1. Dr.Mostafa Mahmoud Kinawy
Water Influx
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Prospecting w/Electric Logs
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Dr.Mostafa Mahmoud Kinawy

2. Nearly all hydrocarbon reservoirs are surrounded by water-bearing rocks called aquifers. These aquifers may be substantially larger than the oil or gas reservoirs they adjoin as to appear infinite in size, or they may be so small in size as to be negligible in their effect on reservoir perfor­mance.
As reservoir fluids are produced and reservoir pressure declines, a pressure differential develops from the surrounding aquifer into the reser­voir. Following the basic law of fluid flow in porous media, the aquifer reacts by encroaching across the original hydrocarbon-water contact. In some cases, water encroachment occurs due to hydrodynamic conditions and recharge of the formation by surface waters at an outcrop.

3. In many cases, the pore volume of the aquifer is not significantly larg­er than the pore volume of the reservoir itself. Thus, the expansion of the water in the aquifer is negligible relative to the overall energy system, and the reservoir behaves volumetrically.
In this case, the effects of water influx can be ignored. In other cases, the aquifer permeability may be sufficiently low such that a very large pressure differential is required before an appreciable amount of water can encroach into the reservoir. In this instance, the effects of water influx can be ignored as well.

4. This chapter focuses on those reservoir-aquifer systems in which the size of the aquifer is large enough and the permeability of the rock is high enough that water influx occurs as the reservoir is depleted.
This chapter also provides various water influx calculation models and a detailed description of the computational steps involved in applying these models.

5. CLASSIFICATION OF AQUIFERS
Many gas and oil reservoirs produced by a mechanism termed water drive. Often this is called natural water drive to distinguish it from artifi­cial water drive that involves the injection of water into the formation. Hydrocarbon production from the reservoir and the subsequent pressure drop prompt a response from the aquifer to offset the pressure decline. This response comes in a form of water influx, commonly called water encroachment, which is attributed to:
Expansion of the water in the aquifer
Compressibility of the aquifer rock
Artesian flow where the water-bearing formation outcrop is located structurally higher than the pay zone

6. Reservoir-aquifer systems are commonly classified on the basis of:
Degree of pressure maintenance
• Flow regimes
Outer boundary conditions
• Flow geometries

7. Degree of Pressure Maintenance
Based on the degree of the reservoir pressure maintenance provided by the aquifer, the natural water drive is often qualitatively described as:
Active water drive
Partial water drive
Limited water drive
The term active water drive refers to the water encroachment mecha­nism in which the rate of water influx equals the reservoir total produc­tion rate. Active water-drive reservoirs are typically characterized by a gradual and slow reservoir pressure decline.

8. Outer Boundary Conditions
•The aquifer can be classified as infinite or finite (bounded). Geologi­cally all formations are finite, but may act as infinite if the changes in the pressure at the oil-water contact are not “felt” at the aquifer boundary. In general, the outer boundary governs the behavior of the aquifer and, therefore:
a. Infinite system indicates that the effect of the pressure changes at the oil/aquifer boundary can never be felt at the outer boundary. This boundary is for all intents and purposes at a constant pressure equal to initial reservoir pressure.
b. Finite system indicates that the aquifer outer limit is affected by the influx into the oil zone and that the pressure at this outer limit changes with time.

9. Flow Regimes
There are basically three flow regimes that influence the rate of water influx into the reservoir. Those flow regimes are:
a. Steady-state
b. Semi-steady (pseudo-steady)-state
c. Unsteady-state

10. Flow Geometries
Reservoir-aquifer systems can be classified on the basis of flow geom­etry as:
a. Edge-water drive
b. Bottom-water drive
c. Linear-water drive

11. Recognition of Natural Water Influx
Normally very little information is obtained during the exploration-devel­opment period of a reservoir concerning the presence or characteristics of an aquifer that could provide a source of water influx during the depletion period.
Natural water drive may be assumed by analogy with nearby producing reservoirs, but early reservoir performance trends can provide clues. A comparatively low, and decreasing, rate of reservoir pressure decline with increasing cumulative withdrawals is indicative of fluid influx.

12. Figure 1. Flow geometries
Linear-water Drive

13. Indications of fluid influx.
Early water production from edge wells is indicative of water encroachment. Such observations must be tempered by the possibility that the early water production is due to formation fractures; thin, high permeability streaks; or to coning in connection with a limited aquifer. The water production may be due to casing leaks.
If the reservoir pressure is below the oil saturation pressure, a low rate of increase in produced gas-oil ratio is also indicative of fluid influx.
Calculation of increasing original oil-in-place from successive reservoir pressure surveys by using the material balance assuming no water influx is also indicative of fluid influx.

14. WATER INFLUX MODELS
Several models have been developed for estimating water influx that are based on assumptions that describe the characteristics of the aquifer.
The mathematical water influx models that are commonly used in the petroleum industry include:
Pot aquifer
Schilthuis’ steady-state
Hurst’s modified steady-state
The Van Everdingen-Hurst unsteady-state
- Edge-water drive
- Bottom-water drive
The Carter-Tracy unsteady-state
Fetkovich’s method
- Radial aquifer
- Linear aquifer

15. The Pot Aquifer Model
The simplest model that can be used to estimate the water influx into a gas or oil reservoir is based on the basic definition of compressibility. A drop in the reservoir pressure, due to the production of fluids, causes the aquifer water to expand and flow into the reservoir. The compress­ibility is defined mathematically as:
V = c V  p …….. (1)
Applying the above basic compressibility definition to the aquifer gives:
Water influx = (aquifer compressibility) (initial volume of water) (pressure drop)

16. or
We = (cw + cf) Wi (pi - p) …………….(2)
where We = cumulative water influx, bbl
cw = aquifer water compressibility, psi-1
cf = aquifer rock compressibility, psi-1
Wi = initial volume of water in the aquifer, bbl
pi = initial reservoir pressure, psi
p = current reservoir pressure (pressure at oil-water contact), psi

17. Calculating the initial volume of water in the aquifer requires the knowledge of aquifer dimension and properties. These, however, are seldom measured since wells are not deliberately drilled into the aquifer to obtain such information. For instance, if the aquifer shape is radial, then:
……………(3(

18. Equation (2) suggests that water is encroaching in a radial form from all directions. Quite often, water does not encroach on all sides of the reservoir, or the reservoir is not circular in nature.
To account for these cases, a modification to Equation (1) must be made in order to properly describe the flow mechanism. One of the simplest modifications is to include the fractional encroachment angle f in the equation, to give:
We = (cw + cf) Wi f (pi - p) …….(4)
……..(5)

19. The above model is only applicable to a small aquifer, i. e
The above model is only applicable to a small aquifer, i.e., pot aquifer, whose dimensions are of the same order of magnitude as the reservoir itself. Dake (1978) points out that because the aquifer is considered relatively small, a pressure drop in the reservoir is instantaneously transmitted throughout the entire reservoir-aquifer system. Dake suggests that for large aquifers, a mathematical model is required which includes time dependence to account for the fact that it takes a finite time for the aquifer to respond to a pressure change in the reservoir.

20. Schilthuis’ Steady-State Model
Schilthuis (1936) proposed that for an aquifer that is flowing under the steady-state flow regime, the flow behavior could be described by Darcy’s equation. The rate of water influx ew can then be determined by applying Darcy’s equation:
……………(6)

21. The last relationship can be more conveniently expressed as:
……….(7)

22. The parameter C is called the water influx constant and is expressed in bbl/day/psi. This water influx constant C may be calculated from the reservoir historical production data over a number of selected time intervals, provided that the rate of water influx ew has been determined independently from a different expression.
If the steady-state approximation adequately describes the aquifer flow regime, the calculated water influx constant C values will be constant over the historical period.
Note that the pressure drops contributing to influx are the cumulative pressure drops from the initial pressure.

23. In terms of the cumulative water influx We, the common Schilthuis expression for water influx is:
Equation (8) may be written in the following form:
…………(8)

24. Hurst’s Modified Steady-State Model
……………….(9)
One of the problems associated with the Schilthuis’ steady-state model is that as the water is drained from the aquifer, the aquifer drainage radius ra will increase as the time increases. Hurst (1943) proposed that the “apparent” aquifer radius ra would increase with time and, therefore the dimensionless radius ra/re may be replaced with a time dependent function, as:
ra/re = at …………………(10)

25. Substituting Equation (10)into Equation (6) gives:
………..(11)
………..(12)
………..(13)

26. The Hurst modified steady-state equation contains two unknown constants a and C, that must be determined from the reservoir aquifer pressure and water influx historical data. The procedure of determining the constants a and C is based on expressing Equation (11) as a linear relationship.
………..(14)

27. Figure 2: Modified Steady State Water Influx Model
Equation (14) indicates that a plot of (pi - p)/ew versus ln(t) will be a straight line with a slope of 1/C and intercept of (1/C)ln(a), as shown schematically in Figure (2).
Determination of C and n

28. Everdingen-Hurst Unsteady-State Model
The mathematical formulations that describe the flow of crude oil system into a wellbore are identical in form to those equations that describe the flow of water from an aquifer into a cylindrical reservoir, as shown in Figure (3)
When an oil well is brought on production at a constant flow rate after a shut-in period, the pressure behavior is essentially controlled by the transient (unsteady-state) flowing condition. This flowing condition is defined as the time period during which the boundary has no effect on the pressure behavior.
The dimensionless form of the diffusivity equation, is basically the general mathematical equation that is designed to model the transient flow behavior in reservoirs or aquifers.

29. Figure 3. Water influx into a cylindrical reservoir.

30. In a dimensionless form, the diffusivity equation takes the form:
Van Everdingen and Hurst (1949) proposed solutions to the dimensionless diffusivity equation for the following two reservoir aquifer boundary conditions:
Constant terminal rate
Constant terminal pressure
……………(15)

31. For the constant terminal rate boundary condition, the rate of water influx is assumed constant for a given period; and the pressure drop at the reservoir-aquifer boundary is calculated. For the constant terminal pressure boundary condition, a boundary pressure drop is assumed constant over some finite time period, and the water influx rate is determined.
In the description of water influx from an aquifer into a reservoir, there is greater interest in calculating the influx rate rather than the pressure.

32. This leads to the determination of the water influx as a function of a given pressure drop at the inner boundary of the reservoir-aquifer system.
Van Everdingen and Hurst solved the diffusivity equation for the aquifer-reservoir system by applying the Laplace transformation to the equation. The authors’ solution can be used to determine the water influx in the following systems:
Edge-water-drive system (radial system)
Bottom-water-drive system
Linear-water-drive system

33. Edge-Water Drive
Figure 4. shows an idealized radial flow system that represents an edge-water-drive reservoir. The inner boundary is defined as the interface between the reservoir and the aquifer.
The flow across this inner boundary is considered horizontal and encroachment occurs across a cylindrical plane encircling the reservoir. With the interface as the inner boundary, it is possible to impose a constant terminal pressure at the inner boundary and determine the rate of water influx across the interface.

34. Figure 4. Idealized radial flow model

35. Van Everdingen and Hurst proposed a solution to the dimensionless diffusivity equation that utilizes the constant terminal pressure condition in addition to the following initial and outer boundary conditions:
Initial conditions:
p = pi for all values of radius r
Outer boundary conditions
• For an infinite aquifer
p = pi at r = ∞
• For a bounded aquifer

36. Van Everdingen and Hurst assumed that the aquifer is characterized by:
Uniform thickness
Constant permeability
Uniform porosity
Constant rock compressibility
Constant water compressibility
The authors expressed their mathematical relationship for calculating the water influx in a form of a dimensionless parameter that is called dimensionless water influx WeD. They also expressed the dimensionless water influx as a function of the dimensionless time tD and dimensionless radius rD, thus they made the solution to the diffusivity equation generalized and applicable to any aquifer where the flow of water into the reservoir is essentially radial.

37. Figure 5. Dimensionless water influx WeD for several values of ra/re
Figure 5. Dimensionless water influx WeD for several values of ra/re. (Van Everdingen and Hurst WeD. Permission to publish by the SPE.)
The authors presented their solution in tabulated and graphical forms .

38. The two dimensionless parameters tD and rD are given by:
(16(
(17)
(18(

39. The water influx is then given by:
(19)
(20)

40. Table 10-1

41. Table 10-2

42. Equation (20 )assumes that the water is encroaching in a radial form
Equation (20 )assumes that the water is encroaching in a radial form. Quite often, water does not encroach on all sides of the reservoir, or the reservoir is not circular in nature. In these cases, some modifications must be made in Equation (20) to properly describe the flow mechanism. One of the simplest modifications is to introduce the encroachment angle to the water influx constant B as:
Ә is the angle subtended by the reservoir circumference, i.e., for a full circle Ә = 360° and for semicircle reservoir against a fault Ә =180°, as shown in Figure
(21)
(22)

43. Figure 6: Water Drive Reservoir