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Water influx models
Water influx models are mathematical models that simulate and predict aquifer performance. Most importantly, they predict the cumulative water influx history. When successfully integrated with a reservoir simulator, the net result is a model that effectively simulates performance of water drive reservoirs.
Aquifer models
There are several popular aquifer models:
- van Everdingen-Hurst (VEH) model^{[1]}
- Carter-Tracy model^{[2]}
- Fetkovich model^{[3]}
- Schilthuis model^{[4]}
- Small- or pot-aquifer model^{[5]}
The first three models are unsteady-state models and are the most realistic. They attempt to simulate the complex pressure changes that gradually occur within the aquifer and between the aquifer and reservoir. As pressure depletion proceeds, the pressure difference between the reservoir and aquifer grows rapidly and then abates as the aquifer and reservoir eventually equilibrate. This pressure interaction causes the water influx rate to start at zero, grow steadily, reach a maximum, and then dissipate. This particular water influx rate history behavior applies to initially saturated oil reservoirs; the behavior for initially undersaturated oil reservoirs is often slightly but distinctly different. The effects of undersaturation on the water-influx performance are discussed below. The unsteady-state models are far more successful at capturing the real dynamics than other models. In contrast, Schilthuis’ steady-state model assumes the aquifer pressure remains constant. The small-aquifer model, however, assumes the aquifer and reservoir pressures are equal.
The VEH model is the most sophisticated of all these models. Its main advantage is its realism. Originally, its main disadvantage was its cumbersome nature. Charts or tables had to be consulted repeatedly to execute a single calculation. To address this limitation, the Carter-Tracy and Fetkovich models were alternatives that were free of tables and charts. These models, however, were only approximations to and simplifications of the VEH model. Since the VEH charts and tables were digitized, ^{[6]}^{[7]}^{[8]} the need for alternatives has diminished.
Allard and Chen^{[9]} proposed an aquifer model specifically for bottomwater drives. This model included 2D flow. In comparison, the VEH model considered only 1D flow. Simulation practitioners, however, have found that the VEH model is satisfactory in simulating bottomwater drives. ^{[10]}
van Everdingen-Hurst (VEH) model
van Everdingen and Hurst considered two geometries: radial- and linear-flow systems. The radial model assumes that the reservoir is a right cylinder and that the aquifer surrounds the reservoir. Fig. 1 illustrates the radial aquifer model, where:
- r_{o} = reservoir radius
- r_{a} = aquifer radius
Flow between the aquifer and reservoir is strictly radial. This model is especially effective in simulating peripheral and edgewater drives but also has been successful in simulating bottomwater drives.^{[9]}
In contrast, the linear model assumes the reservoir and aquifer are juxtaposed rectangular parallelepipeds. Fig. 2 shows examples. Flow between the aquifer and reservoir is strictly linear. This model is intended to simulate edgewater and bottomwater drives. The model definition depends on the application. For edgewater drives, the thicknesses of the reservoir and aquifer are identical; the widths of the reservoir and aquifer are also the same, and the aquifer and reservoir lengths are L_{a} and L_{r}, respectively (Fig. 2a). For bottomwater drives, the width of the reservoir and aquifer are identical; the length (L) of the reservoir and aquifer are also the same; the aquifer depth is L_{a}, and the reservoir thickness is h (Fig. 2b).
van Everdingen and Hurst solved the applicable differential equations analytically to determine the water influx history for the case of a constant pressure differential at the aquifer/reservoir boundary. This case assumes the reservoir pressure is constant. They called this case the "constant terminal pressure" and reported their results in terms of tables and charts. This solution is not immediately applicable to actual reservoirs because it does not consider a declining reservoir pressure. To address this limitation, van Everdingen and Hurst applied the superposition theorem to a specific reservoir pressure history. This adaptation usually requires that the reservoir’s pressure history be known. The first step in applying their model is to discretize the time and pressure domains.
Discretization
The time domain is discretized into (n + 1) points (t_{0}, t_{1}, t_{2}, ...., t_{n}), where t_{0} < t_{1} < t_{2} < .... < t_{n} and t_{0} corresponds to t = 0. The average reservoir pressure domain also is discretized into (n + 1) points , where is the initial pressure p_{i}. The time-averaged pressure between levels j and j – 1 is
The time-averaged pressure at level j = 0 is defined as the initial pressure p_{i}. Table 1 shows discretization of t, , and . The time-averaged pressure decrement between levels j and j – 1 is
No value is defined for j = 0. Table 1 shows the complete discretization of t, , , and Δp.
Cumulative water influx
The cumulative water influx at kth level is
where U is the aquifer constant and W_{D} is the dimensionless cumulative water influx. This equation is based on the superposition theorem. The term W_{D} (t_{Dk} – t_{Dj}) is not a product but refers to the evaluation of W_{D} at a dimensionless time difference of (t_{Dk} – t_{Dj}). If we apply Eq. 3 for k = 1, 2, and 3, we obtain
The length of the equation grows with the time. The aquifer constant, U, and the dimensionless cumulative water influx, W_{D}(t_{D}), depend on whether the radial or linear model is applied.
Radial model
The radial model is based on the following equations. The effective reservoir radius is a function of the reservoir PV and is
where:
- r_{o} is expressed in ft
- V_{pr} is the reservoir PV expressed in RB
- ϕ_{r} is the reservoir porosity (fraction)
- h is the pay thickness in ft
The constant f is θ/360, where θ is the angle that defines the portion of the right cylinder. Fig. 3 illustrates the definition of θ for a radial aquifer model. The dimensionless time is
where:
- k_{a} = aquifer permeability (md)
- μ_{w} = water viscosity (cp)
- c_{t} = total aquifer compressibility (psi–1)
- ϕ_{a} = aquifer porosity (fraction)
- t is expressed in years
where U is in units of RB/psi if h is in ft, r_{o} is in ft, and c_{t} is in psi^{–1}.
The dimensionless aquifer radius is
The dimensionless water influx, W_{D}, is a function of t_{D} and r_{eD} and depends on whether the aquifer is infinite acting or finite.
Infinite radial aquifer
The aquifer is infinite acting if r_{e} approaches infinity or if the pressure disturbance within the aquifer never reaches the aquifer’s external boundary. If either of these conditions is met, then W_{D} is
where a_{7} = 4.8534 × 10^{–12}, a_{6} = –1.8436 × 10^{–9}, a_{5} = 2.8354 × 10^{–7}, a_{4} = –2.2740 × 10^{–5}, a_{3} = 1.0284 × 10^{–3}, a_{2} = –2.7455 × 10^{–2}, a_{1} = 8.5373 × 10^{–1}, a_{0} = 8.1638 × 10^{–1}, or
Marsal^{[7]} presented Eqs. 8 and 10. Walsh^{[11]}^{[8]} presented Eq. 9.
Finite radial aquifer
For finite aquifers, Eqs. 8 through 10 apply if t_{D} < t_{D}*, where
If t_{D} > t_{D}*, then
Marsal^{[7]} gave Eqs. 11 through 13. These equations are effective in approximating the charts and tables by van Everdingen and Hurst. Minor discontinuities exist at some of the equation boundaries. A slightly more accurate but much more lengthy set of equations has been offered by Klins et al.^{[6]} Fig. 4 shows W_{D} as a function of t_{D} for r_{eD} = 5, 7.5, 10, 20, and ∞. These equations simplify the application of the VEH model enormously.
A finite aquifer can be treated effectively as an infinite aquifer if
where t_{Dmax} is the maximum value of t_{D}. These equations follow from Eq. 11. For example, if t_{Dmax} is 540 and corresponds to a time of 8 years, then Eq. 15 yields ≥ r_{eD} = 38. Therefore, if the aquifer has a dimensionless radius greater than 38, then the aquifer acts indistinguishably from and equivalent to an infinite aquifer at all times less than 8 years.
Linear aquifer
The aquifer size in the linear model is given in terms of the aquifer/reservoir pore-volume ratio, V_{pa}/V_{pr}.
The aquifer constant is
For edgewater drives, the aquifer length is
where L_{a} and L_{r} are defined in Fig. 2a. For bottomwater drives, the aquifer depth is
where L_{a} is defined in Fig. 2b. The dimensionless time is
Eqs. 20 and 5 use the same units except L_{a} is given in ft. One difference between the linear and radial models is that t_{D} is a function of the aquifer size for the linear model, whereas t_{D} is independent of the aquifer size for the radial model. This difference forces a recalculation of t_{D} in the linear model if the aquifer size is changed. The dimensionless cumulative water influx is
Eq. 21 is by Marsal, ^{[7]} and Eq. 22 is by Walsh. ^{[11]}^{[8]} Fig. 5 shows W_{D} as a function of t_{D}. The aquifer can be treated as infinite if the aquifer length is greater than the critical length.
where t_{max} is the maximum time expressed in years and L_{ac} is in units of ft. Eqs. 20 and 23 use the same units. Alternatively, the aquifer is infinite-acting if t_{D} ≤ 0.50. If infinite-acting and an edgewater drive, W_{e} can be evaluated directly without computing W_{D} and is
where the units in Eq. 5 apply, and W_{e} is in units of RB and h and w are in units of ft.
Calculation procedure
- Discretize the time and average reservoir pressure domains and define t_{j} and for (j = 0, 1, ..., n) according to Table 1.
- Compute the time-averaged reservoir pressure for (j = 1, 2, ..., n) with Eq. 1. Note that = p_{i}.
- Compute the time-averaged incremental pressure differential Δp_{j} for (j = 1, 2, ...., n) with Eq. 2.
- Compute t_{Dj} for (j = 0, 1, ..., n) with Eq. 5 for radial aquifers or with Eq. 20 for linear aquifers.
- Steps 5 through 9 create a computational loop that is repeated n times. The loop index is k, where k = 1, ..., n. For the kth time level, compute (t_{Dk} – t_{Dj}) for (j = 0, ..., k – 1).
- For the kth time level, compute W_{D}(t_{Dk} – t_{Dj}) for (j = 0, ..., k – 1).
- For the kth time level, compute Δp_{j + 1} W_{D}(t_{Dk} – t_{Dj}) for (j = 0, ..., k – 1).
- For the kth time level, compute W_{ek} with Eq. 3.
- Increment the time from level k to k + 1, and return to Step 5 until k > n.
This procedure is highly repetitive and well suited for spreadsheet calculation. The example below illustrates the procedure.
Determining water-influx model parameters
The minimum parameters that need to be specified in the radial model are the aquifer constant, U, the time constant, k_{t}, and the dimensionless aquifer radius, r_{eD}. The time constant combines a number of constants, is the proportionality constant between the dimensionless and real time, and is defined by
Physically, the time constant represents the aquifer conductivity. In summary, U and k_{t} are defined as
Eq. 26 assumes the same units as Eq. 6, and V_{pr} is given in res bbl. Eq. 27 assumes the same units as Eq. 5.
The minimum parameters that need to be specified in the linear model are the aquifer constant, time constant, and aquifer/reservoir PV ratio (V_{pa}/V_{pr}). The aquifer constant and time constant are
Eq. 29 assumes the same units as Eq. 5.
There are three common methods to estimate model parameters:
- Direct measurement
- History matching
- Material balance
The first two methods are described in the following sections. The material-balance method through the McEwen method,^{[12]} is described on the material balance in water drive reservoirs page.
Direct measurement
This method estimates model parameters from direct measurement of the independent constants. Though ideally preferred, this method is rarely possible because of the uncertainty of some of the constants.
For the radial model, the model parameters (U, k_{t}, and r_{eD}) are a function of the following constants:
- r_{e}
- r_{o}
- k_{a}
- h, f
- ϕ_{a}
- c_{t}
- μ_{w}
These constants follow from inspection of Eqs. 6, 7, and 27. The uncertainty among these constants varies. Of these constants, r_{e}, r_{o}, and k_{a} are perhaps the most uncertain. Qualitatively, these constants are related to the aquifer size, reservoir size, and aquifer conductivity (i.e., V_{pa}, V_{pr}, and k_{t}).
For the linear model, the model parameters (U, k_{t}, and V_{pa}/V_{pr}) are a function of the following constants:
- V_{pa}
- V_{pr}
- k_{a}
- ϕ_{a}
- c_{t}
- μ_{w}
- L_{a}
This list follows from inspection of Eqs. 17 and 29. Of these constants, V_{pa}, V_{pr}, L_{a}, and k_{a} are the most uncertain. If h is approximately known, then V_{pa} and L_{a} are not independent but related through Eqs. 18 or 19. Thus, V_{pa}, V_{pr}, and k_{a} are the most uncertain independent constants. Qualitatively, these constants are related to the aquifer size, reservoir size, and aquifer conductivity (i.e., V_{pa}, V_{pr}, and k_{t}, respectively). Note the similarity between the radial and linear models.
In summary, because of the uncertainty of the aquifer size and conductivity and reservoir size, it is difficult to estimate reliably the water-influx model parameters. Nevertheless, every attempt should be made to estimate the median, variance, and range of each constant and the model parameters. This information is helpful in the history-matching method.
History matching
If the water influx history can be estimated, then model parameters can be estimated from history matching. When history matching is used, only the most uncertain constants should be treated as adjustable parameters: preferably only:
- r_{e}, r_{o}, and k_{a} for the radial model
or
- V_{pa}, V_{pr}, and k_{a} for the linear model
Unless only one adjustable parameter exists, history matching is usually complicated by nonuniqueness. ^{[13]}^{[12]}^{[14]}^{[15]}^{[16]} Nonuniqueness, however, can be minimized by limiting the range of parameter adjustment to realistic ranges. The example below illustrates the history-matching procedure.
Example: History matching water influx
Table 2 summarizes the cumulative water influx and average reservoir pressure as a function of time for an initially saturated, black oil reservoir. Areally, the reservoir is approximately semicircular, bounded on one side by a sealing fault and the other side by an aquifer. Fig. 6a shows a schematic representation of the reservoir. Assume the reservoir and aquifer properties in Table 3 apply.
Assume a radial-flow aquifer with f = 0.50. Find the optimal aquifer size (V_{pa} and r_{eD}) that best simulates the water-influx performance. Plot and compare the actual and predicted water-influx histories.
Find the optimal aquifer size (V_{pa}/V_{pr} and L_{a}) that best matches the water-influx performance assuming a linear-flow aquifer. Assume the reservoir width is w = 2r_{o} and length is L_{r} = πr_{o}/4, where r_{o} is given by Eq. 4. Fig. 6b schematically shows the areal interpretation. Plot and compare the actual and predicted water-influx histories. Which model (linear or radial) best matches the data?
Solution. Compute the effective reservoir radius from
where f = 0.50. The total compressibility is the sum of the rock and water compressibilities or c_{t} = 5.88 × 10^{–6} psi^{–1}. The time constant, k_{t}, is given by Eq. 27 and is k_{t} = 0.8682 years^{–1}. U is given by Eq. 26 and is 3,955 RB/psi. Table 4 tabulates t_{D}, , and Δp.
The solution procedure varies r_{eD} until the best match is obtained. Tables 5 through 7 give the details of the calculation for r_{eD} = 5.0. Table 5 tabulates (t_{Dk} – t_{Dj}) for k and j = 0, 1, ..., 7. Table 6 tabulates W_{D}(t_{Dk} – t_{Dj}) for k and j = 0, 1, ..., 7. Table 7 tabulates Δp_{j + 1}W_{D}(t_{Dk} – t_{Dj}) for k and j = 0, 1, ..., 7. Table 4 tabulates ΣΔp_{j+ 1}W_{D}(t_{Dk} – t_{Dj}) for k = 1, 2, ..., 7 for r_{eD} = 5 and W_{ej} for j = 0, 1, ..., 7 for r_{eD} = 5. Fig. 7 plots W_{e} vs. t and compares model and actual results; excellent agreement is noted. These calculations were repeated for other values of r_{eD} but r_{eD} = 5.0 was found to give the best fit of the actual and model water-influx histories. Notice how well the radial model matches the history. A dimensionless aquifer radius of r_{eD} = 5.0 corresponds to V_{pa}/V_{pr} = 20.8. This means that the aquifer is 20.8 times larger than the reservoir.
To determine whether this aquifer can be treated as infinite acting, we evaluate Eq. 15 with t_{Dmax} = 26.08. This calculation yields r_{eD} = 8.1. Because this value of r_{eD} is greater than the history-matched value of r_{eD} = 5.0, this aquifer cannot be treated as infinite.
For the linear aquifer model, the geometry dictates that L_{r} = 11,696 ft if L_{r} = πr_{o}/4 = π (14,892)/4.With the same trial-and-error procedure as used for the radial aquifer, the linear aquifer yields V_{pa}/V_{pr} = 12 for the best match between the actual and predicted water-influx data. This value of V_{pa}/V_{pr} yields L_{a} = 161,145 ft, U = 27,271 RB/psi, and k_{t} = 0.0074 years^{–1}. Fig. 7 compares the predicted and actual data and shows that the match is poor. This comparison reveals that the linear model is not preferable to simulate water influx for this reservoir.
Aquifer performance
The aquifer performance is described in terms of the:
- Delivery rate
- Average aquifer pressure
- Cumulative water-influx volume as a function of time
The aquifer pressure characteristically lags behind the reservoir pressure and is estimated by
The aquifer delivery rate is q_{w} = ∂W_{e}(t)/∂t, which is determined from the slope of the W_{e} vs. t curve.
Fig. 8 shows the aquifer pressure and delivery rate history for the data in the example above. This figure includes the reservoir pressure history for comparison. The qualitative results in Fig. 8 are representative of many aquifers. The water-delivery rate is initially zero and increases rapidly. It peaks after approximately 12 to 14 years and then slowly decreases. The aquifer and reservoir pressures start at equivalent values. The reservoir pressure declines more quickly than the aquifer pressure. The pressure differential between the aquifer and reservoir grows and is approximately 250, 350, and 500 psia, respectively, after 2, 5, and 10 years. The pressure differential peaks after 12 to 14 years and then begins to dissipate. The pressure differential and delivery rate decline together.
The aquifer performance noted in Fig. 8 is not without exception. The qualitative performance in Fig. 8 is characteristic of an initially saturated reservoir. Aquifers feeding initially undersaturated reservoirs may behave quite differently. The difference stems from the difference in the reservoir pressure histories. The reservoir pressure in initially undersaturated oil reservoirs initially declines much more quickly than in initially saturated reservoirs. Consequently, initially undersaturated reservoirs create a substantial pressure differential between the reservoir and aquifer much sooner than initially saturated reservoirs. Of course, this distinction depends on the degree of undersaturation. If the reservoir is significantly undersaturated, a large pressure differential between the reservoir and aquifer is quickly established. This large pressure differential, in turn, promotes water influx; consequently, the water-influx rate increases more rapidly in initially undersaturated reservoirs than initially saturated reservoirs. Once the bubblepoint is reached, the pressure differential between the aquifer and reservoir may decline temporarily. Later, the pressure differential may increase, reminiscent of an initially saturated reservoir, as in Fig. 8. The net effect is that water recharge rate may oscillate in an initially undersaturated oil reservoir.
Nomenclature
Subscripts
i | = | initial condition |
j | = | index |
k | = | index |
References
- ↑ van Everdingen, A.F. and Hurst, W. 1949. The Application of the Laplace Transformation to Flow Problems in Reservoirs. Trans., AIME 186, 305.
- ↑ Carter, R.D. and Tracy, G.W. 1960. An Improved Method for Calculating Water Influx. Trans., AIME 219: 415.
- ↑ Fetkovich, M.J. 1971. A Simplified Approach to Water Influx Calculations—Finite Aquifer Systems. J Pet Technol 23 (7): 814–28. SPE-2603-PA. http://dx.doi.org/10.2118/2603-PA
- ↑ Schilthuis, R.J. 1936. Active Oil and Reservoir Energy. Trans., AIME 118: 33.
- ↑ Coats, K.H. 1970. Mathematical Methods for Reservoir Simulation. Presented by the College of Engineering, University of Texas at Austin, 8–12 June 1970.
- ↑ ^{6.0} ^{6.1} Klins, M.A., Bouchard, A.J., and Cable, C.L. 1988. A Polynomial Approach to the van Everdingen-Hurst Dimensionless Variables for Water Encroachment. SPE Res Eng 3 (1): 320-326. SPE-15433-PA. http://dx.doi.org/10.2118/15433-PA
- ↑ ^{7.0} ^{7.1} ^{7.2} ^{7.3} Marsal, D. 1982. Topics of Reservoir Engineering. Course Notes, Delft U. of Technology.
- ↑ ^{8.0} ^{8.1} ^{8.2} Walsh, M.P. 1996. A Generalized Approach to Petroleum Reservoir Engineering. Austin, Texas: Petroleum Recovery Research Inst.
- ↑ ^{9.0} ^{9.1} Allard, D.R. and Chen, S.M. 1988. Calculation of Water Influx for Bottomwater Drive Reservoirs. SPE Res Eng 3 (2): 369-379. SPE-13170-PA. http://dx.doi.org/10.2118/13170-PA
- ↑ Field, M.B., Givens, J.W., and Paxman, D.S. 1970. Kaybob South - Reservoir Simulation of a Gas Cycling Project with Bottom Water Drive. J Pet Technol 22 (4): 481-492. SPE-2640-PA. http://dx.doi.org/10.2118/2640-PA
- ↑ ^{11.0} ^{11.1} Walsh, M.P. and Lake, L.W. 2003. A Generalized Approach to Primary Hydrocarbon Recovery. Amsterdam: Elsevier.
- ↑ ^{12.0} ^{12.1} 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
- ↑ 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
- ↑ Woods, R.W. and Muskat, M.M. 1945. An Analysis of Material-Balance Calculations. Trans., AIME 160: 124.
- ↑ 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
- ↑ 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
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