PEH:Waterflooding

Publication Information

Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume V – Reservoir Engineering and Petrophysics

Edward D. Holstein, Editor

Copyright 2007, Society of Petroleum Engineers

Chapter 11 – Waterflooding

H.R. (Hal) Warner Jr., SPE, Warner Consulting Services

Pgs. 1037-1102

ISBN 978-1-55563-120-8
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This chapter concerns the use of water injection to increase the production from oil reservoirs, and the technologies that have been developed over the past 50+ years to evaluate, design, operate, and monitor such projects. Use of water to increase oil production is known as "secondary recovery" and typically follows "primary production," which uses the reservoir’s natural energy (fluid and rock expansion, solution-gas drive, gravity drainage, and aquifer influx) to produce oil.

The principal reason for waterflooding an oil reservoir is to increase the oil-production rate and, ultimately, the oil recovery. This is accomplished by "voidage replacement"—injection of water to increase the reservoir pressure to its initial level and maintain it near that pressure. The water displaces oil from the pore spaces, but the efficiency of such displacement depends on many factors (e.g., oil viscosity and rock characteristics). In oil fields such as Wilmington (California, U.S.A.) and Ekofisk (North Sea), voidage replacement also has been used to mitigate additional surface subsidence. In these cases, the high porosity of the unconsolidated sandstones of the Wilmington oil field’s reservoirs and of the soft chalk reservoir rock in the Ekofisk oil field had compacted significantly when the reservoir pressure was drawn down during primary production.

This chapter discusses various aspects of waterflooding briefly, and it mainly covers the fundamental considerations involved when engineering, designing, operating, and monitoring a waterflood. Waterflooding technology is a large and well-covered topic in the literature, and the ability of this chapter to cover all aspects of these technologies is limited. Over the past 40 years, SPE has published three significant and in-depth books written by Craig, Willhite, and Rose et al. that address waterflooding technology.

SPE Reprint Series Vol. 2a contains the most significant SPE technical papers that describe waterflooding technology as of 1973. A recent search of the SPE eLibrary using keyword "waterflood" identified more than 5,500 SPE technical papers, including 650+ whose titles contain "waterflood."

Overview

Historic Context

In the early days of the oil industry, saline water or brine frequently was produced from a well along with oil, and as the oil-production rate declined, the water-production rate often would increase. This water typically was disposed of by dumping it into nearby streams or rivers. In the 1920s, the practice began of reinjecting the produced water into porous and permeable subsurface formations, including the reservoir interval from which the oil and water originally had come. By the 1930s, reinjection of produced water had become a common oilfield practice.

Reinjection of water was first done systematically in the Bradford oil field of Pennsylvania, U.S.A. There, the initial "circle-flood" approach was replaced by a "line flood," in which two rows of producing wells were staggered on both sides of an equally spaced row of water-injection wells. In the 1920s, besides the line flood, a "five-spot" well layout was used (so named because its pattern is like that of the five spots on a die).

Much of waterflooding’s technology and common practice developed in the U.S. between 1940 and 1970. By the mid-1940s, the onshore U.S. oil industry was maturing and primary production from many of its reservoirs had declined significantly, whereas most reservoirs elsewhere in the world were in the early stages of primary production. Also, in the U.S., thousands of wells had been drilled that were closely spaced, so that the effects of water injection were more obvious and so were more quickly understandable.

In addition to the need to dispose of saline water that was produced along with the oil, several other factors made waterflooding a logical and economical method for increasing recovery from oil fields. Very early on, it was recognized that in most reservoirs, only a small percentage of the original oil in place (OOIP) was being recovered during the primary-production period because of depletion of the reservoirs’ natural energy. Additional recovery methods were needed to produce the large quantity of oil that remained. Water injection’s early success in lengthening the oil-production period by years made waterflooding the natural step after primary production to recover additional oil from reservoirs whose oil-production rate had declined to very low levels.

Other key factors that drove waterflooding’s development and increasing use were that water is inexpensive; that water generally is readily available in large quantities from nearby streams, rivers, or oceans, or from wells drilled into shallower or deeper subsurface aquifers; and that water injection effectively made production wells that were near the water-injection wells flow or be pumped at higher rates because of the increased reservoir pressure. Concurrently, the scientific reasons behind waterflooding’s success were identified (i.e., that water has viscosity, density, and wetting properties, compared to oil, that affect how efficiently it will displace various oils from reservoir rock).

By the 1970s, most onshore oil fields in the U.S., USSR, and China, for which waterflooding was the logical recovery process, were being produced by use of this technology in various well-pattern arrangements. Some U.S. offshore oil fields and oil fields elsewhere in the world were receiving water injection as deemed appropriate by their owners and operators. Since then, many large-scale water-injection projects have been applied to oil reservoirs in locations ranging from far offshore in the North Sea to the Arctic regions to desert areas.

Chapter Topics

This chapter presents the subject of waterflooding in nine sections and follows the logic sequence used by Willhite:

• Microscopic efficiency of immiscible displacement.
• Macroscopic displacement efficiency of a linear waterflood.
• Reservoir-geology considerations in the design and operation of waterfloods.
• Immiscible displacement in two dimensions—areal.
• Vertical displacement in linear and areal models.
• Waterflood design.
• Waterflood monitoring.
• Field case studies: waterflood examples.
• Summary and conclusions.

The sections begin with a discussion of unit-displacement efficiency, which is how water displaces oil from a porous and permeable reservoir rock on a microscopic scale. This is the level of analysis that is applied when water-/oil-flow measurements are made on small core-plug samples in a laboratory. Calculations for determining how well waterflooding will work on a reservoir scale must include the effects of geology, gravity, and geometry (vertical, areal, and well-spacing/-pattern arrangement). The formula for overall waterflood oil-recovery efficiency ER might be simply stated as the product of three independent terms:

....................(11.1)

where E_D = the unit-displacement efficiency, E_I = the vertical-displacement efficiency, and E_A = the areal-displacement efficiency. Of course, assuming independence of these three factors is not valid for real oil reservoirs.

Also, a waterflood is a dynamic process that lasts for several decades; hence, after the project has been initiated, there are opportunities to modify the original waterflood design and operating guidelines on the basis of analysis of the actual field production data. This is why real-time monitoring of waterflood performance is required, both at the injection wells and at the production wells.

Throughout this chapter, keep in mind that the most important aspect of evaluating a field waterflooding project is understanding the reservoir rocks. This understanding begins with knowing the depositional environment at the pore and reservoir levels and possibly also several levels in between. Second, the diagenetic history of the reservoir rocks must be ascertained. Then, the structure and faulting of the reservoir must be determined to understand the interconnectivities among the various parts of the reservoir, particularly the injector/producer connectivity. Finally, the water/oil/rock characteristics need to be understood because they control wettability, residual oil saturation to waterflooding, and the oil relative permeability at higher water saturations. Because of these needs, there always should be a developmental geologist on the waterflood-evaluation team.

This chapter does not include a separate section on oil properties and their impact on waterflooding recovery efficiency, but oil properties are important to technical and economic success and are discussed as appropriate throughout this chapter. The key oil properties are viscosity and density at reservoir conditions. In a porous medium, the mobility of a fluid is defined as its endpoint relative permeability divided by its viscosity; hence, a fluid with a low viscosity (≤ 1 cp) has a high mobility unless its relative permeability is very low. Similarly, a low-API crude oil (≤ 20°API) has a high viscosity and a very low mobility unless it is heated to high temperatures. Because water’s viscosity at reservoir temperatures generally is much lower than or, at best, equal to that of the reservoir oil, the water-/oil-viscosity ratio is generally much greater than 1:1. As discussed in some detail later in this chapter, the water-/oil-mobility ratio is a key parameter in determining the efficiency of the water/oil displacement process, with the recovery efficiency increasing as the water-/oil-mobility ratio decreases.

Topics Covered Elsewhere in This Handbook

Topics that concern typical water-injection operations but that are not discussed in this chapter are (1) aquifer or bottomwater drives with water injected into underlying aquifer intervals; (2) use of numerical reservoir simulators to analyze waterflood performance; and (3) enhanced-oil-recovery (EOR) methods that involve either continuous or alternating water injection. Discussions of these and other aspects of waterflooding technologies can be found elsewhere in this Handbook, including in the numerical-reservoir-simulation chapter in the Reservoir Engineering and Petrophysics volume, in the General Engineering volume chapters about crude-oil properties and water properties, in the Production Operations Engineering volume chapters about handling of oil and water production during waterflooding operations, and in the Facilities and Construction Engineering volume chapters about surface facilities that are required for waterflood operations.

Limitations of Waterflood Technology

Waterflooding can increase the volume of oil recovered from a reservoir; however, it is not always the best technology to use and it can have complicating factors. When evaluating how best to produce a particular oil reservoir, a petroleum engineer should include waterflooding in the options that are analyzed, both technically and economically. Those evaluations should include such potentially complicating factors as compatibility of the planned injected water with the reservoir’s connate water; interaction of the injected water with the reservoir rock (clay sensitivities, rock dissolution, or generally weakening the rock framework); injection-water treatment to remove oxygen, bacteria, and undesirable chemicals; and the challenges involved in separating and handling the produced water that has trace oil content, naturally occurring radioactive materials (NORMs), and various scale-forming minerals.

Microscopic Efficiency of Immiscible

This section discusses the conceptual aspects of the displacement of oil by water in reservoir rocks. Fig. 11.1 is a schematic diagram of the water/oil displacement process. At the pore level (i.e., where the water and oil phases interact immiscibly when moving from one set of pores to the next), wettability and pore geometry are the two key considerations. The interplay between wettability and pore geometry in a reservoir rock is what is represented by the laboratory-determined capillary pressure curves and water/oil relative permeability curves that engineers use when making OOIP and fluid-flow calculations. The sections below discuss these basic concepts and their implications for initial water- and oil-saturation distribution, for relative permeability, and for how initial gas saturation will affect water/oil flow behavior.

• 

  Fig. 11.1 – Saturation profile during a waterflood.

Wettability

Wettability is defined in terms of the interaction of two immiscible phases, such as oil and water, and a solid surface, such as that of the pores of a reservoir rock. For understanding wettability concepts and for simple laboratory determinations, the solid surface is taken as a smooth flat surface. Fig. 11.2 illustrates two styles of wettability: water-wet and oil-wet. Eq. 11.2 describes the force relationship that is in balance for the drop of water that is on the solid surface and is surrounded by oil. The interfacial tension (IFT) between the oil and water phases varies depending on the compositions of the phases but generally is relatively high, in the 10- to 30-dyne/cm range. The contact angle θ is used to define which fluid phase is more wetting—for low contact angles, the water phase is more wetting, whereas for high contact angles, the oil phase is more wetting.

....................(11.2)

where σ_os = the IFT between the oil and solid phases, σ_ws = the IFT between the water and solid phases, and σ_ow = the IFT between the oil and water phases.

• 

  Fig. 11.2 – Wettability of oil/water/solid system.

The particular contact angle depends on many variables, including the composition of the crude oil and the amount of gas in solution; the salinity and pH of the connate brine; the mineralogy of the rock surfaces; and the salinity and pH of the injected water that is used for waterflooding. The concentration of surface-active components (e.g., asphaltenes) that are in the crude oil and that can adsorb on the rock surfaces affects wettability.

Reservoir rocks typically are described as being water-wet, oil-wet, or intermediate-wet. A water-wet rock surface is one that has a strong preference to be coated, or "wetted," by the water phase, so that there will be a continuous water phase on the rock surfaces. Oil-wet rocks prefer to be coated with oil instead of water. Strongly oil-wet rocks have been created for laboratory studies but, as discussed below, are unlikely to exist in real reservoirs. Intermediate-wet reservoir rocks have been found in several oil reservoirs. The term "dalmatian wetting" describes reservoir rocks that have both oil-wet and water-wet surfaces. Fig. 11.3 illustrates two styles of intermediate-wetting.

• 

  Fig. 11.3 – Relationship of mineralogy to wetting conditions: (a) dalmation wetting and (b) mixed wetting.

Two types of laboratory measurements commonly are used to estimate wettability. First, the crude-oil/brine IFT values can be measured on smooth rock surfaces of various mineralogies. Second, Amott tests can be run on the reservoir rock to determine the extents to which it imbibes oil and brine. When running the Amott tests, it is critical to initialize the core plugs as close to original reservoir conditions as possible either by using well-preserved core samples or by aging the core plugs in the presence of reservoir crude oil. High-quality water/oil capillary-pressure (P_c) and water/oil relative permeability (k_rwo) data, both of which are strongly affected by rock wettability, are needed as input to waterflood calculations, whether using simple engineering methods or complex numerical reservoir simulators.

Pore Geometry

The pore geometry for any reservoir rock is the result of its depositional and diagenetic history. The depositional environment determines a rock’s grain size and sorting. Post-depositional diagenetic changes caused by various types of cementation, leaching, and clay alteration will impact a rock’s pore characteristics whether the rock is primarily silica or carbonate. A chapter in the General Engineering volume of this Handbook discusses many of these factors and how they affect the single-phase permeability of a reservoir rock.

Figs. 11.4 and 11.5 show photomicrographs and k_rwo curves for a sandstone with large, well-connected pores and for one with small, well-connected pores, respectively. These illustrate just one of many possible differences in pore geometry. Pore distributions in carbonate rocks often are more complicated because of vug networks and fractures. Also, there are many scales of pore-geometry heterogeneities; a core plug has one scale of pore-size variation, but other important variations are found at each higher scale.

• 

  Fig. 11.4 – Photomicrograph (a) and water/oil relative permeability curve (b) for a sandstone with large, well-connected pores. k_a = air permeability, md; k_ro = relative permeability to oil, fraction; and k_rw = relative permeability to water, fraction.

• 

  Fig. 11.5 – Photomicrograph (a) and water/oil relative permeability curve (b) for a sandstone with small, well-connected pores.

Capillary Pressure

Capillary-pressure concepts are discussed in detail in a chapter of the General Engineering volume of this Handbook. This chapter considers the characteristics of and differences between the drainage and imbibition capillary-pressure/water-saturation (P_c/S_w) curves. Capillary pressure affects waterflood performance and engineering calculations because the extent to which the water/oil flood front is vertically and horizontally "smeared out" during the waterflood is controlled by the P_c/S_w imbibition curve.

Reservoir rocks are considered to be water-wet initially because all reservoir rocks were deposited in water-filled environments or were immersed in water soon after deposition, when their overlying sediments were deposited. The drainage P_c/S_w curve describes the drainage process, or the P_c/S_w relationship while the nonwetting-fluid phase (oil) displaces the wetting-fluid phase (brine) from various parts of the pore system, thus decreasing the wetting-phase saturation. If during the displacement process the process is reversed and the wetting-phase saturation increases, it is known as imbibition (the imbibing of the wetting phase).

Fig. 11.6 shows the drainage and imbibition P_c/S_w characteristics of a strongly water-wet rock. The minimum wetting or water saturation from the drainage process is termed the connate (or irreducible) water saturation. The maximum water saturation from the imbibition process defines the minimum nonwetting-phase saturation, or (for waterflooding considerations) the residual oil saturation to waterflooding S_orw. Figs. 11.7 and 11.8 show the drainage and imbibition P_c/S_w curves from the laboratory tests of an oil-wet rock and a rock with intermediate wettability, respectively. Fig. 11.8 includes both the spontaneous (number 2 on curve) and the forced (number 3 on curve) portions of the imbibition curve. Spontaneous imbibition occurs without any pressure being applied to the test apparatus, whereas obtaining the forced imbibition portion of the curve requires an external pressure to be applied. Note that P_c = 0 does not define the S_orw.

• 

  Fig. 11.6 – Capillary pressure characteristics for a strongly water-wet rock. Curve 1 represents drainage, and Curve 2 represents imbibition.

• 

  Fig. 11.7 – Water/oil capillary pressure characteristics for Tensleep Sandstone oil-wet rock. Curve 1 represents drainage, and Curve 2 represents imbibition.

• 

  Fig. 11.8 – Water/oil capillary-pressure characteristics for intermediate wettability. Curve 1 represents drainage, Curve 2 represents spontaneous imbibition, and Curve 3 represents forced imbibition.

Initial Water-/Oil-Saturation Distribution

An oil field’s initial water-/oil-saturation distribution depends on its hydrocarbon history and has a significant effect on its waterflooding potential. The pore system in a reservoir rock contains a very large number of pore bodies whose filling by oil is controlled by the diameters of the pore throats that link them.

During the oil-filling process, the oil first enters through the largest pore throats, and all other parts of the pore system remain filled with connate brine. As more oil enters the reservoir trap, the oil column lengthens downward. Just above the oil/water contact, only the pores that are accessible from the largest pore throats fill with oil. At the top of the oil column, where the capillary pressure is greatest, not only the largest pores are oil-filled, but also some that have smaller pore throats. The very fine pore spaces remain filled with connate brine.

This process continues until the oil column reaches its maximum length. This whole process is the drainage cycle of the P_c/S_w curves. At this point in the process, oil is filling the largest pores and water is filling the smallest pores; however, the P_c/S_w drainage curve governs the percentage of each. Connate brine will remain as films on the surfaces of the largest pores, but surface-active components of the crude oil might adsorb on some of the pore surfaces, rendering them oil-wet. Hence, the overall system can have mixed-wet characteristics.

There are oil fields that, although initially filled through a drainage process, when discovered were on the imbibition cycle because of a complicated hydrocarbon or structural history. Portions of several west Texas San Andres carbonate reservoirs and the Prudhoe Bay field of Alaska are examples of such oil fields.

This original water-/oil-saturation distribution is important to understand for waterflooding because it controls the efficiency of the waterflood in portions of the reservoir. It also relates directly to the residual oil saturation that can be achieved at the end of a waterflood.

Relative Permeability

Relative permeability (k_r) concepts are discussed in detail in a chapter of the General Engineering volume of this Handbook. For the purposes of this chapter, their important aspect is the characteristics of imbibition oil/water k_r curves because these govern the nature and efficiency of the waterflood displacement and how much of the OOIP will be recovered before the waterflood economic limit is reached.

The shapes of the imbibition water/oil k_r curves depend on pore geometry and wettability. As noted earlier, Figs. 11.4 and 11.5 show the differences between these curves for a sandstone with large, well-connected pores and one with small, well-connected pores. The k_rw is greatly reduced for the sandstone with small pores at all saturation levels. Fig. 11.9 shows the effect of wettability, as measured by the U.S. Bureau of Mines (USBM) Amott wettability index, on the water/oil k_r curves. As is expected for a change from water-wet to oil-wet in such laboratory tests, the water k_r curve rises with increasing oil-wetness and the oil k_r curve decreases.

• 

  Fig. 11.9 – Oil and water relative permeabilities for Squirrel-sandstone cores for water-wet and oil-wet conditions.

Most importantly, laboratory-determined water/oil k_r data should be obtained at the best approximation of reservoir conditions. Salathiel describes the importance of this to actual field oil/water displacement. Fig. 11.10 shows the results of Salathiel’s laboratory experiments that relate to the East Texas oil field. These curves show that the oil relative permeability for water-wet conditions is significantly different than for mixed-wet conditions. In water-wet conditions, the oil phase becomes discontinuous and loses its mobility quickly. In mixed-wet conditions, the oil maintains phase continuity by means of the oil-wetted rock surfaces and slowly drains to significantly lower oil saturations. The comparison of the laboratory results to the actual field production data and the residual-oil-saturation-pressure core data showed that the reservoir had mixed-wettability, yielding S_orw values of < 10% PV in many portions of the reservoir.

• 

  Fig. 11.10 – Comparison of waterflood behavior for mixed-wet and water-wet cores. Insert shows extension of mixed-wet-core flooding data. PV = pore volume.

Residual Oil Saturation

For waterflooding, the two most important numbers for a reservoir rock are the connate-water saturation S_wc and the S_orw. The S wc determines how much oil initially is in each unit volume of rock when the reservoir is discovered. The S_orw is how much of the OOIP will remain in rock that will be well swept by injected-water volumes. Assuming that the oil-formation-volume factor is the same at the beginning and the end of the waterflood, the equation for the unit-displacement efficiency is:

....................(11.3)

where S_oi = initial oil saturation (1 – S_wc).

The S_orw is the endpoint of the water/oil imbibition k_ro curve, which was discussed above; however, for simple waterflood calculations this value is the most critical one. Table 11.1 compares Salathiel’s S_orw results for the water-wet conditions to those for the mixed-wet conditions. The S_orw for the mixed-wet samples generally was 10% PV lower than for the water-wet samples. In the water-wet conditions, more of the oil phase gets "snapped off" and therefore trapped and immobilized as isolated oil globules by the increasing water saturation. Jerauld and Rathmell found similar results for the Prudhoe Bay field.

• 

  Table 11.1

S_orw can be measured several ways. It can be determined as part of all relative permeability laboratory studies. Historically, short core-plug "floodpot" tests have been run in the laboratory, and only the rock sample’s porosity, absolute air permeability, S_wc, S_orw, and permeability at the two endpoint saturations have been reported. It is important to ensure that these laboratory tests are conducted long enough for the displacement to be taken to its true endpoint. They can be performed either as displacement tests or by using a centrifuge to measure these data. Displacement tests historically have been used, but because of improvements in centrifuge technology, the centrifuge approach is becoming more common. Usually, floodpot-test times are inadequate to reach a true S_orw. Imbibition capillary pressure measurements obtain more-reliable values for water-wet porous media.

Generally, S_orw is inversely related to S_wi. This can be understood in terms of the pore spaces that become filled with water and oil. While the S_wi decreases (or the S_oi increases), the oil phase occupies more of the pores and fills more of the smaller pore spaces. When water displaces the oil, the advancing waterfront traps more of the oil, especially if the rock is water-wet.

The performance of a waterflood depends on the impact of viscous and capillary forces on S_orw and k_r. At reservoir flow rates, the viscous forces do not vary enough to make a significant difference in k_r and S_orw; however, under laboratory conditions, viscous and capillary forces are major considerations because short core-plug displacement tests actually measure pressure drops and fluid-production volumes as a function of time that include large capillary end effects. These data must be entered into interpretative calculations to derive the water/oil P_c/S_w and k_r curves that are used later in field waterflood calculations. The laboratory personnel must choose what length of core plugs to test, what flow rates and pressure drops to apply, whether to make the measurements at steady-state or unsteady-state conditions, and how to interpret these data.

Initial Gas Saturation S_gi

In many oil reservoirs, a free-gas saturation formed during the early production period because the waterflood was not initiated before the reservoir pressure had dropped through the oil bubblepoint pressure. For many years, the effect of this gas saturation on S_orw has been a subject of considerable technical interest. Fig. 11.11 summarizes the experimental results of several investigators and shows the impact of S_gt on S_orw for water-wet rocks. The S_orw decreased as S_gt increased. Because gas is the most nonwetting of the fluid phases, the residual gas phase occupies the center of the pore bodies and hence can reduce the volume of oil that is trapped.

• 

  Fig. 11.11 – Effect of trapped-gas saturation on waterflood oil recovery for preferentially water-wet rocks.

Other Considerations

Historically, most laboratory tests have been run at surface temperature and pressure conditions using dead crude oils and constant brine salinity when measuring water/oil P_c/S_w and k_r data. Over the past decade, U.S. researchers at the U. of Wyoming and the U. of Texas have published papers concerning studies of the effect of temperature, salinity, and oil composition on wettability and waterflood oil recovery. Those studies show that oil recovery increases with higher temperature, and generally also with variation in salinity.

Mobility Ratio

The mobility of a phase (Eq. 11.4) is defined as its relative permeability divided by its viscosity. Hence, mobility combines a rock property (relative permeability) with a fluid property (fluid viscosity). The water/oil relative permeability is assumed to depend only on the saturations of the two fluid phases.

....................(11.4)

where λ_i = mobility of fluid phase i, k_i = relative permeability of fluid phase i, and μ_i = viscosity of fluid phase i. Mobility relates to the amount of resistance to flow through a reservoir rock that a fluid has at a given saturation of that fluid. Because viscosity is in the denominator of this equation, low-viscosity fluids generally have high mobility and high-viscosity fluids generally have low mobility.

The mobility ratio M generally is defined as the mobility of the displacing phase (for waterflooding, water) divided by the mobility of the displaced phase (oil). Eqs. 11.5 and 11.6 present two forms of the mobility-ratio equation:

....................(11.5)

where μ_w = viscosity of water, cp; μ_o = viscosity of oil, cp; k_rw = relative permeability to water; and k_ro = relative permeability to oil.

The mobility ratio also can be expressed as the product of the two fluids’ relative permeability and viscosity ratios.

....................(11.6)

Mobility ratios are considered to be either "favorable" or "unfavorable." A favorable mobility ratio is a low value (≤ 1); this means that the displaced phase (oil) has a higher mobility than does the displacing phase (water). An unfavorable mobility ratio (> 1) is the other way around. In practical terms, a favorable mobility ratio means that the displaced oil phase can move more quickly through the reservoir rock than can the displacing water phase.

For simple waterflooding calculations, the mobility ratio is calculated at the endpoint relative permeability values for the two phases. Hence, the equation to be used for the waterflood mobility ratio is:

....................(11.7)

where k_rwe = relative permeability to water at the endpoint (S_orw) and k_roe = relative permeability to oil at the endpoint (S_wi). This mobility ratio assumes a plug-like displacement between the oil phase at connate-water saturation before the flood front and the water phase at residual oil saturation behind the flood front.

Because, in most reservoir situations, water’s viscosity is lower than oil’s, the viscosity ratio is unfavorable for water to displace oil efficiently; however, as Figs. 11.4, 11.5, and 11.9 show, the relative permeability of water at residual oil saturation is lower by a factor of two to eight than that of oil at connate-water saturation. Hence, for many reservoirs, the mobility ratio is close to unity (favorable) if the oil viscosity is greater than the water viscosity at reservoir conditions only by a factor of five.

Macroscopic Displacement Efficiency of a Linear Waterflood

This section discusses the mathematical aspects of water/oil displacement for homogeneous linear systems. The presentation here is brief and does not include the intermediate steps of the mathematical derivation of the key equations. The details of these mathematical derivations are available in Willhite.

The displacement of oil by water from a porous and permeable rock is an unsteady-state process because the saturations change with time and distance from the injection point (see schematic diagram of Fig. 11.1). These saturation changes cause the relative permeability values and pressures to change as a function of time at each position in the rock. Fig. 11.12 illustrates the various stages of an oil/water displacement process in a homogeneous linear system.

• 

  Fig. 11.12 – Saturation distribution during different stages of a waterflood. L = length, ft; x = x-direction length, ft; and x/L is dimensionless and varies from 0 to 1.

The mathematical derivation of fluid-flow equations for porous media begins with the simple concept of a material-balance calculation: accumulation equals fluid in minus fluid out. This equation is written for the whole system and for each of the phases: water, oil, and gas. Eqs. 11.8 and 11.9 are the equations for the conservation of mass for a water/oil homogeneous linear system:

....................(11.8)

and

....................(11.9)

where x = position in x-coordinate system, ft; ρ_o = oil density, lbm/ft³ or g/cm³; u_ox = oil velocity in the x direction, ft/day; t = time, days; S_o = oil saturation, fraction PV; ϕ = porosity, fraction BV; ρ_w = water density, lbm/ft³ or g/cm³; u_wx = water velocity in the x direction, ft/day; and S_w = water saturation, fraction.

Assuming that the oil and water are incompressible and that the porosity is constant, these equations become:

....................(11.10)

and

....................(11.11)

where q_o = oil-production rate, B/D; A = cross-sectional area available for flow, ft²; and q_w = water-production rate, B/D.

Next, the equations for fractional flow of oil and water are incorporated into these equations. The three fractional-flow equations are:

....................(11.12)

....................(11.13)

and

....................(11.14)

where f_o = fractional flow of oil; q_t = the total production rate, B/D; and f_w = fractional flow of water.

Substituting Eq. 11.13 into Eq. 11.11 yields:

....................(11.15)

Buckley-Leverett Solution

Further mathematical manipulation of these equations obtains the Buckley-Leverett equation (Eq. 11.16), or frontal-advance equation. To derive this equation, it is assumed that the fractional flow of water is a function only of the water saturation and that there is no mass transfer between the oil and water phases.

....................(11.16)

This equation shows that in a linear displacement of water displacing oil, each water saturation moves through the rock at a velocity that is computed from the derivative of the fractional flow with respect to water saturation.

The general form of the fractional-flow equation for water is:

....................(11.17)

where k_o = permeability to oil, darcies; g = gravity constant; α = reservoir dip angle, degrees; and k_w = permeability to water, darcies. This equation includes terms for capillary pressure variation (as a function of saturation) in the linear direction and for the linear system possibly dipping at angle α.

Assuming that the gradient in P_c with position is very small and that the linear system is horizontal reduces Eq. 11.17 to:

....................(11.18)

Fig. 11.13 presents a typical fractional-flow curve that would be calculated from Eq. 11.18. This figure also shows a tangent to the fractional-flow curve that originates at the initial water saturation. The tangent point defines the "breakthrough" or "flood-front" saturation S_wf. This saturation is equivalent to the saturation that Buckley and Leverett obtained through intuitive arguments. It subsequently was recognized that this tangent intersects the fractional-flow curve at the saturation that is common to the stabilized and the nonstabilized zones.

• 

  Fig. 11.13 – Determination of flood-front saturation. f_wf = fractional flow of water at the flood front, = average water saturation behind the flood front at water breakthrough, and S_wf = water saturation at the flood front.

The frontal-advance equation (Eq. 11.16) cannot predict the saturation profile between the connate-water saturation and the breakthrough saturation. An approximation that was developed from the Buckley-Leverett solution considers the saturation change to be a step increase ("shock") from the connate-water saturation S_wc to the flood-front saturation S_wf. Fig. 11.14 shows this saturation profile. The shock occurs because all saturations that are less than S_wf travel at the velocity of the flood front. Saturations that are greater than S_wf travel at velocities that are determined from Eq. 11.16 by calculating the derivative of the fractional-flow curve at each S_w value.

• 

  Fig. 11.14 – Saturation profile computed from Buckley-Leverett solution.

That the Buckley-Leverett solution is reasonable has been experimentally verified. Fig. 11.15 compares experimental results with calculated values for two oils that have nearly a hundred-fold difference in viscosity.

• 

  Fig. 11.15 – Comparison of waterflooding response with results calculated from frontal-advance equation.

Fig. 11.16 shows the viscosity ratio’s effect on the water fractional-flow behavior. The viscosity ratio is a key parameter; the efficiency of the linear displacement process of water displacing oil changes and is substantially different when the oil’s and the water’s viscosity is the same compared to when the oil’s viscosity is much higher than the water’s.

• 

  Fig. 11.16 – Effect of viscosity ratio on fractional flow.

Reservoir-Geology Considerations in the Design and Operation of Waterfloods

This section briefly discusses the geologic considerations in assessing waterflood performance and then describes the areal and vertical aspects of waterflood performance and analysis. (See Chap. 1 in this volume of the Handbook for a review of the geologic factors required for detailed reservoir engineering of oil and gas fields.) This section considers the macroscopic aspects of the geology that affects waterfloods; the microscopic aspects were discussed earlier in Sec. 11.2.

All oil reservoirs are heterogeneous rock formations. The primary geological consideration in waterflooding evaluation is to determine the nature and degree of heterogeneities that exist in a particular oil field. Reservoir heterogeneities can take many forms, including

• Shale, anhydrite, or other impermeable layers that partly or completely separate the porous and permeable reservoir layers.
• Interbedded hydrocarbon-bearing layers that have significantly different rock qualities—sandstones or carbonates.
• Varying continuity, interconnection, and areal extent of porous and permeable layers throughout the reservoir.
• Directional permeability trends that are caused by the depositional environment or by diagenetic changes.
• Fracture trends that developed because of regional tectonic stresses on the rock and the effects of burial and uplift on the particular rock layer.
• Fault trends that affect the connection of one part of an oil reservoir to adjacent areas, either because they are flow barriers or because they are open conduits that allow unlimited flow along the fault plane.

The structure of the reservoir and how it affects waterflood performance is another geological consideration. Structure creates dipping beds that dip at various angles. The interplay between the bed angle, gravity, and the oil/brine density difference at reservoir conditions significantly affects the relative vertical and horizontal flow behaviors. Structural considerations also can include whether the oil column has an underlying aquifer or an overlying gas cap, either of which can significantly affect the likelihood of successfully waterflooding the oil column.

Geologists and geophysicists must assess such geological and structural aspects of a reservoir. Geologists use cores and routine-core-analysis data to develop an understanding of the depositional environment and post-depositional diagenesis and to characterize the reservoir’s internal architecture. Using seismic data, geophysicists can discern the major faults, as well as trends in rock quality, since cores and well logs are essentially pin pricks into the overall reservoir.

The technical team that is evaluating and monitoring waterflood performance should include a geologist and a geophysicist. Including a geostatistician on the technical team, as well, will help to ensure that the geoscientists’ reservoir description is properly translated into engineering calculations, whether those are simpler calculations or are detailed numerical reservoir simulations.

For a waterflood, the reservoir description must be developed on the scale that is required for the quantitative evaluation (i.e., it must be "fit-for-purpose"). A variety of approaches (e.g., object- and pixel-based techniques) can be used. The "flow unit" is a concept that frequently is used by geologists and that would be useful to engineers. "A flow unit is a volume of the total reservoir rock within which geological and petrophysical properties that affect fluid flow are internally consistent and predictably different from properties of other rock volumes (e.g., flow units)."

The process of evaluating a reservoir’s geology begins when the reservoir is discovered and is placed on primary production. After a waterflood has been initiated, the production- and injection-well data provide additional insight into the internal characteristics of the rock volume that is being flooded. In fact, the waterflood production-well data (the water and oil rates as a function of time) are critical because they are the first data that relate directly to the interwell connectivity within the reservoir and that validate or cause modification of the geoscientists’ concepts of the various levels of reservoir heterogeneities.

During a waterflood, tracers can be injected to track which injector/producer pairs are well connected and which are poorly connected. (See the chapter on tracers in this volume of the Handbook.) Other monitoring techniques include the use of specially drilled observation wells and 4D-seismic interpretations to track the directionality and shape of the higher-pressure water-swept reservoir areas that are centered on the injection wells.

Immiscible Displacement in Two Dimensions-Areal

Historically, when computer capabilities were limited, the 3D aspects of a waterflood evaluation were simplified so that the technical problem could be treated as either a 2D-areal problem or a 2D-vertical problem. This section reviews the methods for treating the waterflood analysis as a 2D-areal technical problem. To simplify 3D to 2D areal, either the reservoir must be assumed to be vertically a thin and homogeneous rock interval (hence having no gravity considerations) or one of the published techniques to handle the vertical heterogeneity and expected gravity effects within the context of a 2D-areal calculation must be used.

The primary areal considerations for a waterflood involve the choices of the pattern style (see Fig. 11.17) and the well spacing. Maximizing the ultimate oil recovery and economic return from waterflooding requires making many pattern- and spacing-related decisions when secondary recovery is evaluated. This has been particularly true for onshore oil fields in the U.S. in which a significant number of wells were drilled for primary production. Many SPE technical papers have been published addressing these issues. (Sec. 11.9.4 West Texas Carbonate Waterfloods discusses these topics further and references several relevant papers.) For offshore oil fields where the maximum number of wells is limited, the optimal waterflood injection-well/production-well layout is best determined by the use of 3D numerical reservoir simulation.

• 

  Fig. 11.17 – Common waterflood-pattern configurations.

The discussion below describes various factors that affect waterflood performance and discusses some of the 2D-areal calculation methodologies that have been developed. This discussion does not cover the use of modern numerical reservoir-simulation models in a 2D mode for analyzing a reservoir’s waterflood performance, either for history matching or for future-performance projections. See Chap. 17 in this volume of the Handbook for a review of numerical reservoir simulation.

Waterflood Displacement in a Five-Spot Pattern

Because it has been studied extensively, taking a look at the five-spot pattern (Fig. 11.17) provides an effective review of the areal aspects of waterflooding. Fig. 11.18 plots the waterflood performance of a five-spot experimental homogeneous sandpack for range of water/oil viscosity ratios from 0.83 (slightly favorable) to 754.0 (very unfavorable). Fig. 11.19 shows the X-ray shadowgraphs of a waterflood of two five-spot homogeneous sandpacks, one whose mobility ratio is favorable ( M = 0.40) and one whose mobility ratio is unfavorable ( M = 1.43). Fig. 11.20 plots the breakthrough areal-sweep efficiency as a function of M for a five-spot-pattern flood.

• 

  Fig. 11.18 – Oil recovery from waterflooding five-spot pattern models.

• 

  Fig. 11.19 – X-ray shadowgraphs of flood progress in experimental scaled five-spot pattern models. (WOR = instantaneous producing water/oil ratio.)

• 

  Fig. 11.20 – Correlation of areal sweep efficiency at breakthrough with mobility ratio for miscible and immiscible displacement in five-spot-pattern floods. On this figure, the mobility M has been calculated using the average water saturation in the water-contacted portion of the reservoir. The horizontal lines indicate what the range of mobility ratios M would be if it were calculated using the water mobility at floodout conditions (the right extremity of the horizontal bar) or using the water mobility at the flood front (the left extremity of the bar), with the arrows indicating that the values were off the scale of this plot.

Figs. 11.18 through 11.20 show that the water/oil viscosity ratio is critical to the efficiency of the waterflood displacement. When the oil is more viscous than the water, the water areally displaces the oil less efficiently. When the oil is less viscous than the water, the water areally displaces the oil very efficiently.

Steamtube Waterflood Calculations

In the 1960s, Higgins and Leighton published a technique for analyzing waterfloods using what has been termed streamtubes, or stream channels, and using the concept of streamlines. Streamlines are the paths that fluid particles follow when they move from the injector to the producer; a streamtube is the region between two streamlines. The Higgins and Leighton technique approximates the displacement problem by use of a set of streamtubes in which fluid flows from the injection well to the production well and in which no fluid flows perpendicular to the streamlines (Fig. 11.21). Streamlines for the steady flow of a single phase (unit mobility ratio) have been determined for regular displacement patterns. Streamlines for arbitrary arrangements of injection and production wells can be determined by superposition of numerical solutions (see Fig. 11.22).

• 

  Fig. 11.21 – Streamlines and equipotential lines for single-phase flow in a quadrant of a five-spot pattern. For illustrative purpose, for this homogeneous quarter-of-a-five-spot pattern and single-phase flow, a set of streamlines has been shown as (1) through (5); the numbers along the axes are lines of equal potential scaled from 100 at the injection point to zero at the production point. By definition, streamlines are perpendicular to lines of equal potential.

• 

  Fig. 11.22 – Geometries of streamtubes of a heterogeneous multiwall case (113 x 113 grid, heterogeneous, M = 10).

Streamtube techniques originally were developed for areal waterflood analysis. These techniques have continued to be popular with advancements in the mathematical techniques and computing power. (Also see the discussion about streamtube techniques in the Simulation chapter in this volume of the Handbook). The more advanced streamtube models can handle stratified-layer systems. Many universities and oil companies have developed sophisticated streamtube models and applied them to waterflood calculations. Such models also are available from commercial software providers to the oil industry.

Any 2D approach to waterflood analysis is an approximation. How accurate an approximation it is will depend on how vertically stratified the reservoir is and how significant the gravity considerations are in the real reservoir compared to what the numerical modeling assumes about them. For many situations, 2D-streamtube models have been used successfully to model historic waterflood performance and to project future waterflood performance.

Waterflood-Pattern and Well-Spacing Considerations

As Fig. 11.17 shows, a variety of geometric injector/producer pattern layouts can be used when waterflooding an oil reservoir. These geometric layouts are designed to produce an efficient waterflood for the whole of the reservoir, assuming the rock is homogeneous. The producer/injector ratio typically is chosen on the basis of the expected injection rates for the water injectors and the total fluid-production rates for the production wells. The goal is to have voidage replacement, with the injected volume equal to the produced volume. Injection and produced volumes will depend on the transmissibility of the system, the maximum pressure that can be applied at the injector, and the minimum pressure that can be achieved at the producer. It has been customary to limit injection pressures to lower than what would cause fracturing. As will be discussed later, this guideline can be relaxed in some circumstances.

Other considerations with the injector/producer-pattern layout are the anisotropic permeability in the reservoir and the orientation of any natural-fracture systems that are within the reservoir interval. Fig. 11.23 shows the correct and incorrect orientation of lines of injectors and producers in this respect. The goal is to make these natural aspects of the reservoir improve the areal sweepout as the water displaces the oil, rather than allowing them to affect it negatively.

• 

  Fig. 11.23 – Correct and incorrect pattern alignment with antisotropic permeability, or an oriented fracture system.

The choice of waterflood well spacing is primarily a function of four considerations: incremental oil recovery; increased oil-production rate; reduced operating costs (producing more of the oil at lower water cuts); and costs to drill additional wells, including additional platform space at offshore locations. The initial well spacing typically is chosen on the basis of engineering studies, assuming that there will be a logical phased-development plan for the oil field, including possibilities for various levels of infill drilling and pattern realignments. Low-permeability reservoirs typically have closer well spacings than do high-permeability ones because of their reduced ability to move fluids rapidly between widely spaced injectors and producers. The initial well spacing usually is somewhat wider than what is likely to be used toward the end of field life. The number of injection and production wells also is tied to the capacity of the injection facilities and production-fluid-separation facilities.

The incremental oil recovery can be a function of the well spacing and depends on the continuity of the porous and permeable reservoir-rock layers. If the reservoir layers are highly continuous, then reducing the well spacing will not have much effect on the ultimate oil recovery by waterflooding; however, for layers that are discontinuous over the reservoir, closer well spacing will provide more reservoir continuity between injector/producer pairs. This aspect of waterflooding has been studied in depth for a variety of U.S. oil fields, mainly west Texas carbonate reservoirs, in which reservoir-layer continuity is a major issue. Many papers have been published on this topic.

For example, Barber et al. reviewed nine sandstone and carbonate reservoirs’ responses to infill drilling. Figs. 11.24 and 11.25 show aspects of reservoir discontinuity in west Texas Clearfork and San Andres carbonate reservoirs. Fig. 11.25 shows that the San Andres reservoirs have significantly greater reservoir continuity at a 10-acre well spacing than at 20- and 40-acre well spacings. For these reservoirs, "[c]ontinuity calculations made after infill drilling indicated the pay zones to be more discontinuous than when calculations were made before infill drilling." From the experience in those nine fields, the authors concluded that "the ultimate well density in any given field can be determined only after several years of field performance provide sufficient information on reservoir continuity and recovery efficiencies."

• 

  Fig. 11.24 – Type cross section for Fullerton Clearfork reservoir.

• 

  Fig. 11.25 – Continuity progression for Means San Andres unit. The lines on this plot show the estimated percent continuity before drilling infill wells. After each set of infill wells was drilled, the actual reservoir continuity was found to be less than predicted. This is why the “After 10-acre wells” line is the lowest of the three lines. The 3, 4, and 14% vertical bars indicate the expected increase in percent continuity estimated for a 10-acre well spacing from the 40-acre well-spacing data (3%) and from the 20-acre well-spacing data (4%), compared to what actually was found after the 10-acre infill wells were drilled (14%).

Another paper on this topic is by researchers at Texas A&M U., who spent several years analyzing the impact of well-spacing reduction on west Texas Clearfork and San Andres waterflood performance from various geographic areas of the Permian Basin. The authors found infill drilling to be more effective for the San Andres units than for the Clearfork units. In the San Andres units, they determined the infill drilling to be more effective for the units in the Northern Shelf area than for those in the Central Basin Platform area. In those analyses, they calculated a 2 to 5% additional recovery of OOIP for a 9-acre well spacing, compared to a 22-acre well spacing.

As noted above, many more papers have been published on this topic, but these two papers identify the key well-spacing issues. As both studies conclude, one does not know a priori what the optimal well spacing will be. The technical team must analyze the available data before starting a waterflood, then continue to evaluate the production and injection data to determine the waterflood’s efficiency and the extent to which infill-well locations should be drilled or patterns realigned.

Horizontal Wells, Multilateral Wells, and Fracture Orientation

Until the 1980s, all waterflood analysis assumed that the wellbores would penetrate the reservoir vertically and that water should not be injected above the formation parting pressure. Accordingly, mathematical formulations assumed areally that injection and production wells were point sources and sinks. This style of analysis is evident in the streamline patterns in Fig. 11.22.

The advent of horizontal-well and multilateral-well technologies has brought with it the ability to create line injection-well sources and line production-well sinks, which has changed how a waterflood pattern might be developed. Theoretical calculations show that parallel lines of horizontal injectors and horizontal producers will increase oil recovery and in the limit are the perfect line-drive pattern arrangement.

When fracturing injection wells, a concern has been that a fracture might extend from the oil-reservoir intervals into adjacent porous and permeable layers, into which considerable injection brine could be lost (i.e., thief zones); however, given that the principal orientation of any hydraulic fracture is known and that the fractures can be restricted to the oil-reservoir interval, hydraulic fractures can improve the areal sweepout during waterflooding in much the same way that horizontal wells can. This is true of fractures from the injection wells and the production wells, given that an appropriate pattern style is used. On the other hand, if they are oriented so that the fracture tips are significantly closer to each other than the vertical wellbores are, hydraulic fractures from the injectors and producers will yield poorer areal sweepout of the reservoir during waterflooding than would unfractured vertical wells.

An excellent example of a situation in which fracturing was needed is the waterflooding of diatomite reservoirs in California, U.S.A. These reservoirs have high porosity, but permeabilities of 0.1 md or less. Conventional injection methods yielded uneconomic rates; however, positioning the injectors and producers normal to the induced-fracture orientation established reasonable rates without significant loss of injected water to adjacent porous and permeable intervals and without premature water breakthrough.

Vertical Displacement in Linear and Areal Models

The previous section’s discussion of waterflood displacement behavior assumed that the vertical saturation distribution was homogeneous at all areal locations. This section discusses the impact of vertical variations in permeability and the effect of gravity on simple 2D reservoir situations in which the areal effects are ignored. Gravity effects always are present because for any potential waterflood project, oil always is less dense than water, even more so after the gas is included that is dissolved in the oil at reservoir conditions.

Three particular situations are discussed here:

• Stratified systems with noncommunicating layers for various mobility ratios.
• Homogeneous systems with gravity (including dipping beds).
• Stratified systems with communicating layers and assumed vertical fluid equilibrium.

The discussion below does not include the P_c effects on vertical saturation distributions. Through countercurrent imbibition, P_c effects help to counteract nonequilibrium water/oil saturation distributions. The mathematics of including P_c effects makes the problems too complicated for inclusion in this chapter. Standard numerical reservoir simulators—which are needed for a complete analysis of real reservoir situations—account for the effects of countercurrent imbibition caused by P_c effects, as well as for the water/oil density and viscosity differences that lead to injection-water gravity underrunning and the layer-by-layer permeability variations, with or without communicating layers.

Before presenting some of the technical-literature techniques for studying the vertical displacement characteristics of water/oil displacement, one must first define some measure of the vertical permeability variations. Dykstra and Parsons developed a method that is based on routine-core-analysis data. In that approach, the routine-core-analysis permeability data for the pay intervals are arranged in descending order, and the percent of the total number of values that exceeds each entry is calculated. The values then are presented as a log-probability plot (see Fig. 11.26). A reasonably straight line is drawn through the data, with the points in the 10-to-90% range being more heavily weighted. This straight line is a measure of the dispersion and the heterogeneity of the reservoir rock. What has come to be known as the "Dykstra-Parsons coefficient of permeability variation" V is defined as:

....................(11.19)

where k₅₀ = median permeability value, md, and k_84.1 = permeability at 84.1% probability (one standard deviation), md.

• 

  Fig. 11.26 – Plot of permeability data on log-normal paper.

Fig. 11.27 shows the relationship of the Dykstra-Parsons V values to varying degrees of rock heterogeneity. Note that in Fig. 11.27, the V for most reservoirs ranges from 0.5 to 0.9.

• 

  Fig. 11.27 – Characterization of reservoir heterogeneity by permeability variation V.

Stratified Systems With Noncommunicating Layers

Over the years, several waterflood prediction methods have been proposed and published that account for the vertical variations in rock properties, particularly permeability. These simple methods assumed that every rock layer acts independently of all other rock layers (even at 1-ft increments in the reservoir) and that each rock layer is continuous from the injection well to the production well. These early methods were developed when the ability to make detailed, complicated engineering calculations was limited. They focused on how to account for (1) the effect of the vertical permeability variation with minimal consideration of the mobility ratio and (2) the effect of vertical permeability variation and mobility ratio, assuming constant pressure at the injection and production wells.

Stiles developed one of the earliest methods, for which only the permeability-thickness (kh) distribution of the vertical reservoir interval and the mobility ratio at endpoint conditions need to be known (see Eq. 11.6). The water/oil ratio (WOR) F_wo after water breakthrough as a function of the fraction of the total flow capacity C represented by layers having water breakthrough is defined as:

....................(11.20)

where B_o = the oil formation-volume factor, RB/STB.

A more sophisticated method that is widely used is that of Dykstra and Parsons. Their method is based on calculations for linear layered models and assumes no crossflow and the use of the results of more than 200 floodpot tests that were performed on more than 40 California oil-reservoir core samples. This Dykstra-and-Parsons method takes into account initial fluid saturations, mobility ratios, producing WORs, and fractional oil recoveries. The permeability variation was taken into account by use of V, as defined in Eq. 11.19 above. Figs. 11.28 through 11.31 plot the results of the Dykstra-and-Parsons technique as V vs. M for four WOR levels (1, 5, 25, and 100).

• 

  Fig. 11.28 – Permeability variation plotted against mobility ratio, showing lines of a constant E_R(1 - S_w) for a producing WOR of 1.

• 

  Fig. 11.29 – Permeability variation plotted against mobility ratio, showing lines of a constant E_R (1 – 0.72S_w) for a producing WOR of 5.

• 

  Fig. 11.30 – Permeability variation plotted against mobility ratio, showing lines of a constant E_R (1 – 0.52S_w) for a producing WOR of 25.

• 

  Fig. 11.31 – Permeability variation plotted against mobility ratio, showing lines of a constant E_R (1 - ).40S_w) for a producing WOR of 100.

Homogeneous Reservoirs Subject to Gravity Effects

In essentially all reservoirs, even those with close well spacings, the horizontal distance between an injector well and a producer well is very long relative to the vertical thickness of the reservoir pay interval. This means that gravity plays a major role in the water/oil-displacement process, given that the fluids can move vertically within the pay interval. For conceptual and calculation purposes, the limiting case is to assume that gravity forces dominate the water/oil-displacement process, that gravity segregation of the oil and water is complete, and that the system is in "vertical equilibrium." This means that vertically the gravity and capillary forces are in balance and that the vertical saturation distribution is governed by the P_c/S_w function.

The first and simplest homogeneous reservoir situation described here is a reservoir whose permeability is constant throughout the pay interval. Craig studied a set of scaled laboratory vertical models experimentally and developed a correlation between the sweep efficiency at breakthrough and the values of the scaling parameter:

....................(11.21)

where (Δp)_h = pressure difference in the horizontal direction, psi; (Δp)_V = pressure difference in the vertical direction, psi; u_t = horizontal Darcy velocity, ft/D; k_x = permeability in the x direction, darcies; and Δρ = water/oil density difference, lbm/ft³.

As Fig. 11.32 shows, the sweep efficiency as it is related to the scaling parameter is a strong function of mobility ratio. Fig. 11.33 compares the fractional flow of water for a homogeneous system with vertical equilibrium to the fractional flow of water calculated from the original laboratory water/oil relative permeability curves. The effect of the water moving along the base of the reservoir interval because of the gravity effects—but with the P_c/S_w curve controlling the vertical distribution of the water and oil saturations—is that the water breaks through earlier and the WOR rises more slowly.

• 

  Fig. 11.32 – Correlation of scaled-model experiments to determine the effect of gravity segregation on volumetric sweep efficiency in uniform linear systems.

• 

  Fig. 11.33 – Comparison of the fractional-flow curve for vertical equilibrium with the fractional-flow curve obtained from fluid and rock properties. (h = reservoir thickness, ft, and S_wz = thickness-averaged water saturation for vertical equilibrium case with h = 40 ft, fraction PV.)

Another reservoir situation that involves gravity effects is a homogeneous reservoir with dipping beds. If the rate of water injection in a waterflood is too low for vertical equilibrium to occur, there will be gravity-stabilized flow between the water and the oil. Dietz has derived a relationship to predict the critical velocity q_c required to propagate a stable interface through a linear system in which gravity forces dominate, but in which pistonlike displacement occurs and P_c effects are neglected:

....................(11.22)

where ρ_o = oil density, lbm/ft³; ρ_w = water density, lbm/ft³; and α = dip angle, degree.

When the oil/water interface is stable, the velocities of oil and water are equal at every point in the interface. The interface is linear and will move at a constant velocity through the system as long as q < q_c. The stable linear interface will not necessarily be flat; however, it will be stable with a slope β, as defined by Eq. 11.23:

....................(11.23)

where y = the position in y-coordinate system, ft, and G is dimensionless and defined by Eq. 11.24:

....................(11.24)

Figs. 11.34a and 11.34b depict gravity-stable situations for two different mobility ratios. Fig. 11.34c depicts the unstable situation for an unfavorable mobility ratio where the displacement rate is too high for the water and oil to maintain vertical equilibrium.

• 

  Fig. 11.34 – Stable [(a) and (b)] and unstable (c) displacement in gravity-segregated displacement. For (a), G > M – 1, M > 1, and β > α. For (b), G > M – 1, M < 1, and β > α. For (c), G < M – 1.

Stratified Systems With Communicating Layers and Assumed Vertical Equilibrium

One of the systems that have been analyzed with simple calculations is that of water/oil displacement with vertical permeability variations and gravity effects, but with capillary pressure neglected. Dake explores this in his reservoir-engineering textbook. Dake’s illustrative example assumes a three-layer system. He assumes the permeability variation to be highest to lowest from top to bottom, and then compares those results with results from assuming the reverse, the layer with highest permeability variation then on the bottom. Fig. 11.35 lists the properties of the three layers, and Fig. 11.36 presents the averaged relative permeability curves. Fig. 11.37 shows the pseudocapillary pressure (a) and fractional-flow curves (b). Note that Fig. 11.37b includes smoothed and unsmoothed versions of the fractional-flow curve. The smoothed version is the curve that would be used for a Welge-type fractional-flow calculation.

• 

  Fig. 11.35 – Individual layer properties for a three-layer example (after Dake). (ϕ = porosity, fraction BV.)

• 

  Fig. 11.36 – Averaged relative permeability curves for a three-layer example (after Dake).

• 

  Fig. 11.37 – Pseudocapillary pressures (a) and fractional-flow curves (b) for a three-layer example (after Dake).

Dake’s example shows that in waterflooding where gravity effects are significant, having the high-permeability layers at the top of the reservoir interval allows a much more efficient oil displacement than when the high-permeability layers are at the bottom of the reservoir interval. This is because gravity causes the water to slump, and when the lower-permeability layers are the base, the water must move more slowly than the oil from the injector to the producer.

So far, the chapter discussions have highlighted the development of techniques for understanding and analyzing key aspects of oil/water displacement. These techniques predate modern computers; hence, they were developed to simplify the real reservoir problem sufficiently to allow various engineering calculations to be made.

Of course, the availability of modern computers and advanced numerical-reservoir-simulation software has rendered many of these simplifying assumptions unnecessary when quantifying waterflood-type water/oil displacements in real reservoirs. Nevertheless, these historic techniques have been discussed here to provide an understanding of the dynamics of the water/oil-displacement process and the primary variables that influence the recovery efficiency, as background to the discussion of waterflood design in the next section.

Waterflood Design

The design of a waterflood has many phases. First, simple engineering evaluation techniques are used to determine whether the reservoir meets the minimum technical and economic criteria for a successful waterflood. If so, then more-detailed technical calculations are made. These include the full range of engineering and geoscience studies. The geologists must develop as complete an understanding as possible of the internal character of the pay intervals and of the continuity of nonpay intervals. This preflood understanding often is limited because the injector/producer wells’ connectivity has not been determined quantitatively. Interference testing can provide insight into connectivity when its cost is justifiable. Data gathered from smart wells can be particularly helpful in determining connectivity in high-cost environments where there is a limited number of wellbores. Analogs also can prove useful. Otherwise, little definitive data will be available until after there has been significant fluid movement from the injectors toward the producers.

The engineer will make a number of reservoir calculations to determine the well spacing and pattern style that will be used in a particular flood. These choices are based on the available understanding of the reservoir geology, the proposed design of surface facilities (particularly water-injection volumes), and any potential limits on the numbers of injectors and producers. Such factors are interrelated in terms of capital and operating costs and oil-, water-, and gas-producing rates to define the overall economics of the project. In making these preliminary calculations, facility capacities need to be flexible because as the waterflood progresses, there almost certainly will be modifications to the original designs and operating plans.

Fig. 11.17 shows a variety of injector/producer pattern layouts that can be considered. In reality, the existing wellbore locations might limit the pattern layout to a nonsymmetrical arrangement like that shown in Fig. 11.38. Also, as shown in Fig. 11.23, the orientation of the rows of producers and injectors must take into account any permeability anisotropy and natural-fracture orientation. At offshore locations, the number of well slots on the drilling platforms limits the number of producers and injectors and their layout.

• 

  Fig. 11.38 – Irregular five-spot pattern layout.

In this section, a number of waterflood design considerations will be discussed briefly. (Rose et al. is entirely devoted to this topic.) The design aspects discussed below include

• Injection-water sensitivity studies.
• Injection wells, injectivity, and allocation approaches, including well fracturing.
• Pilot waterflooding.
• Production wells.
• Surface facilities for injection water.
• Surface facilities for produced fluids.

Injection-Water-Sensitivity Studies

The factors to which injection-water-sensitivity studies relate are water-source and -volume options, source-water/connate-water compatibility, and source-water/reservoir-rock interactions. After the preliminary reservoir evaluation indicates that waterflooding is likely to be economically justified and that it will increase significantly the volume of oil recovered, the next consideration is to find an acceptable source from which to obtain enough water for the proposed waterflood project. Fig. 11.39 schematically shows the variety of natural sources for such water. Onshore locations typically obtain injection water from subsurface aquifer intervals or nearby streams or rivers. Nearshore and offshore waterflood projects typically use seawater.

• 

  Fig. 11.39 – Possible injection-water sources.

Source-water/connate-water compatibility mainly concerns whether mixing the two waters causes any precipitation of insoluble carbonate or sulfate compounds that might impair reservoir permeability. Although permeability impairment typically is not a major consideration, precipitation and scale buildup in pumps and other surface water-handling equipment can cause costly downtime and repairs.

Potential sensitivity of the reservoir pay intervals to the injection water is a major consideration. For sandstone reservoirs that contain various types of clay, the key consideration is whether there exists clay sensitivity to the difference between the connate-water salinity and the injection-water salinity, particularly for freshwater injection-water sources. Such sensitivity can occur either as clay swelling or as mobilization of clay fines, both of which can reduce reservoir permeability significantly. For high-porosity chalk reservoirs, the injection-water/reservoir-rock interaction might weaken the rock framework and cause pore collapse and surface subsidence.

Another aspect of injection-water sensitivity is the amount and size of suspended particulate being carried by the injection water. This is a concern mainly when using surface water sources for the injection water. An example of where this is a significant consideration is the Kuparuk oil field on the North Slope of Alaska, U.S.A., where nearshore ocean water is the waterflood injection water. There, the spring runoff down the rivers from the Brooks Mountains can cause the nearshore ocean water to contain unacceptable amounts of solid particulate for several weeks of the year. Similar problems occur in the Gulf of Mexico in fields near the mouth of the Mississippi River. Also in the Gulf of Mexico, water that is drawn from too near the surface often contains organic matter that can reduce injectivity.

Injection Wells, Injectivity, and Allocation Approaches, Including Well Fracturing

Several aspects of the design and operation of water-injection wells are critical to their success. The first is that these wells must have sufficient injectivity to flow the desired volume of water into the reservoir each day. The expected injectivity can be calculated on the basis of routine core analysis, special core analysis and/or log data, and the existing production wells’ productivity; however, well injectivity often is not known until water actually is injected into the reservoir interval. This is because the near-wellbore "skin" (a rock volume of reduced permeability around the wellbore) is not known until an actual well test is conducted. Injection wells can be fractured to eliminate positive skin in the near-wellbore region; however, fracturing must be done carefully to avoid fracturing out of the reservoir interval and into adjacent porous and permeable intervals into which injection water can be lost.

An aspect of well injectivity that has been studied during the last 20 years is the change in rock stresses that is caused by the cooling effect of the injection water on the near-wellbore region around injectors. This happens particularly in Arctic and offshore waterflood operations, where the injection-water temperatures can be considerably below the reservoir temperature (i.e., more than 100°F difference). Perkins and Gonzalez have studied this phenomenon and found that the cooling effect reduces the earth stresses by several hundred psi. Hence, in the reservoir, a small area around water injectors’ wellbores will fracture more easily, giving that area enhanced permeability (or negative skin). For the Prudhoe Bay field on the North Slope of Alaska, U.S.A., the fracture gradient was reduced to as low as 0.50 psi/ft from the original fracture gradient of 0.60 to 0.70 psi/ft.

Another critical aspect of water-injection-well design and operation is the allocation of water to zones being waterflooded. Having the ability to allocate injection water as desired to the various waterflooded intervals is important for waterflood success because the overall waterflood is controlled primarily at the injection wells, not at the production wells. This is not an issue if there is only one reservoir interval, but in many oil fields, there are multiple reservoir intervals being waterflooded at the same time. If possible, the injection-well bottomhole-tubing, packer, and perforation configuration should be designed to allow control of the relative volumes of water that are injected into the various intervals being waterflooded. This can be accomplished if each injection well is perforated in only one reservoir interval, but one reservoir interval per injector is unlikely to be cost-effective compared to the alternative of fewer wells with more-complicated arrangements of chokes, tubing strings, and packers, particularly if there are multiple pay intervals stacked on top of each other.

Optimum completion design is site-specific and must be based on mechanical and reservoir characteristics for the project at hand.

Pilot Waterflooding

Pilot waterfloods seldom are used today because of the wealth of experience in waterflooding; however, in many situations, they have been conducted to provide more quantitative data on the potential for successful waterflooding on a fieldwide scale. Such pilot waterfloods definitely provide useful data concerning water injectivity, tendencies for early water breakthrough, and additional recovery potential. Determining recovery potential requires a pilot waterflood that is designed to represent what will happen in a full-scale application. Too often, one-pattern pilot waterfloods have been conducted that do not represent the confined injection/production relationship that is needed. Also, if the pilot waterflood is conducted on a well spacing that is considerably smaller than that used for the full-field waterflood (so that injector/producer connectivity data can be obtained sooner), the information it provides might be misleading about the injector/producer connectivity on the larger well spacing of the full-field waterflood. Thus, definitive objectives of a pilot waterflood should be established, and the pilot project should be designed and operated accordingly.

Production Wells

In many cases, the water-injection wells are drilled as new wells; however, the production wells typically are those that already are producing from the oil field. For waterflooding, producers should be completed in the same intervals in which the injection wells are completed. If the production wells are completed in several reservoir intervals, it is best to have sufficient length between the perforated reservoir intervals to allow workover operations to shut off those intervals that are producing much water and little oil by either cement-squeeze operations or by setting a packer in the production tubing.

Surface Facilities for Injection Water

Maintaining high water quality is important for sustaining injectivity, reducing corrosion-related costs, and minimizing equipment plugging. The American Petroleum Institute (API) has published recommendations for analysis of oilfield waters and for biological analysis of injection waters. The industry also has adopted standardized procedures for membrane/filterability tests.

The water-injection surface facilities prepare the water chemically for injection and pressure the water to the desired wellhead injection pressure. Depending on the source of the injection water, the water might need treatment to remove oxygen, prevent scale and corrosion, and chelate the iron. It also might need microbiological treatment and to be filtered to remove particulates. What injection-water preparation techniques are used will vary from one waterflood project to the next. (See chapters in the Facilities and Construction Engineering volume for extensive information on preparing water for injection.) This section specifically discusses surface and produced waters, but the techniques that are covered here also are applicable to water that is produced from aquifers.

One major consideration in injection-water treatment is to prevent the reservoir from being "inoculated" with sulfate-reducing bacteria that can cause a reservoir to develop an in-situ H₂S concentration during the waterflood. This particularly is a problem when using ocean water, which contains both the sulfate-reducing bacteria and the sulfate ions that are their food supply. Once the sulfate-reducing bacteria have been introduced into a reservoir, they are essentially impossible to kill; however, they can be controlled with the injection of bactericides such as formaldehyde.

Pressuring water to the desired injection pressure is the final step before it is piped to the injection wells. Chapters in the Facilities and Construction Engineering volume of this Handbook cover the types of equipment required for increasing the pressure of the injected water. The wellhead injection pressure is calculated by subtracting the weight of the injection-water column from the desired bottomhole pressure, and then adding friction-flow pressure losses down the wellbore.

In a few reservoir situations, "dumpflooding" has been practiced. This is where a water-bearing formation above or below the oil reservoir is perforated, as is the oil-reservoir interval in those same wellbores. Water then is allowed to flow directly from the water-bearing formation into the oil-bearing formation, without ever bringing that water to the surface for any treating or pumping. This is a very simple approach to waterflooding, but generally it has been unsuccessful because the rate of water injection is uncontrolled and limited to the pressure difference between the two formations, which decreases with time as the water-bearing interval is depleted, particularly near the wellbore, and as the oil reservoir interval near the wellbore pressures up.

Surface Facilities for Produced Fluids

The facilities for handling produced fluids for a waterflood must be designed with considerable flexibility. These facilities must handle a wide range of gas-, oil-, and water-production rates over the course of the waterflood, typically a period of several decades. Several chapters in the Facilities and Construction Engineering volume of this Handbook cover these issues in detail.

Initially, the production wells are likely to handle only oil and gas, without water production. When water breakthrough occurs, the water volumes will increase and, over time, water will become the great majority of the produced fluids. Accordingly, a variety of water issues must be considered. First is whether the produced fluids can be separated easily or must be treated with heat and/or chemicals in the surface equipment to achieve the desired level of separation. Second is whether the precipitation of scale in the producing wells or the production surface facilities is causing complications. Regarding scaling tendencies and because of increasing environmental concerns, the handling of NORMs has become an issue with respect to produced-water discharges.

Over the duration of a waterflood and as produced-water volumes increase, there is likely to be the need and desire to reinject the produced water. In this situation, the produced water must be treated so that its oil and particulate content is sufficiently small that, when the water is reinjected, these very small oil droplets will not reduce the injectivity of the water injectors. Oil fields in the North Sea and on Alaska’s North Slope have had to reinject large volumes of produced water. Regarding injectivity losses, experimental coreflood data tend to be more pessimistic than is actual injector performance in the field. In all cases, to reinject produced water successfully, that water must be treated to meet specifications determined to minimize those injectivity losses.

Waterflood Monitoring

Because a waterflood project spans several decades, it is monitored continuously and routinely by engineers who are responsible for its operations, as well as periodically using more-detailed and specialized technical studies (e.g., full-field numerical-reservoir-simulation studies). There are many opportunities to modify and improve the waterflood as data are acquired and analyzed.

The basics of a waterflood analysis center on material-balance concepts. Applying material-balance concepts means that initially there is "reservoir fill-up" if the reservoir previously had some years of primary production. During this period, the reservoir is repressured to its original reservoir pressure because the injected-water volumes will be substantially greater than the produced-fluid volumes. Thereafter, the waterflood will be operated as a voidage-replacement process.

The earliest waterflood-monitoring techniques were developed soon after the first field applications of waterflooding; they were based on simple plots, maps, and calculations. Among these were the plots published by Dyes et al. that estimated the water breakthrough and post-breakthrough behavior of various waterflood pattern configurations. Figs. 11.40 and 11.41 are examples of these plots for a five-spot pattern and a direct-line-drive pattern, respectively, and can be used to make "first-estimate" waterflood calculations.

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  Fig. 11.40 – Effect of mobility ratio on oil production for five-spot pattern. (Ψ_s = fraction of total flow coming from the swept portion of the pattern.)

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  Fig. 11.41 – Effect of mobility ratio on oil production for the direct-line-drive (square pattern). In this case, the distance between adjacent rows of wells is equal to the distance between like wells in a row.

In this section, the aspects of waterflood monitoring that are discussed are

• Data acquisition: routine data gathering.
• Special-data acquisition: infill and observation wells and 4D-seismic data.
• Simple waterflood-analysis techniques: X-plot, log(WOR) vs. cumulative-oil-production plots, and decline curves.
• Sophisticated waterflood-monitoring techniques.

Data Acquisition-Routine Data Gathering

Waterflood monitoring begins with the acquisition of the routine data that are necessary for engineering calculations. The routine data include well-by-well daily oil-, gas-, and water-production rates; well-by-well water-injection rates; injection wellhead pressures; and production-well pressure data. Often, the well-by-well daily rate data are back-calculated from the gathering center’s total produced volumes of these three phases, and then are allocated back to the individual wells on the basis of periodic individual well tests.

Next, the production- and injection-well data are allocated to the individual reservoir intervals, if multiple reservoir intervals are commingled. This well-by-well data allocation requires that spinner surveys (or their equivalent) be run periodically in the individual wells to determine how much of the fluid is coming from each of the perforated intervals. The spinner surveys should be run both with the well flowing and with the well shut in. Data from these surveys, along with pressure-buildup and -falloff data, help in the estimation of the reservoir pressure in the various reservoir intervals. Also, a variety of production logs should be run to estimate changes in fluid saturations in the near-wellbore region as the waterflood progresses.

The field engineers can use all these data in the types of calculations that are described below.

Special-Data Acquisition: Infill and Observation Wells and 4D-Seismic Data

During the waterflood, there are likely to be opportunities to gather additional data away from the injection and production wellbores. These data can take several forms.

Infill wells are likely to be drilled in locations where oil should not have been displaced by the injected water. Consequently, these locations are not good locations for determining how effectively the various portions of the reservoir intervals are being swept by injected water. After drilling, the hydrocarbon distribution actually present at an infill-well location is evaluated using openhole or cased-hole logging. Where formation waters have a sufficiently high salinity, pulsed-neutron thermal-decay time and resistivity logging may be used to evaluate the residual oil saturations. In lower-water-salinity conditions, carbon/oxygen and resistivity logging are used. Using these techniques, the locations of fully flooded, partly flooded, and unflooded reservoir intervals can be determined in new wells and in existing producing wells. Multiple logging runs over time in a producing well allow the monitoring and management of a waterflood. If wells are drilled later during the waterflood, then the residual-oil-saturation distribution can be obtained by use of special coring procedures or special tracer tests, as described elsewhere in this volume of the Handbook.

Special observation wells sometimes are drilled at a location in the oil reservoir where the water/oil flood front should be detectable as it passes. Historically, most of these wells were cored but had steel casing, such that standard logging methods could not be used; however, before the development of through-steel-casing resistivity logging, fiberglass casing together with induction-resistivity logging occasionally were used to observe the water/oil displacement process over time.

Recently, the 4D-seismic technique has been developed to determine in what directions the water is moving from the injection wells. The 4D-seismic technique compares 3D-seismic data that were obtained before the start of waterflooding to a second or third 3D survey that was conducted some years later. This allows an areal visualization of where there are high-pressure areas caused by water injection and where there are low-pressure areas caused by production (typically from the presence of some free-gas saturation near the production wellbores). Also, the 4D picture might show which portions of the reservoir pay intervals are well connected and which are not.

Simple Waterflood-Analysis Techniques: X-Plot, Log(WOR) vs. Cumulative-Oil-Production Plots, and Decline Curves

The initial engineering analysis of waterflood performance is done to ensure that the water is being injected into the wells at the desired well-by-well rates and, if several oil-reservoir intervals are perforated in each injector, that each of those reservoir intervals is taking its appropriate share of the injected water from each of the injection wells. This may appear trivial, but many waterfloods have had significant problems in this regard. Before the waterflood begins, the engineers must estimate how much water should be flowing into each injector. Those estimates are based on fill-up and voidage-replacement calculations for that area of the oil reservoir and for the reservoir as a whole.

A "bubble map" can be used for visualizing the advance and relative volumes of the injected water. Fig. 11.42 shows the injection wells’ bubble map for one of the staggered line-drive pattern elements of one of the oil-reservoir intervals of the Long Beach Unit (LBU) area of the Wilmington oil field in California, U.S.A. Bubble maps are created for each injector by dividing the volume of water injected by the movable oil per vertical reservoir interval [thickness × porosity × (1-S_wc-S_orw) ] to calculate the area that should have been swept by that volume of injected water; then presenting these areas on a map as circles of various sizes centered on each of the injection-well locations.

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  Fig. 11.42 – Layer injection-bubble map for Voidage Block 3 FO sand interval – LBU of the Wilmington oil field in California, U.S.A.

Another aspect of waterflood monitoring is to track the performance of the production wells. As noted earlier, some oil reservoirs will have had a considerable period of primary production during which the reservoir pressure was drawn down below the bubblepoint pressure and a gas saturation developed. These reservoirs first will "fill up," which means that they will go through a period during which the injection water displaces mobile gas and increases the reservoir pressure to force free gas back into solution in the oil. During the early part of the fill-up period, the production wells will see minimal response because of water injection. When the reservoir pressure has returned to its original value, the water injection becomes a voidage-replacement operation. During the voidage-replacement period, the engineer must make sure (1) that the volume of injected water equals the reservoir voidage of oil, gas, and water production, and (2) that the various areas or patterns of the reservoir are being balanced to maximize the oil recovery and to minimize water-handling operating costs.

A number of graphs of the production and injection data can be prepared to help analyze the waterflood performance. For example, a conformance plot is a plot of cumulative oil recovery (or oil-recovery efficiency) vs. net displaceable hydrocarbon PV injected. A waterflood-performance envelope is defined by drawing an obtuse triangle that is bounded by recovery at the start of waterflooding, the maximum oil recovery at 100% of the x-axis, and a third point, F_pvg, which is related to the net injection required to displace the existing gas saturation at the start of the waterflood and is defined as:

....................(11.25)

where S_g = gas saturation, fraction.

Because actual performance cannot fall outside the performance envelope, this plot is a check on fluid allocation for a pattern. This plot also can indicate when injection water is being lost to thief zones. When slope changes are noted in this plot, the possible causes should be investigated. Fig. 11.43 is a conformance plot for the Kuparuk River oil field on Alaska’s North Slope.

• 

  Fig. 11.43 – Example of a conformance plot.

Other plots that typically are made after water breakthrough are the X-plot and its close cousin, the log(WOR) vs. cumulative-oil-production plot. The X-plot technique, developed by Ershaghi and coworkers, is based on the leaky-piston-displacement concepts of Buckley and Leverett and assumes that the plot of log (k_rw/k_ro) vs. S_w is a straight line. The X-plot is a graph of recovery, E_R vs. X, where . This plot’s underlying assumption suggests that its usefulness theoretically is limited to higher water cuts.

The "cut-cum" plot, or plot of log(WOR) vs. cumulative oil production, easily can be made using production data for each well and for reservoirs as a whole. These plots generally are useful predictors of future waterflood performance because there is a considerable period of straight-line behavior on these plots for many wells and reservoirs when the waterflood is fully developed without major variations in field operations. Analysis of log(WOR) vs. cumulative-oil-production plots for waterflood analysis has been conducted using numerical reservoir simulation. These simulations show that the linear trends on this type of plot are found even at low WOR values and also are found for a variety of reservoir layering, flood configurations, and operational changes.

Decline curves also have been used for waterflood analysis. Where waterfloods were initiated in depleted sandstone reservoirs, empirical correlations were developed to estimate the likely oil-rate increases during fill-up and while the oil bank moves toward the production wells, and then to estimate the oil-rate declines as the WOR increased later in the life of the waterflood. Fig. 11.44 shows the production response for this situation. For the oil-rate-increase and -decline periods, the plot has exponential oil-rate-vs.-time characteristics.

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  Fig. 11.44 – Production response of a fully developed waterflood with water input to all injection wells at the beginning of the flood.

Hall plots and Hearn plots often are used to monitor injection wells. Use of these plots helps to maximize water-injection rates, which accelerates oil production from offsetting producers. On a Hall plot, the bottomhole injection pressure is plotted vs. cumulative water injection to monitor reservoir fill-up and average-reservoir-pressure increase. On a Hearn plot, the inverse injectivity index is plotted vs. cumulative water injection. Both types of plots also can be used to determine whether the water-injection rates are being kept below those allowed by the fracture-parting pressure.

A combination of these simple plot and calculation techniques typically is used for waterflood analysis. By plotting both oil rate vs. time and log(WOR) vs. cumulative oil, the engineer can better understand how well the waterflood is performing compared to original estimates and can determine what changes, if any, are needed.

Keep in mind that most of these techniques are premised on continued current operations. If there are significant changes in the allocation of injection water or if there are well workovers or pattern realignments, then the trends on these plots might not remain straight lines from which future waterflood performance is easy to predict.

Sophisticated Waterflood-Monitoring Techniques

The modern numerical reservoir simulator is the best tool for performing waterflood analysis, including history matching of past performance and projection of future performance for continued current operations or for various operational and well changes. However, this tool generally is used with an updated history match only every 5 to 10 years.

Between major studies, the field engineers typically use simpler surveillance methods that have been upgraded by the availability of the notebook computers that have sizeable hard disks and rapid computing capabilities. New software packages have been developed that analyze trends and can handle large amounts of electronically acquired data.

One example of the use of this approach is the set of techniques used for the LBU area of the Wilmington oil field, which has 1,200 wells, multiple oil reservoirs, and 27 years of waterflood history. Fig. 11.45 shows the logic used for the surveillance calculations. As Fig. 11.45 shows, a large waterflood involves massive amounts of data that must be handled logically and consistently so that the engineers can obtain useful results.

• 

  Fig. 11.45 – Surveillance-overview logic for a layered waterflood.

The routine waterflood-analysis calculations that are used for the Kuparuk River oil field are another example of this approach. The Kuparuk River oil field covers more than 200 square miles and contains 600 patterns, with two separate reservoir intervals. Again, a massive amount of well data has been gathered over more than a decade of waterflooding and primary production. All of the simple calculation procedures that were discussed in earlier sections of this chapter [decline curves, log(WOR) vs. cumulative-oil-production plots, and X-plots] are used to evaluate well-by-well and pattern performance. Figs. 11.46 and 11.47 show relationships among the various calculation and database modules within the material-balance shell and the object relationships.

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  Fig. 11.46 – Relationship of relational database management system (RDBMS) tables to material-balance shell logic. (HCPV = hydrocarbon pore volume, and SQL = structured query language.)

• 

  Fig. 11.47 – Material-balance object relationships.

These two examples are of very large oil fields. Oil fields with fewer injection and production wells will require less-extensive waterflood-monitoring calculations; however, all the types of material-balance calculations will be required to determine whether the waterflood is performing as expected and whether significant operational changes are needed.

Also keep in mind that the usefulness of all these calculations depends greatly on the quality of the input data. The original field data must be reviewed for completeness and accuracy before entering them to these types of calculations. Where gaps in the data or clearly erroneous numbers are found, the engineers must judge how to edit the data or adjust calculations for such data problems. Where total field data has been allocated to the individual wells, the allocation procedures should be checked, including whether they have changed over time and thus caused changes in the slopes of some of plotted data—changes that are not real, but that are artifacts of the data-allocation procedures.

Waterflood Field-Case-Studies Examples

Several oil fields have been discussed in this chapter as examples of the use of various waterflood concepts and calculation procedures. The SPE technical literature contains many papers about major fields that have been waterflooded for a decade or more. In this section, a few of these oil fields are described briefly to highlight these waterflooding projects from their inception to the present, including how they were modified as they proceeded and as detailed engineering analyses were completed, and what complications were encountered. As noted earlier, many technical papers are available on these topics, but here only one to three publications are referenced for each oil field.

The waterfloods in these examples involve oils that range from light (30 to 40°API, with reservoir viscosities of 0.2 to 0.4 cp) to heavy (15 to 20°API, with reservoir viscosities of 30 to 70 cp). In all cases, the waterfloods have maintained reservoir pressure and increased oil recovery very successfully. A few other waterfloods also will be discussed briefly to highlight additional waterflooding considerations. The fields and waterfloods that are discussed here are Ekofisk (North Sea, offshore Norway), Wilmington (southern California, U.S.A.), Kuparuk River (Alaska North Slope, U.S.A.), west Texas carbonate waterfloods (U.S.A.), and Kirkuk (Iraq).

Three of these fields are various types of carbonate, and two are sandstone. Geographically, one is in the North Sea; one in the Long Beach, California, harbor area; one in the Arctic on Alaska’s North Slope; and two are in onshore oilfield areas. In some of these oil fields, the waterflood has been followed by EOR that uses miscible- or immiscible-gas injection in which some water injection has been continued to provide mobility control.

The reservoir-management chapter in this volume of the Handbook also describes the performance of several fields that are being waterflooded. That discussion’s emphasis is on reservoir management; however, it does include some information on recovery performance and depletion strategies.

Ekofisk(North Sea)

The Ekofisk oil field is in the North Sea, south of Norway. It is a large, carbonate reservoir [(6.4 billion bbl stock tank original oil in place (STOOIP)] that has two zones, Ekofisk and Tor, that are high-porosity, fractured chalks with matrix permeabilities of approximately 1 md and effective permeabilities that range from 1 to 50 md. Discovered in 1969, the Ekofisk field was found at very high pressure [7,120 psia at 10,400 ft true vertical depth subsea (TVDSS)] but with an initial bubblepoint pressure that was 1,600 psi below initial reservoir pressure. Ekofisk’s oil is 38°API, has a viscosity of approximately 0.25 cp, and has a solution gas/oil ratio (GOR) of more than 1,500 scf/STB. Primary production began in June 1971 and peaked in 1976 at 350,000 BOPD from 30 production wells (with 8 gas reinjection wells). Fig. 11.48 shows the structure of the Ekofisk field and its injection- and production-well locations as of early 2003. Laboratory-test results indicated that waterflooding by water imbibition into the low-permeability chalk was favorable for the Tor formation, but the laboratory results for the Ekofisk formation were variable. A waterflood pilot of the Tor formation was initiated in 1981, and favorable results were obtained by 1983. A 30-slot water-injection platform with an injection capacity of 375,000 BWPD started up in 1987. A second waterflood pilot was run in the lower Ekofisk formation from 1985 to 1987, and its results were positive. Further laboratory studies of the upper Ekofisk formation were somewhat negative, but a water-injection test into this zone and a coring of a sidetrack well 6 months later indicated that the upper Ekofisk formation also could be waterflooded successfully. By the early 1990s, all three intervals were receiving injected water.

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  Fig. 11.48 – Areal map of Ekofisk field with injection- and production-well locations (image provided by courtesy of ConocoPhillips).

Surface subsidence was a major issue at Ekofisk. By 1984, the seabed had subsided approximately 10 ft, prompting a major project to jack up the offshore platforms. A major field study in 1992 concluded that using water-injection pressure maintenance to arrest the reservoir pressure decline could minimize future seabed subsidence. Voidage replacement was achieved in 1993. Additional laboratory studies found that water injection had induced shear in the chalk. Shear failure and water-weakening of the rock matrix causes additional deformation of the chalk, even under conditions of constant or decreasing stress levels. Despite the use of voidage-replacement waterflooding, seabed subsidence continued until 1998, when the subsidence rates slowed dramatically because the water-weakening effect was expended and the reservoir pressure had increased.

In 1997, production began from the 50-slot production platform 2/4 X, and in 1998 full processing of the Ekofisk fluids was handled by the 2/4 J processing platform. These new "Ekofisk II" facilities replaced the aging original facilities and were designed to increase operational efficiency and to allow safe and economical production until at least 2028, the end of the current license period. The current best estimate of the ultimate recovery factor from the start of production through waterflooding is 44% of OOIP.

During the past 20 years of waterflooding, many operational changes have been made at Ekofisk. The changes were a logical progression that was based on laboratory studies, field pilot tests, and engineering analyses of field production and pressure data. All of this has led to a very successful waterflood project, but with a few unexpected complications that the engineers had to handle in the course of the waterflood project.

Wilmington Oil Field (California)

The LBU area of the Wilmington oil field (southern California, U.S.A.) is mainly under the Long Beach harbor and contains more than 3 billion bbl of OOIP. This oil field is a large anticline that is crosscut by several faults with displacements of 50 to 450 ft. It consists of seven zones between 2,500 and 7,000 ft TVDSS, the upper six of which are turbidite deposits of unconsolidated to poorly consolidated sandstone (1 to 1,000 md and 20 to 30% BV porosity) interbedded with shales. The gross thickness of 3,300 ft contains 900 ft of sandstone. In the largest zone, the Ranger (> 2 billion bbl of OOIP), the oils range from 14 to 21°API gravity and from 20- to 70-cp viscosity; hence, any waterflood would operate under a very unfavorable mobility ratio, and early water breakthrough would be expected.

From its discovery in 1936 to the 1950s, most of the onshore portion of this oil field (the non-LBU area of the Wilmington oil field) was produced using the pressure-depletion oil-recovery mechanism. Because of this, there was significant surface subsidence—up to 29 ft, with some areas dropping from several feet above sea level to below sea level (but protected by dikes). Development of the LBU area was delayed until an agreement with the government was reached that required voidage-replacement waterflooding to be implemented from its beginning to prevent further subsidence. LBU was developed in the mid-1960s, with the directional drilling of more than 1,000 wells from four artificial islands and from the nearby pier area. The early completions consisted of gravel-packed slotted liners that were up to 1,000 ft long in the injectors and the producers.

The LBU Ranger-zone waterflood was a 3:1 staggered line drive on a 10-acre well spacing (see Fig. 11.49); peripheral waterflooding has been used in the other zones. Oil production peaked in 1969 at 150,000 BOPD. The oil production has declined slowly since then, and the water production rate has increased steadily over the years. Water has been injected at rates of up to 1 million BWPD. Current oil production is approximately 32,000 BOPD at an average water cut of approximately 96%. To date, total production is > 940 million bbl of oil.

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  Fig. 11.49 – Areal map of injection and production wells in the Ranger-zone waterflood, LBU, Wilmington oil fields.

As years of oil-production and water-production and -injection data were acquired, the field engineers determined that some of the initial well-completion practices needed to be changed. The early completion techniques caused thief zones to develop in the higher-permeability sands, so that the deeper sands could not be pumped off. Between 1979 and 1986, by the drilling of 450 new wells into the lower portions of the Ranger and Terminal zones, this massive interval was redeveloped as two or three separate intervals. This subzoning added 160 million bbl of oil reserves and increased the field rate by 30,000 BOPD.

In 1991, an optimized waterflood program was undertaken to reduce the volume of produced water, recomplete wells in sands that were not being well swept by the waterflood, and drill new wells in selected locations to improve the performance of the waterflood. Pattern surveillance was used to quantify areas with low water throughputs and to guide the selection of new well locations. To aid the optimization studies, detailed studies of the geology of the field were undertaken and some 3D-seismic data were acquired and interpreted. The 3D-seismic data significantly changed the understanding of orientation of some of the faulting patterns within the field area; the previous fault patterns had been developed from the data gathered during drilling of 1,200 directional wells and from interpretation of early 2D-seismic data. Also in thicker sands, more than 50 horizontal wells have been drilled to capture bypassed "attic" oil and banked oil along faults. This optimized waterflood program has added 125 million bbl of oil reserves and has the potential to add another 90 million bbl of reserves.

Over the past 30 years, the field engineers have monitored and modified the original waterflood design using the full variety of waterflood-analysis techniques, including bubble maps, log(WOR) vs. cumulative-oil–production plots, X-plots, streamtube models, and numerical reservoir simulation for selected intervals within the Ranger zone over most of the LBU area.

One last point: Because of its very unfavorable mobility ratio, the LBU waterflood has undergone several decades of water injection and production with 80 to 97% water cuts (5 to 25 bbl water produced per bbl oil). The injected water has stripped considerable amounts of the dissolved solution gas from the oil in the reservoir sands. This is observed in the producing GORs, which—if all the gas were assumed to be from the oil—would indicate that free gas is being produced along with the oil and water. Instead, what is happening is that each barrel of "dead" injected water that cycles through the reservoir sands is extracting 5 to 10 scf of gas per STB of reservoir oil. This causes the reservoir oil to contain less solution gas and to increase in viscosity as the waterflood progresses.

Estimated ultimate-recovery factors for the LBU as of 2005 are shown in Table 11.2.

• 

  Table 11.2

Overall, the LBU area of the Wilmington oil field has been a very successful waterflood of a lower-API-gravity, more-viscous oil. For much of the waterflood period, the water cuts have been high, 80 to 97%. The LBU waterflood has successfully prevented further surface subsidence.

Kuparuk River (Alaska North Slope)

The Kuparuk River oil field is west of the supergiant Prudhoe Bay oil field on Alaska’s North Slope and was discovered in 1969. It has approximately 5.9 billion bbl of STOOIP and covers more than 200 sq. miles (see Fig. 11.50). The sandstone reservoir consists of two zones [A (62% of STOOIP) and C (38% of STOOIP)] that are separated by impermeable shales and siltstones. Sales oil is approximately 24°API with a viscosity at reservoir conditions of approximately 2.5 cp. The reservoir oil was approximately 300 to 500 psi undersaturated at the original reservoir pressure of approximately 3,300 psia. The reservoir is broken into segments by several north-to-south faults (density of approximately three faults per mile) that have sufficient throw to totally offset adjacent portions of the reservoir. The two major stratigraphic flow units exhibit considerably different properties, with the lower A zone having lower permeability, the C zone having considerably higher permeability, and the difference between them being approximately an order of magnitude.

• 

  Fig. 11.50 – Areal map of reservoir sands, Kuparuk River oil field. (CPF = central production facility. On this map, CPF 1, CPF 2, and CPF 3 refer to the areas of the Kuparuk River field that are producing into these three sets of facilities.)

The oil field was developed in stages, starting with the initially discovered eastern portion. Initial development was on 160-acre well spacing, and production started in 1981. The waterflood began in 1983. Expansion of the water-handling and -injection facilities led to full-field waterflooding in 1985. To date, more than 600 patterns have been developed with approximately 850 wells from 42 drillsites. The pattern style—a 320-acre, nominally 1:1 east/west line drive—takes into consideration the fault alignment. Annual average production peaked at more than 320,000 BOPD in 1992. Water production began in 1983, and the WOR slowly but steadily increased over the years to a value > 1 by 1990. The two intervals have been flooded at different rates because of their different reservoir properties. Practically all of the wells have been hydraulically fractured to enhance well productivity and injectivity.

The most significant aspects of the Kuparuk River waterflood are the field pattern development taking into account the reservoir-fault alignment, the allocation of water injection into the dually completed water-injection wells, and the souring of what was originally an H₂S-free oil reservoir. Produced water is treated and reinjected with make-up seawater to balance pattern withdrawals. During the waterflood, the reservoir became inoculated with sulfate-reducing bacteria. Because of the sulfate content of the seawater, these bacteria flourished and multiplied at reservoir conditions, so that the produced gas began to contain H₂S.

Estimated ultimate-recovery factors for the Kuparuk River oil field are given in Table 11.3.

• 

  Table 11.3

The Kuparuk River waterflood has been very successful because of the field engineers’ constant monitoring and active intervention over the past two decades. Over the years, EOR by immiscible water-alternating-gas (IWAG) and miscible water-alternating-gas (MWAG) injection has been used in various areas of the oil field to gain additional oil recovery (10% for Zone C and 6% for Zone A). Large-scale MWAG injection started in 1996 and has expanded to include more than 50% of the patterns.

West Texas Carbonate Waterfloods

San Andres and Clearfork are two carbonate reservoir intervals that are present over a considerable area of the Permian Basin in west Texas. These reservoirs (e.g., Wasson, Slaughter, Seminole) contain several billion bbl of approximately 30°API oil. They are very-layered, heterogeneous carbonates and dolomites that have large variation in permeability from layer to layer. Interestingly, because of the complex hydrocarbon-accumulation history of this basin, much of this area has an underlying interval that contains residual oil saturation.

Most of these reservoirs were discovered in the late 1930s and the 1940s. Even where both the San Andres and the Clearfork were found to be oil-bearing, surface-lease and unitization considerations required the reservoirs to be developed by drilling separate sets of wells. These reservoirs had approximately two decades of primary production.

From the 1960s through the 1980s, almost all of these reservoirs underwent waterflooding. During this period, many SPE technical papers addressed aspects of these waterfloods. Of particular interest was the continuity, or discontinuity, of the pay intervals and the extent to which infill drilling from 40-acre well spacing to 20 acres, and possibly to 10 acres, could be economically justified by increased oil recovery. Another issue was what pattern style was the best to use. The recovery factors for the sum of primary production and waterflooding ranges from 25 to 60% of OOIP; the variation is caused mainly by geological factors in these different geographical areas and by differences between the San Andres and Clearfork reservoirs.

Pressure coring and sponge coring are technologies that were developed specifically for these waterfloods to determine the residual oil saturation to waterflooding in these carbonate reservoirs. These techniques were used to determine the variation in oil saturation from well-swept higher-permeability layers to poorly swept lower-permeability layers.

The west Texas carbonate waterfloods have proved quite successful and have recovered significant additional oil. Also, the San Andres reservoirs’ waterfloods have provided much of the data and related technical studies that were needed to justify implementation of EOR by using CO₂ injection. During the tertiary EOR projects, water injection in the WAG mode has continued to control the mobility of the low-viscosity CO₂.

Kirkuk (Iraq)

There is only one SPE technical paper about the supergiant reservoir, the Kirkuk oil field of Iraq. This field is given as an example here because its reservoir’s geology is quite different from the other reservoirs discussed above.

The primary pay interval for the Kirkuk field is the 1,200-ft-thick Main Limestone. This interval consists of a series of extensively fractured limestones, some porcelaneous and some dolomitized. These limestones were deposited in a variety of environments—back-reef/lagoonal, fore-reef, and basinal—and have a wide range of porosity and permeability properties. The oil is contained both in an extensive, extremely permeable but low-capacity fracture system and in a low-permeability but high-capacity, matrix-pore system. Also, the reservoir is underlain by a fieldwide aquifer. The oil gravity is approximately 36°API and was approximately 500 psi undersaturated at the original reservoir pressure of 1,100 psia.

Kirkuk began production in 1934, and 2 billion bbl of oil were produced before water injection was implemented in 1961. From 1961 to 1971, 3.2 billion bbl of oil were produced under pressure maintenance by waterdrive using river water. The 1971 production rate was approximately 1.1 million BOPD. Since then, the field has continued to produce large volumes of oil by voidage-replacement water injection; however, few production details for recent years appear in the technical literature.

The interesting technical aspects of this type of reservoir are the determination of the ultimate oil recovery from the matrix and the time scale of matrix oil recovery. Laboratory experiments can be run using matrix rock samples to determine the water/oil imbibition behavior; however, what matters is the actual reservoir’s matrix/fracture interaction because the fracture density varies considerably. The early water injection showed that within the fracture network there was rapid communication over a distance of more than 20 miles. Water injection initially was peripheral; however, because of low injectivity caused by lack of downdip fracturing, injection was shifted to seven injection wells in the saddle area between the two principal domes of this oil field, one of which had an injection capacity of more than 400,000 BWPD.

A 90-day temporary production stoppage in 1967 allowed unique field data to be acquired regarding the matrix/fracture interaction because of the observed changes in the oil/water contact (OWC). It was observed that the OWCs fell in the areas where they were the highest and rose in the areas where they were the lowest. These OWC changes were the result of the countercurrent imbibition process between the fracture network and matrix pore system. From these data, the time-delay function could be calculated on the basis of observed field data. Depending on the assumptions, the half-life was estimated to be 3 to 5 years and the ultimate recovery was estimated at 30 to 45% of the OOIP.

Summary and Conclusions

This chapter has described the technical aspects of waterflooding, but only briefly compared to the vast amount of SPE technical literature on this subject. It also presented several field case studies for a few very large oil fields. Note, though, that reservoirs of all sizes have been waterflooded: very large fields having hundreds to 1,000+ wells; medium-sized fields; and small fields, for which the total waterflood is only a few patterns.

The conclusions concerning waterflooding are:

• Waterflooding is the most commonly used secondary-oil-recovery method. This is because water is inexpensive and readily available in large volumes and because water is very effective at substantially increasing oil recovery.
• The level of effectiveness of a waterflood depends on the mobility ratio between the oil and water, and the geology of the oil reservoir. Waterflooding is effective because almost all reservoir rocks are either water-wet or mixed-wet. The depositional and diagenetic characteristics of a reservoir control major aspects of the water/oil displacement process. These characteristics can either enhance waterflood performance or have detrimental effects on the WOR as a function of time. Often, the details of a reservoir’s internal geology are not known until production wells start producing injected water.
• Gravity effects—i.e., the interplay between the gravity/density effects and the geologic layering of a reservoir—are important in waterfloods because at reservoir conditions, oil always is less dense than connate brine or injected water. This interplay can either help or hurt waterflood performance relative to a homogeneous system.
• Waterflooding is a process that typically takes several decades to complete. Hence, continuous, routine field production and pressure data must be taken for monitoring and analyzing waterflood performance. Occasionally, more-expensive, special-data acquisition programs (i.e., 3D- or 4D-seismic data) are run to assist the evaluation process. A variety of engineering tools have been developed to analyze waterflood performance, ranging from simple plots of field production data to full-field numerical-reservoir-simulation models.
• Waterfloods are dynamic processes the performance of which, as production wells respond to the injection of water, can be improved by modification of operations by the technical team. Such modifications include changing the allocation of injection water among the injection wells and the waterflooded intervals, drilling additional wells at infill locations, and/or modifying the pattern style.
• Waterflooding has been used successfully in oil fields of all sizes and all over the world, in offshore and onshore oil fields.

Nomenclature

A	=	cross-sectional area available for flow, ft2
Bo	=	oil formation-volume factor, RB/STB
C	=	total flow capacity
EA	=	areal displacement efficiency
ED	=	unit-displacement efficiency
EI	=	vertical displacement efficiency
ER	=	waterflood oil-recovery efficiency
EV	=	volumetric sweep efficiency
EVbt	=	volumetric sweep efficiency at breakthrough
fo	=	fractional flow of oil
fw	=	fractional flow of water
	=	average fractional flow of water
fwf	=	fractional flow of water at flood front
Fwo	=	water/oil ratio
Fpvg	=	fraction of displaceable pore volume that is gas saturated
g	=	gravity constant
G	=	value as defined by Eq. 11.24, dimensionless
h	=	reservoir thickness, ft
i	=	fluid phase i
k	=	absolute permeability, darcies
k50	=	median permeability value, md
k84.1	=	permeability at 1 standard deviation above mean value, md
ka	=	air permeability
ki	=	permeability of fluid phase i , darcies
ko	=	permeability to oil, darcies
kr	=	relative permeability
kro	=	relative permeability to oil, fraction
kroe	=	relative permeability to oil at endpoint (Swi)
krw	=	relative permeability to water, fraction
krwe	=	relative permeability to water at endpoint (Sorw)
krwo	=	water/oil relative permeability
kw	=	permeability to water, darcies
kx	=	permeability in the x direction, darcies
L	=	length, ft
M	=	mobility ratio
Pc	=	capillary pressure, psia
qc	=	critical rate, B/D
qo	=	oil-production rate, B/D
qt	=	total production rate, B/D
qw	=	water-production rate, B/D
Sgi	=	initial gas saturation, fraction PV
Sgt	=	trapped gas saturation, fraction PV
So	=	oil saturation, fraction PV
Soi	=	initial oil saturation or (1 – Swc), fraction PV
Sorw	=	residual oil saturation to waterflooding, fraction PV
Sw	=	water saturation, fraction PV
	=	averaged water saturation behind the flood front at breakthrough, fraction PV
Swc	=	connate-water saturation, fraction PV
Swf	=	breakthrough or flood-front water saturation, fraction PV
Swi	=	initial water saturation, fraction PV
Swz	=	thickness-averaged water saturation for vertical equilibrium, fraction PV
t	=	time, days
uox	=	oil Darcy velocity in the x direction, ft/day
ut	=	total Darcy velocity, ft/day
uwx	=	water Darcy velocity in the x direction, ft/day
V	=	Dykstra-Parsons coefficient of permeability variation
x	=	position in x -coordinate system, ft
X	=	the X-plot value X, which is ln
y	=	position in y-coordinate system, ft
α	=	reservoir dip angle, degrees
β	=	slope, degrees
(Δp)h	=	pressure difference in horizontal direction, psi
(Δp)V	=	pressure difference in vertical direction, psi
Δρ	=	water/oil density difference, lbm/ft3 or g/cm3
θ	=	contact angle, degrees
λi	=	mobility of fluid phase i, darcies/cp
μi	=	viscosity of fluid phase i, cp
μo	=	oil viscosity, cp
μw	=	water viscosity, cp
ρo	=	oil density, lbm/ft3 or g/cm3
ρw	=	water density, lbm/ft3 or g/cm3
σos	=	interfacial tension between the oil phase and solid phase, dyne/cm
σws	=	interfacial tension between the water phase and solid phase, dyne/cm
σow	=	interfacial tension between the oil phase and water phase, dyne/cm
ϕ	=	porosity, fraction BV
ψs	=	fraction of total flow coming from the swept portion of the pattern

Acknowledgements

SPE members have published numerous papers concerning waterflood technology over the past 60 years. SPE has several major publications about waterflood technology, including two monographs, one textbook, and one Reprint Series volume. I have relied on these major publications and on a number of these technical papers in preparing this waterflood chapter for SPE’s Petroleum Engineering Handbook. I have particularly relied on Willhite’s Waterflooding textbook. From the outset, I recognized that much of the summary of waterflood technologies presented here could only paraphrase portions of Willhite’s textbook. Willhite describes the waterflood technical calculations in much more detail than is appropriate for this Handbook chapter. Also, I have used a number of the same figures as Willhite did; for the reader’s convenience, I have referenced them to his textbook and not to the many SPE technical papers of the past 50 years in which they originally were published. Appendix A cross-references to Willhite all the tables, figures, and equations from it that are used in this chapter.

References

Appendix A – Cross-Referencing to Willhite²

Table 11A.1 cross-references to Willhite² all the tables, figures, and equations from it that are used in this chapter.

• 

  Table 11A.1

SI Metric Conversion Factors

acre	×	4.046 873	E + 03	=	m2
°API		141.5/(131.5 + °API)		=	g/cm3
bbl	×	1.589 873	E − 01	=	m3
cp	×	1.0*	E − 03	=	Pa•s
dyne	×	1.0*	E − 02	=	mN
dyne/cm	×	1.0*	E − 00	=	mN/m
°F		(°F − 32)/1.8		=	°C
ft	×	3.048*	E − 01	=	m
ft/D	×	3.527 778	E − 03	=	mm/s
ft2	×	9.290 304	E − 02	=	m2
ft3	×	2.831 685	E − 02	=	m3
in.	×	2.54*	E − 00	=	cm
in.2	×	6.451 6*	E + 00	=	cm2
in.3	×	1.638 706	E + 01	=	cm3
lbm	×	4.535 924	E − 01	=	kg
lbm/ft3	×	1.601 846	E + 01	=	kg/m3
mile	×	1.609 344*	E − 00	=	km
psi	×	6.894 757	E − 00	=	kPa
psi/ft	×	2.262 059	E + 01	=	kPa/m
scf/STB	×	1.801 175	E − 01	=	std m3/m3
sq mile	×	2.589 988	E + 00	=	km2

*

Conversion factor is exact.