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Reservoir Models (IPR Models)

Inflow Performance Relationship

Six different inflow models are available; three for oil and three for gas. These are Oil (combined Darcy-Vogel) and Gas deliverability (Forchheimer). In addition, the model allows for user defined PI (productivity index) and user defined IPR curve for both oil and gas. (IPR) [1]

The IPR curve is the relation between the flowing bottom-hole pressure PwfP_{wf} and liquid production rate qq. Undersaturated oil reservoirs exist as single-phase reservoirs where pressures are above the bubble point pressure. The linear IPR model is given as,

q=J(PresPwf)q = J (P_{res} - P_{wf})

where JJ is the productivity index to describe the trends of the IPR curve and PresP_{res} represents the reservoir pressure. The solution gas escapes from the oil and becomes free gas when the flowing bottom-hole pressure PwfP_{wf} is below the bubble point pressure PbP_b [1]. Therefore, Vogel established an empirical equation for two-phase reservoirs in 1968 [2], and it is still widely used in the industry.

If the reservoir pressure Pres<PbP_{res} < P_b,

q=qmax(10.2PwfPres0.8PwfPres2)q = q_{max} (1 - 0.2 \frac{P_{wf}}{P_{res}} - 0.8 \frac{P_{wf}}{P_{res}}^2)

and,

qmax=JPres1.8q_{max} = \frac{JP_{res}}{1.8}

Otherwise the following is being used,

q=J(PresPb)+JPb1.8(10.2PwfPb0.8PwfPb2)q = J (P_{res} - P_b) + \frac{JP_{b}}{1.8} (1 - 0.2 \frac{P_{wf}}{P_{b}} - 0.8 \frac{P_{wf}}{P_{b}}^2)

As the productivity index JJ is an unknown variable, it can be estimated according to different flow types.

Oil

Productivity index for vertical and deviated wells in rectangular drainage areas with constant pressure boundaries.

J=kh18.7Boμo(0.5log2.2xeyecarw2+S)J = \frac{kh}{18.7 B_o \mu_o (0.5 \log \frac{2.2 x_e y_e}{c_a r_w^2} + S ) }

Gas Deliverability

Transient productivity index (i.e. productivity index of a well which has not yet seen any of the boundaries (radial flow) is used in this part. Most DST/WFT fall into this category) which can be used to determine the infinite-acting period.

J=kh21.5Bgμg(logktporoμgcrw23.1+0.87S)J = \frac{kh}{21.5 B_g \mu_g (\log \frac{kt}{\textsf{poro} \mu_g c r_w^2} - 3.1 + 0.87S ) }

Forcheimer's Model

High velocity flow in porous media and fractures is modeled by the Forchheimer equation in gas reservoir when the reservoir pressure exceeds a cut-off value numerically.

Pwf=Presaqbq2P_{wf}= P_{res} - a \cdot q - b \cdot q^2

Forchheimer equation can be performed for the gas systems, where the nonlinear flow is much more significant, due to the lower gas viscosity which will give high numbers for the same velocity as in liquid systems.