HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts and Stephan Matthai Mathematics Research Report No. MRR 003{96,
Mathematics Research Report No. MRR 003{96, HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts 1 and Stephan Matthai 2 3rd February, 1996 Abstract In Petroleum exploration it is often important to understand the ow of liquids through porous media, which can be described by conservation of uid, and the assumption (Darcy' Law) that the uid ux is given by arp where P is the pressure and a is the permeability of the porous media. Three major problems are associated with solving this problem in the situation of interest. First, the permeability a can range over many orders of magnitude (10 orders being typical). Second, the distribution of the permeability can form complicated geometric stuctures associated with geological stuctures such as faults and permeable layers with dierent ow characteristics. Finally, the permeability data and description of the problem can come in arbitrary sizes depending on the experimental data which is available. Toovercome these problems wehave used a nite element method with an algebrain multigrid solver. We present the results of some simulations of ow through faults associated with oil migration through sand-shale structures. In particular we will demonstrate how uid ow patterns at fault-sand-shale intersections vary with fault permeability. 1991 Mathematics Subject Classication. primary 76S05 secondary 65N55 65N60 65M55 65M30. Key words and phrases. Flow, Porous Media, Finite Element Method, Multigrid, Oil Exploration. Postal addresses: 1. School of Mathematical Sciences, ANU, Canberra ACT 0200, Australia. 2. Department of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305, USA.
1. Introduction In petroleum exploration it is often important to understand the ow of liquids through porous media consisting of a complicated structure of permeable layers and faults. In many cases it is possible to understand the ow eld by using the conservation of uid, together with the assumption (Darcy's Law) that the uid ux is given by Krp where p is the pressure and K is a measure of the permeability of the porous media and is dependent on position (the viscosity and uid density has been absorbed into K). Three major problems are associated with solving this problem in the situation of interest: 1. The permeability K can range over many orders of magnitude (10 orders being typical). This usually leads to slow convergence of the numerical methods. 2. The distribution of the permeability can form complicated geometric structures associated with geological structures such as faults and permeable layers with dierent ow characteristics. We would like our numerical scheme to \adapt" to this inherent structure. 3. The permeability data and description of the problem can come in arbitrary sizes depending on the experimental data which is available. The highly complicated nature of the geometry of our problem and the large range in the permeability has led us to use a standard nite element method with triangular elements and piece-wise linear test functions in which the permeability is constant in each triangular element. We have been working with reasonably large problems with approximately 500k variables (850 by 600 data sets). A very ecient choice of solution method for such problems is the multigrid method. This method has been shown to provide optimal convergence [1] (bounded number of iterative steps independent of problem size) even for problems such as ours in which the permeability may abruptly change. We will present the results of some simulations of ow through faults associated with oil migration through sand-shale structures. In particular we will demonstrate how uid ow patterns at fault-sand-shale intersections vary with fault permeability. 2. The Simplied Mathematical Model We would like to study the ow of uid through porous media. If we assume that: 1
The density of the uid is constant, The uid ux satises Darcy's Law (ux proportional to the negative of the gradient of the pressure), and The porosity of the material depends linearly on pressure (hence storativity is constant). Then the pressure for our model problem satises the evolution equation: S @p @t = rkrp (1) where p(t x y) denotes the pressure as a function of time t and position (x y) 2, a two dimensional domain. K(x y) measures the permeability and S(x y) the storativity of the material. Of particular interest to us is the situation of layers of sand and shale in which oil is released under the inuence of heating. In this case we assume just one uid species (oil). Typically the layers are intersected by a complicated collection of faults (see Figure 1). In this case we are interested in approximating the steady state solution of equation (1). Also of interest is the situation in which the permeability of the faults changes, for instance after earthquakes. In this case the permeabilities of the faults may change abruptly, causing the pressure eld to diuse over time in response to the new permeability data. It should be noted that the ow of water through rocks at depths of several kilometres can also be modelled by equation (1). Once again earthquakes can cause large changes in the permeability of the faults. This in turn can setup situations in whichwater from dierent locations (and hence containing dierent dissolved minerals) can mix and cause precipitation of minerals in the fault regions. We will consider the simple case in which the domain of the problem,, is a rectangle. The pressure at the top and bottom boundaries are assumed to be constant, with the vertical boundary pressures calculated assuming that dp dy (x y) / 1 K(x y) : Here we are assuming that the permeability structure is independent of x near the vertical boundaries. It is also possible to apply zero ux boundary conditions on the vertical boundaries. 2
Figure 1: Typical sand shale fault structure for an oil reservoir. Notice how the layers have been displaced along faults. 3. The Finite Element Method Equation (1) is a fairly standard equation to solve. In our situation though we need to take note of the fact that the permeability K ranges over many orders of magnitude and is discontinuous (see Figure 1) and the distribution of K is associated with complicated geological structures and so our numerical method needs to be able to deal with these complicated geometries (see Figure 1). To deal with these complications we decided to use a Finite Element Method (FEM), based on triangular elements, with piece-wise linear test functions to discretize the spatial derivatives. For the time derivative we chose the backward Euler method to ensure stability for the associated sti system of equations. We will assume that the reader has a knowledge of the FEM philosophy (see Johnson's introductory book on FEM's [3]) To paraphrase, the philosophy is \Multiply the Partial Dierential equation by test functions, integrate and then use Green's Theorem to move one derivative to the test function". After this procedure we are led to a variational problem which is \equivalent" to equation (1). The Variational Problem is: Find p(t x y) such that p(t ) 2 H 1 () (the space of functions with square 3
integrable spatial derivatives) such that p satises the specied boundary and initial conditions, and S @p @t da = ; Kr rp da for all test functions 2 H0(). 1 It is important to note that we do not have to make any smoothness restrictions on the permeability K or Storativity S other than to ensure that the previous integrals make sense. In particular we can work with discontinuous permeabilities. Also this new equation only needs the function p to have integrable rst derivatives (as opposed to second derivatives in the original Partial Dierential Equation). We can discretize our Variational Problem by restricting the test functions and approximate solution to appropriate nite dimensional spaces of functions. The nite dimensional space that we use is the space of piece-wise linear functions associated with a triangulation of the domain. A triangulation T consists of a nite collection of closed triangles ( ), such that: 1. If K L 2T then K \ L is either empty, a common vertex of K and L, a common edge of K and L, or K = L. 2. S K2T K =. As our applications are often associated with complicated permeability structures, the use of triangulations (as opposed to meshes) allows for a more reasonable matching of the permeability structure to the numerical structure. Indeed we try to adapt the triangulation to the underlying permeability structure, and in particular to the fault structure. For a particular triangulation T, we can consider the associated space of continuous piece-wise linear functions. To explicitly form our discrete equations we need to nd abasisfor our piece-wise linear functions. Let fx i : i = 1 ::: ng denote the set of vertices of the triangulation T, numberedinsuchaway that x i 2 @ if i>n. Let i be the unique continuous piece-wise linear function which satises i (x j )= ij (see Figure 2 for a typical basis function). The set of functions f i : i =1 ::: ng forms a basis for the space of piece-wise linear functions associated with the triangulation T. The discrete equations can now be written explicitly as: Find ^p(t) = P n j=1 P j(t) j such that i S d^p dt da = ; Kr i r^p da for all i =1 ::: n. 4
Figure 2: A typical piecewise linear basis function. Here P j (t) denotes the time dependent coecients of the approximate solution, with respect to the given basis for the piece-wise functions. To simplify the exposition we will neglect the specication of the pressure boundary conditions. As described the function ^p will satisfy a zero ux condition on the boundary. Expanding in terms of the basis functions we obtain a system of Ordinary Dierential Equations (ODE's) for the coecients. nx j=1 dpj S i j da dt (t) =; n X j=1 Dene two n n matrices via A ij = K r i r j da and B ij = K r i r j da P j (t) for all i =1 ::: n. S i j da: If we let P (t) = (P 1 (t) ::: P n (t)) 0 denote the coecient solution vector, then P (t) satises the ODE B dp dt = ;AP: Finally if we use the backward Euler method to discretize this ODE we obtain our numerical scheme: Find P n =(P n 1 (t) ::: P n N(t)) 0 such that B(P n ; P n;1 )=;kap n 5
where k denote the time-step for the method and P n denotes the numerical approximation of P (nk). In particular, to calculate P n we must solve the linear equation, (B + ka)p n = P n;1 : (2) The matrices A and B are positive denite (as S > 0andK > 0) and so this linear equation has a unique solution (up to a constant function). On the other hand for complicated permeability structures the matrices may be very large (10 6 by10 6 ) but sparse. In the next section we will provide a very basic description of the multigrid technique used to solve this equation. The pressure boundary conditions can now be specied by changing the lower n ; N n submatrix of B + ka to [0jI] and specifying the explicit boudary conditions in the appropriate components of the righthand side vector. 4. The Algebraic Multigrid Method For a large matrix problem like equation (2), derived from an elliptic Partial Dierential Equation (PDE), it is natural to consider the multigrid method (see [2], [4]). Multigrid methods are optimal in the sense that a solution accurate to the level of the truncation error can be obtained in O(N ) operations, N being the number of unknowns. Also it has been shown by Bramble et al. [1] that this optimal behaviour is maintained even if the underlying PDE has discontinuous permeabilities. Multigrid methods are typically associated with solving PDE's on a hierarchy of rened grids or triangulations. To specify a particular multigrid method, it is necessary to dene a way of transferring approximate solutions between ne and coarse levels (restriction and interpolation operators), specication of approximate solvers on the individual levels (the smoother) and a way of specifying the discrete problem on each level. Multigrid methods then work with the hierarchy of levels and build up a solution on the nest level by recursively solving problems on progressively coarser levels and then smoothly transferring the coarse level solutions back to the ner levels. In this way both local and global information about the solution can be obtained in one iterative sweep of the multigrid method. In our particular case the permeability structure is often given to us in arbitrary sizes, and so there is usually no inherent hierarchical structure to the problem. For this reason we have used the Algebraic Multigrid method (AMG) developed by Ruge and Stuben [5]. We have kindly been provided with the fortran code AMG1R5 written by Ruge, Stuben and Hempel. The 6
AMG method takes the matrix associated with the discretization of the problem. Using the connection information inherent in the matrix, an internal hierarchy of matrix problems are produced. A standard Gauss-Seidel smoother is used on each of these matrix problems. The AMG method seems to provide a method which is optimal in the sense described above for standard Multigrid. The major disadvantage of the method is the explicit formation of the discretization matrix and the need for a large amount of work space (the AMG method requires approximately 30N words of storage for the sparse matrix and the workspace, where N is the number of unknowns). In the future we plan to investigate more traditional multigrid method solvers which can be used as preconditioners for our problem. This should reduced the storage requirements markedly. 3.00 3.10 32 34 36 38 40 42 44 Depth, km 3.20 46 48 50 52 3.30 54 56 58 60 62 64 66 68 3.40 0.00 0.10 0.20 0.30 0.40 Width, km -20-19 -18-17 -16 log Permeability, m2 Figure 3: Flow through permeable layers joined by faults of similar permeability. 5. Numerical Results As mentioned previous, we would like to be able to study the inuence of changes of permeability in faults on the over all ow and pressure distribution of a simulation. This may be of use to conclude the existence of oil reservoirs, or to approximate the inuence of mixing on the the precipitation of minerals. To give a simple example of a typical situation, in Figures 3 and 4 we have calculated the pressure and velocity eld for a simple conguration of layers and faults. The only dierence between to two situations is the permeabilities of the faults. In Figure 3 the permeability of the fault is the same as the layer 7
3.00 3.10 32 34 36 38 40 42 Depth, km 3.20 44 46 48 50 52 54 56 3.30 58 60 62 64 66 68 3.40 0.00 0.10 0.20 0.30 0.40 Width, km -20-18 -16-13 -11 log Permeability, m2 Figure 4: Flow through permeable layers joined by highly permeable faults. (10 ;16 ), where as the permeability of the faults in Figure 4 are 10 ;11. As can be seen from the plots, the ow through the permeable layers in Figure 4 are appreciable greater than for those of Figure 3. Hence our simulations can provide useful information about the response of the oil reservoir to changes in the permeability of the faults due to say earthquakes. We are presently studying the time evolution of pressure changes caused by similar changes in fault permeabilities. We are also investigating the consequences of permeability of faults on the position of deposition of minerals due to mixing induced by concentration of ow through faults. It should be noted that at present the solution of these ow problems associated with grids of approximately 500 by 800 take about 30min on our SPARC 10 workstation, and need about 150 MB of memory. The large amounts of memory are due to the large overheads incurred in using the algebraic multigrid method. 6. Acknowledgements Wewould like toacknowledge support from the Cooperative Research Centre for Advanced Computational Systems. 8
References [1] Bramble J. H., Pasciak J. E., Wang J., and Xu H. Convergence Estimates for Multigrid Algorithms without Regularity Assumptions Math. Comput., 5723{45, 1991 [2] Briggs W. L. A Multigrid Tutorial SIAM, 1987 [3] Johnson C. Numerical Solution of Partial Dierential Equations by the Finite Element Method Cambridge University Press, Cambridge, 1990 [4] McCormick S. F., editor Multigrid Methods Frontiers in Applied Mathematics. SIAM, 1987 [5] Ruge J. W. and Stuben K. Algebraic Multigrid In S. F. McCormick, editor, Multigrid Methods, Frontiers in Applied Mathematics SIAM, 1987 9