Summary. An approach for discretizing the integral form of the flow equations in porous media is proposed. The method has the advantage of providing the correct block volume calculations and steady-state flow in curvilinear coordinates. The harmonic mean for permeability is also obtained from the discretization. The equations are developed for a general orthogonal curvilinear grid system in terms of the components of the metric tensor. Introduction The conventional approach for discretizing the reservoir flow equations consists of expanding the partial-differential equations in a Taylor series. Because practical applications of reservoir simulators require material-balance calculations, the discretize equations are then transformed into material-balance equations by multiplying the equations by appropriate factors. The disadvantage of a Taylor series approach is the failure to provide correct block volume calculations and steady-state flow in provide correct block volume calculations and steady-state flow in curvilinear coordinates, and special handling of block boundaries has to be introduced for cylindrical coordinates, as shown by Settari and Aziz. Instead of discretizing the flow equations directly and recasting the resultant finite-difference equations into material-balance equations, this paper proposes an approach for discretizing the integral form of the partial-differential equations to obtain the material-balance equations directly. The resultant equations yield correct block volume calculations and steady-state flow in any-orthogonal curvilinear grid system. Furthermore, the harmonic mean for permeability is also obtained in the discretization process. permeability is also obtained in the discretization process. Flow Equations In Orthogonal Curvilinear Grid Details on orthogonal curvilinear grid systems can be found in Ref. 3. Only the equations relevant to subsequent derivations are reported here. Let um, m= 1,2,3, be an orthogonal curvilinear grid system and let em, m = 1,2,3, be the positively oriented vectors tangential to the respective um. From vector analysis, the following equations can be derived:andwherehm = the square root of the component of the metric tensor associated with um, dV = volume element, dom = surface element on the surface, um = constant, and V = gradient operator.The continuity equation and Darcy, s law give the following partial-differential equations partial-differential equationsandIt is assumed that u1, u2, and u3 are the principal axes of the permeability tensor, k. permeability tensor, k. Integration of Eq. 4 over a control volume, V, giveswhere i= vivdV represents the mass injection rate into the control volume. Using the divergence theorem, the first term of Eq. can be rewritten aswhere denotes the integration over the total surface, S, of V, and where n is the outward unit vector to S. Substituting Eq. 7 into Eq. 6 yieldsThe next section shows how Eqs. 8 and 5 are discretized. The technique is different from the conventional approach, where Eq. 4 is discretized directly. Eq. 8 has also been used by Narasimhan and Witherspoon in their integrated finite-difference method (IFDM). However, the goals of IFDM and the approach presented in this paper are different. IFDM is a generalized finite-difference scheme for gridblocks with an arbitrary number of faces, whereas the approach here uses Eq. 8 to obtain the discretized equations for a general orthogonal curvilinear grid system. Discretization of the Flow Equations Grid Definition. In finite-difference calculations, the reservoir is divided into blocks. Let i, j, and k be the indices corresponding to the axes u1, u2, and u3, respectively. The grid node (i, j, k) has therefore the coordinates (u1, i, u2, j, u3, k). SPERE P. 685