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A Model for Tracking Fronts of Stress-Induced Permeability Enhancement

Using an analogy to the classical Stefan problem, we construct evolution equations for the fluid pore pressure on both sides of a propagating stress-induced damage front. Closed form expressions are derived for the position of the damage front as a function of time for the cases of thermally-induced damage as well as damage induced by over-pressure. We derive expressions for the flow rate during constant pressure fluid injection from the surface corresponding to a spherically shaped subsurface damage front. Finally, our model results suggest an interpretation of field data obtained during constant pressure fluid injection over the course of 16 days at an injection site near Desert Peak, NV.


1
The formation and propagation of subsurface stress-induced damage zones is of great as well as an approximately uniform initial stress field. In the second case, rock failure is 48 driven thermally due to a large temperature difference between fluid and the surrounding 49 medium (see, e.g., [1] and [16]). While in this case there is no assumed threshold pressure, and where γ i is the total compressibility (liquid plus porous medium) in region i. We impose 123 the boundary conditions where R(t) is the position of the damage zone at time t, p 2 (r → ∞, t) is an abbreviation for 126 the value of p 2 as |r| approaches infinity, and p D is defined as the pore pressure at the damage where the damage front is located at x = X. The partial differential equation plus boundary 146 and initial conditions given above can be solved exactly as in [3], pg. 285, with the substi- where erf is the error function, defined as The first two boundary conditions from (2.7) yield (2.14) 157 so that it only remains to find the constants C 1 and C 2 . The third of equations (2.7) yields The first and middle expressions can only be equal to the constant on the right if X = λ √ t 160 for some constant λ. Substituting this expression for X into (2.15) allows one to solve for 161 both C 1 and C 2 as functions of the undetermined constant λ. Application of condition (2.11) 162 then results in the equation This approximation improves as p D approaches p H . In region two, the pressure effect from the 184 damage front propagates to roughly the distance √ a 2 t in time t. Therefore, an approximation 185 similar to (2.18), using the distance √ a 2 t instead of X, can be used to estimate ∂p 2 /∂x; the 186 approximate pressure gradient in region two is given as See Figure 2 (2.20) 191 We search for a solution of the form X = λt n for some undetermined n. Putting this 192 expression into (2.20) yields The only way that this equation can be satisfied for all times is for the powers of t to equal 194 one another; the only value of n for which such is the case is n = 1/2. X therefore takes the 195 form λ √ t and (2.20) becomes There is only one positive root of this equation, leading to the approximate damage front (2.23) Table 1

208
The expression for λ can be further simplified if there is a large contrast in the porosity 209 and permeability on crossing from one side of the damage front to the other. To affect the 210 simplification, we first re-write the approximate expression for λ as The second term on the right side is greater than so if the absolute value of the first term on the right side of (2.24) is much less than this 215 quantity, it may be neglected. This condition may be written as Hence, if the contrast in material properties between regions one and two is large enough to  In the present case, equation (2.9) becomes The ratio of the second term on the right hand side to the first is where r 0 is the radius of the injection well, i.e., p 1 (r 0 ) = p H . Using these expressions, 288 condition (2.9) becomes To affect a solution, we consider the case where k 2 /k 1 ≪ 1 and r 0 /R ≪ 1. .
(2.46) 295 2.7. Thermally induced damage. When damage is driven by thermal effects rather than 296 over-pressure, it is no longer reasonable to assume that the pressure at the damage front 297 is approximately constant. We will now explore the consequences of letting p D vary, from 298 p D = p H when the front is at the injection source to p D = p L as the front approaches infinity.

299
The simplest assumption consistent with this behavior is that p H −p D increases linearly with 300 R − r 0 . That is, where R max is the distance at which ∆p 1 = p H − p ∞ ≡ ∆p. In the following, we will only 303 consider the system behavior for r 0 < R max . In the case of spherical geometry, we substitute 304

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(2.47) into (2.44) and again assume that k 2 /k 1 ≪ 1, obtaining which has the solution If t max is the time at which R = R max , we may solve for R max in terms of this time as where we have assumed that r 0 /R max ≪ 1. Figure 3 shows a comparison between front 311 positions predicted via (2.49) versus (2.46), using the parameters shown in Table 2 The pressure gradient near the injection point is thus Neglecting gravitational effects and integrating the volumetric fluid flux over the surface of 332 a sphere of fixed radius r 0 yields the flow rate Therefore, the flow rate is expected to approach the constant where R is given by equation (2.46). This flow rate approaches (3.4) as R → ∞, regar where κ is the thermal diffusivity, regardless of the particular form that R takes. In the case of thermally induced damage, substituting (2.47) into (3.5) yields where R is now given by (2.49). In this case we note that However, if R ≫ r 0 and R max ≫ r 0 then   In the case of zero damage, i.e., when p D = p H , the above equation gives λ = 0 as expected. where ∆X ≡ x 2 − x 1 . This pressure increase, by (2.4), leads to an increase in density Hence, the discrete form of condition (2.11), when modified to include this density variation, In the limit as ∆t → 0, equation (4.5) reduces to (2.11). transition to the other. The weight function is taken as  Table 2 gives the values of the parameters used to fit the field data. Figure  The velocity of a purely diffusive temperature front is roughly where κ is the thermal diffusivity. Dividing 4.8 by 4.9 yields the dimensionless number Assuming that 10 −2 ≤ ∆φ ≤ 10 −1 and using the values in Table 2 Dividing (A-1) by the term proportional to ∇ 2 p leads to the dimensionless equation Now consider a small vertical section of porous material of height ∆z over which the pressure     [4] during constant pressure injection of fluid at T ≈ 100 • C into rock at T ≈ 190 • C (red circles); data past eleven days shifted three days backward, and points overlapping with the earlier data removed (blue circles).
29 Figure 5. Comparison between the model solution (4.7, blue), flow rates expected in the case of no damage (black), flow rates expected from an expanding spherical damage front starting at five days (green), and the Desert Peak field data (red dots).