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# 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

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.

and

## 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^{[1]} were among the first to study this problem and they noted that the solution was complicated by nonuniqueness. Others, too, have noted nonuniqueness. ^{[2]}^{[3]}^{[4]}^{[5]} 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^{[3]} 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

This equation is an abbreviation of **Eq. 6**. The summation ΣΔ*pW*_{D} is a function of only *r*_{eD} and *k*_{t}. McEwen noted that *U* is related to *N*_{foi} and *G*_{fgi} through

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

where

and

For the case of an initially undersaturated oil reservoir,

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

The sum of the squares of the residual is

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:

- Estimate a limited range of realistic values for
*k*_{t}if possible. - Compute
*F*for each data point with**Eq. 15**if saturated or**Eq. 16**if undersaturated. - Guess
*k*_{t}and*r*_{eD}. - Compute
*E*_{ow}for each data point with**Eq. 11**. - Compute
*N*with least-squares linear regression or graphically from the slope of an*F*-vs.-*E*_{ow}plot. - Compute
*R*_{i}for each data point with**Eq. 13**. - Compute
*R*_{ss}with**Eq. 14**. - If the
*R*_{ss}is minimized, then go to Step 9; otherwise, return to Step 3. - Compute the aquifer constant with
**Eq. 7**. - 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.

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. ^{[6]}^{[7]} 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

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

and ΣΔ*pW*_{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.*, ^{[8]} for instance, proposed plotting *F*/*E*_{o} vs. (ΣΔ*pW*_{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^{[9]} popularized this method and modified it to include the aquifer size (*r*_{eD}) as an additional unknown and determinable parameter. Dake^{[10]} also advocated this method. Chierici *et al.*^{[4]} proposed a variation of this method with a *F*/(ΣΔ*pW*_{D}) vs. *E*_{o}/(ΣΔ*pW*_{D}) plot. McEwen^{[3]} 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^{[2]} presented a systematic analysis of these methods and confirmed McEwen’s conclusions. Wang and Hwan^{[11]} confirmed Tehrani’s findings. Sills^{[5]} presents a review and comparison of the McEwen, Havlena-Odeh, and van Everdingen-Timmerman-McMahon methods.

To help diagnose waterdrives, Campbell^{[12]} proposed plotting *F*/*E*_{owf} vs. *N*_{p} for initially undersaturated oil reservoirs. This method is analogous to Cole’s^{[13]} 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, ^{[3]} Tehrani, ^{[2]} Wang and Hwan, ^{[11]} and Walsh^{[7]} 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.

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.

*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

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.

## Example 2: determining water-influx parameters and OOIP

van Everdingen *et al.*^{[8]} 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.

van Everdingen *et al.*^{[8]} 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.

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

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

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 ΣΔ*pW*_{D}, which is a function of *r*_{eD} and *k*_{t}. The overall solution procedure contains the following steps:

- Assume values of
*r*_{eD}and*k*_{t}. - Compute
*t*_{D}for each historical data point with**Eq. 27**. - Compute ΣΔ
*pW*_{D}for each historical data point with the VEH model. - Compute
*E*_{ow}for each historical data point with**Eq. 10**. - Plot
*F*vs.*E*_{ow}. - Determine the OOIP from the slope.
- Compute the residual,
*R*_{i}, for each data point with**Eq. 13**and compute*R*_{ss}with**Eq. 14**. - Change
*r*_{eD}and*k*_{t}and return to Step 1 until a minimum*R*_{ss}is obtained.

**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 ΣΔ*pW*_{D} are omitted for the sake of brevity. The example in Water influx models illustrates these calculations.

**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.

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.

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.

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.

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

## References

- ↑ Woods, R.W. and Muskat, M.M. 1945. An Analysis of Material-Balance Calculations. Trans., AIME 160: 124.
- ↑
^{2.0}^{2.1}^{2.2}Tehrani, D.H. 1985. An Analysis of a Volumetric Balance Equation for Calculation of Oil in Place and Water Influx. J Pet Technol 37 (9): 1664-1670. SPE-12894-PA. http://dx.doi.org/10.2118/12894-PA - ↑
^{3.0}^{3.1}^{3.2}^{3.3}McEwen, C.R. 1962. Material Balance Calculations With Water Influx in the Presence of Uncertainty in Pressures. SPE J. 2 (2): 120–128. SPE-225-PA. http://dx.doi.org/10.2118/225-PA - ↑
^{4.0}^{4.1}Chierici, G.L. and Ciucci, G.M. 1967. Water Drive Gas Reservoirs: Uncertainty in Reserves Evaluation From Past History. J Pet Technol 19 (2): 237-244. http://dx.doi.org/10.2118/1480-PA. - ↑
^{5.0}^{5.1}Sills, S.R. 1996. Improved Material-Balance Regression Analysis for Waterdrive Oil and Gas Reservoirs. SPE Res Eval & Eng 11 (2): 127–134. SPE-28630-PA. http://dx.doi.org/10.2118/28630-PA - ↑ Walsh, M.P. and Lake, L.W. 2003. A Generalized Approach to Primary Hydrocarbon Recovery. Amsterdam: Elsevier.
- ↑
^{7.0}^{7.1}Walsh, M.P. 1999. Effect of Pressure Uncertainty on Material-Balance Plots. Presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, 3-6 October 1999. SPE-56691-MS. http://dx.doi.org/10.2118/56691-MS - ↑
^{8.0}^{8.1}^{8.2}van Everdingen, A.F., Timmerman, E.H., and McMahon, J.J. 1953. Application of the Material Balance Equation to a Partial Water-Drive Reservoir. Trans., AIME 198: 51. - ↑ Havlena, D. and Odeh, A.S. 1963. The Material Balance as an Equation of a Straight Line. J Pet Technol 15 (8): 896–900. SPE-559-PA. http://dx.doi.org/10.2118/559-PA
- ↑ Dake, L.P. 1978. Fundamentals of Reservoir Engineering. Amsterdam: Elsevier Scientific Publishing Co.
- ↑
^{11.0}^{11.1}Wang, B. and Hwan, R.R. 1997. Influence of Reservoir Drive Mechanism on Uncertainties of Material Balance Calculations. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 5-8 October 1997. SPE-38918-MS. http://dx.doi.org/10.2118/38918-MS - ↑ Campbell, R.A. and Campbell, J.M. 1978. Mineral Property Economics, Vol. 3: Petroleum Property Evaluation. Norman, Oklahoma: Campbell Petroleum Series.
- ↑ Cole, F.W. 1969. Reservoir Engineering Manual. Houston, Texas: Gulf Publishing Co.

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