Effects of the Quadratic Gradient Term in Steady-State and Semisteady-State Solutions for Reservoir Pressure

Abstract

Summary Constant-rate analytical solutions of the one-phase radial-flow equation in two dimensions, including effects of the quadratic gradient term, are derived for an oil reservoir with constant diffusivity and compressibility. The combinations of compressibility contributions to the various terms are analyzed. It is shown that the standard condition allowing the quadratic gradient term to be neglected (cp less than 1) is incorrect; pressure, p, should be replaced by a function of the production rate and other reservoir parameters except absolute pressure. The effect of the quadratic gradient term, for which quantitative expressions are given for steady-state and semisteady-state flow, may amount to several percent of the pressure drawdown at the wellbore. Introduction Considerable emphasis has recently been placed on nonlinear effects in reservoir flow. In particular, a whole literature on effects of a variable diffusivity has appeared. One exception is effects of the quadratic gradient term, which always appears if integral transforms of pressure are not introduced. Virtually the only work on this subject addresses the conditions for linearizing the flow equation by neglecting the term. No rigorous results for the quadratic gradient term's influence on the solution in terms of pressure have been presented. In an era of increasing sophistication of the prediction methods for reservoir flow, the effects of the quadratic gradient term also deserve quantitative consideration. We hope to contribute here toward this goal by solving analytically the one-phase-flow equation for two-dimensional (2D) flow with the quadratic gradient term included, which allows us to calculate the effect on the solution directly. The problem was solved in principle for the case of a constant compressibility by Muskat, who showed that the Darcy-continuity equation would be linearized (except for the nonlinear diffusivity) if it were expressed consistently in terms of density instead of pressure. It can indeed be checked that the transformations introduced here are equivalent to the introduction of density. In Ref. 2, however, the emphasis was not on nonlinear contributions to the solutions in terms of pressure, which were therefore never derived explicitly. The different ways that the various terms in the equation depend on contributions to the total compressibility are also clarified. Conditions are noted for which the compressibility difference has consequences for 2D simulators. Nonlinear Terms In the 2D Radial-Flow Equation Assumptions. We study the case of isothermal radial flow of a single, slightly compressible phase toward one well in a 2D horizontal reservoir, where effects of gravity and inhomogeneities can be neglected. Reservoir height, h, porosity, viscosity, and total permeability, k, can he functions of pressure, but are not assumed to depend on position and time in any other way. Wellbore-storage effects and skin effects are not considered. The flow equation is obtained by introducing Darcy's law into the equation expressing mass conservation. Textbook versions usually assume several of the parameters-h, and k-to be constants. This approach obscures one point made here, so we use the more general approach of Ref. 5. Allowing the reservoir height to depend on pressure essentially corresponds to allowing for a bulk-volume variability with pressure. The bulk volume is not allowed to expand radially in our approach. From a solid mechanics point of view, this corresponds to a plane-stress condition. One way of partially accounting for pressure dependence in reservoir parameters is to introduce integral transforms of the pressure: e.g., what is done when the pseudopressure representation for gas flow is formulated. To compare with the standard procedure for analytical treatment of oil flow, however, we keep pressure, p, as the dependent variable. Quadratic Gradient Term. The basic flow equation in 2D is (1) where p = fluid density. For circular symmetry, it can be rewritten (2) The generalized isothermal compressibilities, are given generically by (3) They are defined for constant reservoir overburden pressure, (a plane-stress condition). Here, we consider all compressibilities, to be constants. An interesting point to be made is that if is nonzero. a nonlinearity is by definition introduced at the right side of Eq. 2, because the porosity, must vary with p. Likewise, if and/or, are nonzero, nonlinearities appear at the right side. (A variable porosity should imply variations in permeability, which, according to the Carmen-Kozeny law, could be quite violent; we do not pursue that question further here.) If one's aim is to study effects of the nonlinear terms as small corrections to the linearized solutions of Eq. 2, then contributions from the quadratic gradient term on the left side and from the pressure dependence of the diffusivity can be studied separately by the use of lowest-order perturbation theory. The theme of this paper concerns the solutions of Eq. 2 when the quadratic gradient term is kept, assuming constant diffusivity. (Formally, this corresponds to the limit where and are much larger than the other compressibilities; see Eq. 10.) Notably, the combination of compressibilities in its coefficient is not the same as in the diffusivity, which warrants a check of relative-compressibility magnitudes together with the search for an analytical solution. Contributions to Compressibility. The PV compressibility, is defined by (4) where, and = pore, grain, and bulk volumes, respectively, the last being proportional to h. SPEFE P. 413^

A solitary wave in a porous medium

“…The notation ch represents the bulk volume compressibility for a reservoir which can expand or contract in a vertical direction perpendicular to flow for D = 1 and D = 2 (for D = 3, in our approach, Ch = 0). With r and t being the radial (linear) and time variables, Finjord and Aadnoy (1989) have shown that the Darcy continuity equation (Dake, 1978) has the form {~p~2 1 a (ap) In what follows, we will treat the diffusivity as a constant and also assume that el > 0. We therefore specialize to the limit where CO and ch are much more dominant than the other compressibilities.…”

“…Usually, the latter term has been neglected when analytical solutions have been sought. Exact analytical solutions, including the quadratic gradient-term effects, have been found by Finjord and Aadnoy (1989) for steady-state and semisteady-state flow for the case where the variability of the diffusivity can be neglected. This is allowed for particular parameter values and also in the limit of small nonlinear effects where a first-order perturbative approach is allowed.…”

Perturbation Analysis of Stress-Sensitive Reservoirs

“…3 with the assumption of pressure-dependent rock properties, we obtain 02p +~ op +~ ok (Op)2 + kp CL(Op)2 = c/>p. (cL +c ma ) oP, ~~ (rk :)= 2:h' ............................. (8). 3 with the assumption of pressure-dependent rock properties, we obtain 02p +~ op +~ ok (Op)2 + kp CL(Op)2 = c/>p. (cL +c ma ) oP, ~~ (rk :)= 2:h' ............................. (8).…”

Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy’s law

“…For conventional Darcy's flow models, using a Laplace transform, Odeh and Badu [52] presented analytical solutions to nonlinear partial differential equations (PDEs) taking into consideration the quadratic pressure gradient term, describing the seepage flow of a slightly compressible fluid; it was concluded that the nonlinear solutions showed the pressure difference for injection and pumping conditions, in comparison with the generally accepted solutions of linearized equations. Finjord et al [53] obtained constant-rate analytical solutions of a one-phase radial flow equation, considering the effect of the quadratic pressure gradient term, in an oil reservoir with constant diffusivity and compressibility. Wang and Dusseault [54] developed an analytical solution for pore pressure coupled with deformation in porous media; the quadratic pressure gradient term was taken into account, and it was shown that existing solutions deviated when the pressure gradient was high.…”