Reservoir Rock Properties and Fluid Flow

Wettability and capillary pressure

Wettability describes which of two immiscible fluids adheres more strongly to a solid surface. The phase that is attracted to the solid more strongly is the wetting phase, and the other is the non-wetting phase. The non-wetting phase interacts only weakly with the solid, so its own cohesive, intra-molecular forces dominate and it keeps away from the grain surfaces. In a water-wet reservoir, water coats the grains and the oil therefore occupies the central, larger parts of the pore space.

• Contact angle — the angle measured through the denser phase at the three-phase line where liquid, vapour and solid meet. An angle below 90° means the liquid wets the solid. A goniometer, or contact angle meter, measures it by imaging the profile of a sessile drop.
• Young's equation — the force balance at the contact line on an ideal smooth solid gives \gamma_{AS} = \gamma_{SL} + \gamma_{LA}\cos\theta, so \cos\theta = (\gamma_{AS}-\gamma_{SL})/\gamma_{LA}.
• Methods for interfacial tension — the pendant drop method uses the drop shape under gravity, the Du Noüy ring method uses the detachment force, the Wilhelmy plate method uses the wetting force, and the spinning drop method uses centrifugal force in a rotating tube.

Capillary rise in a tube follows Jurin's law, h = \dfrac{2\gamma\cos\theta}{\rho g r}. A measured column height in a capillary of known radius and fluid density therefore gives the surface tension, \gamma = \dfrac{h\rho g r}{2\cos\theta}. Remember to convert SI units to dyn/cm, where 1 N/m equals 1000 dyn/cm.

If a capillary is cut shorter than its equilibrium rise height, a perfectly wetting liquid does not overflow. The meniscus at the new top takes a larger radius of curvature so that 2\gamma/R = \rho g l holds for the shorter column, so the meniscus radius changes with the column length.

The capillary pressure in a circular pore is P_c = 2\gamma\cos\theta/r. The pressure needed to move a trapped droplet through a throat is set by the difference between its two menisci. For a zero contact angle, a droplet with a rear meniscus of radius r_A and a forward meniscus of radius r_B requires a pressure drop \Delta P = 2\gamma\left(1/r_B - 1/r_A\right).

Capillary pressure curves plot P_c against water saturation. The entry pressure is the value at which the non-wetting phase first enters the rock. A uniform pore size distribution gives a steep rise after entry followed by a nearly flat plateau, whereas a broad spread of pore sizes gives a gradual slope.

The Leverett J-function, J = \dfrac{P_c}{\gamma\cos\theta}\sqrt{\dfrac{k}{\phi}}, is dimensionless. For the same rock and wettability, J is the same at laboratory and reservoir conditions, so a laboratory capillary pressure can be scaled to reservoir conditions using the ratio of interfacial tensions and the ratio of \sqrt{k/\phi} values.

Capillary pressure also sets the transition zone above the free-water level. The height of a fluid contact above the free-water level is h = P_c/(\Delta\rho\,g), where \Delta\rho is the water-oil density difference and P_c is the entry pressure. Since depth increases downward, the free-water level lies at the observed contact depth plus this height. When working in CGS units, use 1 psi \approx 68\,950 dyn/cm².

Darcy's law

Darcy's original experiment on fully saturated sand columns showed that the volumetric flow rate is proportional to the cross-sectional area and to the hydraulic gradient, Q = -KA\,dh/dl, where the hydraulic conductivity is K = k\rho g/\mu. The driving force is therefore the pressure or hydraulic gradient, while viscosity and permeability control the resistance to flow.

Darcy's law is a macroscopic continuum relation. It describes the average flow over a representative elementary volume that contains many pores, so it does not describe the flow within a single pore or at molecular scale.

In SI units the Darcy (superficial) velocity is u = \dfrac{k}{\mu}\dfrac{\Delta p}{L}. The actual interstitial velocity in the pore space is v = u/\phi, which is always larger than u because only the porosity fraction of the cross section carries flow. Note that 1 darcy is approximately 10^{-12} m².

When the flow path is inclined, gravity enters the flow potential. The potential gradient is d\Phi/dL = \Delta p/L + \rho g\sin\theta, so gravity adds to or subtracts from the pressure gradient depending on whether the flow is downdip or updip. The actual velocity changes with the dip angle even when the pressure drop is unchanged.

In oilfield units, single-phase linear flow is written as q = 0.001127\,\dfrac{kA\,\Delta p}{\mu L}, where q is in bbl/day, k in mD, A in ft², \Delta p in psi, \mu in cP and L in ft. Use 1 bbl \approx 5.615 ft³ when converting between volumetric units.

Productivity index and inflow

The absolute open flow potential (AOF) of a well is the flow rate obtained when the sandface pressure is reduced to zero, found by extrapolating the inflow relation. It is the theoretical maximum deliverability of the well, not a minimum rate.

The productivity index is J = \dfrac{q}{\bar p_r - p_{wf}}, in STB/day/psi. For a vertical well in pseudo-steady radial flow, J = \dfrac{kh}{141.2\,\mu B\,\left(\ln(r_e/r_w)+s\right)}, where s is the skin factor. Hydraulic fracturing and longer perforated intervals raise J by improving the effective wellbore radius or contacted thickness. A higher skin increases the denominator and lowers J, while reducing the oil viscosity, for example by steam heating, raises J.

For a single-phase oil reservoir with a linear inflow performance relationship, q = J(\bar p_r - p_{wf}). Two measured points of rate and flowing pressure give J from the slope, and the average reservoir pressure follows from the intercept. The rate at any other flowing pressure is then found from the same straight line.

Transient flow diffusivity equation

Transient flow is described by the diffusivity equation written in Cartesian coordinates x or cylindrical coordinates r. In cylindrical form, \dfrac{1}{r}\dfrac{\partial}{\partial r}\left(r\dfrac{\partial p}{\partial r}\right) = \dfrac{1}{\eta}\dfrac{\partial p}{\partial t}, where \eta = k/(\phi\mu c_t) is the diffusivity constant. In Cartesian form, the 1/r term is absent. For an incompressible fluid in a rigid rock, the time-derivative term vanishes, so the equation reduces to a steady Laplace-type form. The time-dependent term remains for a slightly compressible fluid, with c_t the total compressibility of fluid and formation. Squared pressure-gradient terms arise only in the fuller nonlinear forms.

The Theis solution for an infinite-acting radial reservoir requires three conditions. The initial pressure is uniform, P(r,0)=P_i, the wellbore is treated as a line source, and the pressure approaches P_i as r\to\infty at all times. The pressure gradient is not constant, because the solution depends on the similarity variable u = r^2/(4\eta t), which varies with both radius and time.