uy Chavent, Jer^ome Jare and Sophie Jegou-Jan The generalized Darcy law says that for each phase the Darcy velocity is proportional to the gradient of

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1 Estimation of Relative Permeabilities in Three-Phase Flow in Porous Media uy Chavent 1;2, Jer^ome Jare 2 and Sophie Jan-Jegou 3 1. Universite Paris-Dauphine, CEREMADE, Paris, France 2. INRIA-Rocquencourt, BP 105, Le Chesnay Cedex, France 3. LA, UFR MI, Universite Paul Sabatier, 118 route de Narbonne, Toulouse Cedex 04, France Abstract Three-phase ow in porous media is modeled as a system of two nonlinear partial dierential equations where the nonlinearities depend on the relative permeability functions. e show how the three relative permeabilities can be estimated as functions of two variables from data obtained in laboratory experiments. Keywords: multiphase ow in porous media, inverse problem. 1 A model for three-phase ow in porous media Flow in a petroleum reservoir is a complicated physical process which may involve various physical eects. In the resulting models there are many coecients which represent rock and ow properties. They have to be determined to be put into a reservoir numerical simulator. Determining these coecients is an elaborate process which includes several inverse problems at various scales. Data are gathered from laboratory experiments performed on core samples, which are extracted when drilling wells, from well logging experiments and from producing well records. ther information is given by geostatistical models which give small scale data which needs to be upscaled to the scale of the discretization cells used in the reservoir simulator. Here we consider three-phase ow in a cylindrical core sample subject to laboratory experiments. The three phases are the three uids, water, oil and gas, assumed to be immiscible and referred to by the indices w; o; g. The ow is assumed to be incompressible, one-dimensional and as we consider an horizontal core sample, we can neglect gravity eects. Conservation of each phase is written as S l t + ' l x = 0; l = w; o; g; (1) where is the porosity of the rock, ' l and S l are the Darcy velocity and the saturation of phase l. Saturations satisfy S w + S o + S g = 1; 0 S l 1; l = w; o; g; (2) which means that the whole pore volume is occupied by the uids. This work has been partially supported by the French Institute of Petroleum

2 uy Chavent, Jer^ome Jare and Sophie Jegou-Jan The generalized Darcy law says that for each phase the Darcy velocity is proportional to the gradient of the pressure in the phase: ' l = K kr l P l ; l = w; o; g: (3) l x Here K is the absolute permeability of the rock, l the viscosity of phase l and kr l its relative permeability. Since we neglect capillary eects all phase pressures are equal and we can drop the index l for the pressure: P = P l ; l = w; o; g. The relative permeabilities kr l are functions of the saturations and are indeed the nonlinear coecients that we would like to estimate from data gathered in laboratory experiments. Since the sum of the three saturations is equal to 1 (see equation (2)), the relative permeabilities are functions of the two saturations S w and S g, and their domain of denition is an equilateral triangle as shown on gure 2. Moreover the relative permeability functions are always positive, and for phase l the relative permeability kr l increases with respect to the saturation S l. This means that the larger the saturation of phase l is, the more easily this phase ows. Since we assumed that the ow is incompressible, summing equations (1) and using (2), we obtain that the divergence of the total ow rate ' T = ' w + ' o + ' g vanishes. Therefore we use the following system of equations to model the ow: In these equations the coecients l and k T S l t + ' l x = 0; ' l = l ' T ; l = w; g; (4) ' T x = 0; ' P T = Kk T x : expressed in terms of the mobilities of the phases k l = kr l l are functions of the saturations which can be k T = k T (S w ; S g ) = k w (S w ; S g ) + k o (S w ; S g ) + k g (S w ; S g ); l = l (S w ; S g ) = k l(s w ; S g ) ; l = w; g: k T (S w ; S g ) To equations (4) we add boundary and initial conditions which depend on the experiment. In standard engineering the three-phase relative permeability functions are constructed from two-phase relative permeability functions, which are functions of only one saturation. There are several methods, all variations of that originally proposed by H.L. Stone, for constructing three-phase relative permeability functions [14, 1]. From the mathematical point of view the resulting models have been studied and shown not to be strictly hyperbolic [2, 12] though it is possible to construct three-phase relative permeabilities which ensure that the resulting system of equations is hyperbolic (see [5]). However, from the practical point of view, these constructions give three-phase relative permeability functions which are not always satisfactory in the sense that there are some experimental data which are dicult to match [11]. Therefore it is natural to try to estimate directly these functions from three-phase experiments and we will now describe one way to proceed. A similar idea using dierent techniques is presented in [13]. 2 as

3 Estimation of Relative Permeabilities in Three-Phase Flow 2 The parameter estimation problem In order to estimate the relative permeability functions, experiments are set up so the following measurements are available: 1. Local measurements : saturations S m l;k;i are measured at dierent times t k and dierent locations x i for each phase l = w; g; 2. lobal measurements : for each phase l = w; o; g cumulated productions at the production end of the core Q m l;k = dierent times t k. Z tk 0 ' l (t)dt, and pressure drops P k are measured at Saturation measurements are not common but they have been shown experimentally to be necessary in order to expect uniqueness of two-phase relative permeabilities [3, 4]. They can be expected to be even more necessary in the case of three-phase ow where there is a larger number of parameters to estimate. The problem of estimating the relative permeability functions is set as the problem of minimizing the least square error function J : J(kr w ; kr o ; kr g ) = X l;k;i ws l;k;i (Sl;k;i c X Sl;k;i) m 2 + wq l;k (Q c l;k Q m l;k) 2 (5) l;k + X k w k p(p c k P m k ) 2 : Here the superscripts m and c refer respectively to \measured" and \calculated", and w l;k;i wq l;k and wp k are weights given to the measurements. The function J evaluates the dierence between the measured quantities and those calculated with the model using the current parameters kr l ; l = w; o; g. In the following we will consider only experiments with a given pressure drop so the third term in the denition of J may be dropped (wp k = 0). Laboratory experiments can be of several types. Here we consider as an example the displacement of water and oil by injection of gas. s, 3 Practical implementation 3.1 Parameterization To provide smooth relative permeability functions, these functions are discretized in such a way that they are continuously dierentiable. Moreover the positivity and monotonicity constraints are simple to write. Furthermore the parameterization has been chosen in order that multiscale optimization is allowed. As we use the same method for the parameterization of all relative permeabilities, let us describe that relative to water. It is represented in a system of two orthogonal axes. ne of these axes is the direction in which the parameter must be monotone (S w for kr w ). The search for the relative permeability function kr w (S w ; S g ) is then replaced by the search of its 3

4 uy Chavent, Jer^ome Jare and Sophie Jegou-Jan derivative in the direction of monotonicity kr w S w (S w ; S g ) and the boundary value kr w (0; S g ) : Z Sw kr w (S w ; S g ) = (S g ) + (u; S g )du; 0 (S w ; S g ) = kr w S w (S w ; S g ); (S g ) = kr w (0; S g ) (6) and the monotonicity constraint (kr w must be an increasing function of S w ) has been replaced by a positivity constraint ((S w ; S g ) 0) which is easier to take into account. Now, for the parameterization of the new functions (S w ; S g ) and (S g ), we have chosen a tensorial product of a C 0 nite element in the direction of monotonicity and a Hermite C 1 nite element in the other direction. See gure 1 and [9] for details. ith this choice we obtain C 1 relative permeability functions which allow us to rewrite the optimization problem as follows : where A is an m n matrix with m n. min J(x) Ax b; x x x; -s -s -s 6 S w s- s- s- s- -s - S g s- s- -s - S g s function values - derivatives in the horizontal direction Figure 1: Degrees of freedom of (left) and (right) in expression (6) of kr w. 3.2 ptimization ith the chosen parameterization, it turns out that the projection on the admissible set can be computed relatively easily (in a nite number of operations). This would allow the use of a method of gradient with projection. However, in the following numerical results, a penalization method has been used. A new objective function is introduced : J (x) = J(x) + mx i=1 2 (Ax b) + i ; where x + denotes the positive part of x and is the penalization parameter. 4

5 Estimation of Relative Permeabilities in Three-Phase Flow Then, as the Hessian of J would be too costly to compute, we used a rst order method of quasi-newton [6, 7] type to minimize the function J. This method is based on a quadratic approximation of J : J (x k ) + rj (x k ) T (x x k ) (x x k) T M k (x x k ); where x k is the current iterate and M k is a symmetric positive denite matrix which approximates the Hessian of J. 3.3 Automatic generation of codes To minimize the function J it is more ecient to use optimization methods which use the exact gradient of J with respect to the parameters. This exact gradient is calculated using the adjoint method which introduces the adjoint equations whose solution is the adjoint state. This calculation must be performed on the discretized problem and then coded into a program. Such a process is very tedious and error prone. Therefore a fast and reliable way to do it is to use the symbolic dierentiation and Fortran code generation capabilities of the software Maple. For this, the RADJ code generator (see [10]), written in Maple, has been extended in such a way that it could handle our problem. It takes as input the discretized version of the direct equations (4) and the least square function J (6). The output of RADJ is two Fortran programs. The rst one solves the direct equations and evaluates J, whereas the second one solves the adjoint equations and computes the gradient of J. These programs are then given as arguments to a minimization program such as M2QN1 [8]. e have used RADJ for our parameter estimation problem and we are convinced that it is a reliable tool to develop codes for parameter estimation in systems governed by PDE's. 4 Numerical results e have performed parameter estimations from synthetic data. First, we have tried to identify one relative permeability using data from only one displacement of oil and water by gas. The results were not completely satisfactory, but we realized that they could be improved by using simultaneously data from several displacements. 4.1 Estimation using data from only one experiment Here we consider a rst estimation using data from one experiment of displacement of oil and water by gas. e identied only the water relative permeability kr w so there were 16 active parameters to be estimated. In this numerical experiment the function J has been divided by Actually the measured and calculated observations are not shown as they superimpose. There were 23 calculations of J and of its gradient counted together (a calculation of J includes solving the direct equations and a calculation of its gradient includes solving the adjoint equations). The computational time was around 5 minutes on a work station. In gure 2 the results are shown in the left column: the rst triangle on top shows the true parameter, the second one, below, shows the chosen initial parameter before optimization, and the third one, on bottom, shows the result of the parameter estimation. The thick curve that 5

6 uy Chavent, Jer^ome Jare and Sophie Jegou-Jan is drawn in the second picture shows the saturations that are obtained during the experiment. e call this curve the saturation experiment curve. e can observe that, even though there is a good overall agreement between the true kr w and that identied, the agreement is not as good in the left part of the triangle. However this can be easily understood when noticing that our experiment does not cover at all this area. Along the saturation experiment curve the agreement is actually perfect to graphical accuracy. This leads to the idea that to obtain a good parameter estimation it is necessary to design experiments producing saturations which cover better the triangular domain of denition of the saturations. This is what is presented now in the next subsection. 4.2 Estimation using data from three experiments Now we try to identify the three relative permeabilities using simultaneously data from three displacements of water and oil by gas. There were 16 active parameters for each water and gas relative permeability and 12 active parameters for the oil relative permeability. The total is thus 44 parameters to be estimated. In this numerical experiment the function J has been divided by There were 458 calculations of J and of its gradient counted together. The computational time was around 36 hours on a work station. The results for kr w are shown in gure 2 (right). Analogous gures could have been shown for kr o and kr g. The three thick curves drawn on the middle triangle are the saturation experiment curves corresponding to the three experiments. As we can see the triangle is now well covered and we obtained a very good agreement between the true kr w and that identied. Actually looking carefully at the gure one can notice that near the bottom left of the triangle the optimal kr w does not match very well the true parameter. This is due to the fact that the region near this vertex is still not covered by the experiments. The French Institute of Petroleum is working on improving local measurements of saturations. In a close future, with this new technology, we hope we will be able to perform identication of three-phase relative permeabilities from real data. 5 Conclusion e presented a method for estimating three-phase relative permeabilities functions on which depend nonlinear coecients involved in the model of three-phase ow in porous media. These functions are functions of two saturations. Based on least square minimization, the method uses an exact calculation of the gradient and we presented several tools which allow for an ecient implementation: automatic generation of Fortran codes, smooth multiscale parameterization, quasi-newton optimization with penalization of the constraints. The method gives satisfactory results for synthetic data when we consider simultaneously several experiments. e have shown that the better the saturation measurements cover the triangular domain of the saturations, the more accurate the estimation is. Actually the relative permeabilities can be estimated only along these curves so it is important to design experiments where the saturations cover as well as possible the triangular domain of the saturations. 6

7 Estimation of Relative Permeabilities in Three-Phase Flow References [1] K. Aziz and Settari, Petroleum Reservoir Simulation, Applied Science, [2] J. Bell, J. Trangenstein, and. Shubin, Conservation laws of mixed type describing three-phase ow in porous media, Siam J. on Applied Mathematics, 46 (1986), pp. 1000{1017. [3] C. Chardaire,. Chavent, J. Jaffre, J. Liu, and B. Bourbiaux, Simultaneous estimation of relative permeabilities and capillary pressure, SPE Formation Evaluation, 7 (1992), pp. 283{289. [4] C. Chardaire-Riviere, P. Forbe, J. Zhang,. Chavent, and R. Lenormand, Improving the centrifuge technique by measuring local saturations, paper SPE 24882, in proceedings of the 67th SPE Annual Technical Conference and Exhibition, ashington, DC, U.S.A, ctober [5]. Chavent and J. Jaffre, Mathematic Models and Finite Elements for Reservoir Simulation, North-Holland, [6] J. Dennis and R. Schnabel, Numerical Methods for Unconstrained ptimization and Nonlinear Equations, Prentice-Hall, Englewood Clis, [7] R. Fletcher, Practical Methods of ptimization, John iley & Sons, [8] J. ilbert and C. Lemarechal, Some numerical experiments with variable storage quasi-newton algorithms, Mathematical Programming, 44 (1989), pp. 407{435. [9] S. Jegou, Estimation des permeabilites relatives dans des experiences de deplacements triphasiques en milieu poreux. These de Doctorat de l'universite Paris-Dauphine, [10] S. Jegou, Using Maple for symbolic dierentiation to solve inverse problems, Maple Tech, 4 (1997), pp. 32{40. [11] F.-M. Kalaydjian, Rapport bibliographique: modelisation des ecoulements triphasiques en milieu poreux, tech. report, Institut Francais du Petrole, march [12] B. Keyfitz, Change of type in three-phase ow: a simple analogue, Journal of Dierential Equations, 80 (1989), pp. 280{305. [13]. Mejia, T. atson, and J. Nordtvedt, Estimation of three-phase ow functions in porous media, AIChE Journal, 42 (1996), pp. 1957{1967. [14] H. Stone, Probability model for estimating three-phase relative permeability, Journal of Petroleum Technology, 22 (1970), pp. 214{218. 7

8 uy Chavent, Jer^ome Jare and Sophie Jegou-Jan True kr w Initial kr w ptimal kr w Figure 2: The parameter kr w : true (top), initial (middle) and optimal (bottom), estimated with one experiment (left) and three experiments (right). 8