Equations: Reservoir conditions and development tasks

The equations of the “Reservoir conditions and development tasks” lessons in “Reservoir engineering”, with their conditions and units, and what each symbol means in them.

Where the oil is left after waterflooding

\frac{N_p}{N} = E_D\,E_AE_V,\qquad \frac{N_{or}}{N} = E_AE_V\,(1 - E_D),\qquad \frac{N_u}{N} = 1 - E_AE_V

fractions; volumes in the same units

Holds when: The displacement and sweep efficiencies are those of the lesson Why water is injected: pressure maintenance and oil displacement. Their product is the recovery factor. The pressure is maintained, the oil formation volume factor hardly changes and there is no free gas. So the shares of the oil in place equal the shares of the oil-saturated pore volume. In the bypassed part the oil saturation is the initial one, and in the swept part it is the residual one. In reality, partly swept zones with an in-between saturation lie between them. So the split is a simplified picture, and it exaggerates the contrast between the parts.

Reservoir engineering: tasks across the life of a field →

Exponential decline

q = q_i\,e^{-Dt},\qquad \Delta N_p = 365\,\frac{q_1 - q_2}{D}

rate in m³/d, time in years, production in m³

Holds when: The rate falls by the same fraction in equal intervals of time. The operating mode does not change: no new wells, no moving of wells to other layers (recompletion), no changes of injection and no limits set by the facilities. The factor 365 converts the product “m³/d × year” into cubic metres.

Reservoir engineering: tasks across the life of a field →

Hyperbolic decline

q = \frac{q_i}{(1 + bD_it)^{1/b}},\qquad N_p = 365\,\frac{q_i}{(1 - b)\,D_i}\left[1 - \left(\frac{q}{q_i}\right)^{1-b}\right]

as in the equation for exponential decline

Holds when: $0 < b < 1$ . As $b \to 0$ the formulas turn into the exponential ones, and at $b = 1$ the production is $365\,(q_i/D_i)\ln(q_i/q)$ . We fit the exponent $b$ to a long history. Without a long history, the choice of $b$ is only an assumption.

Reservoir engineering: tasks across the life of a field →

Absolute pressure from a gauge reading

p_{\text{abs}} = p_{\text{gauge}} + p_{\text{atm}}

any pressure units, the same for all three quantities

Holds when: For the conversion we usually take the standard atmospheric pressure. The actual pressure depends on altitude and weather. At 1000 m above sea level it is about 0.09 MPa. Pressure differences do not depend on the choice of zero. Examples are drawdown (the pressure drop that pushes fluid into the well), the pressure drop across a choke (a restriction that controls the flow) and a pressure gradient.

Systems of units: SI, oilfield and metric oilfield →

Temperature conversion

T\,[\text{K}] = t\,[{}^\circ\text{C}] + 273.15,\qquad t\,[{}^\circ\text{F}] = 1.8\,t\,[{}^\circ\text{C}] + 32,\qquad T\,[{}^\circ\text{R}] = t\,[{}^\circ\text{F}] + 459.67 = 1.8\,T\,[\text{K}]

given in square brackets

Holds when: We convert temperature differences without the shift (the added constant): 1 K = 1 °C, 1 °R = 1 °F, 1 K = 1.8 °R. °R in oilfield formulas is the degree Rankine. In Russian physics reference books it is written °Ra, because there °R is the obsolete degree Réaumur. In older Russian-language literature the name Rankine is also transliterated as “Renkin”. A geothermal gradient (how fast the temperature rises with depth) of 1 °F per 100 feet equals 1.823 °C per 100 m.

Systems of units: SI, oilfield and metric oilfield →

Conversion of a gas volume to other reference conditions

V_2 = V_1\,\frac{T_2}{T_1}\,\frac{p_1}{p_2}

temperatures absolute, pressures absolute and in the same units

Holds when: The amount of gas is the same. At reference conditions we treat the gas as ideal: the gas deviation factor (Z, which shows how far a real gas departs from an ideal one) is close to one, and its small difference almost cancels. The moisture content is the same: under GOST 2939-63 the gas is dry, and water vapour is not included in the volume.

Systems of units: SI, oilfield and metric oilfield →

Solution gas–oil ratio and producing gas–oil ratio: m³/m³ and scf/STB

R\,[\text{scf/STB}] = 5.6146\,R\,[\text{m}^3/\text{m}^3]\qquad\text{or}\qquad R\,[\text{scf/STB}] = 5.529\,R\,[\text{m}^3/\text{m}^3]

gas — standard cubic metres or cubic feet, oil — cubic metres or barrels at standard conditions

Holds when: We use 5.6146 when the gas on both sides is referred to the same conditions. It is simply the number of cubic feet in a barrel. We use 5.529 when the cubic metres of gas are referred to 20 °C and 101.325 kPa, and the cubic feet to 60 °F and 14.696 psia. The volume of oil between 20 °C and 60 °F changes by about 0.35%. We usually neglect this.

Systems of units: SI, oilfield and metric oilfield →

Specific gravity of oil from its density at 20 °C

\rho_{60} = \rho_{20}\left[1 + \beta\,(20 - 15.56)\right],\qquad \gamma_o = \frac{\rho_{60}}{999.0}

kg/m³; 999.0 kg/m³ is the density of water at 60 °F

Holds when: We assume linear expansion over a narrow temperature range. $\beta \approx 8\cdot10^{-4}$ 1/°C is a typical value for medium oils, and the lessons of the library use it. For commercial transactions, the conversion is made with standard tables. The density is that of stock-tank oil at atmospheric pressure.

Systems of units: SI, oilfield and metric oilfield →

Barrels per tonne of oil

n_{\text{bbl/t}} = \frac{6.2898}{\rho\,[\text{t/m}^3]}

6.2898 is the number of barrels in a cubic metre

Holds when: The oil is stock-tank oil. The density and the volume are at the same temperature.

Systems of units: SI, oilfield and metric oilfield →

The darcy in SI units

1\ \text{D} = \frac{(10^{-6}\ \text{m}^3/\text{s})\cdot(10^{-3}\ \text{Pa·s})\cdot(10^{-2}\ \text{m})}{(10^{-4}\ \text{m}^2)\cdot(101{,}325\ \text{Pa})} = 0.9869233\cdot10^{-12}\ \text{m}^2

Holds when: The definition of the darcy: 1 cm³/s of a liquid of viscosity 1 cP through a sample of cross-section 1 cm² and length 1 cm at a pressure difference of 1 atm. The atmosphere here is the standard atmosphere, exactly 101,325 Pa. With the technical atmosphere (1 kgf/cm²) the number would be different.

Systems of units: SI, oilfield and metric oilfield →

Pressure in a water-bearing layer

p_w(z) = p_{\text{atm}} + \rho_w\,g\,z + C

SI, absolute pressure

Holds when: The water is at rest and the column is continuous. The water density is constant. (If it changes with depth, we add up the pressure interval by interval.) $C$ is the same throughout one hydrodynamically connected system (a system in which pressure is passed on freely).

Reservoir pressure and temperature →

Pressure gradient in a continuous phase

\frac{\dd p}{\dd z} = \rho\,g

Holds when: The fluid is at rest (static equilibrium) and forms a continuous phase between the measurement points. The density is constant with height — a good approximation for water and oil; for gas see below.

Reservoir pressure and temperature →

Excess of the pressure at the top of the accumulation over the water pressure

\Delta p = (\rho_w - \rho_o)\,g\,h_o + (\rho_w - \rho_g)\,g\,h_g

Holds when: The accumulation is in hydrostatic equilibrium and the phases are continuous. If there is no gas cap, $h_g = 0$ ; for a purely gas accumulation $h_o = 0$ .

Reservoir pressure and temperature →

Bringing pressure to a datum

p_{\text{datum}} = p_{\text{meas}} + \rho\,g\,(z_{\text{datum}} - z_{\text{meas}})

Holds when: The reservoir between the two depths is hydrodynamically connected and the fluid is at rest. If a contact lies between the depths, we make the conversion section by section, each with its own density.

Reservoir pressure and temperature →

Temperature at depth from the geothermal gradient

T(z) = T_0 + G_T\,z

°C and m (the gradient is often given in °C/100 m)

Holds when: We take the gradient as constant through the section. This is a rough approximation, because the gradient depends on the thermal conductivity of the rocks (how well they conduct heat) (advanced section). The temperature is the stabilised one, undisturbed by drilling and development.

Reservoir pressure and temperature →

Extrapolation of the bottomhole temperature to the stabilised temperature (Horner method)

T_{\text{bh}}(\Delta t) = T_{\text{res}} - m\,\lg\frac{t_c + \Delta t}{\Delta t}

time in the same units (hours); the logarithm is decimal (lg)

Holds when: This is an empirical method, by analogy with the pressure build-up curve. It describes the recovery of temperature around the wellbore with a line heat source model (heat released along a line, the wellbore). We know the duration of circulation only roughly. We need at least three measurements, preferably from different runs to the same depth.

Reservoir pressure and temperature →

Geothermal gradient from heat flow

G_T = \frac{q}{\lambda}

Holds when: Steady one-dimensional vertical heat flow. The layers are horizontal. Above and below salt domes, the heat flow shifts sideways and the picture is more complicated.

Reservoir pressure and temperature →

Pressure in a static column of gas

p_{\text{bh}} = p_{\text{wh}}\,\exp\left(\frac{M\,g\,H}{\bar Z\,R\,\bar T}\right)

SI ( $g = 9.81$ m/s²)

Holds when: The gas is at rest and there is no liquid in the wellbore. We replace $Z$ and $T$ by their averages over the wellbore. We find $\bar Z$ at the average pressure, so we repeat the calculation once or twice.

Reservoir pressure and temperature →

Symbols

Symbol	Meaning and context
$\Delta p$	difference between the hydrocarbon pressure at the top and the water pressure extended along its gradient to the same depth, PaReservoir pressure and temperature
$\bar Z,\ \bar T$	gas deviation factor at the average pressure, and the average absolute temperature in the wellbore, —, KReservoir pressure and temperature
$\dd p/\dd z$	pressure gradient — the increase in pressure per unit of true vertical depth, Pa/mReservoir pressure and temperature
$10^{-2}\ \text{m};\ 10^{-4}\ \text{m}^2$	length of 1 cm and cross-sectional area of 1 cm², m, m²Systems of units: SI, oilfield and metric oilfield
$10^{-3}\ \text{Pa·s}$	viscosity of 1 cP, Pa·sSystems of units: SI, oilfield and metric oilfield
$10^{-6}\ \text{m}^3/\text{s}$	flow rate of 1 cm³/s, m³/sSystems of units: SI, oilfield and metric oilfield
$101{,}325\ \text{Pa}$	pressure difference of 1 atm, PaSystems of units: SI, oilfield and metric oilfield
$b$	exponent of the curve: 0 is exponential, 1 is harmonic decline, —Reservoir engineering: tasks across the life of a field
$C$	constant of the pressure anomaly (the departure from normal pressure) for a hydrodynamically connected system, PaReservoir pressure and temperature
$D$	nominal decline rate, 1/yearReservoir engineering: tasks across the life of a field
$D_i$	initial nominal decline rate, 1/yearReservoir engineering: tasks across the life of a field
$E_AE_V$	sweep efficiency — the product of the sweep over the area (areal) and over the thickness (vertical), fractionReservoir engineering: tasks across the life of a field
$E_D$	displacement efficiency — the fraction of the oil displaced from the swept pores, fractionReservoir engineering: tasks across the life of a field
$g$	acceleration due to gravity, 9.81 m/s², m/s²Reservoir pressure and temperature acceleration due to gravity, m/s²Reservoir pressure and temperature
$G_T$	geothermal gradient, °C/mReservoir pressure and temperature geothermal gradient in the layer, K/mReservoir pressure and temperature
$H$	true vertical depth of the bottom (the middle of the reservoir), mReservoir pressure and temperature
$h_o,\ h_g$	heights of the oil part (from the free water level to the gas–oil contact) and of the gas part (from the contact to the top), mReservoir pressure and temperature
$m$	slope of the line (per logarithmic cycle), °CReservoir pressure and temperature
$M$	molar mass of the gas, $28.97\,\gamma_g$ , kg/kmolReservoir pressure and temperature
$N$	oil initially in place, m³ (standard conditions)Reservoir engineering: tasks across the life of a field
$n_{\text{bbl/t}}$	number of barrels in a tonne of oil, bbl/tSystems of units: SI, oilfield and metric oilfield
$N_{or}$	residual oil in the swept (contacted) pores, m³Reservoir engineering: tasks across the life of a field
$N_p$	oil produced by waterflooding, m³Reservoir engineering: tasks across the life of a field production from the initial moment until the rate equals $q$ , m³Reservoir engineering: tasks across the life of a field
$N_u$	oil outside the sweep — in the parts of the reservoir that the water has not reached, m³Reservoir engineering: tasks across the life of a field
$\Delta N_p$	production while the rate falls from $q_1$ to $q_2$ , m³Reservoir engineering: tasks across the life of a field
$p_1,\ p_2$	absolute pressures of the reference conditions, the same unitsSystems of units: SI, oilfield and metric oilfield
$p_{\text{abs}}$	absolute pressure — measured from vacuum, MPa, psiaSystems of units: SI, oilfield and metric oilfield
$p_{\text{atm}}$	atmospheric pressure: the standard value is 0.101325 MPa = 14.696 psi = 1.01325 bar, MPa, psiSystems of units: SI, oilfield and metric oilfield atmospheric pressure, 0.101 MPa, PaReservoir pressure and temperature
$p_{\text{bh}},\ p_{\text{wh}}$	absolute pressures at the bottom and at the wellhead of the shut-in well, MPaReservoir pressure and temperature
$p_{\text{datum}}$	pressure brought to the datum depth, PaReservoir pressure and temperature
$p_{\text{gauge}}$	gauge pressure — the excess over atmospheric pressure, MPa, psigSystems of units: SI, oilfield and metric oilfield
$p_{\text{meas}}$	measured pressure, PaReservoir pressure and temperature
$p_w$	absolute pressure in the water at depth $z$ , PaReservoir pressure and temperature
$q$	heat flow density (the heat passing through each square metre) from the interior; in sedimentary basins usually 40–80 mW/m², W/m²Reservoir pressure and temperature
$q,\ q_i$	rate at time $t$ and at the initial time, m³/dReservoir engineering: tasks across the life of a field
$R$	solution gas–oil ratio or producing gas–oil ratio — the volume of gas per volume of oil at standard conditions, m³/m³; scf/STBSystems of units: SI, oilfield and metric oilfield gas constant, 8314 J/(kmol·K), J/(kmol·K)Reservoir pressure and temperature
$t$	time from the initial moment, yearsReservoir engineering: tasks across the life of a field temperature on the Celsius or Fahrenheit scale, °C, °FSystems of units: SI, oilfield and metric oilfield
$T$	absolute temperature — in kelvins or degrees Rankine, K, °RSystems of units: SI, oilfield and metric oilfield
$T_0$	mean annual surface temperature (offshore, the temperature at the sea floor), °CReservoir pressure and temperature
$T_1,\ T_2$	absolute temperatures of the reference conditions, K or °RSystems of units: SI, oilfield and metric oilfield
$T_{\text{bh}}(\Delta t)$	bottomhole temperature measured a time $\Delta t$ after circulation stopped, °CReservoir pressure and temperature
$t_c$	duration of circulation at this depth before it stopped, hReservoir pressure and temperature
$T_{\text{res}}$	stabilised reservoir temperature — the value at $\lg(\ldots) = 0$ , that is, as $\Delta t \to \infty$ , °CReservoir pressure and temperature
$T(z)$	temperature of the rocks at depth $z$ , °CReservoir pressure and temperature
$V_1,\ V_2$	volume of the same amount of gas at conditions 1 and 2, m³, ft³Systems of units: SI, oilfield and metric oilfield
$z$	true vertical depth from the level where the water column begins (on land roughly from the ground surface, offshore from sea level), mReservoir pressure and temperature true vertical depth, mReservoir pressure and temperature
$z_{\text{datum}},\ z_{\text{meas}}$	true vertical depths of the datum and of the measurement point, mReservoir pressure and temperature
$\beta$	volumetric thermal expansion coefficient of the oil, 1/°CSystems of units: SI, oilfield and metric oilfield
$\gamma_o$	specific gravity of the oil relative to water at 60 °F, —Systems of units: SI, oilfield and metric oilfield
$\lambda$	thermal conductivity of the rocks of the layer: shales 1–2, sandstones and carbonates 2–4, rock salt about 5–6 W/(m·K), W/(m·K)Reservoir pressure and temperature
$\rho$	density of the oil at the temperature to which the barrel volume is referred (60 °F), t/m³Systems of units: SI, oilfield and metric oilfield fluid density at reservoir conditions, kg/m³Reservoir pressure and temperature density of the fluid that fills the reservoir between these depths, at reservoir conditions, kg/m³Reservoir pressure and temperature
$\rho_{20},\ \rho_{60}$	density of stock-tank oil at 20 °C and at 60 °F (15.56 °C), kg/m³Systems of units: SI, oilfield and metric oilfield
$\rho_w$	density of the formation water, kg/m³Reservoir pressure and temperature
$\rho_w,\ \rho_o,\ \rho_g$	densities of water, oil and gas at reservoir conditions, kg/m³Reservoir pressure and temperature