogeneity increases the value of the Lorenz coefficient Lc. Typical values of the Lorenz coefficient are in the range 0:2 Lc 0:6.
Porosity and Permeability
65
C B Fm
A Hm
Figure 4.6 The Lorenz plot.
An estimate of the Lorenz coefficient is obtained by assuming that all of the permeabilities have equal probability so that the trapezoidal rule can be used to estimate area. The result is Lc ¼
Xn i 1 hXn Xn jk k j = i¼1 ki i j i¼1 j¼1 2n
ð4:8:6Þ
It is not necessary to order permeabilities using this estimate. Notice that in the homogeneous case, all of the permeabilities are equal so that we have the relationship ki ¼ kj. Substituting this equality into Eq. (4.8.6) gives the Lorenz coefficient Lc ¼ 0. In the ideal homogeneous case, the Lorenz coefficient is zero, as just indicated.
4.9
Darcy’s Law with Directional Permeability
Permeability can be a complex function of spatial location and orientation. Spatial and directional variations of a function are described in terms of homogeneity, heterogeneity, isotropy, and anisotropy. If the value of a function does not depend on spatial location, it is called homogeneous. The function is heterogeneous if its value changes from one spatial location to another. If the value of a function depends on directional orientation—that is, the value is larger in one direction than another— then the function is anisotropic. The function is isotropic if its value does not depend on directional orientation. Permeability is a function that can be both heterogeneous and anisotropic. To account for heterogeneity and anisotropy, the simple 1-D form of Darcy’s law must be generalized. In general, flow occurs in dipping beds. To account for the effect of gravity, we define a variable called the potential of phase ‘ as F‘ ¼ P‘ g‘ ðDzÞ where Dz ¼ depth from a datum P‘ ¼ pressure of phase ‘ g‘ ¼ pressure gradient associated with the gravity term
ð4:9:1Þ
66
Integrated Reservoir Asset Management
If we write Darcy’s law for single-phase flow in the form q¼
0:001127KA dF m dz
ð4:9:2Þ
we find that no vertical movement can occur when dF/dz ¼ 0. Thus, Eq. (4.9.2) expresses the movement of fluids in a form that accounts for gravity equilibrium. Darcy’s law in one dimension says that rate is proportional to pressure gradient. This can be extended to three dimensions using vector notation. Darcy’s law for single-phase flow in three dimensions is qx ¼ 0:001127K
A qF m qx
qy ¼ 0:001127K
A qF m qy
qz ¼ 0:001127K
A qF m qz
ð4:9:3Þ
where the gradient of potential accounts for gravity effects. In vector notation we have A q ¼ 0:001127K rF m
!
ð4:9:4Þ
Equation (4.9.3) can be written in matrix notation as 2
3 2 3 qx qF=qx 4 qy 5 ¼ 0:001127K A 4 qF=qy 5 m qz qF=qz
ð4:9:5Þ
where permeability K and cross-sectional area A are treated as constants with respect to direction. A more general extension of Eq. (4.9.5) is 2
3 2 Kxx qx 4 qy 5 ¼ 0:001127 A 4 Kyx m K qz zx
Kxy Kyy Kzy
32 3 Kxz qF=qx Kyz 54 qF=qy 5 Kzz qF=qz
ð4:9:6Þ
where permeability is now treated either as a 33 matrix with nine elements or as a tensor of rank two (Fanchi, 2006c). The diagonal permeability elements {Kxx, Kyy, Kzz} represent the usual dependence of rate in one direction on pressure differences in the same direction. The off-diagonal permeability elements {Kxy, Kxz, Kyx, Kyz, Kzx, Kzy} account for the dependence of rate in one direction on pressure differences in orthogonal directions. Expanding Eq. (4.9.6) gives the corresponding set of equations (Eq. (4.9.7)) that demonstrates this dependence:
Porosity and Permeability
67
2 3 A4 qF qF qF5 Kxx þ Kxy þ Kxz qx ¼ 0:001127 m qx qy qz 2 3 A4 qF qF qF5 qy ¼ 0:001127 Kyx þ Kyy þ Kyz m qx qy qz
ð4:9:7Þ
2 3 A 4 qF qF qF5 Kzx þ Kzy þ Kzz qz ¼ 0:001127 m qx qy qz In many practical situations it is mathematically possible to find a coordinate system {x0 , y0 , z0 } in which the permeability tensor has the diagonal form 2
Kx0 x0 4 0 0
0 Ky0 y0 0
3 0 0 5 Kz 0 z 0
The coordinate axes {x0 , y0 , z0 } are called the principal axes of the tensor, and the diagonal form of the permeability tensor is obtained by a principal axis transformation. The flow equations along the principal axes are 2 3 A4 qF5 Kx0 x0 0 qx0 ¼ 0:001127 m qx 2
3
A qF qy0 ¼ 0:001127 4Ky0 y0 0 5 m qy
ð4:9:8Þ
2 3 A4 qF5 Kz0 z0 0 qz0 ¼ 0:001127 m qz The principal axes in a field can vary from one point of the field to another because of permeability heterogeneity. The form of the permeability tensor depends on the properties of the porous medium. The medium is said to be anisotropic if two or more elements of the diagonalized permeability tensor are different. The permeability of the medium is isotropic if the elements of the diagonalized permeability tensor are equal—that is, Kx0 x0 ¼ Ky0 y0 ¼ Kz0 z0 ¼ K
ð4:9:9Þ
If the medium is isotropic, permeability does not depend on direction. If the isotropic permeability does not change from one position in the medium to another, the
68
Integrated Reservoir Asset Management
(a)
(b)
Figure 4.7 The effect of permeability anisotropy on a drainage area: (a) isotropic (Kx ¼ Ky) and (b) anisotropic (Kx 6¼ Ky).
medium is said to be homogeneous in permeability. On the other hand, if the values of the elements of the permeability tensor vary from one point in the medium to another, both the permeability tensor and the medium are considered heterogeneous. Virtually all reservoirs exhibit some degree of anisotropy and heterogeneity, but the flow behavior in many reservoirs can be approximated as homogeneous and isotropic. In Figure 4.7, (a) is the drainage area of four production wells with isotropic permeability, and (b) is the drainage area of four production wells with anisotropic permeability.
4.9.1
Vertical Permeability
Permeability for flow in a direction that is perpendicular to gravity is horizontal permeability. By contrast, vertical permeability is the permeability for flow in the direction aligned with the direction of the gravitational field. Vertical permeability can be measured in the laboratory or in pressure transient tests conducted in the field. In many cases vertical permeability is not measured and must be assumed. A rule of thumb is to assume vertical permeability to be approximately one-tenth of horizontal permeability. These are reasonable assumptions when there is no data to the contrary. It is preferable from a technical point of view to make direct measurements of all relevant reservoir data. Sometimes it to be difficult to justify the cost or logistics of obtaining direct measurements. If it is necessary to use a rule of thumb or data from an analogous formation to estimate a particular variable, the sensitivity of the shared earth model to changes in the estimated variable should be considered.
CS.4 Valley Fill Case Study: Permeability The permeability distribution in the Valley Fill reservoir is assumed to be isotropic and homogeneous because the spatial dependence of permeability is not known. Horizontal permeability is 150 md based on a pressure transient test. Vertical permeability was not measured and is therefore assumed to be one-tenth of horizontal permeability.
Porosity and Permeability
69
Exercises 4-1. Suppose gross thickness includes 4 feet of impermeable shale and 16 feet of permeable sandstone. What are the gross thickness, net thickness, and net-to-gross ratio? 4-2. Consider a three-layer reservoir with the following permeabilities: Layer 1: k ¼ 35 md Layer 2: k ¼ 48 md Layer 3: k ¼ 126 md Calculate the arithmetic average, the harmonic average, and the Dykstra-Parsons coefficient for a log normal permeability distribution. 4-3. Consider a three-layer reservoir with the following permeabilities: Layer 1: k ¼ 35 md Layer 2: k ¼ 48 md Layer 3: k ¼ 126 md Estimate the Lorenz coefficient using Eq. (4.8.6). 4-4A. Consider a linear flow system with area ¼ 25 ft2. End point A is 5 feet higher than end point B, and the distance between end points is 15 feet. Suppose the system contains oil with viscosity ¼ 0.8 cp, gravity ¼ 35 API (go ¼ 0.85), and FVF ¼ 1.0 RB/ STB. If the end point pressures are PA ¼ PB ¼ 20 psia, is there flow? If so, how much flow is there and in what direction? Use Darcy’s law with the gravity term and dip angle a:
q ¼ 0:001127
kA PA PB þ rg sina L mB
4-4B. Using the data in Exercise 4-4A, calculate the pressure PB that would prevent fluid flow. 4-5A. Suppose the pressure P1 of a water-bearing formation at depth z1 ¼ 10,000 feet is 4,000 psia. If the pressure gradient for water is 0.433 psia/ft, calculate the pressure P2 at depth z2 ¼ 11,000 feet. Calculate the phase potentials F1 and F2 at depths z1 and z2, respectively. 4-5B. Use the pressures and potentials in Exercise 4-5A to estimate the derivatives dP/dz and dF/dz. Will there be vertical flow? 4-6. Estimate the volumetric flow rate using Darcy’s law that is given by Eq. (4.5.1). Assume the permeability in a cylindrical core is 100 md, the length of the core is 6 inches, the diameter of the core is 3 inches, the pressure drop across the core is 10 psi, and the viscosity of liquid passing through the core is 2 cp. Express volumetric flow rate in bbl/day or cc/s. 4-7. A barrier island is a large sand body. Consider a barrier island that averages 3 miles wide, 10 miles long, and 30 feet thick. The porosity of the sand averages almost 25 percent. What is the pore volume of the barrier island? Express your answer in bbl and m3.
4.1
Bulk Volume and Net Volume
The bulk volume of a cell that is shaped like a parallelepiped, shown in Figure 4.1, is given by the triple scalar product ! ! ! VB ¼ U ðV W Þ !
!
ð4:1:1Þ
!
The vectors fU , V , W g are aligned along the axes of the parallelepiped and have magnitudes that are equal to the lengths of the sides of the parallelepiped. In Cartesian coordinates, we have the result VB ¼ Lx i^ Ly j^ Lz k^ ¼ Lx Ly Lz
ð4:1:2Þ
in terms of the lengths Lx , Ly , Lz of the sides of the Cartesian cell. The vectors ^ j, ^ k^ are unit vectors in Cartesian coordinates. i, Bulk volume VB of a Cartesian cell is the product of cross-sectional area A times gross thickness H: VB ¼ AH
ð4:1:3Þ
The area of a cell is the product of the x-direction cell length Dx and the y-direction cell length Dy; thus A ¼ Dx Dy. Gross thickness H is the cell length in the z-direction. Bulk volume is a measure of the gross volume in the system. It includes both rock volume and pore volume. To determine the volume of the system that is commercially significant, the gross volume must be adjusted by introducing the concept
Integrated Reservoir Asset Management. DOI: 10.1016/B978-0-12-382088-4.00004-9 Copyright # 2010 Elsevier Inc. All rights reserved.
50
Integrated Reservoir Asset Management
U V
W
Figure 4.1 Volume of a parallelepiped.
of net thickness. Net thickness h is the thickness of the commercially significant formation. The net-to-gross ratio Z is the ratio of net thickness h to gross thickness H: Z ¼ h=H, 0 Z 1
ð4:1:4Þ
Net thickness is always less than or equal to gross thickness.
4.2
Porosity and Grain Volume
Porosity is the fraction of a porous medium that is void space. The bulk volume VB of a porous medium is the sum of pore volume VP and grain volume VG; thus VB ¼ VP þ V G
ð4:2:1Þ
Porosity is the ratio of pore volume to bulk volume: f ¼ VP =VB
ð4:2:2Þ
Dividing Eq. (4.2.1) by VB and using the definition of porosity expresses the grain volume in terms of porosity as VG ¼1f VB
4.3
ð4:2:3Þ
Effective Pore Volume
Most porous media contain a fraction of pores that are not in communication with the flow path. This pore volume is ineffective. Effective pore volume ðVP Þeff is the interconnected pore volume that communicates with a well. Effective porosity is defined as the ratio of effective pore volume to bulk volume; thus feff
ðVP Þeff VB
ð4:3:1Þ
Porosity and Permeability
51
Unless stated otherwise, any further discussion using porosity will assume that the porosity of interest is effective porosity, and the subscript “eff ” is not written.
4.4
Porosity Compressibility
Porosity compressibility is a measure of the change in porosity f as a function of fluid pressure P. It is defined as cf ¼
1 df f dP
ð4:4:1Þ
If f0 is porosity at pressure P0 and f is porosity at pressure P, the integral of Eq. (4.4.1) yields the relationship 2
3
ðP
6 7 f ¼ f0 exp4 cf dP5
ð4:4:2Þ
P0
If porosity compressibility is constant with respect to pressure, the integral in Eq. (4.4.2) can be evaluated and gives f ¼ f0 exp cf DP
ð4:4:3Þ
where DP ¼ P – P0. The first-order approximation to Eq. (4.4.3) is f f0 1 þ cf DP ¼ f0 1 þ cf ðP P0 Þ
ð4:4:4Þ
Equation (4.4.4) is used in many reservoir flow simulators to calculate the change in porosity with respect to changes in fluid pressure. A first-order approximation is reasonable in many cases because reservoir rock compressibility is typically on the order of cf ¼ 3 106 =psia and a typical pressure change is on the order of P P0 1000 psia, so the bracketed term in Eq. (4.4.4) is on the order of 1.003. There are reservoirs where the approximation is not as valid because the product of rock compressibility and pressure change can be an order of magnitude larger.
4.5
Darcy’s Law and Permeability
The basic equation that describes fluid flow in porous media is Darcy’s law. Darcy found that flow rate was proportional to pressure gradient. Darcy’s equation for calculating volumetric flow rate q for linear, horizontal, single-phase flow is q ¼ 0:001127
KA DP m Dx
ð4:5:1Þ
52
Integrated Reservoir Asset Management
The units of the physical variables determine the value of the constant (0.001127) in Eq. (4.5.1). The constant 0.001127 corresponds to variables expressed in the following oil field units: q K A P m Dx
¼ volumetric flow rate (bbl/day) ¼ permeability (md) ¼ cross-sectional area (ft2) ¼ pressure (psi) ¼ fluid viscosity (cp) ¼ length (ft)
Figure 4.2 illustrates the terms in Darcy’s law for a cylindrical core of rock. The movement of a single-phase fluid through a porous medium depends on crosssectional area A that is normal to the direction of fluid flow, pressure difference DP across the length Dx of the flow path, and viscosity m of the flowing fluid. The minus sign indicates that the direction of fluid flow is opposite to the direction of increasing pressure: the fluid flows from high pressure to low pressure in a horizontal (gravity-free) system. The proportionality constant in Eq. (4.5.1) is permeability. If we rearrange Eq. (4.5.1) and perform a dimensional analysis, we see that permeability has dimensions of area (L2), where L is a unit of length: K¼
rate viscosity length ¼ area pressure
L3 time
force time L L2
¼ L2 force L2 L2
ð4:5:2Þ
The areal unit (L2) is physically related to the cross-sectional area of pore throats in rock. A pore throat is the opening that connects two pores. The size of a pore throat depends on grain size and distribution. For a given grain distribution, the cross-sectional area of a pore throat will increase as grain size increases. Relatively large pore throats imply relatively large values of L2 and correspond to relatively large values of permeability. Permeability typically ranges from 1 md (1.0 1015 m2) to 1 Darcy (1,000 md or 1.0 1012 m2) for commercially successful oil and gas fields. Permeability can be much less than 1 md in unconventional reservoirs such as tight gas and shale gas reservoirs. Advances in well stimulation technology and increases in oil and gas prices have improved the economics of low-permeability reservoirs. Area Flow q
Core
Length Δx
Figure 4.2 Darcy’s law.
Porosity and Permeability
53
Darcy’s law shows that flow rate and pressure difference are linearly related. The pressure gradient from the point of fluid injection to the point of fluid withdrawal is found by rearranging Eq. (4.5.1):
DP q m ¼ Dx 0:001127A K
4.5.1
ð4:5:3Þ
Superficial Velocity and Interstitial Velocity
Superficial velocity is the volumetric flow rate q in Darcy’s law divided by the cross-sectional area A⊥ normal to flow (Bear, 1972; Lake, 1989); thus u ¼ q=A⊥ in appropriate units. The interstitial, or “front,” velocity v of the fluid through the porous rock is the actual velocity of a fluid element as the fluid moves through the tortuous pore space. Interstitial velocity v is the superficial velocity u divided by porosity f, or v¼
u q ¼ f fA⊥
ð4:5:4Þ
Interstitial velocity is larger than superficial velocity since porosity is a fraction between 0 and 1.
4.5.2
The Validity of Darcy’s Law
Darcy’s law is valid when fluid flow is laminar. Laminar fluid flow represents one type of flow regime. Three types of flow regimes may be defined: laminar flow regime with low flow rate; inertial flow regime with moderate rate; and turbulent flow regime with high flow rate. Flow regimes are classified in terms of the dimensionless Reynolds number (Fancher and Lewis, 1933). The Reynolds number is the ratio of inertial (fluid momentum) forces to viscous forces. It has the form N Re ¼ 1488
rvD dg m
ð4:5:5Þ
where r vD dg m
¼ ¼ ¼ ¼
fluid density (lbm/ft3) superficial velocity (ft/sec) average grain diameter (ft) absolute viscosity (cp)
The flow regime is determined by calculating the Reynolds number. A Reynolds number that is low corresponds to laminar flow, and a high Reynolds number corresponds to turbulent flow. Govier (1978) presented the classification in Table 4.1 that expresses flow regime in terms of the Reynolds number.
54
Integrated Reservoir Asset Management
Table 4.1 Classification of Flow Regimes Flow Regime
Description
Laminar Inertial Turbulent
Low flow rates (NRe < 1) Moderate flow rates (1 < NRe < 600) High flow rates (NRe > 600)
The linear relationship between pressure gradient and rate in Darcy’s law is valid for many flow systems, but not all. Fluid flow in a porous medium can have a nonlinear effect that is represented by the Forchheimer equation (Govier, 1978). Forchheimer observed that turbulent flow in high-flow-rate gas wells has the quadratic dependence
2 DP q m q ¼ þ br Dx 0:001127A K A
ð4:5:6Þ
for fluid with density r and turbulence factor b. A minus sign and conversion unit is inserted in the first-order rate term on the right-hand side of Eq. (4.5.6) to be consistent with the rate convention used in Eq. (4.5.1). Equation (4.5.6) is called the Forchheimer equation. The nonlinear effect becomes more important in high-flowrate environments such as some gas wells and in hydraulic fracturing (Barree and Conway, 2005). Darcy’s law correctly describes laminar flow and may be used as an approximation of turbulent flow. Permeability calculated from Darcy’s law is less than true rock permeability at turbulent flow rates.
4.5.3
Radial Flow of Liquids
Darcy’s law for steady-state, radial, horizontal, single-phase liquid flow in a porous medium is Q¼
0:00708KhðPw Pe Þ mBlnðre =rw Þ
where Q ¼ liquid flow rate (STB/D) rw ¼ wellbore or inner radius (ft) re ¼ outer radius (ft) K ¼ permeability (md) h ¼ formation thickness (ft) Pw ¼ pressure at inner radius (psi) Pe ¼ pressure at outer radius (psi) m ¼ viscosity (cp) B ¼ formation volume factor (RB/STB)
ð4:5:7Þ
Porosity and Permeability
55
The formation volume factor that is in the denominator on the right-hand side of Eq. (4.5.7) converts volumetric flow rate from reservoir to surface conditions. The rate Q is positive for a production well {Pw < Pe} and negative for an injection well {Pw > Pe}. Different procedures may be used to estimate the outer radius re. The outer radius re is equated to the drainage radius of the well when analyzing the pressure at a well. Alternatively, the value of re in a reservoir flow model depends on the size of the grid block containing the flow rate term (Peaceman, 1978; Fanchi, 2006a). The flow rate is less sensitive to an error in the estimate of re than a similar error in a parameter like permeability because the radial flow calculation depends on the logarithm of re. It is therefore possible to tolerate larger errors in re than other flow parameters and still obtain a reasonable value for radial flow rate.
4.5.4
Radial Flow of Gases
Consider Darcy’s law in radial coordinates for a single phase: qr ¼ 0:006328
2prhK dPr m dr
ð4:5:8Þ
where the radial distance r increases as we move away from the well, and qr r h m K Pr
¼ ¼ ¼ ¼ ¼ ¼
gas rate (rcf/d) radial distance (ft) zone thickness (ft) gas viscosity (cp) permeability (md) reservoir pressure (psia)
The cross-sectional area in Eq. (4.5.8) is the cross-sectional area 2prh of a cylinder enclosing the wellbore. Let subscripts r and s denote reservoir and surface conditions, respectively. To convert from reservoir to surface conditions, we divide the gas rate at reservoir conditions by the gas formation volume factor; thus qs ¼
qr Bg
ð4:5:9Þ
where qs ¼ gas rate (scf/d) Bg ¼ gas formation volume factor (rcf/scf)
Gas formation volume factor Bg is a function of pressure P, temperature T, and gas compressibility factor Z from the real gas equation of state: Bb ¼
Ps Tr Zr Pr Ts Zs
ð4:5:10Þ
56
Integrated Reservoir Asset Management
The rate at surface conditions is found by substituting Eqs. (4.5.8) and (4.5.10) into (4.5.9) to get qs ¼ 0:03976
rhK Pr Ts Zs dPr m Ps Tr Zr dr
ð4:5:11Þ
and qs has units of scf/d. If we assume a constant rate, we can rearrange Eq. (4.5.11) and integrate from the inner radius to the outer radius to get ðre qs rw
dr re KhTs Zs ¼ qs ln ¼ 0:03976 rw Ps Tr r
P ðe
Pr dPr mZr
ð4:5:12Þ
Pw
Subscripts w and e denote values at the wellbore radius and external radius, respectively. Equation (4.5.12) can be written in a simpler form by introducing the real gas pseudopressure m(P): ðP mðPÞ ¼ 2 Pref
0
P 0 dP mg Z
ð4:5:13Þ
where Pref ¼ a reference pressure P0 ¼ a dummy variable of integration
The integrand of Eq. (4.5.13) has a nonlinear dependence on pressure. It is often necessary to solve the integral numerically because of the nonlinear dependence of gas viscosity and gas compressibility on pressure. Given m(P), the radial form of Darcy’s law becomes qs ¼ 0:01988
KhTs Zs
½mðPe Þ mðPw Þ re Ps Tr ln rw
ð4:5:14Þ
Specifying the standard conditions Zs ¼ 1 Ps ¼ 14.7 psia Ts ¼ 60 F ¼ 520 R
in Eq. (4.5.14) gives Darcy’s law for the radial flow of gas: qs ¼ 0:703
Kh
½mðPe Þ mðPw Þ re Tr ln rw
ð4:5:15Þ
Porosity and Permeability
57
Solving Eq. (4.5.15) for real gas pseudopressure at the external radius gives 1:422Tr mðPe Þ ¼ mðPw Þ
re ln rw
Kh
qs
ð4:5:16Þ
Equation (4.5.16) shows that m(Pe) is proportional to qs and inversely proportional to permeability.
4.5.5
Klinkenberg’s Effect
Klinkenberg found that the permeability for gas flow in a porous medium depends on pressure according to the relationship
b kg ¼ kabs 1 þ P
ð4:5:17Þ
where kg kabs P b
¼ ¼ ¼ ¼
apparent permeability calculated from gas flow tests true absolute permeability of rock mean flowing pressure of gas in the flow system Klinkenberg’s effect
The factor b is a positive constant for gas in a specific porous medium. Equation 1, we have the inequality kg kabs. (4.5.17) shows that when the factor ð1 þ b=PÞ approaches 1 and the apparent permeAs pressure increases, the factor ð1 þ b=PÞ ability to gas kg approaches the true absolute permeability of the rock kabs. The dependence of kg on pressure is called Klinkenberg’s effect and is attributed to the “slippage” of gas molecules along pore walls. The interaction between gas molecules and pore walls is greater at low pressures than at high pressures. Conversely, slippage along pore walls is greater at high pressures than at low pressures. At low pressures, the calculated permeability for gas flow kg may be greater than true rock permeability. Measurements of kg are often conducted with air and are not corrected for Klinkenberg’s effect. This should be considered when comparing kg with permeability estimates from other sources such as well tests.
4.5.6
Properties of Permeability
A micro scale measurement of grain size distribution shows that different grain sizes and shapes affect permeability. Permeability may be viewed as a mathematical convenience for describing the statistical behavior of a given flow experiment. In this context, transient testing gives the best measure of permeability over a large volume. Despite their importance to the calculation of flow, permeability and its distribution will not be known accurately. Seismic data can help define the distribution of
58
Integrated Reservoir Asset Management
permeability between wells if a good correlation exists between seismic amplitude and a rock quality measurement that includes permeability. Permeability depends on rock type. The two most common reservoir rock types are clastic reservoirs and carbonate reservoirs. The permeability in a clastic reservoir depends on pore size, which is seldom controlled by secondary solution vugs. Compacted and cemented sandstone rocks tend to have lower permeabilities than clean, unconsolidated sands. Productive sandstone reservoirs usually have permeabilities in the range of 10 to 1,000 md. The permeability in tight gas and coalbed methane reservoirs is less than 1 md. Carbonate reservoirs are generally less homogeneous than clastic reservoirs and have a wider range of grain size distribution. The typical matrix permeability in a carbonate reservoir tends to be relatively low. Significant permeability in a carbonate reservoir may be associated with secondary porosity features such as vugs and oolites. The presence of clay can adversely affect permeability. Clay material may swell on contact with fresh water, and the resulting swelling can reduce rock permeability by several orders of magnitude. Natural or manmade fractures can significantly increase flow capacity in both carbonate and clastic reservoirs. An extensive natural fracture system can provide high-flow-capacity conduits for channeling flow from the reservoir matrix to a wellbore. Naturally fractured reservoirs are usually characterized by relatively high-permeability, low-porosity fractures, and by a relatively low-permeability, high-porosity matrix. Most of the fluid is stored in the matrix, while flow from the reservoir to the wellbore is controlled by the permeability in the fracture system.
4.6
Permeability Averaging
Permeability averaging poses a problem in the estimation of a representative average permeability for use in Darcy’s equation. Permeability can be obtained from core plugs and well tests. Core plug permeability and well test permeability are measurements of permeability at different scales. Several techniques exist for estimating an average value of permeability. Some practical averaging techniques are presented below.
4.6.1
Parallel Beds—Linear Flow
Linear flow through parallel beds of differing permeability is shown in Figure 4.3. Pressure is constant at each end of the flow system, and total flow rate is the sum of the rates qi in each layer i: q¼
X i
qi
ð4:6:1Þ
Porosity and Permeability
59
q1 q2
~
~
qn
Figure 4.3 Beds in parallel.
Suppose layer i has length L, width w, net thickness hi, and permeability ki. Applying Darcy’s law for linear flow of a fluid with viscosity m gives kave ht wðp1 p2 Þ X ki hi wðp1 p2 Þ ¼ mL mL i
ð4:6:2Þ
where the sum is over all layers. After canceling common terms, we obtain the expression kave ht ¼
X
k i hi
ð4:6:3Þ
i
where ht ¼ total thickness kave ¼ an average permeability
Solving for kave gives X k i hi i X kave ¼ hi
ð4:6:4Þ
i
The average permeability for parallel flow through beds of differing permeabilities equals the thickness weighted average permeability. If each bed has the same thickness, kave is the arithmetic average.
4.6.2
Parallel Beds—Radial Flow
The average permeability for radial flow in parallel beds is the same relationship as linear flow—namely, the thickness weighted average
kave
X k i hi i ¼ X hi i
ð4:6:5Þ
60
Integrated Reservoir Asset Management
Q
Bed 1
Bed 2
Bed 3
Porous Medium
Figure 4.4 Flow in beds in series.
4.6.3
Beds in Series—Linear Flow
Figure 4.4 illustrates beds in series. The average permeability for beds in series is the harmonic average: X
kave
Li i X ¼ Li =ki
ð4:6:6Þ
i
where bed i has length Li and permeability ki.
4.6.4
Beds in Series—Radial Flow
Radial flow in beds in series treats beds as concentric rings around the wellbore. For a system with three beds, the average permeability for radial flow in beds in series is the harmonic average: kave ¼
lnðre =r w Þ lnðr e =r 2 Þ lnðr 2 =r 1 Þ lnðr 1 =r w Þ þ þ k3 k2 k1
ð4:6:7Þ
where rw ¼ the radius of the well re ¼ the radius to the outer ring
The radius re corresponds to the drainage radius of the system.
4.6.5
Random Flow
For permeability values distributed randomly, the average permeability is a geometric average: kave ¼ k1h1 k2h2 k3h3 . . . knhn
n P
1
hi i¼1
ð4:6:8Þ
Porosity and Permeability
61
where hi ¼ the thickness of interval i with permeability ki n ¼ the number of intervals
4.6.6
Permeability Averaging in a Layered Reservoir
The average permeability for a layered reservoir can be estimated using the following procedure: l
l
Determine the geometric average for each layer. Determine the arithmetic average of the geometric averages, weighted by the thickness of each layer.
Several other procedures exist for determining the average permeability of a layered reservoir. One method that can be applied with relative ease is to perform a flow model study using two models. One model is a cross-section model with all of the geologic layers treated as model layers. The other model is a single-layer model with all the geologic layers combined into a single layer. Flow performance from the cross-section model is compared with the flow performance of the single-layer model. The permeability in the single-layer model is adjusted until the performance of the single-layer model is approximately equal to the performance of the crosssection model. The resulting permeability is an “upscaled,” or average, permeability for the cross-section model.
4.7
Transmissibility
Flow between neighboring blocks is treated as a series application of Darcy’s law. We are concerned with the movement of fluids between two blocks such as those shown in Figure 4.5. If we assume that the conditions needed for Darcy flow are satisfied and simplify the problem by considering single-phase flow for a phase with formation volume factor B‘ , we have 1 Pi1 Pi Q ‘ ¼ KA c m‘ B‘ Dxi1 þ Dxi 2
ð4:7:1Þ
where Q ‘ ¼ the average volumetric flow rate of phase ‘ K ¼ the absolute permeability associated with a pressure drop from xi1 to xi Ac ¼ the cross-sectional area between xi1 and xi
To use Eq. (4.7.1), it is necessary to express the product KA c in terms of known variables—namely, xj, Kj, and Acj, where subscript j refers to blocks i1 and i.
62
Integrated Reservoir Asset Management
Pi -1 Pf Pi Ql x-axis Δxi−1
Δxi
Figure 4.5 Flow between blocks.
We begin by using Darcy’s law to write the flow rates through each shaded volume element. The flow rate of phase ‘ is 1 Pi1 Pf Q‘ ¼ Ki1 Ac, i1 m‘ B‘ Dxi1 2
ð4:7:2Þ
for block i1 and 1 P f Pi Q‘ ¼ Ki Ac, i m‘ B‘ Dxi 2
ð4:7:3Þ
for block i. The pressure Pfrefers to the pressure at the interface between block i1 and block i. Solving for Pi1 Pf and Pf Pi in Eqs. (4.7.2) and (4.7.3), respectively, and then adding the results yields 2
3 Dxi1 1 Dxi 1 6 2 m‘ B‘ 2 m‘ B‘ 7 7 Pi1 Pi ¼ Q‘ 6 4Ki1 Ac, i1 þ Ki Ac, i 5
ð4:7:4Þ
Inserting this expression into Eq. (4.7.1) and solving for KA c gives Dxi1 þ Dxi
Dxi1 Dxi þ Ki1 Ac, i1 Ki Ac, i
KA c ¼
ð4:7:5Þ
Substituting this expression back into Eq. (4.7.1) gives, after some simplification, the flow rate equation
1 2ðKAc Þi1 ðKAc Þi ðPi1 Pi Þ Q‘ ¼ m‘ B‘ Dxi1 ðKAc Þi þ Dxi ðKAc Þi1
ð4:7:6Þ
Porosity and Permeability
63
Equation (4.7.6) has the form 0 Q ‘ ¼ A‘, i1=2 ðPi1 Pi Þ
ð4:7:7Þ
0
where A‘, i1=2 is the Darcy phase transmissibility between blocks i1 and i. Notice that transmissibility depends on properties in both blocks i1 and i. A similar procedure is used to obtain transmissibility values for the y and z directions. Transmissibilities may be used to define sealing or partially sealing faults and to define high-permeability channels.
4.8
Measures of Permeability Heterogeneity
It is often useful to represent permeability heterogeneity with a single number. This number is referred to here as a measure of permeability heterogeneity. Several such measures exist (Lake and Jensen, 1989). The Dykstra-Parsons coefficient and the Lorenz coefficient are described in this section as illustrations. The procedure outlined in the following makes some simplifying assumptions that are not too restrictive in practice but make it possible to calculate permeability heterogeneity measures with relative ease. These measures may be used to verify that the permeability distribution used in a model has comparable heterogeneity to the permeability distribution observed in an analysis of field data.
4.8.1
The Dykstra-Parsons Coefficient
The Dykstra-Parsons coefficient can be estimated for a log normal permeability distribution as h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii VDP ¼ 1 exp lnðkA =kH Þ
ð4:8:1Þ
where kA is the arithmetic average kA ¼
n 1X ki n i¼1
ð4:8:2Þ
and kH is the harmonic average n 1 1X 1 ¼ kH n i¼1 ki
ð4:8:3Þ
The Dykstra-Parsons coefficient should be in the range 0 VDP 1. For a homogeneous reservoir, VDP ¼ 0 because kA ¼ kH . An increase in heterogeneity increases VDP. Typical values of the Dykstra-Parsons coefficient are in the range 0.4 VDP 0.9.
64
Integrated Reservoir Asset Management
As an example of a Dykstra-Parsons coefficient calculation, suppose we have a three-layer case with the following permeabilities: Layer 1: k ¼ 35 md Layer 2: k ¼ 48 md Layer 3: k ¼ 126 md
The arithmetic average is kA ¼
n 1X 1 ki ¼ ð126 þ 48 þ 35Þ ¼ 69:7 n i¼1 3
and the harmonic average is
n 1 1X 1 1 1 1 1 þ þ ¼ ¼ kH n i¼1 ki 3 126 48 35 or kH ¼ 52.3 md. Using these average values, we estimate the Dykstra-Parsons coefficient for a log normal permeability distribution to be h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii VDP ¼ 1 exp lnð69:7=52:3Þ ¼ 0:415
4.8.2
The Lorenz Coefficient
The Lorenz coefficient requires a bit more work than the Dykstra-Parsons coefficient. We begin by calculating the following quantities: X Xm n k h k h ; m ¼ 1, . . . , n ð4:8:4Þ Fm ¼ cum flow capacity ¼ i¼1 i i i¼1 i i and Hm ¼ cum thickness ¼
Xm
h i¼1 i
X
n
h; i¼1 i
m ¼ 1, . . . , n
ð4:8:5Þ
for n ¼ number of reservoir layers. Layers should be arranged in order of decreasing permeability; thus i ¼ 1 has thickness h1 and the largest permeability k1, while i ¼ n has thickness hn and the smallest permeability kn. By definition, cumulative flow capacity should be in the range 0 Fm 1, and cumulative thickness should be in the range 0 Hm 1 for 0 < m < n. The Lorenz coefficient is defined in terms of a plot of Fm versus Hm, shown in Figure 4.6. The Lorenz coefficient Lc is two times the area enclosed between the Lorenz curve ABC in the figure and the diagonal AC. The range of the Lorenz coefficient is 0 Lc 1. For a homogeneous reservoir, the Lorenz coefficient satisfies the equality Lc ¼ 0. An increase in heterogeneity increases the value of the Lorenz coefficient Lc. Typical values of the Lorenz coefficient are in the range 0:2 Lc 0:6.
Porosity and Permeability
65
C B Fm
A Hm
Figure 4.6 The Lorenz plot.
An estimate of the Lorenz coefficient is obtained by assuming that all of the permeabilities have equal probability so that the trapezoidal rule can be used to estimate area. The result is Lc ¼
Xn i 1 hXn Xn jk k j = i¼1 ki i j i¼1 j¼1 2n
ð4:8:6Þ
It is not necessary to order permeabilities using this estimate. Notice that in the homogeneous case, all of the permeabilities are equal so that we have the relationship ki ¼ kj. Substituting this equality into Eq. (4.8.6) gives the Lorenz coefficient Lc ¼ 0. In the ideal homogeneous case, the Lorenz coefficient is zero, as just indicated.
4.9
Darcy’s Law with Directional Permeability
Permeability can be a complex function of spatial location and orientation. Spatial and directional variations of a function are described in terms of homogeneity, heterogeneity, isotropy, and anisotropy. If the value of a function does not depend on spatial location, it is called homogeneous. The function is heterogeneous if its value changes from one spatial location to another. If the value of a function depends on directional orientation—that is, the value is larger in one direction than another— then the function is anisotropic. The function is isotropic if its value does not depend on directional orientation. Permeability is a function that can be both heterogeneous and anisotropic. To account for heterogeneity and anisotropy, the simple 1-D form of Darcy’s law must be generalized. In general, flow occurs in dipping beds. To account for the effect of gravity, we define a variable called the potential of phase ‘ as F‘ ¼ P‘ g‘ ðDzÞ where Dz ¼ depth from a datum P‘ ¼ pressure of phase ‘ g‘ ¼ pressure gradient associated with the gravity term
ð4:9:1Þ
66
Integrated Reservoir Asset Management
If we write Darcy’s law for single-phase flow in the form q¼
0:001127KA dF m dz
ð4:9:2Þ
we find that no vertical movement can occur when dF/dz ¼ 0. Thus, Eq. (4.9.2) expresses the movement of fluids in a form that accounts for gravity equilibrium. Darcy’s law in one dimension says that rate is proportional to pressure gradient. This can be extended to three dimensions using vector notation. Darcy’s law for single-phase flow in three dimensions is qx ¼ 0:001127K
A qF m qx
qy ¼ 0:001127K
A qF m qy
qz ¼ 0:001127K
A qF m qz
ð4:9:3Þ
where the gradient of potential accounts for gravity effects. In vector notation we have A q ¼ 0:001127K rF m
!
ð4:9:4Þ
Equation (4.9.3) can be written in matrix notation as 2
3 2 3 qx qF=qx 4 qy 5 ¼ 0:001127K A 4 qF=qy 5 m qz qF=qz
ð4:9:5Þ
where permeability K and cross-sectional area A are treated as constants with respect to direction. A more general extension of Eq. (4.9.5) is 2
3 2 Kxx qx 4 qy 5 ¼ 0:001127 A 4 Kyx m K qz zx
Kxy Kyy Kzy
32 3 Kxz qF=qx Kyz 54 qF=qy 5 Kzz qF=qz
ð4:9:6Þ
where permeability is now treated either as a 33 matrix with nine elements or as a tensor of rank two (Fanchi, 2006c). The diagonal permeability elements {Kxx, Kyy, Kzz} represent the usual dependence of rate in one direction on pressure differences in the same direction. The off-diagonal permeability elements {Kxy, Kxz, Kyx, Kyz, Kzx, Kzy} account for the dependence of rate in one direction on pressure differences in orthogonal directions. Expanding Eq. (4.9.6) gives the corresponding set of equations (Eq. (4.9.7)) that demonstrates this dependence:
Porosity and Permeability
67
2 3 A4 qF qF qF5 Kxx þ Kxy þ Kxz qx ¼ 0:001127 m qx qy qz 2 3 A4 qF qF qF5 qy ¼ 0:001127 Kyx þ Kyy þ Kyz m qx qy qz
ð4:9:7Þ
2 3 A 4 qF qF qF5 Kzx þ Kzy þ Kzz qz ¼ 0:001127 m qx qy qz In many practical situations it is mathematically possible to find a coordinate system {x0 , y0 , z0 } in which the permeability tensor has the diagonal form 2
Kx0 x0 4 0 0
0 Ky0 y0 0
3 0 0 5 Kz 0 z 0
The coordinate axes {x0 , y0 , z0 } are called the principal axes of the tensor, and the diagonal form of the permeability tensor is obtained by a principal axis transformation. The flow equations along the principal axes are 2 3 A4 qF5 Kx0 x0 0 qx0 ¼ 0:001127 m qx 2
3
A qF qy0 ¼ 0:001127 4Ky0 y0 0 5 m qy
ð4:9:8Þ
2 3 A4 qF5 Kz0 z0 0 qz0 ¼ 0:001127 m qz The principal axes in a field can vary from one point of the field to another because of permeability heterogeneity. The form of the permeability tensor depends on the properties of the porous medium. The medium is said to be anisotropic if two or more elements of the diagonalized permeability tensor are different. The permeability of the medium is isotropic if the elements of the diagonalized permeability tensor are equal—that is, Kx0 x0 ¼ Ky0 y0 ¼ Kz0 z0 ¼ K
ð4:9:9Þ
If the medium is isotropic, permeability does not depend on direction. If the isotropic permeability does not change from one position in the medium to another, the
68
Integrated Reservoir Asset Management
(a)
(b)
Figure 4.7 The effect of permeability anisotropy on a drainage area: (a) isotropic (Kx ¼ Ky) and (b) anisotropic (Kx 6¼ Ky).
medium is said to be homogeneous in permeability. On the other hand, if the values of the elements of the permeability tensor vary from one point in the medium to another, both the permeability tensor and the medium are considered heterogeneous. Virtually all reservoirs exhibit some degree of anisotropy and heterogeneity, but the flow behavior in many reservoirs can be approximated as homogeneous and isotropic. In Figure 4.7, (a) is the drainage area of four production wells with isotropic permeability, and (b) is the drainage area of four production wells with anisotropic permeability.
4.9.1
Vertical Permeability
Permeability for flow in a direction that is perpendicular to gravity is horizontal permeability. By contrast, vertical permeability is the permeability for flow in the direction aligned with the direction of the gravitational field. Vertical permeability can be measured in the laboratory or in pressure transient tests conducted in the field. In many cases vertical permeability is not measured and must be assumed. A rule of thumb is to assume vertical permeability to be approximately one-tenth of horizontal permeability. These are reasonable assumptions when there is no data to the contrary. It is preferable from a technical point of view to make direct measurements of all relevant reservoir data. Sometimes it to be difficult to justify the cost or logistics of obtaining direct measurements. If it is necessary to use a rule of thumb or data from an analogous formation to estimate a particular variable, the sensitivity of the shared earth model to changes in the estimated variable should be considered.
CS.4 Valley Fill Case Study: Permeability The permeability distribution in the Valley Fill reservoir is assumed to be isotropic and homogeneous because the spatial dependence of permeability is not known. Horizontal permeability is 150 md based on a pressure transient test. Vertical permeability was not measured and is therefore assumed to be one-tenth of horizontal permeability.
Porosity and Permeability
69
Exercises 4-1. Suppose gross thickness includes 4 feet of impermeable shale and 16 feet of permeable sandstone. What are the gross thickness, net thickness, and net-to-gross ratio? 4-2. Consider a three-layer reservoir with the following permeabilities: Layer 1: k ¼ 35 md Layer 2: k ¼ 48 md Layer 3: k ¼ 126 md Calculate the arithmetic average, the harmonic average, and the Dykstra-Parsons coefficient for a log normal permeability distribution. 4-3. Consider a three-layer reservoir with the following permeabilities: Layer 1: k ¼ 35 md Layer 2: k ¼ 48 md Layer 3: k ¼ 126 md Estimate the Lorenz coefficient using Eq. (4.8.6). 4-4A. Consider a linear flow system with area ¼ 25 ft2. End point A is 5 feet higher than end point B, and the distance between end points is 15 feet. Suppose the system contains oil with viscosity ¼ 0.8 cp, gravity ¼ 35 API (go ¼ 0.85), and FVF ¼ 1.0 RB/ STB. If the end point pressures are PA ¼ PB ¼ 20 psia, is there flow? If so, how much flow is there and in what direction? Use Darcy’s law with the gravity term and dip angle a:
q ¼ 0:001127
kA PA PB þ rg sina L mB
4-4B. Using the data in Exercise 4-4A, calculate the pressure PB that would prevent fluid flow. 4-5A. Suppose the pressure P1 of a water-bearing formation at depth z1 ¼ 10,000 feet is 4,000 psia. If the pressure gradient for water is 0.433 psia/ft, calculate the pressure P2 at depth z2 ¼ 11,000 feet. Calculate the phase potentials F1 and F2 at depths z1 and z2, respectively. 4-5B. Use the pressures and potentials in Exercise 4-5A to estimate the derivatives dP/dz and dF/dz. Will there be vertical flow? 4-6. Estimate the volumetric flow rate using Darcy’s law that is given by Eq. (4.5.1). Assume the permeability in a cylindrical core is 100 md, the length of the core is 6 inches, the diameter of the core is 3 inches, the pressure drop across the core is 10 psi, and the viscosity of liquid passing through the core is 2 cp. Express volumetric flow rate in bbl/day or cc/s. 4-7. A barrier island is a large sand body. Consider a barrier island that averages 3 miles wide, 10 miles long, and 30 feet thick. The porosity of the sand averages almost 25 percent. What is the pore volume of the barrier island? Express your answer in bbl and m3.