Material balance in waterdrive reservoirs

The objectives of a material-balance analysis include:

• Confirming the producing mechanism
• Estimating the original oil in place (OOIP)
• Estimating the water influx history
• Estimating the water influx model parameters

The water influx model parameters are needed to forecast future water influx and oil recovery.

The analysis depends on the known and unknown constants and variables. Three scenarios are considered:

• Water influx history is known but the OOIP is unknown
• Water influx history is unknown but the OOIP is known
• Both the water influx history and the OOIP are unknown

Water influx known, OOIP unknown

If the water influx history is known and the OOIP unknown, the material balance methods in the previous sections are directly applicable. For instance, if the reservoir is initially undersaturated, then an (F – W_e)-vs.-E owf plot can be used to confirm the mechanism and estimate the OOIP. The water influx model parameters can be determined by history matching.

Water influx unknown, OOIP known

If the OOIP is known and the water influx history is unknown, then the material balance equation can be used to estimate the water influx history. Solving Eq. 1 for W_e yields

$G_{fgi}E_{gwf}+N_{foi}E_{owf}+W_{e}=F,$

(1)

$W_{e}=G_{fgi}E_{gwf}+N_{foi}E_{owf}-F.$

(2)

If the OOIP (N) and OGIP (G) are known, G_fgi and N_foi are computed from Eqs. 3 and 4. Eq. 2 is applied for each historical average pressure measurement to compute the cumulative water influx. Example 1 illustrates this method. Once the water-influx history is estimated, the aquifer parameters can be estimated from history matching.

$G=G_{fgi}\left(1+{\frac {R_{si}B_{gi}}{mB_{oi}}}\right)$

(3)

and

$N=N_{foi}\left(1+{\frac {mB_{oi}R_{vi}}{B_{gi}}}\right).$

(4)

Water influx unknown, OOIP unknown

This case simultaneously determines the OOIP, water-influx history, and water-influx model parameters. This is a challenging problem. Woods and Muskat were among the first to study this problem and they noted that the solution was complicated by nonuniqueness. Others, too, have noted nonuniqueness. Despite these complications, certain techniques have proved useful and some approaches are better than others are. The solution method is based on the work of McEwen and depends on whether the radial or linear version of the VEH model is applied.

Radial aquifer

This method simultaneously determines the OOIP, water influx history, and model parameters r_eD and k_t. The aquifer constant U is then subsequently determined. The water influx is

$W_{e}=U\Sigma \Delta pW_{D}.$

(5)

This equation is an abbreviation of Eq. 6. The summation ΣΔpW_D is a function of only r_eD and k_t. McEwen noted that U is related to N_foi and G_fgi through

$W_{ek}(t)=U\sum _{j=0}^{k-1}\Delta p_{j+1}W_{D}(t_{Dk}-t_{Dj})\qquad {\hbox{for}}\ k=1,2,\ldots ,n.$

(6)

$U={\frac {2(N_{foi}B_{oi}+G_{fgi}B_{gi})c_{t}}{(1-S_{wi})}}\left({\frac {\phi _{a}}{\phi _{r}}}\right).$

(7)

$G_{fgi}E_{gwf}+N_{foi}E_{owf}+W_{e}=F,$

(8)

Substituting this equation into Eq. 5 and substituting this result into Eq. 8 yields

$F=N_{foi}E_{ow}+G_{fgi}E_{gw},$

(9)

where

$E_{ow}=E_{owf}+{\frac {2B_{oi}c_{t}\Sigma \Delta pW_{D}(k_{t},r_{eD})}{(1-S_{wi})}}\left({\frac {\phi _{a}}{\phi _{r}}}\right)$

(10)

and

$E_{gw}=E_{gwf}+{\frac {2B_{gi}c_{t}\Sigma \Delta pW_{D}(k_{t},r_{eD})}{(1-S_{wi})}}\left({\frac {\phi _{a}}{\phi _{r}}}\right).$

(11)

For the case of an initially undersaturated oil reservoir,

$F=NE_{ow}.$

(12)

This equation shows that a plot of F vs. E_ow is a straight line, emanates from the origin, and has a slope equal to N, the OOIP. This observation provides a means to confirm the producing mechanism.

E_ow, however, is a function of k_t and r_eD, and these parameters are unknown a priori. Thus, the problem reduces to one of finding the optimal k_t and r_eD that minimizes the material-balance error. Graphically, this is equivalent to varying k_t and r_eD until the straightest possible line is realized. The slope of the line equals the OOIP, and the OGIP is the product NR_si if the reservoir is initially undersaturated. Mathematically, the material-balance error is minimized when the sum of the squares of the residual is minimized. The residual for point i is

$R_{i}=(F)_{i}-N(E_{ow})_{i}.$

(13)

The sum of the squares of the residual is

$R_{ss}=\sum _{i=1}^{n}(R_{i})^{2},$

(14)

where n is the total number of data points.

In summary, the McEwen method to simultaneously estimate N, r_eD, and k_t is as follows:

$F=G_{ps}\left({\frac {B_{g}-B_{o}R_{v}}{1-R_{v}R_{s}}}\right)+N_{p}\left({\frac {B_{o}-B_{g}R_{s}}{1-R_{v}R_{s}}}\right)+(W_{p}-W_{I})B_{w},$

(15)

$F=N_{p}B_{o}+(W_{p}-W_{I})B_{w}.$

(16)

1. Estimate a limited range of realistic values for k_t if possible.
2. Compute F for each data point with Eq. 15 if saturated or Eq. 16 if undersaturated.
3. Guess k_t and r_eD.
4. Compute E_ow for each data point with Eq. 11.
5. Compute N with least-squares linear regression or graphically from the slope of an F-vs.-E_ow plot.
6. Compute R_i for each data point with Eq. 13.
7. Compute R_ss with Eq. 14.
8. If the R_ss is minimized, then go to Step 9; otherwise, return to Step 3.
9. Compute the aquifer constant with Eq. 7.
10. Compute the water influx for each data point with Eq. 5.

If least-squares linear regression is used to compute N in Step 5, an equation analogous to Eq. 17 is used (where E_ow is substituted for E_owf). This solution method is iterative because the material-balance error must be minimized. This calculation is carried out with a trial-and-error method or a minimization algorithm. Least-squares linear regression and minimization algorithms have become standard features in commercial spreadsheets.

$N={\frac {n\sum \limits _{j=1}^{n}(F)_{j}(E_{owf})_{j}-\sum \limits _{j=1}^{n}(F)_{j}\sum \limits _{j=1}^{n}(E_{owf})_{j}}{n\sum \limits _{j=1}^{n}(E_{owf})_{j}^{2}-\sum \limits _{j=1}^{n}(E_{owf})_{j}\sum \limits _{j=1}^{n}(E_{owf})_{j}}},$

(17)

McEwen’s method also can be applied to initially saturated reservoirs; however, the solution procedure must be expanded and modified slightly. More specifically, the solution procedure is the same as for initially undersaturated reservoirs except Steps 5 and 6. Step 5 must be modified to include the simultaneous calculation of N_foi and G_fgi by multivariate, least-squares (planar) regression. Step 6 must be modified, and the residual for point i is computed with $R_{i}=(F)_{i}-G_{fgi}(E_{gw})_{i}-N_{foi}(E_{ow})_{i}$.

Linear aquifer

McEwen’s method for linear aquifers is very similar to the radial model. The method simultaneously determines the:

• OOIP
• Water influx history
• Model parameters [(V_pa/V_pr) and k_t]

The aquifer constant U is related to V_pa through

$U=\left({\frac {V_{pa}}{V_{pr}}}\right){\frac {(N_{foi}B_{oi}+G_{fgi}B_{gi})c_{t}}{(1-S_{wi})}}.$

(18)

Substituting this equation into Eq. 5 and substituting the result into Eq. 8 gives Eq. 8, where

$E_{gw}=E_{gwf}+\left({\frac {V_{pa}}{V_{pr}}}\right){\frac {B_{gi}c_{t}\Sigma \Delta pW_{D}(k_{t})}{(1-S_{wi})}},$

(19)

$E_{ow}=E_{owf}+\left({\frac {V_{pa}}{V_{pr}}}\right){\frac {B_{oi}c_{t}\Sigma \Delta pW_{D}(k_{t})}{(1-S_{wi})}},$

(20)

and ΣΔpW_D is a function of only k_t and not (V_pa/V_pr). Eqs. 19 and 20 are analogous to Eqs. 10 and 11 in the radial model.

The solution procedure for the linear model is identical to that of the radial model except that k_t and V_pa/V_pr are optimized to minimize the material-balance error. Once V_pa/V_pr is determined, the aquifer constant U is determined from Eq. 18 and W_e is determined from Eq. 5. Example 2 illustrates an example of the McEwen method.

Numerous alternative material-balance methods have been proposed to analyze waterdrive reservoirs. Some are very popular and widely used. While most are theoretically valid, most are also unreliable. van Everdingen et al., for instance, proposed plotting F/E_o vs. (ΣΔpW_D)/E_o. The slope of this plot equals U and the y-intercept equals N. van Everdingen et al. proposed varying k_t until the straightest possible line was obtained. Later, Havlena and Odeh popularized this method and modified it to include the aquifer size (r_eD) as an additional unknown and determinable parameter. Dake also advocated this method. Chierici et al. proposed a variation of this method with a F/(ΣΔpW_D) vs. E_o/(ΣΔpW_D) plot. McEwen studied the method of van Everdingen et al. and noted hypersensivity to pressure uncertainty. He observed unacceptably large errors and deemed the method unreliable. Later, Tehrani presented a systematic analysis of these methods and confirmed McEwen’s conclusions. Wang and Hwan confirmed Tehrani’s findings. Sills presents a review and comparison of the McEwen, Havlena-Odeh, and van Everdingen-Timmerman-McMahon methods.

To help diagnose waterdrives, Campbell proposed plotting F/E_owf vs. N_p for initially undersaturated oil reservoirs. This method is analogous to Cole’s popular method of plotting F/E_gwf vs. G_p for gas reservoirs. In theory, an active waterdrive is indicated if the plot varies appreciably from a horizontal line. The degree of curvature is a qualitative measure of the waterdrive strength. The curve emanates from a y-intercept equal to the OOIP. The shape of the curve mimics and is related to the attending water recharge rate history. Cole and Campbell plots are attractive because of their simplicity and are widely reported. Unfortunately, in practice, they are not always reliable because of hypersensitivity caused by uncertainty. The origin of the hypersensitivity is analogous to the problems noted by McEwen, Tehrani, Wang and Hwan, and Walsh for other types of material-balance plots. The quotient F/E_owf approaches infinity initially because the E_owf approaches zero. A systematic reservoir pressure error of only 1 to 2%, for instance, can lead to erroneous conclusions regarding water-influx diagnosis. These facts complicate the interpretation. For these reasons, Campbell plots should be used cautiously to diagnose water influx and used very cautiously to estimate the OOIP. If they are used, the early-time data should be weighted minimally. The reliability of these plots increases with pressure depletion; however, water influx mitigates pressure depletion and delays reliability. Unfortunately, water-influx diagnoses are sought as early as possible, which further complicates and compromises the use of these plots.

Example 1: estimating water influx with material balance

Table 1 summarizes the cumulative oil and gas production as a function of time and average reservoir pressure for a black-oil reservoir. The discovery (initial) pressure is 1,640 psia, and production data are tabulated through a pressure of 800 psia.

• 

  Table 1 - Reservoir performance data for example

Volumetric measurements estimate an OOIP of 210,420 thousand STB with no initial free gas. Estimate the cumulative water influx (RB) at each pressure in Table 1 with material balance. Assume that the standard PVT parameters in Table 2 apply.

• 

  Table 2 - PVT parameters for West Texas black oil

Solution. Eq. 2 gives the cumulative water influx. Because there is no initial free gas, G_fgi = 0, N_foi = N, and Eq. 2 simplifies to

$W_{e}=NE_{o}-F,$

(21)

$E_{o}=B_{to}-B_{toi},$

(22)

$B_{to}={\frac {B_{o}(1-R_{si}R_{v})+B_{g}(R_{si}-R_{s})}{1-R_{v}R_{s}}},$

(23)

where rock and water expansion are ignored and F and E_o are given by Eqs. 15 and 22, respectively. E_o is a function of B_to, which is given by Eq. 23.

Table 3 tabulates the results. Fig. 1 plots the water influx history.

• 

  Table 3 - Summary of water influx calculations for example 1.

• 

  Fig. 1 – Computed water-influx history of example 1.

Example 2: determining water-influx parameters and OOIP

van Everdingen et al. studied water influx in an initially undersaturated oil reservoir located in the Wilcox formation at a depth of 8,100 ft subsea. The accumulation covered approximately 1,830 acres. The maximum gross and net thicknesses were 37 and 26 ft, respectively. The reservoir fluid exhibited an initial oil FVF of 1.538 RB/STB and a GOR of 900 scf/STB. Table 4 reports the reservoir and aquifer properties.

• 

  Table 4 - Reservoir and aquifer properties for Wilcox formation black oil

van Everdingen et al. estimated the OOIP at 24 to 25 million STB from volumetric measurement. From the oil, gas, and water production data and standard PVT properties, they computed F and E_o as a function of pressure. Production took place from 1942 through 1950. Approximately 6.965 million STB of oil and 7.1 Bscf of gas were produced during this period. Table 5 gives the average reservoir pressure, total fluid withdrawal, and oil-phase expansivity as a function of time.

• 

  Table 5 - Reservoir performance data for example 2

Use McEwen’s method to find the optimal dimensionless aquifer radius, r_eD, and the aquifer time constant, k_t (years^–1). Plot F vs. E_ow. Also, compute the aquifer constant, U (RB/psi). Estimate the OOIP (million STB). Estimate the water delivery rate (RB/D) and average aquifer pressure (psia), and plot the histories.

Solution. Though the time constant, k_t, is unknown, first estimate a realistic range of values based on reservoir and aquifer properties. The time constant is given by Eq. 24. The total compressibility is the sum of the rock and water compressibility or 6.8 × 10^–6 psi^–1. The aquifer porosity is 20.9%. The water viscosity is 0.25 cp. The aquifer permeability is 275 md. The only unspecified quantity on the right side of Eq. 24 is the effective reservoir radius. Although this quantity is unknown because the size of the reservoir is uncertain, it can be estimated from

$k_{t}={\frac {2.309k_{a}}{\phi _{a}\mu _{w}c_{t}r_{o}^{2}}},$

(24)

$r_{o}={\sqrt {\frac {5.6146V_{pr}}{\pi \phi _{r}hf}}}.$

(25)

To use this equation, estimate the reservoir PV, which is given by

$V_{pr}={\frac {N_{foi}B_{oi}+G_{fgi}B_{gi}}{(1-S_{wi})}}.$

(26)

Evaluating this equation on the basis of N_foi = 25 million STB yields V_pr = 45.23 million RB. Evaluating Eq. 25 for f = 1, h = 26 ft, and ϕ_r = 19.9% yields r_o = 3,951 ft. Evaluating Eq. 24 yields k_t = 116 years^–1. Thus, a liberal range for k_t is 10 to 1,000 years^–1.

Next, compute E_ow from Eq. 10. This equation requires computing ΣΔpW_D, which is a function of r_eD and k_t. The overall solution procedure contains the following steps:

1. Assume values of r_eD and k_t.
2. Compute t_D for each historical data point with Eq. 27.
3. Compute ΣΔpW_D for each historical data point with the VEH model.
4. Compute E_ow for each historical data point with Eq. 10.
5. Plot F vs. E_ow.
6. Determine the OOIP from the slope.
7. Compute the residual, R_i, for each data point with Eq. 13 and compute R_ss with Eq. 14.
8. Change r_eD and k_t and return to Step 1 until a minimum R_ss is obtained.

$t_{D}={\frac {0.03110k_{a}t}{\phi _{a}\mu _{w}c_{t}r_{o}^{2}}}.$

(27)

Table 6 summarizes the results for the case of r_eD = 20 and k_t = 17 years^–1. The step-by-step calculations to compute ΣΔpW_D are omitted for the sake of brevity. The example in Water influx models illustrates these calculations.

• 

  Table 6 - Predicted water influx history for radial aquifer if r_eD=20 and k_t=17 years^-1

Fig. 2 plots F vs. E_ow for r_eD = 20 and k_t = 17 years^–1. The slope of this plot is 18.7 million STB, which is an estimate of the OOIP. The material-balance error is given by the R_ss, which is 0.015 million res bbl.

• 

  Fig. 2 – F-vs.- E_ow plot for example 2.

The values of r_eD and k_t were varied until a minimum R_ss was realized. Table 7 summarizes the results for the following values of r_eD: 10, 15, 20, and 30. No values greater than r_eD = 30 were considered because they yielded identical results and were equivalent to an infinite-acting aquifer.

• 

  Table 7 - Material balance error as a function of r_eD and k_t

The results in Table 7 are summarized graphically in Fig. 3. This figure plots the standard error vs. the time constant for r_eD = 10, 20, and 30. The standard error is defined as the square root of R_ss/n, where n is the total number of data points. These results show that a minimum error is realized for the following range of properties: r_eD = 20 to 30 and k_t = 16 to 17 years^–1. Table 8 summarizes the range of constants yielded by this range of values.

• 

  Fig. 3 – Material-balance error vs. time constant for several values of r_eD for Example 2.

• 

  Table 8 - Range of constants

The material-balance OOIP estimate (18.7 to 20.3 million STB) approximately agrees with the volumetric estimate (24 to 25 million STB). Technically, r_eD = 20 and k_t = 17 years^–1 yields a minimum error; however, comparable errors are realized for the stated range of r_eD and k_t. The boundary between infinite- and noninfinite-acting aquifers occurs at r_eD = 20.1. Thus, an aquifer with r_eD = 20 is not quite but very nearly an infinite-acting aquifer. These observations help explain why values of r_eD in the range 20 to 30 yield virtually identical errors and similar results. On the basis of simulating reservoir performance through 9 years, selecting any value of r_eD greater than 20 is practically acceptable. A value of r_eD = 30, however, yields an OOIP estimate (20.2 million STB) slightly closer to the volumetric OOIP estimate than the OOIP estimate (18.8 million STB) for r_eD = 20.On this basis, r_eD = 30 may be preferable. Without additional historical data and history matching, a more conclusive determination of r_eD beyond the stated range is not possible.

Fig. 2 includes an F-vs.-E_ow plot for r_eD = 10 and k_t = 11 years^–1. The F vs. E_ow plots for r_eD = 20 and r_eD = 30 are virtually indistinguishable. The plot for r_eD = 10 exhibits appreciable curvature; in contrast, the plot for r_eD = 20 is linear. The degree of curvature is a measure of the lack of material balance. The upward curvature of the former plot indicates that water influx is underpredicted.

Fig. 4 shows the predicted water-influx history. This figure assumes r_eD = 20. A cumulative water-influx volume of 14.83 million res bbl is predicted after 9 years. This amount of water influx equates to 46.0% of the (reservoir) HCPV (assuming an OOIP of 18.7 million STB and a HCPV of 33.83 million res bbl). This substantial amount of water influx, together with the relatively small pressure decline (from 3,800 to 3,060 psia), suggests a moderate to strong waterdrive.

• 

  Fig. 4 – Predicted water-influx history for example 2.

The pressure data in Table 5 show that the reservoir pressure actually increased after 7 years. This example illustrates another consequence of the VEH model; namely, the model can treat pressure increases as well as pressure decreases. The operators offered no explanation regarding why the reservoir pressure began to increase after seven years. It was noted that the pressure remained approximately constant and increased during periods of marked GOR decline. It was suspected that a secondary gas cap formed as the reservoir pressure declined below the bubblepoint. Upstructure wells reportedly began producing free gas as the gas cap formed and grew. The GOR grew as the free-gas production grew. The GOR declined when production from the high GOR wells was curtailed. Though this scenario explains the GOR behavior and some of the pressure behavior, it does not explain why the reservoir pressure actually increased slightly during the last two years of production.

Nomenclature

a	=	constant
b	=	perforation length, L, ft
Bg	=	gas FVF, RB/scf
Bo	=	oil FVF, RB/STB
Btg	=	two-phase gas FVF, RB/scf
Bto	=	two-phase oil FVF, RB/STB
Btw	=	two-phase water/gas FVF, RB/STB
Bw	=	water FVF, RB/STB
cf	=	rock compressibility, Lt2/m, 1/psi
ct	=	total aquifer compressibility, Lt2/m, 1/psi
Ef	=	rock (formation) expansivity
Eg	=	gas expansivity, RB/scf
Egw	=	expansivity for McEwen method, RB/scf
Egwf	=	composite gas/water/rock FVF, RB/scf
Eo	=	oil expansivity, RB/STB
Eow	=	expansivity for McEwen method, RB/STB
Eowf	=	composite oil/water/rock FVF, RB/STB
Et	=	total expansivity, RB/STB
Ev	=	volumetric sweep efficiency, fraction
Ew	=	water expansivity, RB/STB
f	=	reservoir angle
F	=	total fluid withdrawal, L3, RB
G	=	total original gas in place, L3, scf
Gfgi	=	initial free gas in place, L3, scf
Gi	=	cumulative gas injected, L3, scf
Gp	=	cumulative produced gas, L3, scf
Gps	=	cumulative produced sales gas, L3, scf
h	=	pay thickness, L, ft
J*	=	constant
k	=	permeability, L2, md
ka	=	aquifer permeability, L2, md
kH	=	horizontal permeability, L2, md
kt	=	time constant, 1/t, 1/years
kv	=	vertical permeability, L2, md
K	=	gas/oil equilibrium ratio or constant
La	=	aquifer length, L, ft
Lg	=	perforation gap between uppermost perforation and GOC, L, ft
Lr	=	reservoir length, L, ft
Lw	=	perforation gap between lowermost perforation and WOC, ft, L
m	=	initial free-gas-phase to free-oil-phase volume ratio (gas-cap size)
n	=	total number of data points
N	=	total original oil in place, L3, STB
Nfoi	=	initial free oil in place, L3, STB
Ng	=	dimensionless gravity number
Np	=	cumulative produced oil, L3, STB
p	=	pressure, m/Lt2, psi
${\tilde {p}}$	=	time-averaged pressure, m/Lt2, psi
${\overline {p}}$	=	average reservoir pressure, m/Lt2, psi
${\overline {p}}_{a}$	=	average aquifer pressure, m/Lt2, psi
pe	=	pressure at drainage radius, m/Lt2, psi
pw	=	wellbore pressure, m/Lt2, psi
q	=	producing rate at reservoir conditions (RB/D) or surface conditions (STB/D),v L3/t
qc	=	critical coning rate, STB/D, L3/t
qDc	=	dimensionless critical coning rate
qsg	=	surface producing gas rate, L3/t, scf/D
qso	=	surface producing oil rate, L3/t, STB/D
qw	=	aquifer delivery rate, L3/t, RB/D
ra	=	aquifer radius, L, ft
rDe	=	dimensionless radius including vertical permeability
re	=	reservoir drainage radius
reD	=	ratio of reservoir to aquifer radii
ro	=	effective reservoir radius, L, ft
rw	=	wellbore radius, L, ft
R	=	instantaneous producing GOR, scf/STB
Rj	=	residual for point j, L3, RB
Rp	=	cumulative GOR, scf/STB
Rps	=	cumulative sales GOR, scf/STB
Rs	=	dissolved GOR, scf/STB
Rss	=	sum of squares of the residual, L6, RB2
Rsw	=	dissolved-gas/water ratio, scf/STB
Rv	=	volatilized-oil/gas ratio, STB/MMscf
s	=	skin factor
Sg	=	gas saturation, fraction
Sgrw	=	residual gas saturation to water flow, fraction
Sorw	=	residual oil saturation to water flow, fraction
Swi	=	initial water saturation, fraction
t	=	time, t, years
tmax	=	maximum time, t, years
tD	=	dimensionless time
tD*	=	dimensionless time boundary between infinite-acting and finite aquifers
tDmax	=	maximum dimensionless time
U	=	aquifer constant, L4t2/m, RB/psi
Vfgi	=	initial volume of free gas, L3, RB
Vfoi	=	initial volume of free oil, L3, RB
Vp	=	reservoir PV, L3, RB
Vpa	=	aquifer PV, L3, RB
Vpi	=	initial reservoir PV, L3, RB
Vpr	=	reservoir PV, L3, RB
w	=	reservoir width, L, ft
W	=	initial water in place, L3, STB
WD	=	dimensionless cumulative water influx
We	=	cumulative water influx, L3, RB
WI	=	cumulative injected water, L3, STB
Wp	=	cumulative produced water, L3, STB
δg	=	dimensionless perforation gap below GOC
δw	=	dimensionless perforation gap above WOC
Δp	=	difference of time-averaged pressure, m/Lt2, psi
Δρ	=	density difference, m/L3, lbm/ft3 and g/cm3
Δρog	=	oil/gas density difference, m/L3, g/cm3
Δρwo	=	water/oil density difference, m/L3, g/cm3
ε	=	b/h
ϕa	=	aquifer porosity, fraction
ϕr	=	reservoir porosity, fraction
θ	=	angle
μg	=	gas viscosity, m/Lt, cp
μo	=	oil viscosity, m/Lt, cp
μw	=	water viscosity, m/Lt, cp
ρg	=	gas density, m/L3, g/cm3
ρo	=	oil density, m/L3, g/cm3
ρw	=	water density, m/L3, g/cm3

Noteworthy papers in OnePetro

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External links

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See also

Water drive reservoirs

Water influx models

Material balance in oil reservoirs

Oil fluid characteristics

Oil fluid properties

PEH:Oil_Reservoir_Primary_Drive_Mechanisms