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VA – GP - OA: Numerical Multiphase PTA p 1/29

Numerical Multiphase PTA Vincent Artus - Gérard Pellissier - Olivier Allain

1. Introduction

In 2004-2005, KAPPA developed a multiphase option in Saphir NL. This option allowed 2-phase

flow without exchange between the phases (eg water injection in dead oil, or gas) and complex 3-phase flow with phase exchanges, for black-oil or condensate. In all those situations, it was

soon realized that multiphase simulations could exhibit large oscillations on the loglog derivative. In the case of water injection, we were able to damp these oscillations through the

use of pseudo-kr corrections, but no ‘easy’ solution was found for black-oil or condensate. As a consequence, a complete 3-phase option was not released, and Saphir NL was limited to two-

phase PVT including water.

With the sector model option of Rubis v4.12, 3-phase simulations can de facto be run in Saphir

NL, albeit through a tortuous path. It was thus decided to make the full multiphase option directly accessible in Saphir NL during the course of the v4.20 upgrades, after a detailed

investigation of the nature of the oscillations, and the possible solutions to damp them. This document summarizes the work undertaken; its reading is highly recommended before running

any multiphase Saphir NL cases. It is important to note that the problems described here go unnoticed in numerical simulation, while they are exemplified in PTA by the use of the pressure

derivative, and the focus on short time scale.

Approximately a hundred numerical simulations have been run for this study, covering various

multiphase contexts across a wide range of depletion levels. Although no generic description is possible because visible effects are strongly dependent on the simulation context (PVT, KrPc,

etc…), most of the possible situations are covered by the examples in this document.

Four main cases are presented:

Case 1 (section 2.1): water injection in an oil reservoir. Illustration of the development of oscillations while flooding; the origin of oscillations is explained in section 2.2.

Case 2 (section 2.4): complex history of water injection into an oil reservoir. Validation

of the numerical model against the analytical results. Case 3 (section 3.1): depletion into a black-oil reservoir. The origin of oscillations is

analyzed in section 3.2; section 3.3 is dedicated to the interpretation of the results. Case 4 (section 4.1): depletion of a condensate gas reservoir.

The results are interpreted in section 4.2.

Yellow sections of this document are not strictly required for a global comprehension of the

origin of oscillations with multiphase transient simulations. However, they provide interesting material for a deeper analysis of the complexity of the various physical mechanisms involved

with multiphase processes.

VA – GP - OA: Numerical Multiphase PTA p 2/29

2. Water or gas injection We start with the analysis of injection of one phase into another, without mass transfer

between the two phases. Although we illustrate the effects based on the results of simulations

of water injection into dead oil, the main features described below remain valid for other injection schemes (e.g. water injection into gas, gas injection, etc...).

Two test cases are presented. Case 1 illustrates and explains the development of oscillations,

while Case 2 is a validation of the numerical model against analytical predictions.

2.1. Test Case 1 Test description

We consider a rectangular reservoir of dimensions 10,00010,000 ft², with thickness 100 ft.

The injection well is located at the center of the reservoir. The formation compressibility is

cf=3e-6 psi-1, the permeability is k=1000 mD, and the porosity is =0.2. The reservoir initially

contains dead-oil, with constant properties (o = 0.3 cp, Bo=1). The initial water saturation in

the reservoir is Swi=0.2. Water is injected for 10,000 hr at constant rate Qw=10,000 stb/D,

followed by a 10,000 hr fall-off. Water is also characterized by constant properties (w =

0.3cp, Bw=1). Equal viscosities between oil and water have been chosen so that the mobility

effects are only due to the choice of relative permeability curves. Relative permeabilities have the following properties:

Kromax = 0.8; Krwmax = 0.5; Swr = 0.2; Sor = 0.25

The curve shape is a power curve (Corey type). 3 values of the Corey exponent have been

tested: 1 (called test “Corey 1”), 2 (“Corey 2”) and 3 (“Corey 3”), in order to show the influence of the non-linearity of Kr curves on oscillations.

Results

During the injection phase, the pressure derivative exhibits oscillations on the loglog plot. The

level of these oscillations increases with the non-linearity of the relative permeability curves (Figure 1). This can be related to an increase of the mobility contrasts with the non-linearity of

the curves.

Figure 1: Water injection for 3 values of the exponent of the relative permeability curves

0.01 0.1 1 10 100 1000 10000 Time [hr]

10

100

P re

s s u re

[ p s i]

Corey 1

Corey 2

Corey 3

VA – GP - OA: Numerical Multiphase PTA p 3/29

Looking at the evolution of the water saturation field during the injection, we see that every

oscillation on the loglog plot corresponds to the invasion of a new ring of cells by the water bank (Figure 2). The higher the mobility contrast between the water bank and the initial oil,

the larger the resulting oscillations.

Figure 2: Water saturation map around the well at the end of the injection period

Although the level of the oscillations can be spectacular for strong mobility contrasts (e.g. Figure 1 for Corey 3 curves), it can be greatly reduced by refining the simulation grid. Figure 3

compares the results obtained using a gridding progression ratio of 1.4 with those obtained using ratios of 1.2 and 1.1, in the case of very non-linear curves (Corey 3). It is worth noticing

that for weaker contrasts (Corey 1 and 2), the oscillations almost completely disappear with a gridding progression ratio of 1.2.

Figure 3: Effect of grid refinement on oscillations (gpr = gridding progression ratio)

0.01 0.1 1 10 100 1000 10000 Time [hr]

10

100

P re

s s u re

[ p s i]

Corey 3

Corey 3 refined (gpr=1.2)

Corey 3 refined (gpr=1.1)

VA – GP - OA: Numerical Multiphase PTA p 4/29

On the right side of Figure 3, one can see that reducing the cell size increases the frequency of

oscillations, but also reduces their amplitude. This is explained in the next section.

2.2. Origin of oscillations Let us consider a 1D displacement of oil by water, water being injected at constant rate. Both

fluids are assumed incompressible. From viscosity values and kr curves, the oil, water and total mobility curves can be derived:

Such curves are presented on Figure 4. Note that for the sake of illustration in all the figures

below, we used the same viscosity values as in test case 1, with the “Corey 2” relative permeability curves. It is important to notice that due to the non-linearity of the relative

permeability curves, the total mobility curve is also non-linear. Indeed, as shown on figure 4, the total mobility starts to decrease when the water saturation increases from Swr, before

increasing again as Sw approaches (1-Sor).

Figure 4: Relative permeability and mobility curves

Let us know consider the classical Buckley-Leverett model (Buckley and Leverett, 1942, Marle, 1981) to further investigate the problem. With this model, the 1D, incompressible

displacement in the absence of capillary pressure and gravity can be described with the following hyperbolic equation:

0

t

S

x

S

dS

dfu ww

w

Above, u is the total velocity, is the porosity, and f is the water fractional flow, defined as:

u

u f w

VA – GP - OA: Numerical Multiphase PTA p 5/29

The Buckley-Leverett analysis shows that the fractional flow actually depends on the water

saturation only, and can be expressed:

t

w wSf

Such a fractional flow curve is presented on the left side of Figure 5. Using the method of

characteristics to solve the hyperbolic equation, it can be shown that each saturation plane travels at its own, constant speed:

w

w w

dS

Sdf SV

This leads to the velocity profile shown on the right side of Figure 5. Because low saturation

planes travel slowly compared to some higher saturation planes, a saturation front appears. The value of the front saturation Sf can be deduced from the fractional flow curve through

Welge’s tangency (left of Figure 5), as developed in Marle, 1981.

Figure 5: Fractional flow and velocity profile corresponding to the curves of Figure 4.

From the front s