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
• Carter-Tracy model
• Fetkovich model
• Schilthuis model
• Small- or pot-aquifer model

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, the need for alternatives has diminished.

Allard and Chen 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.

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

• 

  Fig. 1 – Radial aquifer model.

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.

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).

• 

  Fig. 2 – Linear aquifer model for (a) an edgewater drive and (b) a bottomwater drive.

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 (t₀, t₁, t₂, ...., t_n), where t₀ < t₁ < t₂ < .... < t_n and t₀ 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

...............(1)

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

...............(2)

No value is defined for j = 0. Table 1 shows the complete discretization of t, , , and Δp.

• 

  Table 1- Time and pressure domain discretization for van Everdingen-Hurst model.

Cumulative water influx

The cumulative water influx at kth level is

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

(3)

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

......................(4)

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

.....................(5)

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

......................(6)

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.

• 

  Fig. 3 – Definition of angle, θ, for radial aquifer model.

The dimensionless aquifer radius is

.......................(7)

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

........................(8)

........................(9)

where a₇ = 4.8534 × 10^–12, a₆ = –1.8436 × 10^–9, a₅ = 2.8354 × 10^–7, a₄ = –2.2740 × 10^–5, a₃ = 1.0284 × 10^–3, a₂ = –2.7455 × 10^–2, a₁ = 8.5373 × 10^–1, a₀ = 8.1638 × 10^–1, or

........................(10)

Marsal presented Eqs. 8 and 10. Walsh presented Eq. 9.

Finite radial aquifer

For finite aquifers, Eqs. 8 through 10 apply if t_D < t_D*, where

.....................(11)

and .................(7)

If t_D > t_D*, then

.....................(12)

where ...............(13)

Marsal 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. 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.

• 

  Fig. 4 – W_D vs. t_D for a radial aquifer model.

A finite aquifer can be treated effectively as an infinite aquifer if

...............(14)

...............(15)

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.

...............(16)

The aquifer constant is

...............(17)

For edgewater drives, the aquifer length is

...............(18)

where L_a and L_r are defined in Fig. 2a. For bottomwater drives, the aquifer depth is

...............(19)

where L_a is defined in Fig. 2b. The dimensionless time is

...............(20)

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

...............(21)

and ...............(22)

Eq. 21 is by Marsal, and Eq. 22 is by Walsh. 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.

...............(23)

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

...............(24)

where the units in Eq. 5 apply, and W_e is in units of RB and h and w are in units of ft.

• 

  Fig. 5 – W_D vs. t_D for linear aquifer model.

Calculation procedure

1. Discretize the time and average reservoir pressure domains and define t_j and for (j = 0, 1, ..., n) according to Table 1.
2. Compute the time-averaged reservoir pressure for (j = 1, 2, ..., n) with Eq. 1. Note that = p_i.
3. Compute the time-averaged incremental pressure differential Δp_j for (j = 1, 2, ...., n) with Eq. 2.
4. Compute t_Dj for (j = 0, 1, ..., n) with Eq. 5 for radial aquifers or with Eq. 20 for linear aquifers.
5. 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).
6. For the kth time level, compute W_D(t_Dk – t_Dj) for (j = 0, ..., k – 1).
7. For the kth time level, compute Δp_{j + 1} W_D(t_Dk – t_Dj) for (j = 0, ..., k – 1).
8. For the kth time level, compute W_ek with Eq. 3.
9. 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

...............(25)

Physically, the time constant represents the aquifer conductivity. In summary, U and k_t are defined as

...............(26)

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

(27)

and ............(28)

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

...............(28)

and ............(29)

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, 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. 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.

• 

  Table 2 - Reservoir performance data for example.

• 

  Table 3 - Reservoir and fluid properties for example.

• 

  Fig. 6 – Areal schematic of a reservoir partially surrounded by a radial aquifer; (b) interpretation of reservoir geometry when linear aquifer model is applied.

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

...............(30)

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.

• 

  Table 4 - Predicted water influx history for radial aquifer if r_eD=5.0.

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.

• 

  Table 5 - Summary of (t_Dk - t_Dj) calculations for radial aquifer.

• 

  Table 6 - Summary of W_D(t_Dk - t_Dj)calculations for radial aquifer.

• 

  Table 7 - Summary of Δp_j+1W_D(t_Dk - t_Dj)calculations for radial aquifer.

• 

  Fig. 7 – Comparison of actual and predicted water-influx histories for Example 1.

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

.....................(31)

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.

• 

  Fig. 8 – Predicted aquifer pressure and water-influx rate histories for example.

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

cf	=	rock compressibility, Lt2/m, 1/psi
ct	=	total aquifer compressibility, Lt2/m, 1/psi
f	=	reservoir angle
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
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
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
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
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
Δp	=	difference of time-averaged pressure, m/Lt2, psi
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

Subscripts

i	=	initial condition
j	=	index
k	=	index

Noteworthy papers in OnePetro

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External links

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See also

Water drive reservoirs

Material balance in water drive reservoirs

Oil fluid characteristics

Oil fluid properties

PEH:Oil_Reservoir_Primary_Drive_Mechanisms