Sp Interpretation for Forced Convection along a Vertical Fracture Zone

SP interpretation can be ambiguous giving highly non-unique physical models. The ambiguity can be limited by constraining the interpretation. If the SP anomaly arises from forced convection along a vertical fracture zone, an integral equation can be derived from which the SP anomaly may be calculated or inverted to a source. The source arises from divergence of a product of flow parameters, divided by the permeability. The formulation permits an assessment of the non-uniqueness present in SP interpretations for forced convection flow. INTRODUCTION The SP method has been used extensively and successfully to locate geothermally prospective zones (see, for example, Ross et al., 1990). Since the method is responsive to geothermal parameters such as fluid and heat flow and chemical environment, it has been thought to be a natural candidate for geothermal reservoir description and monitoring. With this motivation, the SP response of reservoir models has been numerically simulated by Ishido and Tosha (1998), and Nishi et al. (1998). Yet, as Tripp et al. (1998) observe, great care must be taken to constrain the inversion as much as possible, because of the level of modeling ambiguity present in SP interpretation. In fact, even with a half-space geometry, the flow parameters cannot be uniquely determined from SP data without additional information. From this beginning, determining the parameters for more realistic situations seems problematic. However, some appreciation of the possibilities and limitations of the SP method can be found by examination of severely constrained models. Such a model class will be examined in this paper. First, a brief recapitulation of the mathematical basis of SP interpretation is motivational. The SP interpretation problem has two parts. The first part consists of determining equivalent electrical sources from the SP data. The second part is determining the physical processes and parameters, which give rise to the equivalent electric source. Both of these steps can be illconditioned and ill-posed in the absence of independent geological information. An examination of the pertinent equations will illustrate this point. Determination of equivalent sources for a SP response can be formulated as the solution of a volume integral equation for the DC resistivity problem with the primary field being equal to zero. In this case the relevant equation is given by the equation Φ (xr) = ? G0 (xr,xs)S(xs) dxs, (1) where Φ (xr) is the measured potential and G0 (xr,xs) is the Green's function for the assumed background electrical conductivity distribution. Determining the distribution of S(xs) from Φ (xr) via equation (1) constitutes the first part of the SP interpretation problem. Decomposition of S(xs) into physically significant terms constitutes the second part of the SP interpretation problem. If the source of the SP anomaly is a flow with a potential ζ , then the source term is (Sill, 1983, eq. (7)), S(xs) = — (C /ρ). —ζ (C /ρ) —2 ζ . (2) where C is a cross-coupling term. For the case when a gradient formulation is not possible and the flow is divergence free, the source is (Sill, 1982, eq. (5)), S(xs) = —. (Lvv), (3) where Lv = (Le,m / k) = (σCe,m/k) , k is the matrix permeability, v is the fluid velocity, Le,m is the Onsager cross-coupling coefficient linking fluid flow and the electrical potential, and Ce,m is the streaming potential. Whatever the form of the source, it is apparent that inverting to hydrologic parameters is an under-determined problem. Hence severe constraints are necessary for any meaningful solution to the problem. One geological situation, which is often encountered, is a SP anomaly arising from forced convection along a near vertical fault. This problem, with its constrained geometry, gives a nice test case for examination. Hence, we suppose that forced convection occurs along a fracture zone. The flow nears the surface over a limited area, and then forms a plume, which is dispersed away from the anomaly. We wish to measure the spontaneous potential Φ (xr) on the earth's surface and invert the measurements to give information concerning flow parameters along the fracture zone. The strategy for this problem is to derive an appropriate integral equation by constraining the source term in (1). For any form of the source, we assume that physical properties and measurables vary only with respect to the vertical dimension. The resulting equation in velocity flow parameters is Φ (xr) = ? θ (xr,zs) [ (Le,m vzs / k )/ zs] dzs. (4) where θ (xr,zs) = ? ? G0 (xr,xs) dxsdys (5) is a function of the geometry of the fault crosssection and the electrical background. If we assume that the fracture zone extends from z = a to z = b, then we can integrate (4) by parts. The resulting equation is Φ (xr) = [θ (xr,zs)] [ Le,m vzs / k ]|ab ?ab [ (xr,zs)/ c] [ Le,m vzs / k ] dzs (6) These equations give an appreciation of the level of ambiguity of SP interpretation when the background electrical conductivity distribution and the fault zone geometry are both known. In this case, 1) Equation (4) has the form of a Fredholm Integral Equation of the First Kind in terms of the fault parameters and is ill-posed and illconditioned. The null space contains the functions [ Le,m vzs / k ] = constant. Hence only the vertical derivative of the quantity P = Le,m vzs / k can be derived; 2) If P is known at the ends of the fault zone, then (6) may be solved for P along the fault zone; 3) If P/ zs is determined from (4), then the values can arise from variations in Le,m, vzs, or k; 4) Variations in Le,m can arise due to phase transitions or changes in chemistry; 5) Variations in vertical velocity and permeability are coupled and are related to lithology variations; (6) If we assume that discharge from the fault occurs only at a discrete point or along a line, and that the physical properties and the flow velocity are constant along the fault up to the point of discharge, then (6) becomes Φ (xr) = θ (xr,xs) [ δP(zs)], (7) where δP(zs) is the step magnitude of P(zs) at zs. We will now demonstrate the importance of the initial assumption of background physical properties on the solution of equations (4) and (6). INFLUENCE OF BACKGROUND PHYSICAL STRUCTURE We will first remind the reader of some general relationships between the background physical property structure and the SP response and then discuss relationships which hold true for our specialized geometry. Recall the equation linking gradient driven flows and the SP response: —. (σ—φ) = — (C /ρ) . —ζ (C /ρ) —2 ζ . From this equation, we can conclude: 1) The source terms are inversely proportional to the electrical resistivity ρ. This means that sources are pronounced in conductive areas. However, the effect of the source may be attenuated by the conductive material. 2) Sources may be induced by gradients in either C or ρ which parallel pressure gradients. 3) The SP response φ is invariant to linear scaling of the ρ distribution. In the special case of a resistivity half-space, the SP response is invariant to the resistivity. 4) The SP response φ is invariant with respect to the ratio C /ρ. 5) The SP response φ α (C Γ1 / a Le,m), where Γ1 is the hydrologic flow source, Le,m is the hydrologic conductivity, and a is a scale factor (Sill and Killpack, 1982). In the case of a fluid velocity source, the equation is —. (σ—φ) = — (Ce,m/ρ) . v, and similar conclusions hold. In equation (6) the electrical structure of the earth determines the kernel θ (xr,zs). Thus it determines the surface expression of an SP source. For example, a very conductive surface layer will mask an SP source at depth. Evaluating the effect of the electrical structure is best done numerically, for each individual case. However, some representative calculations illustrating some of the major issues encountered are in order. For our calculations, we will use the 2D earth – 3D pressure source algorithm discussed by Sill ( 1983). Besides being convenient to use, this algorithm is appropriate for situations where convection is occuring along a major fault at a location of increased permeability, say at the point of intersection of a minor fault. This situation is encountered often.