Author: Professor Jon Kleppe Assistant producers: Farrokh Shoaei Khayyam Farzullayev NTNU

Two phase system (oil – water) The equations for multi-phase flow : Where: flow equations for the two phases flow with substituting Darcy's equations: Where:

Oil-Water Relative Permeabilities and Capillary Pressure most processes of interest, involve displacement of oil by water, or imbibition. the initial saturations present in the rock will normally be the result of a drainage process at the time of oil accumulation. Drainage process: Imbibition process: SW = 1 SW =SWir oil water Sw Pc 1 Pc Sw Kr Sw Kr 1 Swir Swir Pcd Swir Swir water 1-Sor water oil oil 1-Sor Sw

Discretization of Flow Equations We will use similar approximations for the two-phase equations as we did for one phase flow. Left side flow terms: Where: Oil transmissibility: Oil mobility: The mobility term is now a function of saturation in addition to pressure. This will have significance for the evaluation of the term in discrete form.

Upstream mobility term Because of the strong saturation dependencies of the two-phase mobility terms, the solution of the equations will be much more influenced by the evaluation of this term than in the case of one phase flow. Buckley-Leverett solution: QW x Sw 1 1-Sor B.L with PC = 0 Swir

In reservoir simulation, upstream mobilities are normally used. In simulating this process, using a discrete grid block system, the results are very much dependent upon the way the mobility term is approximated. Flow of oil between blocks i and i+1: Upstream selection: weighted average selection: QW x Sw 1 1-Sor B.L (PC = 0) Upstream Weighted average Swir In reservoir simulation, upstream mobilities are normally used.

The deviation from the exact solution depends on the grid block sizes used. For very small grid blocks, the differences between the solutions may become negligible. x Sw 1 1-Sor B.L (PC = 0) Small grid blocks Large grid blocks Swir The flow rate of oil out of any grid block depends primarily on the relative permeability to oil in that grid block. If the mobility selection is the weighted average, the block i may actually have reached residual oil saturation, while the mobility of block i+1 still is greater than zero. For small grid block sizes, the error involved may be small, but for blocks of practical sizes, it becomes a significant problem.

Expansion of Discretized equations The right hand side of the oil equation: By: Replacing oil saturation by water saturation. Use a standard backward approximation of the time derivative. the right hand side of the oil equation thus may be written as: Where:

The right hand side of the water equation: By: expression the second term Since capillary pressure is a function of water saturation only Using the one phase terms and standard difference approximations for the derivatives the right side of the water equation becomes: Where:

The discrete forms of the oil and water equations Oil equation: Where: Water equation: Transmissibility and mobility terms are the same as for oil equation, except the subtitles are changed from “o” for oil to “w” for water.

Boundary Conditions Constant water injection rate the simplest condition to handle for a constant surface water injection rate of Qwi (negative) in a well in grid block i: At the end of a time step, the bottom hole injection pressure may theoretically be calculated using the well equation: where: Well constant Drainage radius

The fluid injected in a well meets resistance from the fluids it displaces also. As a better approximation, it is normally accepted to use the sum of the mobilities of the fluids present in the injection block in the well equation. Well equation which is often used for the injection of water in an oil-water system: or Time qinj Injection wells are frequently constrained by a maximum bottom hole pressure, to avoid fracturing of the formation. This should be checked, and if necessary, reduce the injection rate, or convert it to a constant bottom hole pressure injection well. Time Pbh Pmax Pbp

2. Injection at constant bottom hole pressure Injection of water at constant bottom hole pressure is achieved by: Having constant pressure at the injection pump at the surface. Letting the hydrostatic pressure caused by the well filled with water control the injection pressure. The well equation: Capillary pressure is neglected At the end of the time step, the above equation may be used to compute the actual water injection rate for the step.

3. Constant oil production rate for a constant surface oil production rate of Qoi (positive) in a well in grid block i: in this case oil production will generally be accompanied by water production. The water equation will have a water production term given by: Capillary pressure is neglected Around the production well the bottom hole production pressure for the well may be calculated using the well equation for oil:

Time qprod Production wells are normally constrained by a minimum bottom hole pressure, for lifting purposes in the well. If this is reached, the well should be converted to a constant bottom hole pressure well. Time Pbh Pmin WC(%) vs. Time(year) Pbp As the limitation for water cut was 75%, so at this point gridblocks that exceeded allowable water cut had been closed in order to keep the limit. If a maximum water cut level is exceeded for well, the highest water cut grid block may be shut in, or the production rate may have to be reduced.

4. Constant liquid production rate Total constant surface liquid production rate of QLi (positive): If capillary pressure is neglected: and Time qprod Total liquid Oil Water

5. Production at Constant reservoir voidage rate A case of constant surface water injection rate of Qwinj in some grid block. total production of liquids from a well in block i is to match the reservoir injection volume so that the reservoir pressure remains approximately constant. If capillary pressure is neglected: and

6. Production at Constant bottom hole pressure Production well in grid block i with constant bottom hole pressure, Pbhi: and Substituting the flow terms in the flow equations: and The rate terms contain unknown block pressures, these will have to be appropriately included in the matrix coefficients when solving for pressures. At the end of each time step, actual rates are computed by these equations, and water cut is computed.

IMPES Method Discretized form of flow equations: Where: i=1, …, N the primary variables and unknowns to be solved for equations are: Oil pressures Poi, Poi-1, Poi+1 Water saturation Swi Assumption: All coefficients and capillary pressures are evaluated at time=t.

Using the oil equation yields: The two equations are combined so that the saturation terms are eliminated. The resulting equation is the pressure equation: i=1, …, N This equation may be solved for pressures implicitly in all grid blocks by Gaussian Elimination Method or some other methods. The saturations may be solved explicitly by using one of the equations. Using the oil equation yields:

Having obtained oil pressures and water saturations for a given time step, well rates or bottom hole pressures may be computed as q’wi, q’oi and Pbh. The surface production well water cut may be computed as: Required adjustments in well rates and well pressures, if constrained by upper or lower limits are made at the end of each time step, before all coefficients are updated and we can proceed to the next time step.

Limitations of the IMPES method The evaluation of coefficients at old time level when solving for pressures and saturations at a new time level, puts restrictions on the solution which sometimes may be severe. IMPES is mainly used for simulation of field scale systems, with relatively large grid blocks and slow rates of change. It is normally not suited for simulation of rapid changes close to wells, such as coning studies, or other systems of rapid changes. When time steps are kept small, IMPES provides accurate and stable solutions to a long range of reservoir problems.

Questions 1. Make sketches of typical Kro, Krw and Pcow curves for an oil-water system, both for oil-displacement of water and water-displacement of oil, and label all relevant points. 2. Show how the saturation profile (Sw vs. x), if calculated in a simulation model, typically is influenced by the choice of mobilities between the grid blocks (include simulated results with upstream and average mobility terms) (neglecting capillary pressure). Make also a sketch of the exact solution. 3. Write the two flow equations for oil and water on discretized forms in terms ot transmissibilities, storage coefficients and pressure differences. 4. Write an expression for the selection of the upstream mobility term for use in the transmissibility term of he oil equation for flow between the grid blocks (i-1) and (i). 5. List the assumptions for IMPES solutin, and outline briefly how we solve for pressures and saturations.

References Kleppe J.: Reservoir Simulation, Lecture note 6 Snyder and Ramey SPE 1645

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