Sources and References

• Dake, L. P. Fundamentals of Reservoir Engineering. Elsevier, Amsterdam.
• Ahmed, T. Reservoir Engineering Handbook. Gulf Professional Publishing.
• Craft, B. C.; Hawkins, M.; Terry, R. E. Applied Petroleum Reservoir Engineering. Prentice Hall.
• Lake, L. W. Enhanced Oil Recovery. Prentice Hall / SPE.
• McCain, W. D. The Properties of Petroleum Fluids. PennWell.
• Amyx, J. W.; Bass, D. M.; Whiting, R. L. Petroleum Reservoir Engineering: Physical Properties. McGraw-Hill.
• Tiab, D.; Donaldson, E. C. Petrophysics: Theory and Practice of Measuring Reservoir Rock and Fluid Transport Properties. Gulf Professional Publishing.
• Archie, G. E. “The Electrical Resistivity Log as an Aid in Determining Some Reservoir Characteristics.” Trans. AIME, 1942.
• Buckley, S. E.; Leverett, M. C. “Mechanism of Fluid Displacement in Sands.” Trans. AIME, 1942.
• Welge, H. J. “A Simplified Method for Computing Oil Recovery by Gas or Water Drive.” Trans. AIME, 1952.
• Peng, D.-Y.; Robinson, D. B. “A New Two-Constant Equation of State.” Ind. Eng. Chem. Fundam., 1976.
• Stanford University, ERE 265 (Fundamentals of Petroleum Engineering) - public course syllabus and reading list.
• Texas A&M University, PETE 603 (Advanced Reservoir Engineering) - public course syllabus and reading list.

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Chapter 1: The Petroleum System and Production Engineering Overview

1.1 What Petroleum Production Engineering Studies

Petroleum production engineering sits at the crossroads of geology, surface chemistry, fluid mechanics, thermodynamics, and chemical engineering. The central problem is economic: a reservoir is a porous rock volume that originally held hydrocarbons under high pressure, and the engineer’s task is to move as much of that fluid as possible to the surface at a predictable rate and at acceptable cost. Every tool in the course — capillary pressure curves, Archie’s equation, Darcy’s law, the diffusivity equation, material balance, Buckley–Leverett theory, PVT correlations — exists to answer one of three linked questions. How much is down there? How fast will it come up? What can we do to change the answer to the first two?

A petroleum reservoir is not an underground lake. Hydrocarbons occupy micron-scale pore throats between mineral grains in sedimentary rock, usually sandstone or carbonate, overlain by an impermeable seal such as shale or evaporite. The pore network is tortuous, partially saturated with water left over from deposition (connate water), and at pressures of tens of megapascals and temperatures of 50–150 degrees Celsius. Fluids move under pressure gradients, and the solid matrix is not inert — it exerts capillary and wettability forces on the fluid phases that are every bit as important as viscous forces.

1.2 Elements of the Petroleum System

Five geological elements must coincide in time and space for a commercial accumulation. A source rock (organic-rich shale) must have buried deep enough for kerogen to mature thermally into oil or gas. A permeable carrier bed must connect source to reservoir. A reservoir rock must offer porosity and permeability. A seal must prevent upward escape. A trap — anticline, fault block, stratigraphic pinch-out, or salt dome — must create a closed volume. The engineer inherits all of this as given, but the quality of each element controls every later calculation.

1.3 Life Cycle of a Reservoir

Production proceeds in stages. During primary recovery, natural reservoir energy — compressed fluids, dissolved gas, an aquifer, or a gas cap — drives fluids to the wellbore without injection. Primary recovery factors typically run 5–25 percent of original oil in place. Secondary recovery injects water or gas to maintain pressure and sweep oil toward producers, adding perhaps another 15–25 percent. Tertiary or enhanced recovery methods (thermal, miscible gas, chemical) target the remaining residual oil by altering viscosity, interfacial tension, or wettability. Each stage has a different dominant physics, and the notes below are organized so that the early chapters on rock–fluid interaction feed directly into the later chapters on displacement and EOR.

Chapter 2: Surface Chemistry and Wettability

2.1 Interfacial Tension

Two immiscible fluids in contact develop an interface whose molecules experience unbalanced attractive forces. The free-energy cost per unit area of this interface is the interfacial tension \( \sigma \), with units of newton per metre or, equivalently, joule per square metre. For oil–water systems \( \sigma_{ow} \) is typically 20–35 mN/m at reservoir conditions; for gas–water systems it can exceed 70 mN/m. Interfacial tension is the single most important number in EOR design because, as shown later, it sets the scale of capillary trapping.

Interfacial tension depends on composition, temperature, and pressure. Adding a surfactant can reduce \( \sigma_{ow} \) by three or four orders of magnitude to values below \( 10^{-3} \) mN/m (ultralow IFT), and at reservoir temperatures \( \sigma \) generally falls with increasing temperature, a fact exploited in thermal EOR.

2.2 Contact Angle and Wettability

When an oil drop sits on a mineral surface submerged in water, three interfaces meet at a contact line. A force balance along the solid surface gives Young’s equation

\[ \sigma_{os} - \sigma_{ws} = \sigma_{ow} \cos\theta \]

where \( \theta \) is measured through the denser (water) phase. The wettability of the rock is classified by \( \theta \): water-wet if \( \theta < 75^\circ \), oil-wet if \( \theta > 105^\circ \), and intermediate or mixed-wet between. Wettability controls where each phase sits in the pore space. In a water-wet rock water coats the grains and occupies the small pore throats; oil lives in the pore centres. In an oil-wet rock the opposite holds, and oil films drain down the grain surfaces during production.

2.3 Why Wettability Matters to Production

Residual oil saturation after waterflooding of a water-wet core is typically 25–35 percent of pore volume, because non-wetting oil snaps off into disconnected ganglia trapped in pore bodies. In strongly oil-wet rocks, water fingers through oil-filled pore centres and bypasses oil films on the grains, giving very inefficient sweep. Mixed-wet reservoirs, in which the largest pores are oil-wet and the smallest are water-wet, often produce with the most favourable fractional-flow curves and lowest residuals. Amott and USBM indices quantify wettability from imbibition and drainage experiments on core plugs.

Chapter 3: Capillary Pressure and Saturation Distribution

3.1 Capillary Pressure in a Pore

The pressure difference across a curved fluid interface is the capillary pressure. For a circular capillary of radius \( r \) containing a wetting/non-wetting pair, the Young–Laplace equation gives

\[ P_c = P_{nw} - P_w = \frac{2\sigma\cos\theta}{r} \]

Capillary pressure is positive when the non-wetting phase pressure exceeds that of the wetting phase — which is the usual situation for oil displacing water into a small pore, or for mercury being forced into a dry rock. The \( 1/r \) dependence is the origin of nearly every capillary phenomenon in reservoirs: small pores hold the wetting phase fiercely, and a finite pressure difference is needed to displace it.

3.2 Drainage and Imbibition Curves

Measured on a core plug, the relationship \( P_c(S_w) \) takes the form of a saturation-dependent curve. Starting from 100 percent water saturation, increasing \( P_c \) (drainage, non-wetting oil invading a water-wet rock) produces a curve with a sharp entry pressure followed by a plateau, asymptoting at an irreducible water saturation \( S_{wi} \). Reversing the process (imbibition, water displacing oil) follows a different path that ends at residual oil saturation \( S_{or} \); the two curves enclose a hysteresis loop. The area under each curve has units of energy per unit pore volume and represents the reversible work of displacement.

3.3 Leverett J-Function

To compare capillary curves across cores of different porosity and permeability, Leverett introduced the dimensionless group

\[ J(S_w) = \frac{P_c}{\sigma\cos\theta}\sqrt{\frac{k}{\phi}} \]

where \( k \) is permeability and \( \phi \) is porosity. For a given rock type and wettability, \( J(S_w) \) collapses onto a single curve, allowing one laboratory measurement to be rescaled to the reservoir’s properties. This is the workhorse relation for populating reservoir simulators with saturation-height functions above the free-water level.

3.4 Saturation Above the Free-Water Level

At equilibrium in a gravity field, oil and water pressures both vary hydrostatically but with different densities. The free-water level (FWL) is the elevation where \( P_c = 0 \). At a height \( h \) above the FWL,

\[ P_c(h) = (\rho_w - \rho_o) g h \]

Combined with the measured \( P_c(S_w) \) curve, this gives saturation as a continuous function of depth: \( S_w \) is unity below FWL, drops through a transition zone whose thickness depends on \( \sigma \), \( k \), and \( \phi \), and asymptotes to \( S_{wi} \) high in the column. Thin, low-permeability reservoirs may consist almost entirely of transition zone.

Chapter 4: Petrophysics

4.1 Porosity

Porosity \( \phi \) is the ratio of pore volume to bulk volume. Total porosity includes isolated pores; effective porosity counts only interconnected pore space relevant to flow. Laboratory methods include Boyle’s-law helium porosimetry (accurate, uses gas expansion), fluid resaturation (simple, destructive), and grain-density reconstruction from mineral composition. Logs infer porosity from bulk density (assuming mineral and fluid densities), neutron response, or acoustic travel time. Reservoir porosity ranges from roughly 0.05 in tight carbonates to 0.35 in unconsolidated sands.

4.2 Absolute Permeability

Permeability \( k \) is defined by Darcy’s law and has units of m\( ^2 \) (1 darcy ≈ \( 9.87\times10^{-13} \) m\( ^2 \)). It is a property of the rock alone if the fluid is a single, inert, incompressible liquid that does not interact chemically with the rock. Klinkenberg corrections are required for gas measurements because of slip at pore walls at low pressure: the apparent gas permeability

\[ k_g = k_{\infty}\left(1 + \frac{b}{\bar P}\right) \]

extrapolates to the absolute (liquid) permeability \( k_{\infty} \) as mean pressure \( \bar P \) becomes large.

4.3 Pore-Scale Permeability Models

A capillary-bundle model with \( N \) tubes of radius \( r \) per unit area gives Poiseuille flow \( q = N\pi r^4 \Delta P / (8\mu L) \). Matching this to Darcy form yields \( k \propto r^2 \), which is the single most important scaling in petrophysics — permeability scales with the square of a characteristic pore-throat size. The Kozeny–Carman relation refines this to

\[ k = \frac{\phi^3}{c\, \tau^2 S_v^2 (1-\phi)^2} \]

where \( S_v \) is specific surface per unit grain volume, \( \tau \) is tortuosity, and \( c \) is a shape factor. The cube-squared dependence on \( \phi \) and inverse-square dependence on surface area explain why clay-bearing sands lose permeability much faster than they lose porosity.

4.4 Electrical Properties and Archie’s Equation

Reservoir rocks conduct electricity only through the brine in the pore space; dry oil and hydrocarbon gas are insulators. Archie’s first law relates the formation resistivity at 100 percent brine saturation \( R_o \) to brine resistivity \( R_w \) through the formation factor

\[ F = \frac{R_o}{R_w} = \frac{a}{\phi^m} \]

with cementation exponent \( m \approx 2 \) for consolidated sandstones and \( a \approx 1 \). When oil or gas partially displaces brine, resistivity rises; the resistivity index is

\[ I = \frac{R_t}{R_o} = S_w^{-n} \]

with saturation exponent \( n \approx 2 \). Combining these gives the log interpretation formula

\[ S_w = \left(\frac{a R_w}{\phi^m R_t}\right)^{1/n} \]

which is the basis of every quicklook log interpretation. Shaly sands require corrections (Waxman–Smits, dual-water) because clay surface conductance adds a parallel current path.

4.5 Flow Properties: Relative Permeability (preview)

When two fluids share the pore space, each phase experiences an effective permeability smaller than \( k \). Relative permeabilities \( k_{rw}(S_w) \) and \( k_{ro}(S_w) \) are dimensionless curves usually measured by steady-state or unsteady-state (Welge) core experiments. Chapter 9 treats these in detail; they enter the Buckley–Leverett analysis through the fractional-flow function.

Chapter 5: Hydrostatic Pressure Regimes and Reserves Estimation

5.1 Pore Pressure Regimes

A normal pressure gradient follows the hydrostatic gradient of saline water, about 10.5 kPa/m (0.465 psi/ft). Overpressured zones exceed this because rapid burial, hydrocarbon generation, or sealing prevents fluid escape and the pore fluid supports part of the overburden. Underpressured zones, common in depleted reservoirs, fall below hydrostatic. Drilling engineers and reservoir engineers share this vocabulary because pore pressure sets kick risk, casing design, and the initial energy available for primary production.

5.2 Volumetric Reserves Estimate

Original oil in place (OOIP) at reservoir conditions equals the bulk rock volume times porosity times oil saturation. Converted to surface barrels, using the oil formation volume factor \( B_o \) (reservoir volume per surface volume),

\[ N = \frac{V_b \phi (1 - S_{wi})}{B_o} \]

Recoverable reserves equal OOIP times a recovery factor, which depends on drive mechanism (see Chapter 8). Gas in place uses \( B_g \) similarly:

\[ G = \frac{V_b \phi (1 - S_{wi})}{B_g} \]

Volumetric estimates are the earliest available but carry uncertainty from every term; probabilistic ranges (P10, P50, P90) are reported rather than point values.

5.3 Decline Curve Analysis

Once production history exists, empirical Arps decline curves fit rate–time data. The three forms are exponential (constant fractional decline), hyperbolic, and harmonic. The generalized form is

\[ q(t) = \frac{q_i}{(1 + b D_i t)^{1/b}} \]

with initial rate \( q_i \), initial decline \( D_i \), and hyperbolic exponent \( b \) (zero for exponential, one for harmonic). Cumulative production follows by integration. Decline analysis is strictly empirical and presumes boundary-dominated flow at constant bottomhole pressure; extrapolating a transient-period decline gives wildly optimistic reserves.

Chapter 6: Darcy’s Law and Single-Phase Flow

6.1 Darcy’s Experiment and Its Generalization

Henry Darcy’s 1856 Dijon fountain study produced the linear relation between superficial velocity and head gradient in sand filters. Generalized to three dimensions for a Newtonian single-phase fluid,

\[ \vec u = -\frac{k}{\mu}\left(\nabla P - \rho \vec g\right) \]

Here \( \vec u \) is the Darcy (superficial) velocity — volumetric flux per unit bulk area, not interstitial velocity. Dividing by \( \phi \) gives interstitial velocity. Limitations are real: Darcy’s law fails at high Reynolds number (Forchheimer’s quadratic correction applies near gas wells), at very low gradient in clay-rich rocks (threshold gradient), and for non-Newtonian fluids such as polymer solutions.

6.2 Steady-State Linear Flow

For incompressible flow through a core plug of length \( L \) and cross-section \( A \) with pressures \( P_1 \) at the inlet and \( P_2 \) at the outlet,

\[ q = \frac{kA(P_1 - P_2)}{\mu L} \]

For compressible gas at low pressure, conservation of mass with real-gas deviation factor \( Z \) and temperature \( T \) gives, after integration,

\[ q_{sc} = \frac{kA(P_1^2 - P_2^2)}{2\mu \bar Z \bar T L}\cdot\frac{T_{sc}}{P_{sc}} \]

where the squared-pressure form comes from the pressure dependence of gas density.

6.3 Steady-State Radial Flow

The well-bore geometry is nearly always radial. For steady incompressible flow between well radius \( r_w \) and drainage radius \( r_e \) with constant boundary pressures,

\[ q = \frac{2\pi k h (P_e - P_{wf})}{\mu \ln(r_e/r_w)} \]

This is the Dupuit–Thiem equation. The logarithmic dependence on \( r_e/r_w \) is distinctive: most of the pressure drop occurs in the first few metres around the wellbore, which is why near-wellbore damage (skin) is so damaging to productivity.

6.4 Skin Factor and Productivity Index

Real wells deviate from ideal radial flow because of drilling-induced damage, partial penetration, perforation geometry, or stimulation. Hawkins’ skin factor \( s \) adds a dimensionless pressure drop:

\[ q = \frac{2\pi k h (P_e - P_{wf})}{\mu [\ln(r_e/r_w) + s]} \]

Positive \( s \) means damage; negative \( s \) (typically \(-3\) to \(-6\)) means an effective hydraulic fracture or acidized zone. The productivity index \( J = q/(P_e - P_{wf}) \) is the engineering summary used to compare wells and to design artificial lift.

Chapter 7: Transient Flow and the Diffusivity Equation

7.1 Derivation

Combining Darcy’s law, mass conservation, and a slightly compressible fluid equation of state (constant total compressibility \( c_t \)) gives the radial diffusivity equation:

\[ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial P}{\partial r}\right) = \frac{\phi \mu c_t}{k}\frac{\partial P}{\partial t} \]

The group \( \eta = k/(\phi \mu c_t) \) is the hydraulic diffusivity with units of m\( ^2 \)/s. The same equation governs heat conduction, and decades of heat-transfer solutions transfer directly to well testing.

7.2 Flow Regimes

Starting from a uniform-pressure reservoir produced at constant rate, three regimes appear in sequence. Early wellbore-storage effects dominate while the well itself fills or empties. Infinite-acting radial flow follows, during which the pressure transient has not yet felt the boundary; the bottomhole pressure falls as the logarithm of time. Once boundaries are reached, flow becomes pseudo-steady (closed system, constant withdrawal) or steady (aquifer support).

7.3 Line-Source Solution

For constant-rate production from an infinitesimal line source in an infinite reservoir, the solution in terms of the exponential integral is

\[ P(r, t) = P_i - \frac{q\mu}{4\pi k h}\, \mathrm{Ei}\!\left(-\frac{\phi\mu c_t r^2}{4kt}\right) \]

For practical times and the wellbore radius, the argument is small and \( -\mathrm{Ei}(-x) \approx -\ln(x) - \gamma \). Evaluating at \( r = r_w \) and adding skin yields the workhorse form

\[ P_{wf}(t) = P_i - \frac{q\mu}{4\pi k h}\left[\ln\!\left(\frac{kt}{\phi\mu c_t r_w^2}\right) + 0.809 + 2s\right] \]

A semilog plot of \( P_{wf} \) versus \( \ln t \) is linear during infinite-acting radial flow. The slope gives \( kh \); the intercept gives \( s \).

7.4 Pseudo-Steady Flow

Once the pressure disturbance reaches a closed outer boundary, every point in the drainage volume drops at the same rate \( dP/dt = -q / (V_p c_t) \), where \( V_p \) is drainage pore volume. In a circular drainage area of radius \( r_e \),

\[ q = \frac{2\pi k h (\bar P - P_{wf})}{\mu[\ln(r_e/r_w) - 3/4 + s]} \]

where \( \bar P \) is the volume-averaged reservoir pressure. The \( -3/4 \) distinguishes pseudo-steady from steady flow.

Chapter 8: Well Testing

8.1 Why We Test

A pressure transient test perturbs the reservoir briefly and records the pressure response at a single well. From that response we can extract near-wellbore permeability–thickness \( kh \), skin \( s \), average drainage pressure \( \bar P \), and boundary distances. Core measurements yield centimetre-scale information; well tests average over the drainage volume. The two techniques are complementary, not competing.

8.2 Drawdown Tests

In a drawdown test a well at equilibrium is put on constant production and the bottomhole pressure is recorded. During infinite-acting flow a semilog plot of \( P_{wf} \) versus \( \log t \) has slope

\[ m = \frac{162.6\, q\mu B}{k h}\quad(\text{field units}) \]

from which permeability–thickness follows directly. Skin is extracted from the intercept by comparing the measured \( P_{wf} \) at one hour to the ideal value. Drawdown tests are attractively simple but suffer from rate fluctuations and superposition errors.

8.3 Buildup Tests and Horner Analysis

A buildup test is cleaner: produce at constant \( q \) for time \( t_p \), then shut in and record pressure rise. Superposition of a constant-rate producer and an injector started at time \( t_p \) yields Horner’s result that shut-in pressure is linear in the Horner time ratio:

\[ P_{ws}(\Delta t) = P^* - m \log\!\frac{t_p + \Delta t}{\Delta t} \]

The slope \( m \) has the same relation to \( kh \) as in the drawdown case. Extrapolating the straight line to a Horner ratio of unity gives \( P^* \), which for a closed system is corrected to average pressure \( \bar P \) using Matthews–Brons–Hazebroek charts. Skin is found from the value of \( P_{ws} \) at \( \Delta t = 1 \) hour.

8.4 Type-Curve and Derivative Analysis

Modern interpretation uses log–log plots of pressure change and of the pressure derivative \( t\,dP/dt \) against time. The derivative is flat during infinite-acting radial flow, giving a clear visual marker. Different boundary geometries produce characteristic late-time derivative signatures: a unit slope for a closed system, a doubling of the flat value for a sealing fault, a drop to zero for a constant-pressure boundary. Type curves for wellbore-storage-plus-skin (Gringarten) and for hydraulic fractures (uniform-flux, infinite-conductivity) complete the diagnostic toolkit.

Chapter 9: Thermodynamics of Petroleum Fluids

9.1 Black-Oil Model

For engineering calculations that do not need compositional detail, the black-oil model treats reservoir fluids as two pseudo-components, stock-tank oil and surface gas, distributed across two phases at reservoir conditions by the following properties:

• Solution gas–oil ratio \( R_s \): surface standard cubic metres of gas dissolved per standard cubic metre of stock-tank oil at given P, T.
• Oil formation volume factor \( B_o \): reservoir volume of saturated oil per surface volume of stock-tank oil.
• Gas formation volume factor \( B_g \): reservoir volume per surface volume for dry gas.
• Oil compressibility \( c_o \), gas Z-factor, and viscosities \( \mu_o \), \( \mu_g \).

Above the bubble point all gas is dissolved and \( R_s \) is constant. Below the bubble point \( R_s \) falls with pressure and \( B_o \) decreases as gas comes out of solution. Correlations (Standing, Vasquez–Beggs, Glaso) give these quantities from readily measurable properties like API gravity, separator GOR, and temperature.

9.2 Phase Envelope

In a pressure–temperature diagram, a multicomponent hydrocarbon mixture has a two-phase envelope bounded by the bubble-point curve and the dew-point curve meeting at the critical point. Reservoir fluids are classified by where the reservoir condition sits relative to the envelope. A dry gas lies to the right of the envelope and stays single-phase in the reservoir and at the surface. A wet gas lies to the right but crosses the envelope on its way to separator conditions. A retrograde gas-condensate lies between the critical point and the cricondentherm — isothermal pressure depletion first crosses the dew-point and liquid drops out in the reservoir, a peculiar and economically troublesome behaviour. A volatile oil and a black oil lie inside the envelope to the left of the critical point, distinguished by shrinkage and surface GOR.

9.3 Cubic Equations of State

The workhorse models are Peng–Robinson (1976) and Soave–Redlich–Kwong (1972). Peng–Robinson reads

\[ P = \frac{RT}{v - b} - \frac{a(T)}{v(v+b) + b(v-b)} \]

where \( a \) and \( b \) are component parameters functions of critical properties and acentric factor, and \( a(T) \) includes a temperature-dependent \( \alpha \) function. Mixtures use van der Waals one-fluid mixing rules with binary interaction parameters. Flash calculations solve for the compositions of vapour and liquid at equilibrium by requiring fugacity equality of each component in each phase. Compositional reservoir simulators embed these calculations at every grid block and time step.

9.4 Gas Properties

For dry natural gas, the real-gas law is

\[ PV = ZnRT \]

with \( Z(P, T) \) from Standing–Katz charts or correlations in pseudo-reduced coordinates. Gas density and formation volume factor follow immediately. Gas viscosity \( \mu_g \) comes from Lee–Gonzalez–Eakin correlations and rises with pressure at reservoir temperatures.

Chapter 10: Material Balance

10.1 Philosophy

Material balance treats the entire reservoir as a single tank. It is a zero-dimensional conservation statement: the change in pore volume occupied by each fluid phase plus the change in volume of injected/aquifer fluids must equal the volume of fluid withdrawn, all measured at reservoir conditions. No geometry, no pressure gradients — only averages. The power of the method lies in what it can back out from simple production data: original hydrocarbon in place, aquifer strength, and the relative contribution of each drive mechanism.

10.2 The Generalized Equation

In Schilthuis’s compact form, rearranged for oil reservoirs by Havlena and Odeh, the material balance reads

\[ F = N\,E_o + N m E_g + N(1 + m) E_{f,w} + W_e \]

Here \( F \) is cumulative reservoir-volume withdrawal, \( N \) is OOIP, \( m \) is initial gas-cap-to-oil pore-volume ratio, \( W_e \) is cumulative water influx, and the \( E \) terms are expansion factors: \( E_o \) for oil-plus-solution-gas expansion, \( E_g \) for gas-cap expansion, and \( E_{f,w} \) for formation and connate-water expansion. Plotting \( F/E_o \) versus \( E_g/E_o \) or \( W_e/E_o \), depending on the dominant drive, yields straight lines whose slope and intercept give \( N \), \( m \), and aquifer parameters.

10.3 Drive Mechanisms

Primary drives differ in physics and recovery factor.

Drive	Physics	Typical recovery factor
Rock and liquid expansion	Undersaturated oil, compressibility only	3–7%
Solution-gas (depletion) drive	Gas evolving from oil below bubble point	5–20%
Gas-cap drive	Expanding free gas pushing down oil	20–40%
Water drive (aquifer)	Active aquifer supplies influx	35–60%
Gravity drainage	Steeply dipping reservoir, oil drains to lower producers	40–80%

Water drive is the most efficient natural drive because pressure is maintained and sweep is piston-like when the aquifer is strong. Solution-gas drive is the worst because the evolving gas mobility ratio quickly exceeds unity, gas breakthroughs dominate production, and residual oil stays high.

10.4 Aquifer Models

When an aquifer contributes, \( W_e \) is not known directly. It must be modelled. Schilthuis’s steady-state model assumes \( W_e \propto \int(P_i - \bar P)\,dt \). The Hurst–van Everdingen unsteady-state solution treats the aquifer as a radial transient pressure problem; the response is tabulated as a dimensionless water influx function. Fetkovich’s pseudo-steady method lumps the aquifer into a finite reservoir obeying a rate–pressure-squared relation and is popular for simulator coupling.

Chapter 11: Two-Phase Flow and Relative Permeability

11.1 Extended Darcy’s Law

For each flowing phase \( \alpha \in \{w, o\} \), Darcy’s law in the multiphase form is

\[ \vec u_\alpha = -\frac{k\, k_{r\alpha}(S_w)}{\mu_\alpha}\left(\nabla P_\alpha - \rho_\alpha \vec g\right) \]

The relative permeability \( k_{r\alpha} \) is a dimensionless saturation function, zero at and below the phase’s irreducible saturation and approaching an endpoint below unity at maximum saturation. The oil and water pressures differ by the capillary pressure. Common parametric forms are Corey–Brooks power laws

\[ k_{rw}(S_w) = k_{rw}^{\max}\left(\frac{S_w - S_{wi}}{1 - S_{wi} - S_{or}}\right)^{n_w} \]

with analogous form for \( k_{ro} \). Wettability reshapes the curves: in a water-wet rock \( k_{ro} \) decreases rapidly from high saturation and \( k_{rw} \) endpoint is low; in an oil-wet rock the reverse holds, often with crossover saturation above 50 percent.

11.2 Mobility and Mobility Ratio

Phase mobility is \( \lambda_\alpha = k_{r\alpha}/\mu_\alpha \). The end-point mobility ratio for waterflooding is

\[ M = \frac{\lambda_w(S_{or})}{\lambda_o(S_{wi})} = \frac{k_{rw}^{\max}/\mu_w}{k_{ro}^{\max}/\mu_o} \]

Mobility ratio is the single strongest predictor of sweep efficiency. \( M < 1 \) is favourable (viscous fingering suppressed, piston-like displacement). \( M > 1 \) is unfavourable and leads to viscous fingers, channelling, and early breakthrough. Because \( \mu_o \) can exceed \( \mu_w \) by one or two orders of magnitude in heavy-oil reservoirs, \( M \) routinely reaches 10–100 and drives many design decisions.

Chapter 12: Buckley–Leverett and Fractional Flow

12.1 Fractional Flow Function

For 1-D horizontal incompressible two-phase flow with no capillary pressure, the fraction of total volumetric flux that is water is

\[ f_w(S_w) = \frac{1}{1 + \lambda_o/\lambda_w} = \frac{1}{1 + \mu_w k_{ro}/(\mu_o k_{rw})} \]

The sigmoidal curve rises from zero at \( S_{wi} \) to unity at \( 1 - S_{or} \). Adding gravity for dipping reservoirs and capillary pressure for heterogeneous media introduces extra terms but preserves the structure.

12.2 Buckley–Leverett Frontal-Advance Equation

Substituting the extended Darcy law into mass conservation for water in 1-D gives

\[ \phi\frac{\partial S_w}{\partial t} + u\,\frac{df_w}{dS_w}\frac{\partial S_w}{\partial x} = 0 \]

This is a first-order hyperbolic PDE. Along characteristics of constant saturation, the wave speed is \( u f_w'(S_w) / \phi \). Because \( f_w' \) is not monotonic in \( S_w \), characteristics collide and a saturation shock forms.

12.3 Welge’s Construction

Welge showed that the physically correct shock saturation \( S_{wf} \) is determined by drawing a tangent to the fractional-flow curve from the point \( (S_{wi}, 0) \). The tangent point is \( S_{wf} \); the tangent slope is \( f_w'(S_{wf}) \); the fractional flow at the shock is \( f_{w,f} \). Behind the shock saturation rises continuously to the maximum injected-water saturation at the inlet. Average saturation behind the front at breakthrough is

\[ \bar S_w = S_{wf} + \frac{1 - f_{w,f}}{f_w'(S_{wf})} \]

and recovery at breakthrough equals \( (\bar S_w - S_{wi})/(1 - S_{wi}) \) fraction of movable oil. Post-breakthrough, oil continues to be produced at ever-increasing water cut until economic limit.

12.4 What Controls Waterflood Recovery

The Buckley–Leverett picture reveals why mobility ratio matters. Favourable \( M \) produces a sharp front, high shock saturation, high recovery at breakthrough, and low water cut for a long period. Unfavourable \( M \) flattens the fractional-flow curve near \( S_{wi} \), lowers the shock saturation, and causes early breakthrough with a long tail of high-water-cut production. Adding polymer to injected water raises \( \mu_w \), lowers \( M \), and directly improves this picture — the physics basis for polymer flooding.

Chapter 13: Sweep Efficiency

13.1 Volumetric Sweep

Overall displacement efficiency breaks into three factors:

\[ E_{R} = E_D \cdot E_A \cdot E_V \]

\( E_D \) is displacement efficiency (the microscopic fraction of oil moved from the contacted pore volume, set by relative permeability and residual saturation). \( E_A \) is areal sweep (the fraction of pattern area contacted by injectant). \( E_V \) is vertical sweep (the fraction of net pay contacted, degraded by permeability stratification and gravity).

13.2 Areal Sweep

In a homogeneous pattern flood — five-spot, seven-spot, line-drive — areal sweep at breakthrough depends only on mobility ratio. For an unfavourable \( M \) of 10 in a five-spot, areal sweep at breakthrough falls to about 55 percent, compared to 72 percent at \( M = 1 \). Streamline patterns and experiments in scaled models (Craig–Geffen–Morse) quantify the dependence.

13.3 Vertical Sweep

Permeability stratification is the usual culprit for poor vertical sweep. The Dykstra–Parsons coefficient \( V_{DP} \) and the Lorenz coefficient summarize layer heterogeneity. High \( V_{DP} \) (approaching unity) means a few high-permeability streaks take most of the injectant, bypassing the rest. Gravity override in thick reservoirs worsens vertical sweep for injected gas and can improve it for bottom-driven water.

13.4 Pattern Selection

Pattern geometry trades sweep, well count, and timing. Five-spot and nine-spot patterns suit homogeneous, isotropic reservoirs. Line drives orient along major permeability trends or natural fracture strike. Inverted patterns (one injector surrounded by producers) are preferred for expensive injectants like CO\( _2 \) because they concentrate surveillance.

Chapter 14: Enhanced Oil Recovery

14.1 The Capillary Number and Residual Oil

The residual oil left by waterflooding is held by capillary forces in pore bodies. The dimensionless capillary number

\[ N_c = \frac{\mu_w u}{\sigma_{ow}} \]

compares viscous to capillary forces. At typical waterflood values \( N_c \sim 10^{-7} \), and residual saturation is essentially unchanged from laboratory values. Mobilizing trapped oil requires \( N_c \) to exceed roughly \( 10^{-4} \), a three-orders-of-magnitude jump. Reducing interfacial tension by surfactants, increasing viscosity by polymers, or both, is the only practical route.

14.2 Thermal Methods

Heavy oils have viscosities of hundreds to thousands of centipoise. Heating them is the most effective way to lower viscosity — a 10 K temperature rise can halve viscosity. Steam-flooding injects wet or superheated steam; steam-assisted gravity drainage (SAGD) pairs a horizontal injector over a horizontal producer and exploits the high thermal mobility ratio to drain hot, mobilized oil by gravity. Cyclic steam stimulation (“huff and puff”) soaks a single well with steam, shuts it in, and produces the now-hot zone. Recovery factors for thermal EOR in suitable reservoirs reach 50–60 percent.

14.3 Chemical Methods

Polymer flooding (partially hydrolyzed polyacrylamide or biopolymers like xanthan) thickens injected water to lower mobility ratio and improve sweep. It does not reduce residual saturation — it just contacts more of the reservoir. Surfactant flooding drops interfacial tension by three or four orders of magnitude, attacking \( E_D \) directly. Alkaline–surfactant–polymer (ASP) formulations combine mechanisms: alkali reacts with naphthenic acids to generate in-situ surfactant, the surfactant lowers IFT, and the polymer maintains sweep. Chemical costs and adsorption on the rock limit economic application to specific reservoir–crude pairs.

14.4 Miscible Gas Flooding

If the injected gas and reservoir oil mix in all proportions with no interface, the displacement becomes effectively single-phase and residual saturation goes to zero in the swept region. First-contact miscibility requires very high pressures; multiple-contact miscibility develops through repeated equilibration between flowing gas and oil — the vaporizing-gas drive enriches the gas with intermediate components, the condensing-gas drive enriches the oil with injected intermediates, and both mechanisms act in modern CO\( _2 \) and enriched-hydrocarbon floods. The minimum miscibility pressure (MMP) is measured in slim-tube experiments. CO\( _2 \) flooding has the dual appeal of high recovery and geological carbon storage; operational concerns include corrosion, asphaltene precipitation, and gravity override.

14.5 Ranking EOR Options

14.6 Outlook

Low oil prices, decarbonization pressure, and growing CO\( _2 \) supply shift the balance of EOR practice. CO\( _2 \)-EOR coupled to carbon sequestration now has policy support in several jurisdictions. Produced-water management and chemical disposal constrain chemical EOR. Digital twin models, coupling compositional simulation to automated history matching, are becoming standard for EOR planning. Regardless of which method wins, the physical hierarchy remains unchanged: lower capillary forces, lower mobility ratios, and deliver as much injectant energy to untouched rock as possible.